Operator Theory and Indefinite Inner Product Spaces : Presented on the Occasion of the Retirement of Heinz Langer in the Colloquium on Operator Theory,

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Operator Theory and Indeﬁnite Inner Product Spaces Presented on the occasion of the retirement of Heinz Langer in the Colloquium on Operator Theory, Vienna, March 2004

Matthias Langer Annemarie Luger Harald Woracek Editors

Birkhäuser Verlag Basel . Boston . Berlin

Editors: Matthias Langer Department of Mathematics University of Strathclyde 26 Richmond Street Glasgow G1 1XH UK e-mail: [email protected]

Annemarie Luger Harald Woracek Institut für Analysis und Scientific Computing Technische Universität Wien Wiedner Hauptstrasse 8–10 / 101 1040 Wien Austria e-mail: [email protected] [email protected]

2000 Mathematics Subject Classification Primary 46C20, 47B50; Secondary 34L05, 47A57, 47A75

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

ISBN 3-7643-7515-9 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2006 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ' Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN-10: 3-7643-7515-9 e-ISBN: 3-7643-7516-7 ISBN-13: 978-3-7643-7515-7 987654321

www.birkhauser.ch

Heinz Langer

Table of Contents Introduction Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Laudation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Speech of Heinz Langer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Conference Programme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv Bibliography of Heinz Langer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx

Research articles V.M. Adamyan and I.M. Tkachenko General Solution of the Stieltjes Truncated Matrix Moment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo Q-functions of Quasi-selfadjoint Contractions . . . . . . . . . . . . . . . . . . . . . . . .

23

J. Behrndt A Class of Abstract Boundary Value Problems with Locally Deﬁnitizable Functions in the Boundary Condition . . . . . . . . . . .

55

´ P. Binding and B. Curgus Riesz Bases of Root Vectors of Indeﬁnite Sturm-Liouville Problems with Eigenparameter Dependent Boundary Conditions, I . . . . . . . . . . . . .

75

A. Dijksma, A. Luger and Y. Shondin Minimal Models for Nκ∞ -functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A. Fleige, S. Hassi, H.S.V. de Snoo and H. Winkler Generalized Friedrichs Extensions Associated with Interface Conditions for Sturm-Liouville Operators . . . . . . . . . . . . . . . . . . . 135 K.-H. F¨ orster and B. Nagy Spectral Properties of Operator Polynomials with Nonnegative Coeﬃcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 B. Fritzsche, B. Kirstein and A. Lasarow Szeg˝o Pairs of Orthogonal Rational Matrix-valued Functions on the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 M. Kaltenb¨ ack, H. Winkler and H. Woracek Singularities of Generalized Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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L.V. Mikaelyan Orthogonal Polynomials on the Unit Circle with Respect to a Rational Weight Function . . . . . . . . . . . . . . . . . . . . . . . . . 249 D. Popovici Bi-dimensional Moment Problems and Regular Dilations . . . . . . . . . . . . . 257 U. Prells and P. Lancaster Isospectral Vibrating Systems, Part 2: Structure Preserving Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 V. Strauss A Functional Description for the Commutative W J ∗ -algebras of the Dκ+ -class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 F.H. Szafraniec On Normal Extensions of Unbounded Operators: IV. A Matrix Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

337

B. Textorius Directing Mappings in Kre˘ın Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351

Survey Article Z. Sasv´ ari The Extension Problem for Positive Deﬁnite Functions. A Short Historical Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Preface In this volume we present a collection of research papers which mainly follow lectures given at the “Colloquium on Operator Theory”. This conference was held at the Vienna University of Technology in March 2004 on the occasion of the retirement of Heinz Langer. The present volume of the series “Operator Theory: Advances and Applications” is dedicated to him and his scientiﬁc work. The book starts with an introductory part which provides some information about the colloquium itself. We have also included the laudation given by Aad Dijksma and a list of the recent publications of Heinz Langer, which updates his bibliography given in OT 106. The main part of the book consists of ﬁfteen original research papers, which deal with various aspects of operator theory and indeﬁnite inner product spaces. It concludes with a historical survey on the theory of positive deﬁnite functions. It would not have been possible to bring together so many colleagues in Vienna without ﬁnancial support provided by several organizations. We wish to thank the Außeninstitut der TU Wien ¨ ¨ Osterreichische Forschungsgemeinschaft (OFG) Research Training Network HPRN-CT-2000-00116 of the European Union Vienna Convention Bureau ¨ Osterreichisch-Ukrainisches Kooperationsb¨ uro ¨ ¨ Osterreichische Mathematische Gesellschaft (OMG) and everybody who has contributed to make this conference and this book possible. Our special thanks also go to the referees who did a lot of work and in some cases made valuable and essential suggestions, which decisively improved the quality of the papers. Photos were provided by Dr. Mathias Beiglb¨ ock (conference photo) and Peter Geisenberger (photo of Heinz Langer). As organizers of the “Colloquium on Operator Theory” it was a great pleasure for us that so many colleagues participated in the conference; as editors of the present volume we are happy that a large part of them decided to contribute to these proceedings. We greatly acknowledge the interesting experience.

Vienna, July 2005

Matthias Langer Annemarie Luger Harald Woracek (the editors)

Laudation By Aad Dijksma, Vienna, March 4, 2004 Dear Vice-Rector Kaiser, Dean Dorninger, and Professor Inge Troch, dear Heinz, dear colleagues, Ladies and Gentlemen.

Introduction Heinz Langer is a world leading expert in spectral analysis and its applications, in particular in operator theory on spaces with an indeﬁnite inner product. He has moved mathematical boundaries and opened new areas of research. He has the talent to focus on what is fundamental and to give directions to what is possible. By sharing his ideas he has stimulated many to explore unknown mathematical territory. Heinz is co-author of a book with M.G. Krein and I.S. Iokhvidov and he has published more than 170 papers with 45 co-authors from all over the world. He has directed the research of about 25 Ph.D. students. His results are still being applied and quoted in international journals in mathematics and theoretical physics. As one of his collaborators I can attest to the fact that Heinz is still very active. When the organizers of this workshop invited me to give this laudation speech I immediately said yes. I deem it an honor to pay tribute to Heinz. Not only because I admire his work, but also because I have learned a lot from him and still do and because I consider him a close friend. I feel privileged and pleased to speak on this occasion. First I will outline Heinz’s biography and highlight his successful years at the Vienna University of Technology. Then I will try to clarify what I meant when I said that Heinz has moved mathematical boundaries and opened new areas of research by discussing some of the main themes in his work. Finally, I will speak about Heinz’s connection with The Netherlands and especially with Groningen.

Biography Heinz Langer was born in Dresden on August 8, 1935. He lived in Dresden for almost 55 years. He attended the Gymnasium there, studied mathematics at the Technical University of Dresden, where he became an assistant and where he obtained his Ph.D. in 1960 and his Habilitation in 1965. He was appointed professor in mathematics at his Alma Mater in 1966, at the age of 31. At that time there was oﬃcially no research group in analysis, let alone functional analysis, so Heinz joined the group in stochastics headed by his Ph.D. advisor Prof. P.H. M¨ uller. He did research in and lectured on semigroups, one-dimensional Markov processes, and the spectral theory of Krein-Feller diﬀerential operators, but with Ph.D. students and colleagues from abroad he could work in his favorite topic: operator theory. Heinz declined a position at the prestigious Mathematical Institute of the Academy of Sciences of the GDR, because he liked to teach and to work with Ph.D. and post-doctoral students.

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At the beginning of his career Heinz spent two years abroad, ﬁrst as a postdoc in Odessa in 1961/62 upon the invitation of M.G. Krein and later in 1966/67, shortly after his appointment as professor, on a fellowship of the National Research Council of Canada which was arranged by Professor I. Halperin in Toronto. The contact with Krein began with Heinz posting a handwritten manuscript in German, in the mailbox at the central station of Dresden on December 31, 1959. I will say more about the contents of the manuscript when I speak about the main themes in Heinz’s work. The subsequent stay in Odessa and the many shorter visits after that had a tremendous inﬂuence on Heinz, his career in mathematics, and, I think, on his style of doing mathematics. When Heinz mentions Krein it is with great aﬀection and respect, and I know that Krein thought of Heinz as one his most brilliant students and collaborators. In between the two trips abroad Heinz had married Elke and in May 1967 their daughter Henriette was born. In the 1970’s and 1980’s Heinz spent several extended periods away from home in, for example, Jyv¨ askyl¨ a, Stockholm, Uppsala, Link¨ oping, Antwerp, Groningen, Amsterdam, and Regensburg. In all these places Heinz left his mark. He established some form of research co-operation, set up Ph.D. projects and made it possible for students and colleagues to come to Dresden. My account on Heinz’s connection with Groningen later on serves as an example of his stimulating inﬂuence on other mathematicians. Heinz remained in Dresden until 1989. In October of that year, shortly before the fall of the Berlin wall, he and his family gave up hearth and home, left the GDR and went to Regensburg. Turbulent times followed. Thanks to Albert Schneider Heinz obtained a professorship for one year at the University of Dortmund. After that year, with the support of Reinhard Mennicken, Heinz became professor at the University of Regensburg. Finally, in August 1991, Heinz moved to Vienna where he began a new and successful period in his life and in his mathematical career. Heinz was oﬀered the prestigious chair “Anwendungsorientierte Analysis” at the Vienna University of Technology, previously held by Professor Edmund Hlawka, but under a diﬀerent name. The University could hardly have found a more worthy successor. For several years Heinz was chairman of the Institute. He was pragmatic and apparently did his job well because he was re-elected. In no time the Vienna University of Technology became an internationally renowned research center of operator theory with young people doing challenging research. In Vienna Heinz guided 7 Ph.D. and 3 post-doctoral students. The center attracted many visitors from all over the world for visits to do research and for workshops. During the past 12 years Heinz has organized 4 workshops, one in co-operation with the Schr¨ odinger Institute and one in 2001 when the Vienna University of Technology awarded Professor Israel Gohberg from Israel an honorary doctorate. He obtained various long term research grants including funds for Ph.D. and post-doctoral projects. Heinz has clear and practical ideas and a keen sense on what to apply for and on how to formulate it. Two of these grants were from the Austrian “Fonds zur F¨ orderung der Wissenschaftlichen Forschung (FWF),” the last of which was

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awarded very recently for a project on canonical systems. As to a third grant: Heinz is the leader of the Austrian-German team in a Research Training Network of the European Union, in which 10 universities from 9 diﬀerent countries participate. In recognition of his work Heinz was elected corresponding member of the Austrian Academy of Sciences.

Main themes in his work Heinz’s lifelong mathematical interest has been in the theory of operators, in particular operators on indeﬁnite inner product spaces and its applications. This subject was suggested by Professor P.H. M¨ uller, his thesis advisor. Heinz has initiated many new projects and made fundamental contributions to their development. His work draws the attention of several mathematicians and physicists. To illustrate this I elaborate on four of the main themes in his work: 1. From the Ph.D. period: the invariant subspace theorem. 2. From the Habilitation period: deﬁnitizable operators. 3. From the period 1975–1985: extension theory. 4. From the Vienna period: block operator matrices. These main themes are interconnected and overlap in time. The division in periods is made only to facilitate the exposition. The invariant subspace theorem Now I come back to the handwritten German manuscript. For his Ph.D. thesis, so in the late ﬁfties and early sixties, Heinz read the papers by L.S. Pontryagin, M.G. Krein, and I.S. Iokhvidov. Pontryagin in 1944 published his famous theorem that a self-adjoint operator A in a Πκ space with a κ-dimensional negative subspace has a maximal invariant non-positive subspace and that the spectrum of A restricted to this subspace lies in the closed upper half-plane. In 1956 Krein gave a diﬀerent proof of Pontryagin’s theorem using a ﬁxed point theorem but then for unitary operators. Iokhvidov used the Cayley transform to show that the theorems of Pontryagin and Krein are equivalent. Heinz generalized Pontryagin’s theorem to self-adjoint operators on a Krein space. The sole assumption was a simple compact corner condition to have control over the operator when restricted to the possibly inﬁnite-dimensional negative subspace of the Krein space. It was a truly remarkable achievement. Heinz wrote it up by hand and in German and sent the manuscript to Krein on New Year’s eve in 1959. It would later be the main result in Heinz’s Ph.D. thesis. Iokhvidov in Odessa had tried to prove the same theorem before but he had not succeeded. So naturally Krein was very interested in Heinz’s proof and he must have been impressed because in one of his Crimean lecture notes he refers to the generalized invariant subspace theorem as the Pontryagin-Langer theorem. In any case, Krein reacted by inviting Heinz to come to Odessa. The visit marked the beginning of a fruitful co-operation that lasted for more than 25 years, until the death of Krein in 1989, and resulted in 13 joint papers. But the story goes on: One of the ﬁrst things they discovered was a beautiful and unexpected application of the invariant subspace theorem. With the

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help of this theorem they proved that a self-adjoint quadratic operator pencil has a root with some speciﬁed spectral properties. The two papers “On some mathematical principles in the linear theory of damped oscillations”, which deal with this factorization, led to new publications in spectral theory and applications to mechanics and physics. They are probably their most frequently cited joint papers. Definitizable operators In 1962 Krein and Heinz proved that a self-adjoint operator in a Pontryagin space has a generalized spectral function. Krein showed that an integral operator with a positive kernel gives rise to a positive operator on a Krein space and that this operator also has a generalized spectral function. Heinz discovered that these are two examples of a special class of self-adjoint operators on a Krein space, namely the class of deﬁnitizable operators. The concept and the name are due to Heinz. A self-adjoint operator A is deﬁnitizable if it has a nonempty resolvent set and for some polynomial p, p(A) is nonnegative. Like the compact corner condition in the Pontryagin-Langer theorem, deﬁnitizability is a way to keep control over the nonpositivity of the inner product of the Krein space. In his Habilitation submitted in 1965 Heinz shows that a deﬁnitizable operator has a generalized spectral function and he applies his theory to quadratic operator pencils. It is no exaggeration when I say that this work is a genuine corner stone in the spectral theory of operators in spaces with an indeﬁnite metric. Like its Hilbert space counterpart, the spectral function in a Krein space has a wide range of applications such as to quadratic pencils, just mentioned, Sturm-Liouville problems with indeﬁnite weight, elliptic problems, and variational principles. Extension theory The research of Heinz and Krein in the extension theory of symmetric operators in Pontryagin and Krein spaces began with generalizing Krein’s theory of generalized resolvents, resolvent matrices, and entire operators to an indeﬁnite setting with applications involving new classes of meromorphic functions with ﬁnitely many ¨ poles and canonical systems. It culminated in the 4 seminal “Uber einige Fortsetzungsprobleme . . . .” papers published jointly with Krein in the period 1977–85. These papers are still quoted and applied today. In fact, they could be called trendsetters, because there is a growing interest in generalizing, where possible, positive deﬁnite results to an indeﬁnite setting. Of the many examples I only mention the indeﬁnite version of the de Branges theory of entire functions and canonical systems. The research in this area is carried out by the group around Heinz here at the Vienna University of Technology. The results are important and of a high quality. Heinz can be proud of the research team he leaves behind. Block operator matrices This topic was taken up by Heinz jointly with Reinhard Mennicken in Regensburg and came to bloom in Vienna. 2 × 2 block operator matrices are common in the theory of operators on Krein spaces. But now the emphasis is diﬀerent. The problem is to describe the spectral properties of an operator deﬁned on a product of two

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Hilbert spaces and given as a 2×2 block operator matrix in terms of the properties of the operator entries of the matrix. Typical examples come from mathematical physics or system theory, where the entries are diﬀerential operators of diﬀerent order and hence unbounded. One of the ﬁrst problems is to deﬁne the domain of deﬁnition of the block operator matrix. Many papers have appeared since 1991. They concern, for example, the location of the essential spectrum, the solution of a Riccati equation, and block diagonalization. A new concept initiated and further developed jointly with others in the last 5 years is that of the quadratic numerical range. It is a new tool for localizing the spectrum of a block operator matrix.

The Netherland connection Let me begin with how I came to meet Heinz. I wrote a letter to him with some results on diﬀerential operators with eigenvalue depending boundary conditions on February 25, 1980. Heinz responded on March 19, 1980 with detailed answers to my questions. Some years before, Rien Kaashoek had visited Heinz in Dresden and organized a return visit for Heinz to Amsterdam. This took place in October 1981 and Heinz used the opportunity to come to Groningen as well. This visit was an immediate success, we were on ﬁrst name basis within a few minutes, and his visit to my house and family went as smooth as pie. With Henk de Snoo we started to work on classes of meromorphic functions which arose in extension theory of symmetric operators in spaces with an indeﬁnite metric and applied the theory to self-adjoint boundary eigenvalue problems with eigenvalue depending boundary conditions. At one time Heinz predicted that what we were about to start would lead to many publications. Before I realized it, I asked “Oh, how many?” (I dislike vague remarks.) Heinz actually paused to think and answered “About 10.” I was impressed. How could he predict that many? But how right he was: Many papers have appeared since then. First jointly with Henk de Snoo, even more than 10, and later also with others. Since that ﬁrst visit in 1981, we have met at numerous working visits, conferences, and workshops. When near the Mediterranean we would go out to swim. Out of all these contacts grew a friendship which I treasure very much. During his many visits to Groningen, Heinz generously shared many ideas with us and stimulated us to work on them by ourselves. They resulted in a master thesis and a Ph.D. thesis about extension theory and interpolation, some papers with postdoctoral students about extension theory and commutant lifting, and even a book about Schur functions, operator colligations, and reproducing kernel Pontryagin spaces which my co-authors D. Alpay, J. Rovnyak, and H.S.V. de Snoo and I dedicated to him in appreciation, admiration and amity. Right now Heinz and I together with Daniel Alpay (from Israel), Tomas Azizov, and Yuri Shondin (both from Russia) and others are working on two projects. One is about an indeﬁnite version of the Schur algorithm. The other concerns singular perturbations of self-adjoint operators with applications to quantum physics and to, for example, the Bessel and Laguerre diﬀerential operators. I very much enjoy working with Heinz. His enthusiasm for the problem at hand is stimulating

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and I am impressed by the ease with which Heinz ﬁnds the right wording for a paper. Our style of working together depends on where we are. When in Vienna, we sit on a couch in Heinz’s oﬃce, we have scratch paper on our knees, and our hands are blue from writing, so to speak. When in Groningen, we stand before a big green blackboard and our hands are white from the chalk. Upon the invitation of Rien Kaashoek, Heinz regularly visited the Free University in Amsterdam. Heinz took part in the reading and examination committee for some of Rien’s Ph.D. students. He also gave a series of lectures at the Thomas Stieltjes Institute. Heinz was an important participant in the INTAS-projects which Rien arranged. As to mathematics, Heinz’s visits to the Free University resulted in joint works about solutions of the Riccati equation with Andr´e Ran and his students. Heinz was co-promotor for one of them. Heinz’s relation with the Netherlands is not restricted to the Free University and the State University of Groningen only. At present Heinz is a member of the international evaluation committee to evaluate the output over the period 1996–2001 of the Mathematics Departments of all Dutch Universities. It is yet another indication that Heinz is recognized as an authority in mathematics with an international reputation.

Concluding remark I have said a lot and inadvertently I may have omitted things I should have said, but I should ﬁnish. Allow me to make one last remark: some years ago at some place Heinz and I had to ﬁll in some forms and we discovered that at the space where we had to write down our profession we had entered diﬀerent things. I had written “Professor” and Heinz had ﬁlled in “Mathematician.” Heinz may have retired as a professor but I hope and wish that he will not retire as a mathematician for many years to come, for mathematics and for the sake of all of us, I am sure.

Speech of Heinz Langer By Heinz Langer, Vienna, March 4, 2004

Magniﬁzenz, Spectabilis, Inge and Aad, Ladies and Gentlemen! I feel very honoured by all these nice words, thank you very much indeed. And I am deeply touched by the fact that so many friends and colleagues have come to this conference, although at the beginning it was intended to be just a small meeting. My thanks go to the organizers, Institutsvorstand Inge Troch, Harald Woracek, Annemarie Luger, Matthias Langer, and Fritz Vogl, who have put a lot of eﬀorts in bringing us together for these days. Many things have been said about the 45 years of my life as a mathematician. All this may look very smooth from outside, but in fact I am not somebody who likes to make long-term plans for the future. Looking back, I see two special features which have determined my life. The ﬁrst one is that I often made decisions which at the time were rather unpopular and not understandable for others. Let me give three examples. At the time of my diploma at the end of the 1950’s, under the strong inﬂuence of Bourbaki, abstract linear topological spaces and, in particular, locally convex spaces were very much in fashion. However, I chose a more classical topic, namely operator theory in Hilbert spaces which at that time by some colleagues was considered a bit outdated, but from today’s viewpoint it turned out to be a very good decision. And I am very grateful to my teacher P.H. M¨ uller that he proposed to me to study indeﬁnite inner product spaces. Another example is my stay in Odessa at the beginning of the sixties. Krein had invited me to spend a year with him within an existing exchange programme between the GDR and the Soviet Union. Oﬃcially the German-Soviet friendship was a big issue, but many people were feeling diﬀerently, and not all of my friends could understand why I wanted to follow this invitation. But I insisted: even after my ﬁrst application in 1960 had been rejected for political reasons, I tried again and succeeded one year later in 1961. In fact, this year in Odessa was one of the most important turning points of my life. I was deeply impressed by Krein’s personality, and his school of Functional Analysis in Odessa did not only inﬂuence me strongly as a mathematician, but was also the origin of true friendships which last until today. And I am very happy that some of these friends are here. The third example is my decision not to return to the GDR in October 1989 and to give up a secure position for a rather uncertain future. But, of course, the price for this security was very high, and this brings me to the second special feature, which I mentioned before: during a great part of my life I had to ﬁnd ways to cope with the boundary conditions imposed by the political system of the GDR. So it was not always possible to do what you wanted and when you wanted it.

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For example, – I was not allowed to study physics (which was maybe not the worst thing in the end), – I was allowed to go to Odessa only one year later, – contacts to colleagues in the West were very restricted, and permissions to travel abroad were often denied or delayed, like my ﬁrst trip to Canada: I got the invitation from Israel Halperin to Toronto for 1965. This was not so long after the Berlin wall had been built, and so this ﬁrst invitation was not even considered seriously by the authorities. However, Israel Halperin was very insistent and renewed the invitation for the next year, then even by a letter to the minister. And he succeeded. This was somehow a typical situation, and I am very grateful also to other colleagues who made it possible for me to visit them in the West by not giving up after a ﬁrst or even a second failure: Ilppo Simo Louhivaara who at the time was rector in Jyv¨ askyl¨ a, Rudi Hirschfeld in Antwerp who visited the GDR with an ofﬁcial delegation from Belgium and on this occasion established contacts with me, G¨ oran Borg, who was the rector of the Royal Institute of Technology in Stockholm, and Bj¨ orn Textorius, ˚ Ake Pleijel in Uppsala, Hrvoje Kraljevic and the late Branko Najman in Zagreb, Rien Kaashoek and Israel Gohberg in Amsterdam, and ﬁnally Reinhard Mennicken in Regensburg. All these contacts to colleagues from abroad were very important for me, also because they allowed me to continue to work in operator theory while in Dresden I held a professorship in probability theory. And even in the most critical situation, when I arrived in West Germany in October 1989 and did not have a job, colleagues from there, in particular, Reinhard Mennicken and Albert Schneider from Dortmund helped me to keep my period of unemployment to 3 weeks. Now you may wonder how it came to this happy end in Austria. Well, I don’t know myself, and this is one of the big miracles in my life. Certainly, it had nothing to do with the fact that Austria had been present for me already since my childhood like a fairy tale. I am still keeping an Edelweiss in Meyers Ostalpen from my father which he found in 1926 at the Karlingerboden near Kaprun, and in our living room we had a painting called “Im Stubaital”. After the Second world war it was basically impossible to visit Austria from East Germany, and my ﬁrst visit in 1955 via West Germany was illegal from the point of view of GDR and kind of adventurous. Hitchhiking with a friend from Munich to Salzburg, but in separate cars, we had agreed to meet at a bridge over the Salzach mentioned in my Baedeker from the 1920’s. I arrived ﬁrst, but the bridge had disappeared . . . . Only 35 years later, in 1990, I could return to Austria, when I applied for the Chair previously held by Edmund Hlawka. That I ﬁnally got this position was for me like a dream came through. In the past almost 13 years here at the University of Technology in Vienna I had the freedom to follow my own scientiﬁc interests, to build up a research group in operator theory, and to cooperate with colleagues from all over the world. I am

Speech of Heinz Langer

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very grateful to the University and the former Institute of Analysis and Technical Mathematics for giving me this opportunity and supporting me in many ways. It has always been a pleasure to work in the friendly atmosphere here. And I also enjoyed teaching several generations of students of electrical engineering and mathematics. You may now have got the impression that I am just another example of a ¨ Saxonian as described by Franz Grillparzer in his “Loblied auf Osterreich” where it reads S’ist m¨ oglich, daß in Sachsen und beim Rhein Es Leute gibt, die mehr in B¨ uchern lasen than people in Austria, I should add. However, I really enjoy living in Vienna with all its theaters and museums, opera houses and concert halls, caf´es and vineyards, and nearby mountains and valleys. And, being attentive, you can even ﬁnd a number of close cultural ties between Vienna and Dresden, like the Albertina or Gottfried Semper’s architecture. I was happy when I received the Austrian citizenship 13 years ago. Indeed, I do now share the passionate love of the Austrians for this beautiful country and its rich culture. And I have at least tried to adopt the Austrian mentality which in his “Loblied” Grillparzer describes as follows: Allein, was not tut und was Gott gef¨ allt, Der klare Blick, der oﬀ’ne, richt’ge Sinn, ¨ Da tritt der Osterreicher hin vor jeden, denkt sich sein Teil und l¨ aßt die andern reden!

Programme of the “Colloquium on Operator Theory” Thursday, 4 March

9’15

Opening: H. Kaiser (Vice-Rector of the Vienna University of Technology) D. Dorninger (Dean of the Faculty for Mathematics and Geoinformation) I. Troch (Head of the Institute for Analysis and Scientiﬁc Computing) Laudation: A. Dijksma Heinz Langer

10’30 I. Gohberg: Continuous analogue of orthogonal polynomials Coﬀee break Chair: A. Luger 11’30 C. Tretter: Spectral theory of block operator matrices and applications in mathematical physics 12’15 V. Adamyan: Matrix continuous analogues of orthogonal trigonometric polynomials 12’45 Organizational remarks, conference photo Lunch break Chair: R. Mennicken 14’30 P. Lancaster: An inverse quadratic eigenvalue problem 15’00 H. de Snoo: Singular Sturm–Liouville problems nonlinear in the eigenvalue parameter ¨ck: Indeﬁnite canonical systems and 15’30 M. Kaltenba the inverse spectral theorem Coﬀee break Chair: A. Fleige 16’30 F. Szafraniec: q-disease in operator theory: some cases 17’00 B. Textorius: Directing mappings in Krein spaces 17’30 H. Winkler: Isometric isomorphisms between strings and canonical systems

Programme

xxi

Friday, 5 March Chair: B. Kirstein 9’00

A. Dijksma: The algorithm of Issai Schur in an indeﬁnite setting

9’45

¨ rster: On matrix and operator polynomials with K.-H. Fo nonnegative coeﬃcients Coﬀee break

Chair: O. Staﬀans 11’00 A. Ran: Some remarks on LQ-optimal control: asymptotics and the inverse problem 11’30 A. Gheondea: The indeﬁnite Caratheodory problem 12’00 D. Alpay: Rational functions and backward shift operators in the hyperholomorphic case Coﬀee break Chair: B. Textorius 13’00 M. Brown: Inverse resonance problems for the Sturm–Liouville problem and for the Jacobi matrix 13’30 P. Binding: Oscillation of indeﬁnite Sturm–Liouville eigenfunctions ¨ ller: The spectrum of the multiplication operator 14’00 M. Mo associated with a family of operators in a Banach space Afternoon in Vienna Conference dinner

xxii

Programme

Saturday, 6 March, morning Chair: M. Langer 9’00

P. Kurasov: Pontryagin type models for soliton potentials: inverse scattering method for operator extensions Chair: H. de Snoo

9’30

10’00 10’30

D. Volok: De Branges– Rovnyak spaces and Schur functions: the hyperholomorphic case X. Mary: Subdualities and associated (reproducing) kernels

S. Hassi: Boundary relations and Weyl families of symmetric operators

Y. Shondin: Pontryagin space boundary value problems for a singular diﬀerential expression

V. Strauss: On J-symmetric operators with square similar to a bounded symmetric operator

A. Batkai: Polynomial stability of operator semigroups

Coﬀee break Chair: A. Ran 11’30

´ B. Curgus: Indeﬁnite Sturm–Liouville problems with eigenparameter dependent boundary conditions Chair: P. Binding

12’00

12’30

C. Trunk: Spectral points of type π+ for closed operators in Krein spaces J. Behrndt: Finite-dimensional perturbations of locally deﬁnitizable selfadjoint operators in Krein spaces

D. Popovici: Moment theorems for commuting multi-operators H. Lundmark: Direct and inverse spectral problem for a non-selfadjoint third order generalization of the discrete string equation

Lunch break

Programme

Saturday, 6 March, afternoon Chair: C. Tretter 14’30

V. Pivovarchik: Shifted Hermite–Biehler functions Chair: C. Trunk

15’00

G. Wanjala: The Schur transform at the boundary point z = 1

15’30

L. K´ erchy: Canonical factorization of vectors with respect to an operator

A. Lasarow: Some basic facts on orthogonal rational matrix-valued functions on the unit circle L. Mikayelyan: Orthogonal polynomials on the unit circle with respect to a rational weight function

Coﬀee break Chair: H. Woracek 16’30 17’15

´ri: The extension problem for positive deﬁnite functions. Z. Sasva A historical survey M. Kaashoek: Metric constrained interpolation problems and control theory Closing

xxiii

Participants of the “Colloquium on Operator Theory”

28

36

59

34

6

53

26

46

19

4

9

37

54

50

24

33

55

8

42

14

3

15

57

16

25

35

10

39

41

43

49

23

1

40

44

22

27

45

18

51

5

13

7 29

21

30

32 2

12 47 20

38

17

58

31

56

Participants xxv

xxvi

Participants

List of Participants 1. Vadim Adamyan, I.I. Mechnikov Odessa National University, Ukraine, [email protected], [email protected] 2. Daniel Alpay, Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel, [email protected] 3. Andras Batkai, Department of Applied Analysis, ELTE TTK, Budapest, Hungary, [email protected] 4. Jussi Behrndt, Institut f¨ ur Mathematik, TU-Berlin, Germany, [email protected] 5. Christa Binder, Institut f¨ ur Analysis und Scientiﬁc Computing, TU Wien, Austria, [email protected] 6. Paul Binding, Department of Mathematics and Statistics, University of Calgary, Canada, [email protected] 7. Bernhard Bodenstorfer, Frankfurt/Main, Germany, [email protected] 8. Malcolm Brown, Department of Computer Science, Cardiﬀ University, United Kingdom, [email protected] ´ 9. Branko Curgus, Department of Mathematics, Western Washington University, United States, [email protected] 10. Aad Dijksma, Department of Mathematics, University of Groningen, The Netherlands, [email protected] 11. David Eschw´e, Wien, Austria, [email protected] 12. Andreas Fleige, Dortmund, Germany, [email protected] 13. Karl-Heinz F¨orster, Institut f¨ ur Mathematik, TU-Berlin, Germany, [email protected] 14. Aurelian Gheondea, Department of Mathematics, Bilkent University, Ankara, Turkey, [email protected] 15. Israel Gohberg, School of Mathematical Sciences, Tel-Aviv University, Israel, [email protected]

Participants

xxvii

16. Seppo Hassi, Department of Mathematics and Statistics, University of Vaasa, Finland, [email protected] 17. Rien Kaashoek, Department of Mathematics, Vrije Universiteit, Amsterdam, The Netherlands, [email protected] 18. Michael Kaltenb¨ack, Institut f¨ ur Analysis und Scientiﬁc Computing, TU Wien, Austria, [email protected] 19. Victor Katsnelson, Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot, Israel, [email protected] 20. L´aszl´o K´erchy, University of Szeged, Hungary, [email protected] 21. Bernd Kirstein, Mathematisches Institut, Universit¨at Leipzig, Germany, [email protected] 22. Hrvoje Kraljevic, Zagreb, Croatia, [email protected] 23. Pavel Kurasov, Department of Mathematics, Lund Institute of Technology, Sweden, [email protected] 24. Peter Lancaster, University of Calgary, Canada, [email protected] 25. Heinz Langer, Institut f¨ ur Analysis und Scientiﬁc Computing, TU Wien, Austria, [email protected] 26. Matthias Langer, Institut f¨ ur Analysis und Scientiﬁc Computing, TU Wien, Austria, [email protected] 27. Andreas Lasarow, Departement Computerwetenschappen, K. U. Leuven, Heverlee, Belgium, [email protected] 28. Annemarie Luger, Institut f¨ ur Analysis und Scientiﬁc Computing, TU Wien, Austria, [email protected] 29. Hans Lundmark, Department of Mathematics, Link¨ oping University, Sweden, [email protected] 30. Xavier Mary, ENSAE – CREST LS, 3, avenue Pierre Larousse, 92245 Malakoﬀ Cedex, Paris, France, [email protected]

xxviii

Participants

31. Vladimir Matsaev, School of Mathematics, Tel Aviv University, Israel, [email protected] 32. Reinhard Mennicken, Naturwissenschaftliche Fakult¨ at I, Universit¨ at Regensburg, Germany, [email protected] 33. Levon Mikaelyan, Department of Informatics and Applied Mathematics, Yerevan State University, Armenia, [email protected] 34. Manfred M¨ oller, School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa, [email protected] 35. Maria Magdalena Nafalska, Institut f¨ ur Mathematik, TU-Berlin, Germany, m [email protected] 36. Vjacheslav Pivovarchik, Odessa State Academy of Structure and Architecture, Ukraine, [email protected], [email protected] 37. Dan Popovici, Department of Mathematics and Computer Science, University of the West Timi¸soara, Romania, [email protected] 38. Andre Ran, Department of Mathematics/FEW, Vrije Universiteit, Amsterdam, The Netherlands, [email protected] 39. Adrian Sandovici, Department of Mathematics, University of Groningen, The Netherlands, [email protected] 40. Zolt´an Sasv´ari, Institut f¨ ur Mathematische Stochastik, TU Dresden, Germany, [email protected] 41. Wilfried Schenk, Institut f¨ ur Mathematische Stochastik, TU Dresden, Germany, [email protected] 42. Yuri G. Shondin, Faculty of Physics, Nizhny Novgorod State Pedagogical University, Russia, [email protected] 43. Henk de Snoo, Department of Mathematics, University of Groningen, The Netherlands, [email protected] 44. Olof Staﬀans, Abo Akademi, Finland, [email protected]

Participants

xxix

45. Vladimir Strauss, Depto de Matem´aticas Puras y Aplicadas, Universidad Sim´on, Caracas, Venezuela, [email protected] 46. Franciszek Hugon Szafraniec, Instytut Matematyki, Uniwersytet Jagiello´ nski, Krak´ ow, Poland, [email protected], [email protected] 47. Bj¨ orn Textorius, Department of Mathematics, Link¨ oping University, Sweden, [email protected] 48. Igor Tkachenko, Polytechnic University of Valencia, Spain, [email protected] 49. Christiane Tretter, FB 3 – Mathematik, Universit¨ at Bremen, Germany, [email protected] 50. Inge Troch, Institut f¨ ur Analysis und Scientiﬁc Computing, TU Wien, Austria, [email protected] 51. Carsten Trunk, Institut f¨ ur Mathematik, TU-Berlin, Germany, [email protected] 52. Fritz Vogl, Institut f¨ ur Analysis und Scientiﬁc Computing, TU Wien, Austria, [email protected] 53. Dan Volok, Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel, [email protected] 54. Markus Wagenhofer, FB 3 – Mathematik, Universit¨ at Bremen, Germany, [email protected] 55. Gerald Wanjala, Department of Mathematics, University of Groningen, The Netherlands, [email protected] 56. Henrik Winkler, Department of Mathematics, University of Groningen, The Netherlands, [email protected] 57. Monika Winklmeier, FB 3 – Mathematik, Universit¨ at Bremen, Germany, [email protected] 58. Harald Woracek, Institut f¨ ur Analysis und Scientiﬁc Computing, TU Wien, Austria, [email protected] 59. Rosalinde Pohl (Secretary)

Bibliography of Heinz Langer In the volume OT 106 of the series “Operator Theory: Advances and Applications” which appeared in 1998, a bibliography of Heinz Langer up to this year was compiled. The following list updates this bibliography; it collects his recent papers. [131] Direct and inverse spectral problems for generalized strings, Integral Equations Operator Theory 30 (1998), 409–431 (with H. Winkler) [132] A factorization result for generalized Nevanlinna functions, Integral Equations Operator Theory 36 (2000), 121–125 (with A. Dijksma, A. Luger, and Yu. Shondin) [133] Classical Nevanlinna–Pick interpolation with real interpolation points, Oper. Theory Adv. Appl. 115 (2000), 1–50 (with D. Alpay and A. Dijksma) [134] The spectral shift function for certain block operator matrices, Math. Nachr. 211 (2000), 5–24 (with V. Adamjan) [135] A class of 2 × 2–matrix functions, Glas. Matem. Ser. III 35(55) (2000), 149–160 (with A. Luger) [136] Linearization and compact perturbation of self-adjoint analytic operator functions, Oper. Theory Adv. Appl. 118 (2000), 255–285 (with A. Markus and V. Matsaev) [137] Self-adjoint diﬀerential operators with inner singularities and Pontryagin spaces, Oper. Theory Adv. Appl. 118 (2000), 105–175 (with A. Dijksma, Yu. Shondin, and C. Zeinstra) [138] On singular critical points of positive operators in a Krein space, Proc. Amer. ´ Math. Soc. 128 (2000), 2621–2626 (with B. Curgus and A. Gheondea) [139] Variational principles for real eigenvalues of selfadjoint operator pencils, Integral Equations Operator Theory 38 (2000), 190–206 (with D. Eschw´e and P. Binding) [140] Dissipative eigenvalue problems for a Sturm–Liouville operator with a singular potential, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 1237–1257 (with B. Bodenstorfer and A. Dijksma) [141] Diagonalization of certain block operator matrices and applications to Dirac operators, Oper. Theory Adv. Appl. 122 (2001), 331–358 (with C. Tretter) [142] Existence and uniqueness of contractive solutions of some Riccati equations, J. Funct. Anal. 179 (2001), 448–473 (with V. Adamyan and C. Tretter) [143] A spectral theory for a λ-rational Sturm–Liouville problem, J. Diﬀerential Equations 171 (2001), 315–345 (with V. Adamyan and M. Langer) [144] Compact perturbation of deﬁnite type spectra of self–adjoint quadratic operator pencils, Integral Equations Operator Theory 39 (2001), 127–152 (with V. Adamyan and M. M¨ oller)

Bibliography

xxxi

[145] A new concept for block operator matrices: The quadratic numerical range, Linear Algebra Appl. 330 (2001), 89–112 (with A. Markus, V. Matsaev, and C. Tretter) [146] Corners of numerical ranges, Oper. Theory Adv. Appl. 124 (2001), 385–400 (with A. Markus and C. Tretter) [147] Elliptic eigenvalue problems with eigenparameter dependent boundary conditions, J. Diﬀerential Equations 174 (2001), 30–54 (with P. Binding, R. Hryniv, and B. Najman) [148] A relation for the spectral shift function of two self–adjoint extensions, Oper. Theory Adv. Appl. 127 (2001), 437–445 (with V. A. Yavrian and H. de Snoo) [149] The Schur algorithm for generalized Schur functions I: coisometric realizations, Oper. Theory Adv. Appl. 129 (2001), 1–36 (with D. Alpay, T. Ya. Azizov, and A. Dijksma) [150] Invariant Subspaces of inﬁnite-dimensional Hamiltonians and solutions of the corresponding Riccati equations, Oper. Theory Adv. Appl. 130 (2001), 235–254 (with A. C. M. Ran and B. A. van de Rotten) [151] On the Loewner problem in the class Nκ , Proc. Amer. Math. Soc. 130 (2002), 2057–2066 (with D. Alpay and A. Dijksma) [152] Triple variational principles for eigenvalues of self-adjoint operators and operator functions, SIAM J. Math. Anal. 34 (2002), 228–238 (with D. Eschw´e) [153] Variational principles for eigenvalues of block operator matrices, Indiana Univ. Math. J. 51 (2002), 1427–1459 (with M. Langer and C. Tretter) [154] The Schur algorithm for generalized Schur functions II: Jordan chains and transformations of characteristic functions, Monatsh. Math. 138 (2003), 1–29 (with D. Alpay, T. Ya. Azizov, and A. Dijksma) [155] Self-adjoint block operator matrices with non-separated diagonal entries and their Schur complements, J. Funct. Anal. 199 (2003), 427–451 (with A. Markus, V. Matsaev, and C. Tretter) [156] The Schur algorithm for generalized Schur functions III: J-unitary matrix polynomials on the circle, Linear Algebra Appl. 369 (2003), 113–144 (with D. Alpay, T. Ya. Azizov, and A. Dijksma) [157] Continuous embeddings, completions and complementation in Krein spaces, ´ Rad. Mat. 12 (2003), 37–79 (with B. Curgus) [158] A basic interpolation problem for generalized Schur functions and coisometric realizations, Oper. Theory Adv. Appl. 143 (2003), 39–76 (with D. Alpay, T. Ya. Azizov, A. Dijksma, and G. Wanjala) [159] Rank one perturbations at inﬁnite coupling in Pontryagin spaces, J. Funct. Anal. 209 (2004), 206–246 (with A. Dijksma and Yu. Shondin) [160] Continuations of Hermitian indeﬁnite functions and canonical systems: an example, Methods Funct. Anal. Topology 10 (2004), 39–53 (with M. Langer and Z. Sasv´ ari)

xxxii

Bibliography

[161] Solution of a multiple Nevanlinna–Pick problem via orthogonal rational functions, J. Math. Anal. Appl. 293 (2004), 605–632 (with A. Lasarow) [162] Oscillation results for Sturm–Liouville problems with an indeﬁnite weight function, J. Comput. Appl. Math. 171 (2004), 93–101 (with P. Binding and M. M¨ oller) [163] Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions, Linear Algebra Appl. 387 (2004), 313–342 (with D. Alpay and A. Dijksma) [164] The Schur algorithm for generalized Schur functions IV: unitary realizations, Oper. Theory Adv. Appl. 149 (2004), 23–45 (with D. Alpay, T. Ya. Azizov A. Dijksma, and G. Wanjala) [165] A Krein space approach to PT-symmetry, Czechoslovak J. Phys 54 (2004), 1113–1120 (with C. Tretter) [166] Minimal realizations of a scalar generalized Nevanlinna function related to their basic factorization, Oper. Theory Adv. Appl. 154 (2004), 69–90 (with A. Dijksma, A. Luger, and Yu. Shondin) [167] Partial non-stationary perturbation determinants, Oper. Theory Adv. Appl. 154 (2004), 1–18 (with V. Adamyan) [168] Spectrum of deﬁnite type of self-adjoint operators in Krein spaces, Linear and Multilinear Algebra 53 (2005), 115–136 (with M. Langer, A. Markus, and C. Tretter) [169] Spectral problems for operator matrices, Math. Nachr. 278 (2005), 1408– 1429 (with A. Batkai, P. Binding, A. Dijksma, and R. Hryniv) [170] Bounded normal operators in a Pontryagin space, Oper. Theory Adv. Appl. 162 (2006), 231–251 (with F. Szafraniec) [171] Partial non-stationary perturbation determinants for a class of J-symmetric operators, Oper. Theory Adv. Appl. 162 (2006) 1–18 (with V. Adamyan)

Operator Theory: Advances and Applications, Vol. 163, 1–22 c 2005 Birkh¨ auser Verlag Basel/Switzerland

General Solution of the Stieltjes Truncated Matrix Moment Problem Vadim M. Adamyan and Igor M. Tkachenko To Heinz Langer with admiration and gratitude for fruitful co-operation

Abstract. The description of all solutions of the truncated Stieltjes matrix moment problem consisting in ﬁnding all s × s matrix measures dσ (t) on [0, ∞) with given ﬁrst 2n + 1 power s × s matrix moments (Cj )n j=0 is obtained in a general case, when the block Hankel matrix Γn := (Cj+k )n j,k=0 may be non-invertible. Special attention is paid to the description of canonical solutions for which dσ (t) is a sum of at most sn + s point matrix “masses” with the minimal sum of their ranks. Mathematics Subject Classiﬁcation (2000). Primary 30E05, 30E10; Secondary 82C70, 82D10. Keywords. Stieltjes moments problem, matrix functions, Nevanlinna’s formula.

1. Introduction In this paper we consider the following problem: Given a set of Hermitian s × s matrices {C0 , C1 , C2 , . . . , C2n },

n = 0, 1, 2, . . . .

Find all non-negative matrix measures dσ (t) such that ∞ tk dσ (t) = Ck , k = 0, 1, 2, . . . , 2n.

(1.1)

(1.2)

0

This is a matrix version of the well-known Stieltjes truncated moment problem. The Stieltjes problem was considered in diﬀerent settings in many monographs and papers starting from the classical memoirs by Stieltjes himself [14, 15]. Classical The authors appreciate the fair work and helpful suggestions and corrections of the referees of this paper.

2

V.M. Adamyan and I.M. Tkachenko

results on the topic are contained in the books [3, 10, 13] and the papers [12, 11]; more recent developments on the truncated moment problems can be found in [5, 7, 6, 1, 8, 9]. The above matrix problem was considered recently in [2]. The main results of that paper can be summarized in the following two theorems. Theorem 1.1. A system of Hermitian matrices {C0 , C1 , C2 , . . . , C2n }, 0, 1, 2, . . . , admits the representation ∞ tk dσ (t) = Ck , k = 0, 1, 2, . . . , 2n,

n =

(1.3)

0

if and only if a) the block Hankel matrix Γn := (Ck+j )nk,j=0 is non-negative; b) for any set ξ 0 , . . . , ξr ∈ Cs , 0 ≤ r ≤ n − 1, and with (ξ, η) being the standard scalar product in Cs , the condition r (Cj+k ξ k , ξ j ) = 0 (1.4) j,k=0

implies

r j,k=0

(Cj+k+2 ξ k , ξ j ) = 0;

(1.5)

(1)

c) the block Hankel matrix Γn−1 := (Ck+j+1 )n−1 k,j=0 is non-negative and for any set ξ 0 , . . . , ξ r ∈ Cs , 0 ≤ r ≤ n − 1, the condition r (Cj+k+1 ξ k , ξ j ) = 0 (1.6) j,k=0

implies (1.5). In what follows we will denote by R the Nevanlinna class of holomorphic dissipative s × s matrix functions on the upper half-plane, i.e., matrix functions with non-negative imaginary parts and by S the subset of R consisting of all Nevanlinna s × s matrix functions t (z), Imz > 0, which admit the integral representation ∞ t (z) = 0

1 dρ (t) t−z

with a non-decreasing s × s matrix function (Cs operator) ρ (t) such that the condition ∞ d (ρ (t) h, h) < ∞ 0

holds for any h ∈ Cs . Theorem 1.2. Let the conditions of Theorem 1.1 hold and det Γn > 0, and let ∆Ξ (z) be the upper-left block of the matrix function −1 (1) Γn , Γn ΓΞ;n (z) − zΓn

Stieltjes Truncated Matrix Moment Problem where

⎛ ⎜ ⎜ (1) ΓΞ;n (z) := ⎜ ⎝

Cn+1 .. .

(1)

Γn−1 Cn+1

. . . C2n

C2n Ξ (z)−1 + z

3 ⎞ ⎟ ⎟ ⎟ ⎠

and Ξ ∈ S. Then the relation ∞ 1 dσ Ξ (t) = ∆Ξ (z) , Imz > 0, t−z 0

by means of the Stieltjes inversion formula (σ Ξ (β)h, h) − (σ Ξ (α)h, h) = − π1 lim η↓0

β α

Im (∆Ξ (t + iη)h, h) dt,

σ Ξ (α − 0) = σ Ξ (α + 0), σ Ξ (β − 0) = σ Ξ (β + 0), h ∈ Cs , −∞ < α < β < ∞, establishes a one-to-one correspondence between a set of all solutions σ Ξ (t) of the truncated Stieltjes matrix moment problem with the given moments {C0 , . . . , C2n } and the subset of all Nevanlinna matrix functions Ξ from S which satisfy the condition n−1 −1 Ξ (λ) − Cn+j+1 bjk (λ)Cn+k+1 > 0, λ < 0 , (1.7) j,k=0

with s × s matrices bjk (z) deﬁned by the equality −1 (1) Γn−1 − z = (bjk (z))n−1 / [0, ∞). j,k=0 , z ∈ The aim of the present work is to describe all solutions of the Stieltjes truncated matrix moment problem in the degenerate case: det Γn = 0. In Section 2 following ideas of M.G. Krein [11] we show how the truncated matrix moment problems can be reduced to problems of the extension theory for symmetric operators. In Section 3 we consider the so-called completely degenerate truncated moment problem, we ﬁnd here its unique solution in an explicit form. A special class of the canonical solutions of the truncated Stieltjes problem is described in Section 4. For this class of solutions the sought non-decreasing matrix function σ (t) has no more than ns + s points of growth. An algorithm for the construction of such solutions is given here as well. In the last section we give, omitting the detailed proof, the description of all solutions of the problem under consideration.

2. Reduction to the extension theory problem From now on we will assume that the conditions of Theorem 1.1 hold. Notice that due to the conditions a) and c) of this theorem, the quadratic forms of the matrices Cj are non-negative, or brieﬂy Cj ≥ 0, j = 0, . . . , 2n. Without loss of

4

V.M. Adamyan and I.M. Tkachenko

generality we can assume that all matrix moments Cj are invertible, i.e., that Cj > 0, j = 0, . . . , 2n. Indeed, let Nj ⊂ Cs be the null-spaces of Cj . Due to the condition a) of Theorem 1.1 for any vector η ∈ Cs the equality C0 η = 0 implies Cj η = 0, j = 1, . . . , 2n. Besides, since for any vector η ∈ Cs and any solution σ (t) of the moment problem all integrals ∞ tk d(σ (t) η, η), 1 ≤ k ≤ 2n 0

vanish simultaneously, any equality Cj η = 0 implies Ck η = 0, 1 ≤ k ≤ 2n. Hence, N0 ⊂ N1 = Nk , k = 1, . . . , 2n and for a suitable basis in Cs the matrices Cj can be reduced to the form

0 0

j 0 C C , j = 1, . . . , 2n, (2.1) ; C = C0 = j

0 0 0 0 C

j > 0, j = 0, 1, . . . , 2n. where by construction det C

j is solvIf the truncated Stieltjes problem for invertible matrix moments C able, then the initial problem is solvable as well, and its general solution σ (t) can be presented as

(t) σ 0 σ (t) = U U∗ ,

0 ϑ (t) 0 C where U is a ﬁxed s × s-unitary matrix, 0, t ≤ 0, ϑ (t) = 1, t > 0,

(t) runs through the set of solutions of the truncated Stieltjes matrix moment and σ

j. problem for the moments C Let a non-decreasing matrix function σ (t), 0 ≤ t < ∞, be a solution of the truncated Stieltjes matrix moment problem, i.e., ∞ tk dσ (t) = Ck , k = 0, 1, 2, . . . , 2n. (2.2) 0

Consider the set of continuous vector functions f (t) , 0 ≤ t < ∞, with values in Cs , which satisfy the condition ∞ (dσ (t) f (t) , f (t)) < ∞. (2.3) 0

Construct a pre-Hilbert space L of such vector functions taking the bilinear functional ∞ f , g = (dσ (t) f (t) , g (t)) (2.4) 0

as the scalar product. Notice that by (1.2) the vector polynomials f (t) = ξ 0 + t · ξ 1 + · · · + tn · ξn ,

ξ 0 , . . . , ξ n ∈ Cs ,

(2.5)

of degree n belong to L. We will denote the linear subset of these polynomials by Pn .

Stieltjes Truncated Matrix Moment Problem

5

Let L0 be the subspace of L consisting of all vector functions f such that f := f , f = 0. If g = f + f0 , where f ∈ L, f0 ∈ L0 , than, due to the Schwartz inequality f , f0 = 0 and, hence, g = f . Let us denote by L1 the factor space L/L0 . For the class of = f +L0 of this factor space we set elements g gL1 = f . Taking the completion of L1 with respect to this norm, we obtain the Hilbert space L2σ (Cs ). We keep the same symbol ., . for the scalar product in L2σ (Cs ). Let Ln be the subspace of L2σ (Cs ) generated by the subset of vector polynomials Pn . By (2.2) and (2.4) for f , g ∈Pn , f (t) =

n

tr · ξ r ,

g (t) =

l=0

n

tr · η r ,

ξ 0 , . . . , η n ∈ Cs ,

l=0

we have: f , g =

n

(Cj+k ξ k , η j ).

(2.6)

j,k=0

Therefore the scalar products on Ln in the spaces L2σ (Cs ) for all non-decreasing ˜ (t) satisfying matrix functions σ (t) which satisfy (1.2) must coincide. Take any σ

0 of multiplication by the (1.2) and consider the non-negative symmetric operator A independent variable t on the subset of all continuous compact vector functions in

of A

0 is a self-adjoint operator [4]. Let us the related space L2σ (Cs ). The closure A denote by Ln−1 the subspace of Ln generated by vector polynomials of the degree

to Ln−1 is a non-negative symmetric ≤ n − 1. The restriction A0 of the operator A

operator which by deﬁnition of A actually does not depend on the choice of a ˜ (t) solution of the truncated Stieltjes moment problem. Therefore each solution σ

of A0 . of this problem generates some non-negative self-adjoint extension A

t , 0 ≤ t < ∞, be now the spectral function of some simple non-negative Let E

1 of A0 . For the canonical orthonormal basis {e1 , . . . , es } in Cs we extension A introduce the set of classes { e10 , . . . , es0 } ⊂ Ln which contain zero degree vector monomials e10 (t) ≡ e1 , . . . , es0 (t) ≡ es , ˜ (t) = respectively. Let us consider the non-decreasing s × s matrix function σ ( σµν (t))sµ,ν=1 , 0 ≤ t < ∞,

t σ

µν (t) := E eν0 , eµ0 , 1 ≤ µ, ν ≤ s. (2.7) Ln

for the classes { e1k , . . . , esk } ⊂ Ln which contain the By deﬁnition of A0 and A, vector monomials e1k (t) = tk e1 , . . . , esk (t) = tk es , 0 ≤ k ≤ n, 1 A self-adjoint extension A of A acting in a Hilbert space H ⊇ L is called simple if A has no n 0 invariant subspaces in H Ln .

6

V.M. Adamyan and I.M. Tkachenko

it holds:

k eµk = Ak0 eµ0 = A eµ0 , 1 ≤ µ ≤ s, Hence, for 0 ≤ j, k ≤ n and 1 ≤ µ, ν ≤ s we have (Cj+k )µ,ν

0 ≤ k ≤ n.

= (Cj+k eν , eµ )Cs =

k

j eν0 , A eµ0 eµj Ln = A = eνk , Ln ∞ ∞

t eν0 , = tj+k d E eµ0 = tj+k d σµν (t) . Ln

0

0

We proved the following Proposition 2.1. The formula

t ˜ µν (t) = E eν0 , eµ0 σ

Ln

, 0 ≤ t < ∞,

(2.8)

t establishes a one-to-one correspondence between the set of spectral functions E

of A0 and the set of solutions σ ˜ (t) of the of non-negative simple extensions A truncated Stieltjes matrix moment problem.

3. Solution in the completely degenerate case The non-decreasing matrix functions σ (t) which satisfy (2.2) and for which L2σ (Cs ) =Ln are called canonical. Equivalently, solutions of the truncated matrix Stieltjes prob t is the spectral function of lem given by the expression (2.7) are canonical if E

of A0 , i.e., a non-negative some non-negative canonical self-adjoint extension A self-adjoint extension without leaving Ln . Due to (2.6), a canonical σ (t) is a nondecreasing matrix function which has only a ﬁnite number of points of growth and the sum of the ranks of all jumps of σ at such points is ≤ ns. It was proven in [2] that the set of canonical solutions of the truncated matrix Stieltjes moment problem is non-empty whenever this problem is solvable, i.e., whenever the conditions of Theorem 1.1 hold. Formula (2.8) which establishes a one-to-one correspondence between the set of non-negative self-adjoint extensions of A0 without leaving Ln and the set of canonical solutions of the Stieltjes problem, makes it possible to ﬁnd, under the conditions of Theorem 1.1, an explicit algebraic formula for the description of the sought canonical solutions. To this end we can use as a starting point (2.7) and the relation ∞ ∞ 1 1

t d σµν (t) = d E eν0 , eµ0 = (3.1) t−z t−z Ln 0 0

− z)−1 = (A eν0 , eµ0 . Ln

From now on we will assume that det Γn = 0, i.e., we will consider the degenerate case of the above problems.

Stieltjes Truncated Matrix Moment Problem

7

Let Ln (Cs ) denote the (n + 1)s-dimensional linear space of column vectors ξ=

ξ0

···

ξn

, ξ 0 , . . . , ξ n ∈ Cs ,

(3.2)

( stands for the transposition operation) with the scalar product (ξ, η) =

n

j=0

ξj , η j

Cs

.

We will denote as before by Ln the same linear vector space but with the scalar product n ξ, η = (Γn ξ, η) = Cj+k ξk , η j C . j,k=0

s

Ln was considered above as the space of vector polynomials. Let Ln−1 (Cs ) be the subspace of Ln (Cs ) which consists of vectors (3.2) but with ξ n = 0 and let Nn = Ln (Cs ) Ln−1 (Cs ) . We denote by Pn the orthogonal projector in Ln (Cs ) onto Nn . In the natural basis of subspaces of Ln (Cs ) this projector is evidently given as the following (n + 1) × (n + 1) block operator matrix ⎞ ⎛ 0 ... 0 ⎟ ⎜ Pn = ⎝ ... . . . ... ⎠ , 0 ...

I

where I is the s × s unit matrix. Let us denote by Qn the orthogonal projector in Ln (Cs ) onto the null-space Mn of the operator Γn and let Q⊥ n = I − Qn . We call the truncated matrix Stieltjes moment problem completely degenerate if Pn Mn = Nn . We will also use the same deﬁnition for non-negative Hankel matrices Γn having the property: Pn Mn = Nn . In the completely degenerate case for each vector ⎛ ⎞ 0 ⎜ .. ⎟ ⎜ ⎟ eni = ⎜ . ⎟ ⎝ 0 ⎠ ei of the canonical orthonormal basis {en1 , . . . , ens } in Nn there is a non-zero vector ⎛ ⎞ ξ ni,0 ⎜ .. ⎟ ⎜ ⎟ ξni = ⎜ . ⎟ ⎝ ξ ni,n−1 ⎠ ei

8

V.M. Adamyan and I.M. Tkachenko

such that

⎛

⎜ ⎜ Γn ξ ni = ⎜ ⎝

Cn .. .

Γn−1 Cn

. . . C2n−1

C2n−1 C2n

⎞⎛

ξ ni,0 ⎟ ⎜ .. ⎟⎜ . ⎟⎜ ⎠ ⎝ ξ ni,n−1 ei

⎞ ⎟ ⎟ ⎟ = 0, i = 1, . . . , s. ⎠

(3.3) Note that at least some of ξni,k = 0 in (3.3), k = 0, . . . , n − 1. Indeed, otherwise we would have that Cn ei = · · · = C2n ei = 0. But by our assumption all matrices C0 , . . . , C2n are invertible, a contradiction. As follows, the class from Ln containing the monomial tn ei contains also the vector polynomial dni (t) = −ξ ni,0 − · · · − tn−1 ξ ni,n−1 . Hence, in the completely degenerate case Ln = Ln−1 . This equality implies that the symmetric operator A0 from Ln−1 into Ln deﬁned above by relations ek = ek+1 , k = 0, . . . , n − 1, A0 actually is self-adjoint. As follows, in the completely degenerate case the solution of the truncated Stieltjes problem is unique. For vectors ξ n0 , . . . , ξ ns appearing in (3.3) let us introduce vector polynomials φi (z) := ξ ni,0 + · · · + z n−1 ξ ni,n−1 + z n ei , i = 1, . . . , s, and the s × s matrix function D(z), the corresponding columns of which are φi (z). Since n D(z) = z n I + o (|z| ) , (3.4) z→∞

the matrix function is invertible everywhere except of at most sn points. Let σ(D) be the spectrum of the matrix function D(z), i.e., the set of those z ∈ C for which there exists a non-zero h ∈ Cs such that D(z)h = 0. Proposition 3.1. In the completely degenerate case the spectrum σ(A0 ) of the nonnegative self-adjoint operator A0 coincides with σ(D) and for z ∈ / σ(D) the resol−1 vent (A0 − z) of A0 acts on a vector polynomial g(t) of degree ≤ n − 1 representing a class of polynomials from Ln−1 by the formula 1 (A0 − z)−1 g (t) = g(t) − D(t)D(z)−1 g(z) . (3.5) t−z Proof. In the completely degenerate case each vector of Ln can be represented as a vector polynomial of degree n − 1. Correspondingly A0 acts on such vector polynomials as the multiplication operator by t. Let us consider the equation (t − z) f (t) = ((A0 − z)f )(t) = g(t), z ∈ / σ(D), (3.6)

Stieltjes Truncated Matrix Moment Problem

9

and assume that the degree of the representing polynomial g(t) is ≤ n and that the degree of the sought polynomial f (t) is ≤ n − 1. If g(z) = 0 then we put −1

f (t) = (t − z)

g(t).

Otherwise we can substitute on the right-hand side of the equation (3.6) instead of g(t) the equivalent polynomial

(t) = g(t) − D(t)D(z)−1 g(z), g and get as a solution of (3.6) the vector polynomial 1 g(t) − D(t)D(z)−1 g(z) , f (t) = (t − z) the degree of which is, evidently, ≤ n − 1. Hence for z ∈ / σ(D) and any vector polynomial g(·) ∈ Ln the equation (3.6) has a solution in the form of a vector polynomial from Ln−1 . Therefore C \ σ(D) belongs to the resolvent set of A0 and −1 (A0 − z) for z ∈ C \ σ(D) is given by (3.5). On the other hand, if z ∈ σ(D), then there is a non-zero vector h ∈ Cs such that D(z)h = 0. Note that by (3.4) D(t)h = tn h + o(tn ) t→∞

and thus D(t)h = 0 identically. If (t − z)−l+1 D(t)h, 1 ≤ l ≤ n, belongs to the zero class in Ln , but f0 (t) := (t − z)−l D(t)h is a non-zero element from Ln , then f0 (t) is a non-trivial solution of the homogeneous equation (A0 − z) f = 0. Remark 3.2. Since A0 is a non-negative operator, we have σ(D) ⊂ [0, ∞). Let σ 0 (t) be the (unique) solution of the truncated Stieltjes matrix moment problem and let ∞ 1 dσ 0 (t) (D(t) − D(z)) . E(z) = (3.7) t−z 0

Note that if D(z) = z n I + z n−1 An−1 + · · · + zA1 + A0 , then E(z) =

n−1

z k Cn−1−k +

0

n−2

z k Cn−2−k An−1 + · · · + C0 A1 .

(3.8)

0

It stems from formula (3.5) applied to the classes which contain the zerodegree polynomials and the expression (3.1) that the following assertion is true. Proposition 3.3. In the completely degenerate case the unique solution σ0 (t) of the truncated Stieltjes matrix moment problem satisﬁes the relation ∞ 1 dσ 0 (t) = −E(z)D(z)−1 . (3.9) t−z 0

10

V.M. Adamyan and I.M. Tkachenko

The point of Proposition 3.3 is that it allows one to calculate “explicitly” the unique solution σ 0 (t) of the truncated Stieltjes moment problem in the completely degenerate case. To this end it is enough to ﬁnd, following the above prescription, the matrix polynomial D(z), take E(z) from (3.8) and apply the Stieltjes inversion formula to (3.9). By this formula for each h ∈ Cs and β > α such that det D(α) = 0, det D(β) = 0, we have β 1 (σ 0 (β)h, h) − (σ 0 (α)h, h) = − lim Im E(t + iη)D(t + iη)−1 h, h dt. (3.10) π η↓0 α

4. Canonical solutions Now we will omit the assumption that the projection Pn Mn of the null-space Mn onto the subspace Nn necessarily covers all this subspace. Theorem 4.1. Let C0 , C1 , C2 , . . . , C2n be any set of Hermitian s×s matrices which satisfy the conditions of Theorem 1.1 and let σ(t) be any solution of the truncated Stieltjes matrix moment problem for this set. Then the extended set of non-negative Hermitian matrices ∞ ∞ C0 , . . . , C2n , C2n+1 = t2n+1 dσ (t) , C2n+2 = t2n+2 dσ (t) (4.1) 0

0

satisﬁes all conditions of Theorem 1.1 with n replaced by n+1 and for the extended set (4.1) the truncated Stieltjes moment problem is completely degenerate if and only if σ(t) is a canonical solution. Proof. Since the Stieltjes problem for the set (4.1) is evidently solvable, this set satisﬁes the conditions of Theorem 1.1. Suppose that σ(t) is a canonical solution of the Stieltjes problem for the set C0 , . . . , C2n . Recall that the solvability of the Stieltjes problem for this set guarantees the existence of canonical σ(t) [2]. By the deﬁnition of canonical solutions for σ(t) we have Ln+1 =Ln =L2σ (Cs ). Therefore for any non-zero vector h ∈ Cs there is a vector polynomial Ph (t) = tn ξhn + · · · + ξh0 such that the vector polynomials tn+1 h and Ph (t) belong to the same class or, more precisely, the polynomial tn+1 h − Ph (t) belongs to the zero-class in L2σ (Cs ). n+1 Since the block Hankel matrix Γn+1 = (Cj+k )j,k=0 is non-negative deﬁnite and (C2n+2 h, h) − (C2n+1 h, ξhn ) − (C2n+1 ξhn , h) − · · · + (C2n+1 ξh0 , ξh0 ) = ∞ dσ (t) tn+1 h − Ph (t) , tn+1 h − Ph (t) = 0, 0

we see that the vector

⎞ −ξh0 ⎜ .. ⎟ ⎜ . ⎟ ⎟ ⎜ ⎝ −ξhn ⎠ h ⎛

(4.2)

Stieltjes Truncated Matrix Moment Problem

11

belongs to the null-space of Γn+1 . As this is true for any h ∈ Cs we conclude that for the set (4.1) the Stieltjes problem is completely degenerate. Let us assume now that the solution σ(t) is such that for the extended set (4.1) the Stieltjes problem is completely degenerate. Then for each vector h ∈ Cs there is a vector of the form (4.2) from the null-space of Γn+1 or, equivalently, there is a vector polynomial tn+1 h − tn ξhn − · · · − ξh0 from the zero-class in L2σ (Cs ). Therefore, for the given solution σ(t) we have Ln+1 = Ln = L2σ (Cs ) This implies that σ(t) is a canonical solution. Theorem 4.1 gives us a hint for the description of all canonical solutions of the Stieltjes moment problem. To this end it is enough ﬁrst to ﬁnd all those extensions C0 , C1 , . . . , C2n , C2n+1 , C2n+2 of the given set of moments C0 , C1 , . . . , C2n , for which corresponding Stieltjes problems turn out to be completely degenerate, and then to ﬁnd unique solutions of the arising completely degenerate problems as it has been done in the previous section. We start the description of the demanded extensions of a given moment set with the following general proposition. Theorem 4.2. Let A be a bounded non-negative self-adjoint operator in a Hilbert space H1 and B be a bounded operator from H1 into a Hilbert space H2 . Put Q(s) = B(A + s)−1 B ∗ , 0 < s < ∞. The following conditions are equivalent: • there are bounded non-negative self-adjoint extensions of the operator A + B : H1 → H1 ⊕ H2 to the Hilbert space H = H1 ⊕ H2 ; • the operator Q0 in H2 deﬁned in a way that Q0 f = lim Q(s)f, s→0+

f ∈ H2 ,

is bounded.

of If these conditions hold, then bounded non-negative self-adjoint extensions A (A + B)|H1 have the form of block operator matrices A B∗

= , (4.3) A B D where D runs through the set of non-negative self-adjoint operators in H2 and satisﬁes the condition D − Q0 ≥ 0.

(4.4)

Proof. Since the operators A and B are bounded, we have that each bounded self-adjoint extension of (A + B)|H1 to H can be represented in the form (4.3)

is non-negative with a bounded self-adjoint D. Note further that an extension A

− λ is strictly positive. By the if and only if for every λ < 0 the operator A

12

V.M. Adamyan and I.M. Tkachenko

Schur-Frobenius factorization formula we have I 0

A−λ= B(A − λ)−1 I A−λ 0 I × 0 0 D − λ − B (A − λ)−1 B ∗

(A − λ)−1 B ∗ I

(4.5) , λ < 0.

is non-negative if and only if Therefore the extension A W (λ) := D − λ − B (A − λ)

−1

B ∗ > 0, λ < 0.

Let the operator Q0 exist and be bounded. Since the operator function W (λ) is

is non-negative if and only non-increasing on the half-axis (−∞, 0), the operator A if the inequality (4.4) holds. Suppose now that for some f ∈ H2 we have lims→0+ (Q(s)f, f ) = +∞. As D is a bounded operator we conclude that for s small enough and the same f ∈ H2 the inequality (W (−s)f, f ) < 0 holds. Therefore there are no non-negative operators among bounded self-adjoint extensions of (A + B)|H1 . Remark 4.3. Let the operators A and B be as in Theorem 4.2 and the space H1 be ﬁnite-dimensional. Then among bounded self-adjoint extensions of (A + B)|H1 to the Hilbert space H = H1 ⊕ H2 there are non-negative operators if and only if the null-space of A is contained in the null-space of B. Remark 4.4. Let us assume that the operator A in Theorem 4.2 has a bounded inverse. Then the condition D ≥ BA−1 B ∗ for a bounded self-adjoint D in (4.3)

given by the block operator is necessary and suﬃcient for the non-negativity of A matrix (4.3). Proposition 4.5. Let operators A : H1 → H1 and B : H1 → H2 be as in Theorem

0 in the Hilbert 4.2 and assume that A has a bounded inverse. Then the operator A space H = H1 ⊕ H2 given as the block operator matrix A B∗

0 = (4.6) A B B ∗ A−1 B is the unique non-negative self-adjoint extension of (A+B)|H1 to H which satisﬁes the condition: P2 MA 0 = {0} ⊕ H2 , (4.7) where P2 is the orthogonal projector in H from H to {0} ⊕ H2 and MA 0 is the

0 . null-space of A

Proof. By assumptions of the proposition and Remark 4.4 the extension (4.6) is bounded, self-adjoint and non-negative with D = BA−1 B ∗ . Let us take an arbitrary vector h ∈ H2 and consider the vector −A−1 B ∗ h

∈ H. h= h

0 It is evident that A h = 0. Hence, (4.7) holds for the extension (4.6).

Stieltjes Truncated Matrix Moment Problem

13

be any bounded non-negative self-adjoint extension which satisﬁes Now, let A the condition (4.7). Then for any h ∈ H2 there is a vector h1 ∈ H1 such that

that is h = h1 + h ∈ H is a null-vector of A, Ah1 + B ∗ h = 0, Bh1 + Dh = 0.

(4.8)

As A is invertible we deduce from (4.8) that the equality −BA−1 B ∗ + D h = 0 is true for any h ∈ H2 . Hence D = BA−1 B ∗ . Having disposed of these preliminary steps, we can now return to the description of all canonical extensions of the truncated Stieltjes moment problem. Let C0 , C1 , C2 , . . . , C2n be any sequence of Hermitian s×s matrices, which satisfy all the conditions of Theorem 1.1. We begin with the non-degenerate case and assume that the non-negative n block Hankel matrix Γn = (Cj+k )j,k=0 is invertible. This implies that the non(1)

n−1

negative block Hankel matrix Γn−1 = (Cj+k+1 )j,k=0 is invertible as well. Indeed, (1)

(2)

if Γn−1 is non-invertible, then by the conditions of Theorem 1.1 the matrix Γn−1 = (2) (Cj+k+2 )n−1 j,k=0 is also non-invertible. But Γn−1 is a diagonal block of the positive deﬁnite matrix Γn , a contradiction. Put ⎛ ⎞ Cn+1 ⎜ ⎟ .. S1 = ⎝ ⎠. . C2n By Theorem 4.2 and Remark 4.4 for any non-negative deﬁnite s × s matrix W the n × n block Hankel matrix ⎛ ⎞ (1) Γ S 1 n−1 −1 ⎝ ⎠ Γ(1) (4.9) n = (1) S∗1 W + S∗1 Γn−1 S1 (1)

is a non-negative deﬁnite n × n block Hankel extension of the block matrix Γn−1 . Let ⎛ ⎞ Cn+1 .. ⎜ ⎟ ⎜ ⎟ . ⎜ ⎟. S=⎜ ⎟ C ⎝ ⎠ 2n −1 (1) W + S∗1 Γn−1 S1 −1 (1) By taking W + S∗1 Γn−1 S1 as C2n+1 and the matrix S∗ Γ−1 n S as C2n+2 , we construct the (n + 2) × (n + 2) block Hankel matrix S Γn n+1 . Γn+1 = (Cj+k )j,k=0 = S∗ S∗ Γ−1 n S

(4.10)

14

V.M. Adamyan and I.M. Tkachenko

From Proposition 4.5 we can conclude that the matrix (4.10) is the unique nonnegative deﬁnite extension of Γn for which the condition Pn+1 Mn+1 = Nn+1 holds; as before Mn+1 is the null-space of Γn+1 and Pn+1 is the orthogonal projector in the space Ln+1 of column vectors ⎛ ⎞ ξ0 ⎜ .. ⎟ ⎜ ⎟ ξ = ⎜ . ⎟ , ξ 0 , . . . , ξn+1 ∈ Cs , ⎝ ξn ⎠ ξn+1 onto the subspace Nn+1 = Ln+1 Ln of column vectors with ξ 0 = · · · = ξ n = 0. Proposition 4.6. For the set of Hermitian s × s matrices C0 , C1 , . . . , C2n which satisfy the conditions of Theorem 1.1 and such that the block Hankel matrix Γn = n (Cj+k )j,k=0 is invertible, all extensions C2n+1 , C2n+2 that generate completely degenerate truncated Stieltjes matrix moment problems are described by formulas ⎛ ⎞ Cn+1 −1 ⎜ ⎟ (1) .. (4.11) C2n+1 = W + (Cn+1 , . . . , C2n ) Γn−1 ⎝ ⎠, . ⎛

C2n ⎞

Cn+1 ⎟ .. C2n+2 = (4.12) ⎠, . C2n+1 where the “parameter” W runs through the set of all non-negative deﬁnite s × s matrices. (Cn+1 , . . . , C2n+1 ) Γ−1 n

⎜ ⎝

Proof. It remains to prove that (4.11) with (4.12) provides all demanded extensions. Let now C2n+1 , C2n+2 be an extension generating a completely degenerate Stieltjes problem. By Theorem 1.1 this means, in particular, that the extended (1) Hankel matrix Γn should be non-negative deﬁnite. According to Remark 4.4 this implies the expression (4.11) for C2n+1 with some non-negative deﬁnite matrix parameter W. By Proposition 4.5, C2n+2 for a certain appropriate C2n+1 cannot diﬀer from that given by expression (4.11). From now on we assume that the block Hankel matrix Γn is not invertible. The problem of describing all extensions C2n+1 , C2n+2 which transform the given Stieltjes (2n + 1) moment problem into completely degenerate (2n + 3) ones, can be handled in the following way. We can take any sequence of Hermitian s×s matrices G0 , G1 , G2 , . . . , G2n such that the block Hankel matrices ∆n := (Gk+j )nk,j=0 (1)

and ∆n−1 := (Gk+j+1 )n−1 k,j=0 are invertible and positive deﬁnite. Then for any ε > 0 the perturbed Hankel matrices (1)

(1)

(1)

Γn (ε) := Γn + ε∆n , Γn−1 (ε) := Γn−1 + ε∆n−1

(4.13)

Stieltjes Truncated Matrix Moment Problem

15

are invertible and positive deﬁnite. Taking for the set of moments C0 (ε) := C0 + εG0 , . . . , C2n (ε) := C2n + εG2n the solutions (4.11), (4.12) of the extension problem under consideration, we can further obtain the demanded solutions by passing to the limit ε ↓ 0. To this end let us ﬁrst show that under the conditions of Theorem 1.1 there exists a ﬁnite limit of non-negative matrices ⎛

⎛

⎞⎞ Cn+1 (ε) ⎜ ⎜ ⎟⎟ .. ⎝S1 (ε) = ⎝ ⎠⎠ . C2n (ε)

−1 (1) S∗1 (ε) Γn−1 (ε) S1 (ε), as ε ↓ 0. (1)

We denote by P1 (Γn−1 ) the orthogonal projector onto the null-space MΓ(1) ⊂ n−1

(1)

Ln−1 (Cs ) of the non-negative block Hankel matrix Γn−1 := (Ck+j+1 )n−1 k,j=0 and (1)

(1)

set P2 (Γn−1 ) = I − P1 (Γn−1 ). With respect to the representation Ln−1 (Cs ) = M⊥ ⊕ MΓ(1) (1) Γ

n−1

n−1

and by the deﬁnition of MΓ(1) and (4.13) we can write n−1

(1) Γn−1 (ε)

(1)

=

(1)

(1)

(1)

ε∆n−1,12 (1) (1) Γn−1,22 + ε∆n−1,22

ε∆n−1,11 (1) ε∆n−1,21 (1)

(1)

Γn−1,ij + ε∆n−1,ij = Pi (Γn−1 )Γn−1 (ε) |Pj (Γ(1)

n−1 )Ln−1 (Cs )

Put

−1 (1) (1) (1) Υn−1 = lim P2 Γn−1 Γn−1 − λ |M⊥

(1) Γ n−1

λ↑0 (1)

,

, i, j = 1, 2.

.

(1)

Note that the matrices Υn−1 and ∆n−1,11 are invertible. The Schur-Frobenius factorization formula gives: 1 (1) Γn−1 (ε)−1 =

(1)

∆n−1,11 0

−1

0 0

+ ε ⎞ ⎛ −1 −1 −1 (1) (1) (1) (1) (1) (1) (1) (1) − ∆n−1,11 ∆n−1,12 Υn−1 ⎟ ⎜ ∆n−1,11 ∆n−1,12 Υn−1 ∆n−1,21 ∆n−1,11 −1 ⎠ ⎝ (1) (1) (1) (1) Υn−1 ∆n−1,21 ∆n−1,11 Υn−1 +O(ε).

(4.14)

16

V.M. Adamyan and I.M. Tkachenko

Note further that for any h ∈ Cs we have −1 −1 (1) (1) ∗ S1 (ε)h, h = Γn−1 (ε) k(ε), k(ε) , S1 (ε) Γn−1 (ε) ⎞ ⎛ 0 ⎜ .. ⎟ ⎟ ⎜ (2) (2) k(ε) = Γn−1 (ε) ⎜ . ⎟ , Γn−1 (ε) = (Ck+j+2 + εGk+j+2 )n−1 k,j=0 . ⎝ 0 ⎠ h (1)

Recall that by the assumptions of Theorem 1.1, the null space of Γn−1 belongs to the null-space of (2) (2) Γn−1 = Γn−1 (0) . Therefore taking into account (4.14), we get −1 (1) (1) (1) (1) S1 (ε) = S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0). lim S∗1 (ε) Γn−1 (ε) ε↓0

(1)

Note that by Proposition 4.6 for ε > 0 a self-adjoint Hankel extension Γn (ε) of (1) Γn−1 (ε) is non-negative deﬁnite if and only if −1 (1) C2n+1 (ε) = W + S∗1 (ε) Γn−1 (ε) S1 (ε), where W is any non-negative deﬁnite s × s matrix. As follows, for each ﬁxed (1)(0) with the limit non-negative deﬁnite W the limit Hankel matrix Γn (1) (1) (1) (4.15) C2n+1 (0) = W + S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0) is non-negative deﬁnite. On the other hand, as −1 (1) (1) (1) (1) S1 (0), S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0) = lim S∗1 (0) Γn−1 − λ λ↑0

it stems from Proposition 4.6 that only extensions C2n+1 given by (4.15) with (1) W ≥ 0 provide the non-negativity of Γn . To ﬁnd now the set of all appropriate extensions C2n+2 it remains to select among the non-negative deﬁnite s× s matrices W those for which there exist ﬁnite limits of the matrices S∗ (ε)Γn (ε)−1 S(ε) as ε ↓ 0. To this end note that taking any h ∈ Cs and setting ⎛ ⎞ ⎞ ⎛ Cn+1 (ε)h 0 ⎜ ⎟ .. ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ k=⎜ ⎟, ⎜ ⎟ = Γ(1) n (ε)k, (ε)h C 2n ⎟ ⎝ 0 ⎠ ⎜ −1 ⎝ ⎠ (1) h S1 (ε) h W + S∗1 (ε) Γn−1 (ε)

Stieltjes Truncated Matrix Moment Problem we can write

(1) S∗ (ε)Γn (ε)−1 S(ε)h, h = Γn (ε)−1 Γ(1) (ε)k, Γ (ε)k . n n

17

(4.16)

Let P1 (Γn ) be the orthogonal projector onto the null-space MΓn ⊂ Ln (Cs ) of the non-negative block Hankel matrix Γn := (Ck+j )nk,j=0 , P2 (Γn ) = I − P1 (Γn ) and let −1 Υn = lim P2 (Γn ) (Γn − λ) |M⊥ . Γ λ↑0

n

Note that for any ξ ∈ MΓn we have

Γ(1) n (0)ξ

= j

n−1

Cj+1+k ξk = (Γn ξ)j+1 = 0, 0 ≤ j ≤ n − 1.

(4.17)

k=0

Let Pn be the orthogonal projector onto the subspace Pn Mn . Taking into account (1) that the matrices Γn (0), Γn are Hermitian, we deduce from (4.17) that (1) (1) (1) Γ(1) n (0)P1 (Γn )Γn (0) = Γn (0)P1 (Γn )Pn P1 (Γn ) Γn (0).

(4.18)

Applying the same arguments as above to (4.16) and by virtue of (4.18), we obtain that ∗ (1) (1) S (ε)Γn (ε)−1 S(ε)h, h = 1ε Pn P1 (Γn )Γn (0)h, Pn Γn (0)h ε↓0 (4.19) (1) (1) + Υn P2 (Γn )Γn (0)h, P2 (Γn )Γn (0)h + O(ε). Hence, the condition Pn P1 (Γn )Γ(1) n (0)Pn = 0,

(4.20)

or, equivalently, Pn Γ(1) n (0)P1 (Γn )Pn = 0,

is necessary and suﬃcient for the existence of a ﬁnite limit of the matrix function S∗ (ε)Γn (ε)−1 S(ε) as ε ↓ 0. Remark 4.7. The condition (4.20) as well as the condition (2) (0)P (Γ )P = 0 P P (Γ )Γ (0)P = 0 Pn Γ(2) 1 n n 1 n n n n n

(4.21)

are necessary for an extended set of matrices (1) (1) (1) C0 , . . . , C2n , W + S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0), C2n+2 to be, in the degenerate case, the set of the ﬁrst 2n + 3 moments of some nondecreasing measure on the half-axis (0, ∞). Indeed, let the condition of the remark hold. Then by Theorem 1.1, both Hankel matrices Γn and Γn+1 are non-negative deﬁnite. Let ξ ∈ Ln (Cs ) be a non-zero null-vector of Γn . Then for the vector ξ ∈ Ln+1 (Cs ) ξ= 0

18 we have:

V.M. Adamyan and I.M. Tkachenko Γn+1 ξ, ξ = (Γn ξ, ξ) = 0.

ξ is a null-vector of Γn+1 . Therefore, ξ is also Since Γn+1 ≥ 0, this means that (1) a null-vector of the Hankel matrix Γn (0), which implies (4.20). The condition (4.21) can be veriﬁed in the same manner. Let us introduce the linear subspace ⎧ ⎫ ⎞ ⎛ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎜ .. ⎟ ⎜ . ⎟ (4.22) Nn := h ∈ Cs : ⎜ ⎟ ∈ Pn Mn . ⎪ ⎪ ⎝ 0 ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ h By deﬁnition (4.22) for any h ∈ Nn there is n−1

ξ(h) ∈ Ln−1 (Cs ) , ξ(h) = (ξj (h))j=0 such that

ξ(h) h

∈ Mn .

The application of (4.20) and (4.21) to such a null-vector yields (1) (1) (1) W + S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0) h = −Cn+1 h−. . .−C2n h, (4.23) (1) (1) (1) C2n+2 h = −Cn+2 h − . . . − W + S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0) h. (4.24) Thus all admissible extensions (1) (1) (1) C2n+1 = W + S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0), C2n+2 of the moment set must coincide on the subspace Mn . Since admissible matrices C2n+1 , C2n+2 are Hermitian, they can diﬀer only on the subspace N⊥ n = Cs N n . We proved that for any set of Hermitian s × s matrices C0 , C1 , C2 , . . . , C2n which satisfy the conditions of Theorem 1.1, the set of extensions C2n+1 , C2n+2 which generates a completely degenerate Stieltjes problem, is always non-empty and the variety of such extensions is exhausted by the expressions ⎛ ⎞ Cn+1 ⎜ ⎟ (1) (1) (1) .. C2n+1 = S∗1 P2 Γn−1 Υn−1 P2 Γn−1 S1 + W, S1 = ⎝ (4.25) ⎠, . C2n n (1) , Γ(1) (4.26) C2n+2 = Γ(1) n P2 (Γn )Υn P2 (Γn )Γn n = (Cj+k+1 )j,k=0 , nn

where W runs through the set of all non-negative s × s-matrices which satisfy the condition WNn = {0} and (. . .)nn stands for the nnth block of the corresponding block-matrix.

Stieltjes Truncated Matrix Moment Problem

19

Proceeding from the assertion of Theorem 4.1 and the results of this and previous sections, we arrive at the following algorithm for ﬁnding all canonical solutions of the truncated Stieltjes matrix moment problem: • for the set of matrix moments C0 , C1 , C2 , . . . , C2n which satisfy the condi(1) tions of Theorem 1.1, ﬁnd the orthogonal projectors P1 Γn−1 and P1 (Γn ) (1)

onto the null-spaces of Hankel matrices Γn−1 = (Cj+k+1 )n−1 j,k=0 and Γn = n (Cj+k )j,k=0 , respectively; (1)

• ﬁnd Hermitian matrices Υn−1 , Υn which satisfy the equations (1) (1) (1) (1) Γn−1 Υn−1 = P2 Γn−1 = I − P1 Γn−1 , Γn Υn = P2 (Γn ) = I − P1 (Γn ) and the conditions (1) (1) Υn−1 P1 Γn−1 = Υn P1 (Γn ) = 0; • taking any matrix W≥0 which satisﬁes the condition WPn P1 (Γn ) = 0, deﬁne the extension C2n+1 by (4.25) and then the extension C2n+2 by formula (4.26); n+1 • for the Hankel matrix Γn+1 = (Cj+k )j,k=0 determine the (n + 2) × s matrix = X

X I

⎛

⎞ X0 ⎜ ⎟ , X = ⎝ ... ⎠ , Xn

(4.27)

= 0; then with s × s matrices Xj , I, which satisﬁes the equation Γn+1 X determine the (n + 2) × s matrix ⎛

Y0 .. .

⎜ ⎜ Y=⎜ ⎝ Yn−1 Yn

⎞ j ⎟ ⎟ Ck Xn+1−j+k , Xn+1 = I; ⎟ , Yj = ⎠ k=0

• for the matrix polynomials D(z) = z n+1 I +

n k=0

z k Xk , E(z) =

n k=0

z k Yk

20

V.M. Adamyan and I.M. Tkachenko ﬁnd all poles tmin = t1 < t2 < · · · < tν = tmax , ν ≤ ns, of the rational matrix function E(z)D(z)−1 and its residues M1 , . . . , Mν at these poles, and put ⎧ ⎪ 0, t ≤ t1 , ⎪ ⎪ ⎪ ⎪ ⎪ M , t 1 1 < t ≤ t2 , ⎨ σ(t) = M1 + M2 , t2 < t ≤ t3 , ⎪ ⎪ ⎪. . . ⎪ ⎪ ⎪ ⎩M + · · · + M , t > t . 1 ν ν

5. Non-canonical solutions By (4.25), (4.26) the set of all canonical solutions of the truncated matrix Stieltjes moment problem is parametrized in the degenerate but not the completely degenerate case by the set of s × s matrices W ≥ 0 which satisfy the condition WNn = {0}. To ﬁnd the description of all canonical solutions σ W (t), notice that for the extended Hankel matrix Γn+1 the (n + 1) × s matrix X (4.27) is of the form X = X⎛0 + XW⎞ , ⎛ ⎞ X00 Cn+1 ⎜ ⎟ ⎜ ⎟ .. X0 = ⎝ ... ⎠ := Υn ⎝ ⎠, . X0n C2n+1 (1) (1) (1) ∗ C2n+1 = S1 P2 Γn−1 Υn−1 P2 Γn−1 S1 ; ⎛ ⎛ ⎛ ⎞ ⎞ X10 W X10 ⎜ ⎜ ⎜ .. ⎟ ⎜ ⎟ . . XW = ⎝ ⎠ , X1 = ⎝ . ⎠ := Υn ⎜ . ⎝ X1n X1n W Put D0n (z) = z n+1 I + E0n (z) = E1n (z) =

n # k=0 n−1 # k=0

n #

z k X0k , D1n (z) =

k=0

z k Y0k, Y0j = z k Y1k, Y1j =

j # k=0 j #

n #

0 .. . 0 I − Pn

⎞

(5.1)

⎟ ⎟ ⎟. ⎠

z k X1k ;

k=0

Ck X0n+1−j+k , X0n+1 = I,

(5.2)

Ck X0n−j+k .

k=0

It stems from the above results that the Nevanlinna type formula ∞ −1 1 dσ W (t) = − E0n (z) + E1n (z)W D0n (z) + D1n (z)W t−z

(5.3)

0

establishes a one-to-one correspondence between the set of all canonical solutions σ W of the truncated matrix Stieltjes moment problem and the set of s × s matrix W ≥ 0 which satisfy the condition WNn = {0}.

Stieltjes Truncated Matrix Moment Problem

21

Notice also that the matrix functions D0n (z), D1n (z), E0n (z), E1n (z) are independent of the choice of W ≥ 0, WNn = {0}. Let us denote by P n the orthoprojector onto the subspace Nn in Cs . By deﬁnition of these matrix functions or, irrespectively of this, from (5.3) it follows that D0n (z)∗ E0n (z) = E0n (z)∗ D0n (z), D1n (z)∗ E1n (z) = E1n (z)∗ D1n (z); (5.4) D0n (z)∗ E1n (z) − E0n (z)∗ D1n (z) = I − P n , Im z ≥ 0. Besides, by (5.3), for any W ≥ 0, WP n = 0 the matrix function D0n (z) + D1n (z)W is invertible in the upper half-plane and on the half-axis (−∞, 0). By applying these facts and the arguments used in [1], [2] and not adduced here, we come to the following description of all solution of the truncated matrix Stieltjes moment problem in the degenerate case. Theorem 5.1. Let the conditions of Theorem 1.1 hold. Then the formula ∞ 1 dσ Ξ (t) t−z 0 −1 − E0 (z) + E1 (z) (Ξ (z) + zI)(I − P n ) × D0 (z) + D1 (z) Ξ (z)−1 + zI (I − P n ) ,

(5.5)

establishes a one-to-one correspondence between the set of all solutions σ Ξ (t) of the truncated matrix Stieltjes problem and the subset S of the Nevanlinna matrix functions Ξ(z) with values in Cs Nn . Remark 5.2. The subset of canonical solutions is generated by substitution into −1 (5.5) of matrix functions Ξ(z) = (W − zI) with W ≥ 0 such that Nn ⊂ ker W. Acknowledgement The ﬁnancial support of the Polytechnic University of Valencia is gratefully acknowledged. V. Adamyan was also supported by the USA Civil Research and Development Foundation and the Government of Ukraine (CRDF grant UM12567-OD-03).

References [1] V.M. Adamyan, I.M. Tkachenko, Solution of the truncated matrix Hamburger moment problem according to M.G. Krein, Operator Theory: Advances and Applications, 118 (2000), 33–52 (Proceedings of the Mark Krein International Conference on Operator Theory and Applications, Vol.II, Operator Theory and Related Topics), Birkh¨ auser, Basel. [2] V.M. Adamyan, I.M. Tkachenko, Solution of the Stieltjes Truncated Matrix Moment Problem, Opuscula Mathematica, 25 (2005), no.1, 5–24. [3] N.I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner, N.Y., 1965.

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[4] N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Spaces, Frederick Ungar Publishing Co., N.Y., 1966. [5] V. Bolotnikov, Degenerate Stieltjes moment problem and J-inner polynomials, Z. Anal. Anwendungen 14 (1995), no. 3, 441–468. [6] R.E. Curto and L.A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math., 17 (1991), 603–635. [7] H. Dym, On Hermitian Block Hankel Matrices, Matrix Polynomials, the Hamburger Moment Problem, Interpolation and Maximum Entropy, Integral Equations Operator Theory, 12 (1989), 757–812. [8] G.-N. Chen, Y.-J. Hu, A uniﬁed treatment for the matrix Stieltjes moment problem in both nondegenerate and degenerate cases, J. Math. Anal. Appl. 254 (2001), no. 1, 23–34. [9] Y.-J. Hu, G.-N. Chen, A uniﬁed treatment for the matrix Stieltjes moment problem, Linear Algebra Appl. 380 (2004), 227–239. [10] S. Karlin, W.S. Studden, Tschebyscheﬀ systems: with applications in analysis and statistics, Interscience, 1966. [11] M.G. Krein, The theory of extensions of semi-bounded Hermitian operators and its applications, Mat. Sbornik I, 20 (1947), 431–495; II, 21 (1947), 365–404 (in Russian). [12] M.G. Krein, M.A. Krasnoselskii, Fundamental theorem on the extension of Hermitian operators and certain of their applications to the theory of orthogonal polynomials and the problem of moments, Uspekhi. Matem. Nauk (N.S.) 2 (1947), no. 3(19), 60–106 (in Russian). [13] M.G. Krein, A.A. Nudel’man, The Markov moment problem and extremal problems, “Nauka”, Moscow, 1973 (in Russian). English translation: Translation of Mathematical Monographs AMS, 50, 1977. [14] T.J. Stieltjes, Recherches sur les fractions continues, Annales de la Facult´e des Sciencies de Toulouse, 8 (1894), 1–122; 9 (1895), 1–47. [15] T.J. Stieltjes, Collected Papers, G. van Dijk (ed.), Springer, Berlin, 1993. Vadim M. Adamyan Department of Theoretical Physics I.I. Mechnikov Odessa National University 65026 Odessa Ukraine e-mail: [email protected] Igor M. Tkachenko Department of Applied Mathematics Polytechnic University of Valencia 46022 Valencia Spain e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 23–54 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Q-functions of Quasi-selfadjoint Contractions Yury Arlinski˘ı, Seppo Hassi and Henk de Snoo To Heinz Langer on the occasion of his retirement

Abstract. A bounded everywhere deﬁned operator T in a Hilbert space H is said to be a quasi-selfadjoint contraction or (for short) a qsc-operator, if T is a contraction and ker (T − T ∗ ) = {0}. For a closed linear subspace N of H containing ran (T − T ∗ ) the operator-valued function QT (z) = PN (T − zI)−1 N, |z| > 1, where PN is the orthogonal projector from H onto N, is said to be a Q-function of T acting on the subspace N. The main properties of such Q-functions are studied, in particular the underlying operator-theoretical aspects are considered by using some block representations of the contraction T and analytical characterizations for such functions QT (z) are established. Also a reproducing kernel space model for QT (z) is constructed. In the special case where T is selfadjoint QT (z) coincides with the Q-function of the symmetric operator A := T (H N) and its selfadjoint extension T = T ∗ in the usual sense. Mathematics Subject Classiﬁcation (2000). Primary: 47A10, 47A56, 47A64; Secondary 47A05, 47A06, 47B15. Keywords. Symmetric contraction, contractive extension, quasi-selfadjoint operator, Q-function, operator model, resolvent.

1. Introduction The concept of a Q-function was introduced by M.G. Kre˘ın for the case of a densely symmetric operator S in a Hilbert space H with equal defect numbers by means of a selfadjoint extension A of S, cf. [26], [28], [33], and also [29], [30], [31]. Such a function belongs to the class N of Nevanlinna (or Herglotz-Nevanlinna) functions, i.e., Q(z) ∈ N if it is holomorphic in the open upper and lower half-planes and satisﬁes the conditions Q(¯ z ) = Q(z)∗ and (Im z) (Im Q(z)) ≥ 0, z ∈ C+ ∪ C− . The This work was supported by the Research Institute for Technology at the University of Vaasa. The ﬁrst author was also supported by the Academy of Finland (projects 203227, 208057) and the Dutch Organization for Scientiﬁc Research NWO (B 61–553).

24

Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

Q-function plays an essential role in Kre˘ın’s resolvent formula, which describes all (generalized resolvents of) selfadjoint extensions of S. In fact, all generalized resolvents (canonical as well as exit space) were ﬁrst described independently by M.A. Naimark [41] and M.G. Kre˘ın [26]; see also [29] for further historical remarks. A characteristic property of a Q-function Q(z) in the class of Nevanlinna functions is that Im Q(z) is invertible (at some or equivalently at every point z ∈ C+ ∪ C− ): every Nevanlinna function with this property is a Q-function of some simple symmetric operator S and a selfadjoint extension A of S in a Hilbert space H. Moreover, the simple (completely non-selfadjoint) symmetric operator S and its selfadjoint extension A are essentially unique in the sense that the Q-function of S determines S and A uniquely up to unitary equivalence. Another approach for describing selfadjoint as well as non-selfadjoint intermediate extensions of a symmetric operator is via a boundary value space and the corresponding Weyl function, see [21], [19], [18], and the references therein. Two special subclasses of Q-functions, consisting of the so-called Qµ - and QM -functions, which belong to the class N of Nevanlinna functions were deﬁned and investigated by M.G. Kre˘ın and I.E. Ovcharenko in [32]. Here the underlying symmetric operator is a non-densely deﬁned contraction. In a recent paper [7] by the authors some extensions of Qµ - and QM -functions were introduced; in fact, this paper contains also some corrections to the result stated in [32]. Some other type of Q-function associated to a non-densely deﬁned symmetric contraction has been considered in [47], including the resolvent formulas for the selfadjoint (canonical and exit space) extensions. In this paper a class of operator-valued Q-functions associated with a nondensely deﬁned symmetric contraction A and its, in general, non-selfadjoint contractive extensions T is introduced. By deﬁnition a bounded operator T in the Hilbert space H is a quasi-selfadjoint contraction or, for short, a qsc-operator if dom T = H, T ≤ 1, and ker (T − T ∗ ) = {0}. Let T be a qsc-operator and let N be a proper subspace of H which contains ran (T − T ∗ ). Deﬁne the operator-valued function Q(z) as follows Q(z) = PN (T − zI)−1 N,

|z| < 1.

(1.1)

In what follows the function Q(z) in (1.1) will be called a Q-function of T with respect to the subspace N ⊂ H. Observe, that if T is selfadjoint then the function Q deﬁned by (1.1) is an ordinary Q-function associated with T and the symmetric restriction A := T H0 of T , where H0 = H N. However, if T is not selfadjoint this function in general is not a Nevanlinna function. A qsc-operator T may be considered as a contractive, in general, non-selfadjoint extension of the symmetric contraction A = T H0 which is also called a quasi-selfadjoint contractive extension of A; here A is symmetric due to H0 ⊂ ker (T − T ∗ ). Such kind of extensions were parametrized and investigated in [10] and [12]. The special case of selfadjoint contractive extensions was investigated by M.G. Kre˘ın [27] and by M.G. Kre˘ın and I.E. Ovcharenko [32]. In particular, in [32] two special Q-functions of the Nevanlinna class for the symmetric contraction were deﬁned and studied

Q-functions of Quasi-selfadjoint Contractions

25

and the resolvent formulas for selfadjoint contractive extensions (sc-extensions) were established. These formulas were extended in [10] and [12] for qsc-extensions. A boundary value space approach for describing extensions of dual pairs of densely deﬁned operators appears in [37] and for dual pairs of linear relations and their canonical and generalized resolvents in [39], [40], see also [34], [35]. In the present paper the approach can be seen as a non-selfadjoint counterpart of the Q-function approach developed and systematically used in the papers of M.G. Kre˘ın and H. Langer, cf., e.g., [29]–[31]. The contents of this paper will be brieﬂy described. In Section 2 some preliminary notions are introduced. The extension theory for closed symmetric contractions is developed in Section 3. This includes a discussion of minimality of the underlying symmetric operator A and its contractive extensions. The Q-functions for intermediate contractive extensions as in (1.1) are introduced in Section 4, where also a number of associated nonnegative kernels will appear. A resolvent formula for qsc-extensions of a symmetric contraction A is derived in Section 5. It involves a Q-function of the form (1.1) for a given qsc-extension T of A. In Section 6 a model for such Q-functions is constructed by means of a qsc-operator acting in a reproducing kernel Hilbert space and it is proved that two N-minimal qscoperators whose Q-functions in (1.1) coincide are unitarily equivalent. This model is used to establish some characteristic properties of Q-functions of qsc-operators in Section 7. In Section 8 linear fractional transformations of Q-functions are considered. The results in the present paper can be connected with and augmented by the study of a certain class of passive systems. In particular, the Q-functions of quasi-selfadjoint operators investigated in the present paper are in one-to-one correspondence with the transfer functions of so-called passive quasi-selfadjoint systems, which are introduced and investigated in [8]. The work in this paper is a continuation of some investigations done by several authors in the early eighties, including Heinz Langer and Bj¨ orn Textorius, being inﬂuenced by ideas developed by M.G. Kre˘ın and I.E. Ovcharenko. It is a pleasure to record the mathematical indebtedness of the authors of the present paper to Heinz Langer over a period of many years.

2. Preliminaries 2.1. Basic notations The class of all continuous linear operators deﬁned on a complex Hilbert space H1 and taking values in a complex Hilbert space H2 is denoted by L(H1 , H2 ) and L(H) := L(H, H). The domain, the range, and the null-space of a linear operator T are denoted by dom T , ran T , and ker T . For T ∈ L(H) the operators TR = (T + T ∗ )/2, TI = (T − T ∗ )/2i are said to be the real and the imaginary part of T . For a contraction T ∈ L(H1 , H2 ) the defect operator DT of T is deﬁned by DT := (I − T ∗ T )1/2 .

(2.1)

26

Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

It is a nonnegative contraction and satisﬁes the well-known commutation relation T DT = DT ∗ T,

(2.2)

cf. [46]. The closure of the range ran DT is denoted by DT and ρ(T ) stands for the set of all regular points of a closed operator T . If Rl and Rr are two nonnegative operators in L(H) and S0 ∈ L(H), then the symbol B(S0 , Rl , Rr ) denotes the operator ball in L(H) with the center S0 and the left and right radii Rl and Rr , 1/2 1/2 respectively, i.e., the set of all operators in L(H) of the form T = S0 + Rl XRr , where X is a contraction from ran Rr into ran Rl . It is well known, see [43], [44], that a necessary and suﬃcient condition for T ∈ L(H) to belong to B(S0 , Rl , Rr ) is the following: 2

|((T − S0 )f, g)| ≤ (Rr f, f ) (Rl g, g) for all f, g ∈ H.

(2.3)

If Rl = Rr = R the corresponding operator ball is denoted by B(S0 , R). 2.2. Quasi-selfadjoint contractions Recall that T ∈ L(H) is a quasi-selfadjoint contraction (a qsc-operator) if dom T = H,

T ≤ 1, and ker (T − T ∗ ) = {0}.

A qsc-operator T is said to be a quasi-selfadjoint contractive extension or qscextension of a closed symmetric contraction A if dom A ⊂ ker (T − T ∗ ) or equivalently ran (T − T ∗ ) ⊂ (dom A)⊥ , cf. [10], [12]. Clearly, an operator T ∈ L(H) is a qsc-extension of A if and only if A ⊂ T and A ⊂ T ∗ , or, equivalently, if T is an intermediate extension of A. A qsc-operator T has always symmetric restrictions A for which T is a qsc-extension. Namely, with a subspace N ⊃ ran (T − T ∗ ) deﬁne dom A = H N,

A = T dom A.

Then dom A ⊂ ker (T − T ∗). A qsc-operator T is called completely non-selfadjoint if there is no non-zero invariant subspace on which the restriction of T is selfadjoint. Lemma 2.1. [15] A qsc-operator T is completely non-selfadjoint if and only if span { ran T n (T − T ∗ ) : n = 0, 1, . . . } = H. 2.3. The classes C(α) Let α ∈ [0, π/2) and denote by S(α) the following sector of the complex plane: S(α) = { z ∈ C : | arg z| ≤ α } . A linear operator S, in general unbounded, in a Hilbert space H is said to be sectorial with vertex at the origin and semiangle α, if its numerical range W (S) = { (Sf, f ) : f = 1, f ∈ dom S }

Q-functions of Quasi-selfadjoint Contractions

27

is contained in the sector S(α), cf. [25]. This condition is equivalent to |Im (Sf, f )| ≤ (tan α) Re (Sf, f ) for all f ∈ dom S. If the resolvent set of S is not empty then S is called maximal sectorial. A maximal sectorial operator S is densely deﬁned and its adjoint S ∗ is also a maximal sectorial operator. A bounded operator T on a Hilbert space H is said to belong to the class C(α), α ∈ (0, π/2), if T sin α ± i cos α I ≤ 1, (2.4) cf. [3]. Clearly, T belongs to C(α) if and only if T ∗ belongs to C(α). Moreover, it follows from (2.4) that the operators belonging to C(α) are contractive. The condition (2.4) is equivalent to each of the following two conditions: |(TI f, f )| ≤ or

tan α DT f 2 2

for all f ∈ H;

(2.5)

the operator (I − T ∗ )(I + T ) is sectorial with vertex at the origin and semiangle α,

(2.6)

cf. [4]. Note that the linear fractional transformation T = (I − S)(I + S)−1 of a maximal sectorial operator S with vertex at the origin and semiangle α is an operator of the class C(α). Let $

= C {C(α) : α ∈ [0, π/2)}.

were studied in [3], [4]. In particular, Some properties of the operators in the class C

in [3] it was proved that T ∈ C implies that ran DT n = ran DT ∗n = ran DTR ,

n = 1, 2, . . . ,

where TR is the real part of T . Furthermore it was proved in [3] that the subspace DT reduces the operator T , that the operator T ker DT is selfadjoint and unitary, and that T DT is a completely non-unitary contraction of the class C00 , i.e., lim T n f = lim T ∗n f = 0

n→∞

n→∞

for all f ∈ DT ,

cf. [46]. 2.4. The Schur-Frobenius formula for the resolvent of a block operator Let the Hilbert space H be decomposed as H = H1 ⊕ H2 and decompose T ∈ L(H) accordingly: T11 T12 T = , Tij ∈ L(Hi , Hj ). (2.7) T21 T22 Deﬁne the operator-valued functions VT (z) = T21 (T11 − zI)−1 T12 − T22 ,

WT (z) = −zI − VT (z),

z ∈ ρ(T11 ). (2.8)

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Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

By the Schur-Frobenius formula the resolvent (T − z)−1 of T can be rewritten in the block form (T11 − zI)−1 I + T12 WT (z)−1 T21 (T11 − zI)−1 −(T11 − zI)−1 T12 WT−1 (z) WT−1 (z) −WT−1 (z)T21 (T11 − zI)−1 (2.9) for z ∈ ρ(T ) ∩ ρ(T11 ). In particular, −1

PH2 (T − zI)−1 H2 = − (VT (z) + zI)

,

z ∈ ρ(T ) ∩ ρ(T11 ).

(2.10)

2.5. Nevanlinna functions Let N be a Hilbert space. An operator-valued function V (z), z ∈ C \ R, with values in L(N) is said to be a Nevanlinna function or an R-function, cf. [24], if V (z) is holomorphic on C \ R, V ∗ (z) = V (z), and Im z Im V (z) ≥ 0 for all z ∈ C \ R. The subclass of Nevanlinna functions V (z) which are holomorphic on the domain Ext [−1, 1] := C \ [−1, 1] is denoted by NN [−1, 1]. By the general theory of Nevanlinna functions, cf. [24], [15], every function V (z) in NN [−1, 1] has an integral representation of the form 1 V (z) = Γ + −1

dG(t) , t−z

where Γ is a bounded selfadjoint operator on N and the L(N)-valued function G(t) is nondecreasing, nonnegative, normalized by G(−1 − 0) = 0, and has ﬁnite total variation concentrated on [−1, 1]. Clearly, V (∞) := s − lim V (z) = Γ. The next z→∞

result is also well known, cf. [15]. Theorem 2.2. Let N be a Hilbert space and let V (z) ∈ NN [−1, 1]. Then there exist a Hilbert space H, a selfadjoint contraction B on H, and F ∈ L(N, H), such that V (z) = V (∞) + F ∗ (B − zI)−1 F,

z ∈ Ext [−1, 1].

(2.11)

In what follows the subclass of functions V (z) in NN [−1, 1] which have the limit values V (±1) in L(N) plays a central role. In this case Theorem 2.2 can be completed as follows. Theorem 2.3. Let N be a Hilbert space and let V (z) ∈ NN [−1, 1]. If for all f ∈ N the limit values lim (V (x)f, f ) , lim (V (x)f, f ) (2.12) x↑−1

x↓1

are ﬁnite, then there exist a Hilbert space H, a selfadjoint contraction B in H, and an operator G ∈ L(N, DB ), such that 2 (B − zI)−1 G, V (z) = V (∞) + G∗ DB

z ∈ Ext [−1, 1].

(2.13)

Conversely, for every function V (z) of the form (2.13) the limit values (2.12) exist for all f ∈ N and are ﬁnite.

Q-functions of Quasi-selfadjoint Contractions

29

Proof. By Theorem 2.2 V (z) has the representation (2.11), where B is a selfadjoint contraction in a Hilbert space H and F ∈ L(N, H). Since the limits in (2.12) exist for all f ∈ N, one concludes that ran F ⊂ ran (I − B)1/2 ∩ ran (I + B)1/2 . Consequently, ran F ⊂ ran DB and this implies that F = DB G for some operator G ∈ L(N, DB ), cf. [23]. Conversely, if V (z) is of the form (2.13) then ran DB ⊂ ran (B ± I)1/2 and this implies the existence of the limit values (2.12) for all f ∈ N, cf. [32]. It follows from Theorem 2.3 that V (−1) := s − lim V (x) = V (∞) + G∗ (I − B)G ∈ L(N), x↑−1

V (1) := s − lim V (x) = V (∞) − G∗ (I + B)G ∈ L(N),

(2.14)

x↓1

so that V (−1) + V (1) = 2V (∞) − 2G∗ BG,

V (−1) − V (1) = 2G∗ G.

(2.15)

2.6. Nonnegative kernels, reproducing kernel Hilbert spaces, and sectorial kernels An operator-valued function K(z, ξ) : Ω × Ω → L(N), Ω ⊂ C, is said to be a nonnegative kernel [1], [13], [42], if n

(K(wj , wi )fi , fj )N ≥ 0

i,j=1

for every choice of points {wi }ni=1 ⊂ Ω and vectors {fi }ni=1 ⊂ N. With the kernel K(z, ξ) is associated a reproducing kernel Hilbert space HK . It is the completion of the linear space of vectors of the form n

K(·, wi )fi ,

{wi }ni=1 ⊂ Ω,

{fi }ni=1 ⊂ N,

n ∈ N,

i=1

with respect to the inner product ⎛ ⎞ n m ⎝ K(·, wi )fi , K(·, µj )gj ⎠ i=1

j=1

=

HK

m n

(K(µj , wi )fi , gj )N .

i=1 j=1

Then the Hilbert space HK consists of the N-valued functions f (·) such that for every h ∈ N the reproducing property holds: (f (·), K(·, w)h)HK = (f (w), h)N ,

w ∈ Ω.

Observe that an L(N)-valued function V (z) belongs to the Nevanlinna class N(N) if and only if the function K(z, ξ) =

V (z) − V (ξ)∗ , z−ξ

z, ξ ∈ C \ R,

30

Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

is a nonnegative kernel. Also note that the kernel associated with generalized resolvents (of selfadjoint exit space extensions) in a Hilbert space is given by K(z, ξ) =

V (z) − V (ξ)∗ − V (z)V (ξ)∗ , z−ξ

z, ξ ∈ C \ R.

An operator-valued function K(z, ξ) : Ω × Ω → L(N), Ω ⊂ C, is said to be an α-sectorial kernel, if n (K(wj , wi )fi , fj )N ∈ S(α) i,j=1

for every choice of points {wi }ni=1 ⊂ Ω and vectors {fi }ni=1 ⊂ H, i.e., % % ⎛ ⎞ % % n n % % %Im (K(wj , wi )fi , fj )N %% ≤ (tan α) Re ⎝ (K(wj , wi )fi , fj )N ⎠ , % % % i,j=1 i,j=1 cf. [5]. For α = 0 the corresponding kernel is nonnegative.

3. qsc-extensions of closed symmetric contractions 3.1. A decomposition of closed symmetric contractions Let A be a non-densely deﬁned closed symmetric contraction in the Hilbert space H with the domain dom A =: H0 and let N := H dom A. Let P0 and PN be the orthogonal projections in H onto H0 and N, respectively. Then the operator A0 = P0 A is contractive and selfadjoint in the subspace H0 . Let DA0 = (I −A20 )1/2 be the defect operator determined by A0 . The operator A21 = PN A is also contractive. 2 Moreover, it follows from A∗ A ≤ I that A∗21 A21 ≤ DA . Therefore, the identity 0 K0 DA0 f = PN Af,

f ∈ dom A,

deﬁnes a contractive operator K0 from DA0 := ran DA0 into N, cf. [20], [23]. This gives the following decomposition for the symmetric contraction A A0 A = A0 + K0 DA0 = . (3.1) K 0 D A0 3.2. A matrix representation for qsc-extensions Let the closed symmetric contraction A be deﬁned on the subspace H0 = dom A and decompose A according to H = H0 ⊕ N as in (3.1). Let T be a qsc-extension of A, so that A ⊂ T and A ⊂ T ∗ , and decompose T = (Tij ) also with respect ∗ to H = H0 ⊕ N, cf. (2.7). Then clearly T11 = A0 , T12 = T21 = K0 DA0 . The next result gives a parametrization of all qsc-extensions of A and some of its subclasses by means of block formulas, cf. [14], [17], [45], and [10], [12]. For completeness a short, simple proof is presented. Theorem 3.1. Let A be a closed symmetric contraction A in H = H0 ⊕ N with dom A = H0 and decompose A as in (3.1). Then:

Q-functions of Quasi-selfadjoint Contractions (i) the formula A0 T = K 0 D A0

DA0 K0∗ −K0 A0 K0∗ + DK0∗ XDK0∗

31

H0 H0 → : N N

(3.2)

gives a one-to-one correspondence between all qsc-extensions T of the symmetric contraction A = A0 + K0 DA0 and all contractions X in the subspace DK0∗ := ran DK0∗ ⊂ N; (ii) T in (3.2) belongs to the class C(α) if and only if X belongs to the class C(α), α ∈ (0, π/2); (iii) T is a selfadjoint contractive extension of A if and only if X in (3.2) is a selfadjoint contraction in DK0∗ . Proof. (i) Every operator T ∈ L(H) satisfying the conditions A ⊂ T and A ⊂ T ∗ admits the block-matrix representation of the form H0 H0 A0 DA0 K0∗ : → , T = K 0 D A0 D N N where D ∈ L(N). Then I − T ∗ T is given in the block form 2 − DA0 K0∗ K0 DA0 −A0 DA0 K0∗ − DA0 K0∗ D DA ∗ 0 . I −T T = 2 2 ∗ ∗ −K0 DA0 A0 − D∗ K0 DA0 DK ∗ − K0 A0 K0 − D D 0 Contractivity of T means that 0 ≤ DA0 f − A0 K0∗ h2 + DK0∗ h2 − K0 DA0 f + Dh2 ,

(3.3)

ran K0∗

for all f ∈ H0 and h ∈ N. Since ⊂ DA0 and A0 DA0 ⊂ DA0 , there exists a sequence {fn }∞ n=1 ⊂ DA0 such that for a given h ∈ N the equality lim DA0 fn = A0 K0∗ h

n→∞

holds. Hence, it follows from (3.3) that E := K0 A0 K0∗ + D satisﬁes Eh2 ≤ DK0∗ h2 ,

E ∗ h2 ≤ DK0∗ h2 ,

h ∈ N,

(3.4)

where the second inequality follows from the ﬁrst one by taking into account that T ∗ is a contraction, too. By the second inequality in (3.4) there exists a contraction Z ∈ L(N, DK0∗ ) such that E = DK0∗ Z, i.e., D = −K0 A0 K0∗ + DK0∗ Z. By substituting this into (3.3) one obtains 0 ≤ DK0 (DA0 f − A0 K0∗ h) − K0∗ Zh2 + DK0∗ h2 − Zh2 ,

f ∈ H0 ,

h ∈ N, (3.5)

since by means of (2.2) one has −K0(DA0 f − A0 K0∗ h) + DK0∗ Zh2 = −K0 (DA0 f − A0 K0∗ h)2 − Zh2 + K0∗ Zh2 − 2Re (DK0 (DA0 f − A0 K0∗ h), K0∗ Zh) . Due to the inclusion ran Z ⊂ DK0∗ , one can choose a sequence {fn }∞ n=1 ⊂ DA0 such that for a given h ∈ N the equality lim DK0 DA0 fn = DK0 A0 K0∗ h + K0∗ Zh

n→∞

(3.6)

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Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

holds. Now (3.5) shows that Zh2 ≤ DK0∗ h2 for all h ∈ N, so that Z = XDK0∗ for some contraction X in DK0∗ . Therefore, E = DK0∗ Z = DK0∗ XDK0∗ and D = −K0 A0 K0∗ + DK0∗ XDK0∗ .

(3.7)

Conversely, let D be of the form (3.7), where X is a contraction in DK0∗ . 2 ≥ 0 implies that T given by (3.2) satisﬁes Then DX f f , (I − T ∗ T ) h h (3.8) ∗ ∗ 2 2 ∗ ∗ = DK0 (DA0 f − A0 K0 h) − K0 XDK0 h + DX DK0 h ≥ 0, cf. (3.5). Thus, every contraction X in DK0∗ deﬁnes a qsc-extension T of A via (3.2). (ii) It follows from (3.2) and (3.8) that T satisﬁes (2.5) if and only if |(XI DK0∗ h, DK0∗ h)| (3.9) tan α DK0 (DA0 f − A0 K0∗ h) − K0∗ XDK0∗ h2 + DX DK0∗ h2 ≤ 2 holds for all f ∈ H0 , h ∈ N. In view of (3.6) the condition (3.9) is equivalent to tan α DX h2 (3.10) |(XI h, h)| ≤ 2 for all h ∈ DK0∗ . (iii) The statement is clear since T in (3.2) is selfadjoint if and only if X is selfadjoint in DK0∗ . The class of all selfadjoint contractive (sc-) extensions of A in part (iii) of Theorem 3.1 forms an operator interval [Aµ , AM ]. Using the block representation (3.2) the endpoints of [Aµ , AM ] are given by DA0 K0∗ A0 , (3.11) Aµ = 2 K0 DA0 −K0 A0 K0∗ − DK ∗ 0 and

AM =

A0 K 0 D A0

DA0 K0∗ 2 −K0 A0 K0∗ + DK ∗ 0

,

(3.12)

with X = −I DK0∗ and X = I DK0∗ , respectively. From the formulas (3.11) and (3.12) it is seen that Aµ + AM 0 0 AM − Aµ A0 DA0 K0∗ = = , . 2 0 DK ∗ K0 DA0 −K0 A0 K0∗ 2 2 0 This means that all qsc-extensions in (3.2) of the symmetric contraction A form an operator ball Aµ + AM AM − Aµ , B 2 2 with center (Aµ + AM )/2

Q-functions of Quasi-selfadjoint Contractions

33

and equal left and right radii

√ Rl = Rr = (AM − Aµ )1/2 / 2.

The one-to-one correspondence between all qsc-extensions T of A and all contractions X in Theorem 3.1 can be reformulated also as follows 1/2 1/2 Aµ + AM AM − Aµ AM − Aµ T = + X , (3.13) 2 2 2 where the parameters X are contractions in the subspace ran (AM − Aµ ), cf. [10], [11], [12]. It is easy to see from (3.2), (3.11), and (3.12), that if T is a qsc-extension of A such that TR = (T + T ∗ )/2 = Aµ (AM ), then in fact T = Aµ (AM ). Namely, X = XR + iXI satisﬁes 2 0 ≤ X ∗ X = XR + i(XR XI − XI XR ) + XI2 ≤ I, (3.14) ∗ 2 0 ≤ XX = XR − i(XR XI − XI XR ) + XI2 ≤ I, 2 2 so that 0 ≤ XR + XI2 ≤ I and here clearly XR = I implies XI = 0.

The description of all contractive selfadjoint extensions of a symmetric contraction A as the operator interval [Aµ , AM ] is due to M.G. Kre˘ın [27]. In that paper the notion of shorted operators was also introduced and used for instance to establish the following characterization for Aµ and AM : ran (I + Aµ )1/2 ∩ N = {0},

ran (I − AM )1/2 ∩ N = {0},

(3.15)

cf. [7], [22]. Block formulas for describing all contractive extensions of a dual pair appear in [14], [17], [45], a description in Crandall’s form [16] in [2]. The one-toone correspondence between all qsc-extensions T of A of the class C(α) and all operators X in ran (AM − Aµ ) belonging to the class C(α) by means of (3.13) was proved in a diﬀerent way in [12], another proof based on (3.2) was given in [38]. 3.3. Simplicity of the symmetric operator According to [32] a closed symmetric contraction A is said to be simple if there is no non-zero subspace in dom A which is invariant under A. Since A is symmetric, simplicity of A is equivalent to A being completely non-selfadjoint, i.e., to A having no selfadjoint parts. Lemma 3.2. Let the closed symmetric contraction A = A0 +K0 DA0 in H = H0 ⊕N, H0 = dom A, be decomposed as in (3.1) with K0 : DA0 → N. Then A is simple if and only if the subspace & ' Hs0 := span (A0 − zI)−1 K0∗ N : z ∈ ρ(A0 ) (3.16) = span { An0 K0∗ N : n = 0, 1, . . . } coincides with H0 . In this case, DA0 = H0 , K0 : H0 → N, and A0 f < f for all f ∈ H0 \{0}.

34

Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

Proof. Suppose that A is simple. Then clearly ker DA0 = {0} or equivalently A0 f < f for all f ∈ H0 \ {0}, so that DA0 = H0 and K0 : H0 → N. Observe, that the subspace Hs0 in (3.16) and therefore also H0 Hs0 is invariant under A0 = A∗0 . Then the subspace H0 Hs0 is also invariant under DA0 . Moreover, H0 Hs0 = { f ∈ H0 : K0 An0 f = 0, n = 0, 1, . . . } .

(3.17)

It follows that K0 DA0 f = 0 for all f ∈ H0 Hs0 . Hence, in view of (3.1) Af = A0 f for all f ∈ H0 Hs0 . This means that the subspace H0 Hs0 is invariant under A. Since A is a simple, one concludes that Hs0 = H0 . Conversely, assume that Hs0 = H0 . Since ran K0∗ ⊂ DA0 and DA0 is invariant under A0 , the deﬁnition of Hs0 in (3.16) shows that Hs0 ⊂ DA0 . Hence, the assumption implies that H0 = DA0 = ran DA0 , so that ker DA0 = {0}. Now,

0 ⊂ H0 is a subspace which is invariant under A. Then for every suppose that H

0 , so that K0 DA0 f = 0 for all f ∈ H

0

f ∈ H0 one has Af = A0 f + K0 DA0 f ∈ H

0 = A0 H

0 . Hence, H

0 is invariant under A0 and DA . Moreover, since and A H 0

0 . This implies that K0 H

0 is dense in H

0 = {0} ker DA0 = {0} the image DA0 H n n

and since A0 H0 ⊂ H0 one has K0 A0 H0 = {0} for all n = 0, 1, . . ., i.e.,

0 ⊂ { f ∈ H0 : K0 An f = 0, n = 0, 1, . . . } = H0 Hs = {0}, H 0 0

cf. (3.17). Therefore, A is simple.

Let T be a qsc-extension of A in the Hilbert space H = H0 ⊕ N with H0 = dom A. It is evident that the subspace & ' HT := span (T − zI)−1 N : |z| > 1 = span { T n N : n = 0, 1, 2, . . . } , (3.18) is invariant under T , and that the subspace HT := H HT , ∗

is invariant under T . Since N ⊂

HT ,

(3.19)

one obtains

HT ⊂ N⊥ = dom A ⊂ ker (T − T ∗ ). Therefore the restriction of T ∗ to HT is a selfadjoint operator in HT . The restriction T HT (= PHT T HT ) is called the N-minimal part of T . Moreover, T is said to be N-minimal if the equality H = HT holds. If T be a qsc-extension of A then its adjoint T ∗ is also a qsc-extension of A and one can associate with it the subspace HT ∗ and the corresponding N-minimal part of T ∗ . The next result shows the Nminimal parts of T and T ∗ are qsc-extensions of the simple part A Hs0 of A in the same subspace HT = HT ∗ . Proposition 3.3. Let A be a symmetric contraction in H = H0 ⊕ N with H0 = dom A, let T be a qsc-extension of A in H, and let T ∗ be its adjoint. Then the subspaces HT , HT ∗ , and Hs0 of H = H0 ⊕ N as deﬁned in (3.18) and (3.16) are connected by (H :=) HT = HT ∗ = Hs0 ⊕ N. (3.20)

Q-functions of Quasi-selfadjoint Contractions

35

In particular, the symmetric contraction A is simple if and only if the qsc-extension T , or equivalently T ∗ , of A is N-minimal. Proof. It follows from the Schur-Frobenius formula (2.9) that −(A0 − z)−1 DA0 K0∗ N −1 , |z| > 1, (T − z) N = N which implies that & ' span (T − zI)−1 N : |z| > 1 & ' = span (A0 − zI)−1 DA0 K0∗ N : z ∈ ρ(A0 ) ⊕ N & ' = (clos DA0 span (A0 − zI)−1 K0∗ N : z ∈ ρ(A0 ) ) ⊕ N. This shows that

HT = (clos DA0 Hs0 ) ⊕ N. (3.21) ∗ s Since ran K0 ⊂ DA0 and DA0 is invariant under A0 one has H0 ⊂ DA0 . In particular, Hs0 ∩ ker DA0 = {0}, which together with DA0 Hs0 ⊂ Hs0 implies that clos DA0 Hs0 = Hs0 . Hence, (3.21) implies the equality HT = Hs0 ⊕ N. It follows from (T ∗ − zI)−1 − (T − zI)−1 = (T − zI)−1 [T − T ∗ ](T ∗ − zI)−1 ,

|z| > 1,

∗

and the inclusion ran (T − T ) ⊂ N that (T ∗ − zI)−1 N ⊂ (T − zI)−1 N ⊂ HT , HT ∗

|z| > 1.

HT

Therefore, ⊂ and the reverse inclusion follows by symmetry. This completes the proof of (3.20). The last statement is clear from (3.20). For selfadjoint extensions of A the result in Proposition 3.3 has been given in [32]. In the case of closed densely deﬁned symmetric operators A there is an equivalent criterion for the simplicity of A due to M.G. Kre˘ın based on the defect elements: span { ker (A∗ − λ) : λ ∈ C \ R } = H, cf. Lemma 3.2. This characterization has been extended to non-densely deﬁned symmetric operators in [36].

4. Q-functions of qsc-operators Let T be a qsc-operator in a separable Hilbert space H and let N be a subspace of H such that N ⊃ ran (T − T ∗ ). The operator-valued function QT (z) = PN (T − zI)−1 N,

|z| > 1,

(4.1)

where PN is the orthogonal projection in H onto N, is said to be a Q-function associated with T and the subspace N. Clearly, it has the limit value QT (∞) = 0 and the Q-functions of T and T ∗ in N are connected by QT ∗ (z) = QT (¯ z )∗ ,

|z| > 1.

(4.2)

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Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

If T is a selfadjoint contraction then the Q-function QT (z) in (4.1) is a Nevanlinna function of the class NN [−1, 1]. The next result contains some basic properties for the Q-function QT (z) of a qsc-operator T as deﬁned in (4.1). Proposition 4.1. Let QT (z) be a Q-function of a qsc-operator T as deﬁned in (4.1). Then: (i) QT (z) has the following asymptotic expansion: 1 1 1 , z → ∞, (4.3) QT (z) = − I + 2 F + o z z z2 where F = −PN T N; (ii) Q−1 T (z) ∈ L(N) for all |z| > 1; −1 (iii) Q−1 T (z) has strong limit values QT (±1): −1 Q−1 T (−1) = lim QT (x), x↑−1

−1 Q−1 T (1) = lim QT (x); x↓1

(iv) for all f, g ∈ N the following inequality holds: % −1 % % Q (−1) + Q−1 (1) f, g %2 T T −1 −1 ≤ Q−1 QT (−1) − Q−1 T (−1) − QT (1) f, f T (1) g, g ; (v) the function −Q−1 T (z) − F − zI is an operator-valued Nevanlinna function; (vi) QT (z) ∈ NN [−1, 1] if and only if F = F ∗ . Moreover, if T is decomposed as in (3.2) with H0 = H N and A = T H0 , then F = K0 A0 K0∗ − DK0∗ XDK0∗ , ∗ ∗ Q−1 T (−1) = DK0 (X + I)DK0 ,

−Q−1 T (z)

(4.4)

∗ ∗ Q−1 T (1) = DK0 (X − I)DK0 ,

− F − zI = K0 (I −

A20 )(A0

−1

− zI)

K0∗ .

−1

Proof. (i) Clearly, limz→∞ zQT (z)h = limz→∞ zPN (T − zI) h ∈ N. Moreover, for all h ∈ N

(4.6)

h = −PN h for all

lim z(I + zQT (z))h = lim zPN T (T − zI)−1 h = −PN T h.

z→∞

(4.5)

z→∞

(4.7)

Hence, QT (z) admits the asymptotic expansion (4.3). (ii) Let |z| > 1, let f ∈ N, and let ϕ = (T − zI)−1 f . Then f ≤ (1 + |z|)ϕ and % % |(QT (z)f, f )| = % (T − zI)−1 f, f % = |(ϕ, (T − zI)ϕ)| % % |z| − 1 = %(ϕ, T ϕ) − z¯ϕ2 % ≥ f 2 . (|z| + 1)2 Since |(QT (z)f, f )| = |(QT (z)∗ f, f )|, this implies that QT (z)f ≥

|z| − 1 f , (|z| + 1)2

Therefore, Q−1 T (z) ∈ L(N) for all |z| > 1.

QT (z)∗ f ≥

|z| − 1 f . (|z| + 1)2

Q-functions of Quasi-selfadjoint Contractions

37

(iii) Decompose H = H0 ⊕N and write T in block form as in (3.2), where H0 = 1/2 , HN, A = T H0 , A0 = P0 A is a selfadjoint contraction in H0 , DA0 = I − A20 K0 ∈ L (DA0 , N) is a contraction, and X is a contraction in the subspace DK0∗ ⊂ N. The formula (4.4) for F is immediate from (3.2). Write Q−1 T (z) as in (2.10), Q−1 T (z) = −VT (z) − zI, where

|z| > 1,

VT (z) = K0 A0 + (A0 − zI)−1 (I − A20 ) K0∗ − DK0∗ XDK0∗ .

(4.8)

This shows that the limit values Q−1 T (±1) exist and that they are given by (4.5). (iv) It follows from (4.5) that −1 Q−1 T (−1) + QT (1) = DK0∗ XDK0∗ , 2 −1 Q−1 2 ∗ T (−1) − QT (1) = DK ∗ = I − K0 K0 . 0 2

(4.9)

2 It remains to apply the criterion (2.3) with S0 = 0 and Rl = Rr = DK ∗. 0 (v) It follows from (4.4) and (4.8) that (4.6) holds. Clearly, the function in (4.6) is a Nevanlinna function. (vi) If QT (z) ∈ NN [−1, 1], then −QT (z)−1 is a Nevanlinna function and now part (v) implies that F = F ∗ . Conversely, if F = F ∗ then the function VT (z) in (4.8) and −QT (z)−1 = VT (z) + zI are Nevanlinna functions. Therefore, QT (z) ∈ NN [−1, 1].

Let T be a qsc-operator, let QT (z) be deﬁned by (4.1), and let F be deﬁned by F = −PN T N. Associate with QT (z) the following kernels: GT (z, ξ) :=

QT (z) − QT (ξ)∗ − QT (z)(F − F ∗ )QT (ξ)∗ , z − ξ¯

(4.10)

¯ T (z, ξ), MT (z, ξ) := I + zQT (z) + ξQT (ξ)∗ + z ξG

(4.11)

LT (z, ξ) := GT (z, ξ) − MT (z, ξ),

(4.12)

KT (z, ξ) := LT (z, ξ) − QT (z)(F − F ∗ )QT (ξ)∗ ,

(4.13)

and ¯ |z|, |ξ| < 1. The insertion of the deﬁnition of GT (z, ξ) in LT (z, ξ) and with z = ξ, KT (z, ξ) leads to the identities ¯ T (z, ξ) = (1 − z 2 )QT (z) − (1 − ξ¯2 )QT (ξ)∗ (z − ξ)L ¯ T (z)(F − F ∗ )QT (ξ)∗ − (z − ξ)I, ¯ − (1 − z ξ)Q and ¯ T (z, ξ) = (1 − z 2 )QT (z) − (1 − ξ¯2 )QT (ξ)∗ (z − ξ)K ¯ T (z)(F − F ∗ )QT (ξ)∗ − (z − ξ)I. ¯ − (1 + z)(1 − ξ)Q

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Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

Proposition 4.2. Let T be a qsc-operator, let QT (z) be deﬁned by (4.1), and let F be deﬁned by F = −PN T N. Let the kernels associated with QT (z) be given by (4.10), (4.11), (4.12), and (4.13). Then the following equalities hold for every ¯ |z|, |ξ| > 1: z = ξ, GT (z, ξ) = PN (T − zI)−1 (T ∗ − ξI)−1 N, −1

MT (z, ξ) = PN (T − zI)

∗

∗

−1

T T (T − ξI)

N,

(4.14) (4.15)

and

LT (z, ξ) = PN (T − zI)−1 (I − T T ∗)(T ∗ − ξI)−1 N. (4.16) The operator-valued functions GT (z, ξ), MT (z, ξ), and LT (z, ξ) are nonnegative kernels. If, in addition, the operator T belongs to the class C(α) then the function KT (z, ξ) = PN (T − zI)−1 (I + T )(I − T ∗ )(T ∗ − ξI)−1 N

(4.17)

with |z|, |ξ| > 1 is an α-sectorial kernel. Proof. Note that ran (T − T ∗ ) ⊂ N implies that N⊥ ⊂ ker (T − T ∗ ), and hence T − T ∗ = PN (T − T ∗ )PN . Therefore, for every f, g ∈ N, ¯ −1 f, g ((QT (z) − Q∗T (ξ))f, g) = PN (T − zI)−1 f − PN (T ∗ − ξI) ¯ −1 f, g = PN (T − zI)−1 (T ∗ − T )(T ∗ − ξI) ¯ PN (T − zI)−1 (T ∗ − ξI) ¯ −1 f, g + (z − ξ) = (QT (z)(F − F ∗ )QT (ξ)∗ f, g) ¯ PN (T − zI)−1 (T ∗ − ξI) ¯ −1 f, g . + (z − ξ) Hence, it follows that ¯ N (T − zI)−1 (T ∗ − ξI) ¯ −1 N, QT (z) − Q∗T (ξ) = QT (z)(F − F ∗ )Q∗T (ξ) + (z − ξ)P and this proves (4.14). The identity (4.15) follows now from ∗ ∗ ¯ −1 f, T ∗ (T ∗ − z¯I)−1 g = f + ξ(T ¯ ∗ − ξI) ¯ −1 f, g + z¯(T ∗ − z¯I)−1 g T (T − ξI) ¯ ∗ (ξ)f, g) + z ξ(G ¯ T (z, ξ)f, g), f, g ∈ N. = (f, g) + z(QT (z)f, g) + ξ(Q T

Subtracting (4.15) from (4.14) gives immediately the identity (4.16). It is clear from the given formulas (4.14), (4.15), and (4.16), that the functions GT (z, ξ), MT (z, ξ), and LT (z, ξ) are nonnegative kernels. Since T − T ∗ = PN (T − T ∗ )PN , the deﬁnitions of QT (z) and F in (4.1), (4.7) show that ¯ −1 . −QT (z)(F − F ∗ )Q∗ (ξ) = PN (T − zI)−1 (T − T ∗ )(T ∗ − ξI) T

Combining this identity with (4.16) leads to (4.17). It is a consequence of (2.6) that KT (z, ξ) is an α-sectorial kernel.

Proposition 4.3. Let T be a qsc-operator in a Hilbert space H, N ⊃ ran (T − T ∗ ). Suppose that T is N-minimal, i.e., H = span { (T − z)−1 N : |z| > 1 }. Then the following conditions are equivalent: (i) N = H;

Q-functions of Quasi-selfadjoint Contractions

39

(ii) GT (z, z) = QT (z)QT (z)∗ for at least one (and equivalently for every) z with |z| > 1, where QT (z) is Q-function of T deﬁned by (4.1) and GT (z, ξ) is deﬁned by (4.10); (iii) the operator-valued function Q−1 T (z) + zI is constant. Proof. (i) ⇒ (ii) & (iii) If N = H then QT (z) = (T − zI)−1 and the equality GT (z, z) = QT (z)QT (z)∗ for all z, |z| > 1, follows immediately from (4.14). Besides, Q−1 T (z) + zI = T for all z, |z| > 1. (ii) ⇒ (i) Now suppose that GT (z, z) = QT (z)QT (z)∗ for some z, |z| > 1. Then (4.1) and (4.14) yield (T ∗ − z¯I)−1 f = PN (T ∗ − z¯I)−1 f for every f ∈ N. Therefore, (T ∗ − z¯I)−1 N ⊂ N which implies that the subspace N is invariant under T ∗ , and hence also under T , since ran (T − T ∗ ) ⊂ N. Because T is N-minimal, this leads to N = H. (iii) ⇒ (ii) Suppose that Q−1 T (z) + zI is constant for |z| > 1. According to Proposition 4.1 the function −Q−1 T (z) − zI − F has a holomorphic continuation onto Ext [−1, 1] as a Nevanlinna function. Since −Q−1 T (z) − zI − F is constant for |z| > 1, one has −∗ −Q−1 + z¯I + F ∗ = 0, T (z) − zI − F + QT (z)

|z| > 1.

It follows that −∗ − (F − F ∗ ) −Q−1 T (z) + QT (z) = I, z − z¯

|z| > 1,

and thus

−∗ QT (z) −Q−1 − (F − F ∗ ) QT (z)∗ T (z) + QT (z) = QT (z)QT (z)∗ , z − z¯ Therefore G(z, z) = QT (z)QT (z)∗ for all z, |z| > 1.

|z| > 1.

Observe, that the equality (4.14) can be rewritten in the following two equivalent forms: −QT (z)−1 − F − (−QT (ξ)−1 − F )∗ z − ξ¯ (4.18) ¯ −1 QT (ξ)−∗ , = QT (z)−1 PN (T − zI)−1 (T ∗ − ξI) and −QT (z)−1 − F − zI − (−QT (ξ)−1 − F − ξI)∗ z − ξ¯ ¯ −1 QT (ξ)−∗ . = QT (z)−1 PN (T − zI)−1 (I − PN )(T ∗ − ξI)

(4.19)

These formulas show that −QT (z)−1 − F and −QT (z)−1 − F − zI indeed are Nevanlinna functions. In particular, the conditions (i)–(iii) in Proposition 4.3 are equivalent to the right side of (4.19) to vanish.

40

Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

Remark 4.4. The Q-function QT (z) as deﬁned in (4.1) can be interpreted as the Weyl function for a special kind of boundary value space of a dual pair of operators, cf. [37], [39], [40]. To explain this, let A = A0 + K0 DA0 be a Hermitian contraction and let T be a qsc-extension of A, i.e., T is a contractive extension of a dual pair {A, A}. Let A∗ be the adjoint linear relation of A in the Cartesian product H × H. Then A∗ can be represented as follows: ( ) ( ) A∗ = {f, T f + ϕ} : f ∈ H, ϕ ∈ N = {f, T ∗ f + ψ} : f ∈ H, ψ ∈ N . Deﬁne the following bounded linear operators acting from A∗ into N: Γ0 {f, f } = PN f,

Γ1 {f, f } = PN T ∗ f − PN f ,

Γ2 {f, f } = PN T f − PN f ,

where {f, f } ∈ A∗ . Then {N, Γ0 , Γ1 , Γ2 } forms a boundary value space for A∗ . In g = {g, g } ∈ A∗ the following identity holds particular, for all f = {f, f }, (f , g) − (f, g ) = (Γ0 f, Γ1 g ) − (Γ2 f, Γ0 g), and moreover ker Γ1 = T ∗ , ker Γ2 = T , and ( ker Γ0 = {h, A0 h + ϕ} : h ∈ H0 ,

) ϕ∈N .

The corresponding γ-ﬁelds are the following operator functions ⎧ −1 ∗ ⎪ ⎪ γ0 (z)ϕ = −(A0 − zI) K0 DA0 ϕ, ⎨ γ1 (z)ϕ = (T ∗ − zI)−1 ϕ, ⎪ ⎪ ⎩ γ2 (z)ϕ = (T − zI)−1 ϕ, where ϕ ∈ N and |z| > 1. It follows that QT (z) = Γ0 γ2 (z) is given by QT (z) = PN (T − zI)−1 N, and that −Q−1 T (z) = Γ2 γ0 (z) is given by −1 (I − A20 ) K0∗ − DK0∗ XDK0∗ + zI N, −Q−1 T (z) = K0 A0 + (A0 − zI) where T is decomposed as in (3.2); see also Proposition 4.1. In particular, this means that QT (z) can be interpreted as the Weyl function corresponding to the boundary value space {N, Γ0 , Γ1 , Γ2 } in the sense of [39], [40].

5. The resolvent formula for qsc-extensions Let A = A0 + K0 DA0 be a closed symmetric contraction in H and let T be a qscextension of A given by the block matrix (3.2). If QT (z) = PN (T − zI)−1 N is the −1 Q-function of T , then by (4.9) the operator (Q−1 T (−1) − QT (1))/2 is nonnegative on N. Let −1 −1 −1 Q−1 T (−1) + QT (1) QT (−1) − QT (1) BQT := B − , (5.1) 2 2

Q-functions of Quasi-selfadjoint Contractions

41

be the operator ball in L(N) with center −1 ∗ ∗ −(Q−1 T (−1) + QT (1))/2 = −DK0 XDK0

and equal left and right radii −1 2 (Q−1 T (−1) − QT (1))/2 = DK0∗ .

Recall that it is the set of all operators in N of the form −1 1/2 −1 1/2 −1 Q−1 QT (−1) − Q−1 QT (−1) − Q−1 T (−1) + QT (1) T (1) T (1) + Y , − 2 2 2 where Y ≤ 1. Theorem 5.1. Let A be a closed symmetric operator in a Hilbert space H. Then the formula −1

I + QT (z)B (T − zI)−1 = (T − zI)−1 − (T − zI)−1 B PN (T − zI)−1 (5.2) with |z| > 1 gives a one-to-one correspondence between the resolvents of all qsc belonging to the operator ball BQT in (5.1). extensions T of A and all operators B Proof. By Theorem 3.1 every qsc-extension T of A can be written in the block form A0 DA0 K0∗ T = , (5.3) K0 DA0 −K0 A0 K0∗ + DK0∗ Y DK0∗ where Y ≤ 1. This together with (3.2) gives

:= T − T N = −DK ∗ XDK ∗ + DK ∗ Y DK ∗ (5.4) B 0 0 0 0

∈ BQT . It follows from which in view of (4.9) this means that B

N T − zI = T − zI + BP that

(5.5)

N (T − z)−1 , (T − z)−1 = (T − z)−1 + (T − z)−1 BP

|z| > 1,

and compression to N leads to

Q(z).

Q(z) = Q(z) + Q(z)B

Since Q(z) and Q(z) are invertible by part (ii) of Proposition 4.1, one obtains −1

= Q(z)−1 (I + Q(z)B)

= (I + BQ(z))Q(z)

−1 = Q(z)−1 + B . Q(z)

Therefore, the operators

I + Q(z)B

and I + BQ(z),

|z| > 1,

are invertible in N, too. Furthermore, by rewriting (5.5) in the form

N (T − zI)−1 (T − zI), T − zI = I + BP

42

Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

−1

N (T − zI)−1 ∈ L(H) for every |z| > 1 and it is clear that I + BP −1

N (T − zI)−1 (T − zI)−1 = (T − zI)−1 I + BP , |z| > 1.

(5.6)

It also follows from (5.5) that

N (T − zI)−1 . (T − zI)−1 − (T − zI)−1 = −(T − zI)−1 BP

(5.7)

Now using the identities (5.6), (5.7), and −1 −1

N (T − zI)−1

N I + PN (T − zI)−1 BP

N = BP

N I + BP , BP one obtains (T − zI)−1 − (T − zI)−1

−1

N (T − zI)−1

N (T − zI)−1 = −(T − zI)−1 I + BP BP −1

N I + PN (T − zI)−1 BP

N = −(T − zI)−1 BP PN (T − zI)−1 −1

I + QT (z)B = −(T − zI)−1 B PN (T − zI)−1 ,

which gives the required identity (5.2).

∈ BQT , i.e., that B

is given by Conversely, assume that B 1/2 −1 1/2 −1 −1 Q−1 QT (−1) − Q−1 Q−1 T (−1) + QT (1) T (−1) − QT (1) T (1)

+ Y − 2 2 2

= −DK ∗ XDK ∗ + DK ∗ Y DK ∗ . Consider for some Y ≤ 1. By (4.9) one has B 0 0 0 0

of the form (5.3) which the qsc-extension T of A given by the block operator T

= T − T N. As was shown above, the is determined by Y . Then clearly B

is invertible for all |z| > 1 and the resolvent of T takes the operator I + QT (z)B form (5.2). The one-to-one correspondence is clear from the given arguments. Observe, that the Q-function QT (z) of the operator T in (5.2) and the Qfunction QT (z) of T are connected via QT (z) = PN (T − zI)−1 N

−1 QT (z) = QT (z)(I + BQ

T (z))−1 = (I + QT (z)B)

(5.8)

+ Q−1 (z))−1 . = (B T Remark 5.2. The resolvent formula (5.2) established in the theorem for the qscextensions T of A remains true for all quasi-selfadjoint extensions of A, i.e., T and T in Theorem 5.1 can be taken to be bounded, not necessarily contractive, extensions of A. Indeed, by deﬁning the functions QT (z), |z| > T , and QT (z), |z| > T as in (4.1) then they are bounded and boundedly invertible (cf., e.g.,

Q-functions of Quasi-selfadjoint Contractions

43

the proof presented for part (ii) of Proposition 4.1). It remains to repeat the proof of Theorem 5.1 to obtain the resolvent formula (5.2) for two arbitrary bounded quasi-selfadjoint extensions T and T of A in the domain |z| > max{T , T }. The

is still given by (5.4), but now X and Y are not in general contractive parameter B

need not belong to the operator and even if T is a qsc-extension of A (X ≤ 1), B

ball BQT in (5.1). Of course, T can be interpreted as a bounded perturbation of

in (5.4). This allows one to apply the resolvent formula T by the parameter B (5.2) to study, for instance, the behavior of the resolvents under arbitrary small

≤ ε with B

not necessarily belonging to the operator ball BQT perturbations B in (5.1). Descriptions of canonical and generalized resolvents of selfadjoint contractive extensions of a non-densely deﬁned symmetric contraction A were given by M.G. Kre˘ın and I.E. Ovcharenko in [32] by means of Q-functions of the selfadjoint extremal extensions Aµ and AM . Later B. Textorius [47] described this set using

of A and the corresponding special an arbitrary selfadjoint contractive extension A form of its Q-function. In [10], [12], [11] all canonical and generalized resolvents of qsc-extensions were parametrized by means of Qµ - and QM -functions in the sense of Kre˘ın-Ovcharenko in [32]. The proof of the parametrization formula (5.2), which is presented here, is kept very elementary, along the lines of a similar presentation in [9]. An abstract formula for canonical and generalized resolvents of extensions of a dual pair of linear relations in terms of a boundary value space and its Weyl function is derived in [39], [40].

6. A reproducing kernel space model for Q-functions of quasi-selfadjoint contractions Let N be a Hilbert space. An operator-valued function Q(z) with values in L(N) and holomorphic outside the unit disk is said to belong to the class Q(N) if: (i) Q(z) has the expansion 1 1 1 , z → ∞; (6.1) Q(z) = − I + 2 F + o z z z2 (ii) the L(N)-valued function G(z, ξ) =

Q(z) − Q(ξ)∗ − Q(z)(F − F ∗ )Q(ξ)∗ , z − ξ¯

¯ z = ξ,

with |z|, |ξ| > 1, is a nonnegative kernel; (iii) the L(N)-valued function ¯ ¯ − F ∗ )Q(ξ)∗ − (z − ξ)I (1 − z 2 )Q(z) − (1 − ξ¯2 )Q(ξ)∗ − (1 − z ξ)Q(z)(F L(z, ξ) = , z − ξ¯ ¯ |z|, |ξ| > 1, is a nonnegative kernel; with z = ξ,

44

Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

(iv) there exist a complex number z0 , |z0 | > 1, and a vector f ∈ N, such that G(z0 , z0 )f = Q(z0 )Q(z0 )∗ f. If T is a qsc-operator in the Hilbert space H, N is a subspace of H such that N = H and ran (T − T ∗ ) ⊂ N, and QT (z) is its Q-function deﬁned by (4.1), then according to Propositions 4.1, 4.2, and 4.3 the function QT (z) belongs to the class Q(N). The converse statement is also true. Theorem 6.1. Let Q(z) be a function of the class Q(N). Then there exist a Hilbert space H ⊃ N, N = H, and an N-minimal qsc-operator T in H, such that N ⊃ ran (T − T ∗ ) and Q(z) = PN (T − zI)−1 N,

for all |z| > 1.

(6.2)

If, in addition, the L(N)-valued function K(z, ξ) := L(z, ξ) − Q(z)(F − F ∗ )Q(ξ)∗ ¯ ¯ − F ∗ )Q(ξ)∗ − (z − ξ)I (1 − z 2 )Q(z) − (1 − ξ¯2 )Q(ξ)∗ − (1 + z)(1 − ξ)Q(z)(F = z − ξ¯ ¯ |z|, |ξ| > 1, where F is given by (6.1), is an α-sectorial kernel with with z = ξ, α ∈ [0, π/2), then the corresponding operator T belongs to the class C(α).

be the reproducing kernel Hilbert space associated with the Proof. Step 1. Let H

is the completion of nonnegative kernel G(z, ξ), i.e., H span { G(·, w)f : f ∈ N, |w| > 1 } with respect to the norm determined by the inner product (G(·, w)f, G(·, µ)g) H

= (G(µ, w)f, g)N . For all f ∈ N and |w|, |µ| > 1, 2 2 wG(·, w)f − µ ¯ G(·, µ)f 2H

= |w| (G(w, w)f, f ) N + |µ| (G(µ, µ)f, f )N

− µw (G(µ, w)f, f )N − µw (G(w, µ)f, f )N .

(6.3)

In view of (6.1) one has Q(z) = (−1/z)I + o(1/z) as z → ∞, which implies that lim wG(z, w)f = −Q(z)f,

|z| > 1,

w→∞

(6.4)

and moreover that lim µwG(µ, w)f = f,

µ,w* →∞

f ∈ N.

(6.5)

(Here * → stands for the nontangential limit in a sector | arg(z) − π/2| ≤ α < π/2.)

Hence (6.3) and (6.5) imply that the following limit exists in H Kf := − lim wG(z, w)f w* →∞

(6.6)

for which and deﬁnes a linear operator K : N → H 2 2 2 Kf 2H

= lim ||wG(·, w)f ||H

= lim |w| (G(w, w)f, f ) N = ||f ||N . w* →∞

w* →∞

(6.7)

Q-functions of Quasi-selfadjoint Contractions

45

Thus, K is isometric. It follows from (6.4) that (Kf, G(·, µ)g)H

= − lim w (G(·, w)f, G(·, µ)g) H

w* →∞

= − lim w (G(µ, w)f, g)N = (Q(µ)f, g)N , w* →∞

which shows that K ∗ G(·, µ)g = Q(µ)∗ g,

g ∈ N.

(6.8)

by Step 2. Deﬁne the linear relation S in H ++ n , , n n S= G(·, wi )fi , Kfi + w i G(·, wi )fi : fi ∈ N, |wi | > 1 . i=1

i=1

(6.9)

i=1

In fact, S is a contractive linear By deﬁnition the domain of S is dense in H.

operator in H, since -2 -2 n n n - - G(·, wi )fi - − Kfi + wi G(·, wi )fi H

i=1

=

n

−

H

i=1

(G(wj , wi )fi , fj )N −

i,j=1 n

=

i=1

n

(fi , fj )N −

i,j=1 n

wj (Q(wj )fi , fj )N −

i,j=1 n

n

w i (Q(wi )∗ fi , fj )N

i,j=1

wj w i (G(wj , wi )fi , fj )N

i,j=1

(L(wj , wi )fi , fj )N ≥ 0,

i,j=1

where (6.7) and (6.8) have been used. Therefore, the operator S has a unique

and for which the same contractive continuation which is deﬁned everywhere on H notation S is preserved. # Step 3. To calculate the imaginary part of S note that for h = ni=1 G(·, wi )fi the following identities holds ⎛ ⎞ n n n Kfi + w i G(·, wi )fi , G(·, wj )fj ⎠ (Sh, h) = ⎝ i=1

=

n

i=1

j=1

(Q(wj )fi + wi G(wj , wi )fi , fj )N .

i,j=1

Similarly one obtains (h, Sh) =

n i,j=1

(Q(wi )∗ fi + wj G(wj , wi )fi , fj )N .

H

46

Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

Since Q(wj ) − Q(wi )∗ + (w i − wj )G(wj , wi ) = Q(wj )(F − F ∗ )Q(wi )∗ , one obtains ⎛ ⎞ n n ⎝(S − S ∗ ) G(·, wi )fi , G(·, wj )fj ⎠ i=1

=

=

n i,j=1 n

j=1

H

(Q(wj )(F − F ∗ )Q(wi )∗ fi , fj )N ((F − F ∗ )K ∗ G(·, wi )fi , K ∗ G(·, wj )fj )N

i,j=1

⎛

= ⎝K(F − F ∗ )K ∗

n

G(·, wi )fi

,

i=1

n

⎞ G(·, wj )fj ⎠ .

j=1

H

This implies that S − S ∗ = K(F − F ∗ )K ∗ . By the deﬁnition of S in (6.9) one has (S − wI)G(·, w)f = Kf , so that (S − wI)−1 Kf = G(·, w)f,

f ∈ N,

|w| > 1.

(6.10) (6.11)

∗

Step 4. Since K is isometric, ran K is closed. # Let H0 := ker K and deﬁne n H := H0 ⊕ N. Observe, that according to (6.8) h = i=1 G(·, wi )fi belongs to the # n ∗

if and only if

subspace H0 of H i=1 Q(wi ) fi = 0. Now decompose H = H0 ⊕ran K

and deﬁne the operator U : H → H by U(x + y) = x + K ∗ y,

x ∈ H0 ,

y ∈ ran K.

Then U maps H onto H and it is unitary. Hence, the operator T deﬁned by T := US ∗ U−1 is contractive in H and (6.10) shows that ran (T − T ∗ ) ⊂ U(ran K) = N. Furthermore, for f, g ∈ N and |z| > 1 the identities (6.8) and (6.11) yield (T − z)−1 f, g H = (S ∗ − zI)−1 U−1 f, U−1 g H

∗ −1 = (S − zI) Kf, Kg H

= Kf, (S − z¯I)−1 Kg H

= (Kf, G(·, z)g)H

= (Q(z)f, g)N . Thus, Q(z) = PN (T − zI)−1 N, |z| > 1. Moreover, it follows from (6.11) that the operator T is N-minimal. Step 5. Finally it is shown that H0 = {0}. If H0 = {0} then N = H and by Proposition 4.3 the equality G(z, z) = Q(z)Q(z)∗ holds for all |z| > 1. But this is impossible due to the condition (iv) of the deﬁnition of the class Q(N). Therefore H0 = 0, N = H, and T is a qsc-operator whose Q-function QT (z) coincides with Q(z).

Q-functions of Quasi-selfadjoint Contractions

47

As to the last statement observe, that since Q(z) is of the form (6.2), the kernel K(z, ξ) admits the operator representation (4.17) in Proposition 4.2. Since T is N-minimal, it follows from (4.17) and (2.6) that T ∈ C(α). The qsc-operator T constructed in Theorem 6.1 is N-minimal. The next result shows that this model for functions Q(z) belonging to the class Q(N) is essentially unique. Namely, the N-minimal part of a qsc-operator T (and hence also of T ∗ ) is up to unitary equivalence uniquely determined by its Q-function; a fact which is well known in the selfadjoint case. Theorem 6.2. Let H1 = H01 ⊕ N and H2 = H02 ⊕ N be two Hilbert spaces, and let T1 and T2 be qsc-operators in H1 and H2 , respectively, such that ran (T1 − T1∗ ) ⊂ N and ran (T2 − T2∗ ) ⊂ N. If QT1 (z) = QT2 (z) in some neighborhood of inﬁnity then the N-minimal parts of T1 and T2 are unitarily equivalent. Proof. Assume that QT1 (z) = QT2 (z) holds in some neighborhood of inﬁnity, say, for |z| > R > 1. Then these functions coincide everywhere outside the unit disk. It follows from (4.3) and (4.7) that F1 = F2 , while (4.14) implies that PN (T1∗ − ξI)−1 (T1 − zI)−1 N = PN (T2∗ − ξI)−1 (T2 − zI)−1 N, for all |z|, |ξ| > 1; cf. (4.2). Hence, for all f, g ∈ N (T1 − zI)−1 f, (T1 − ξI)−1 g = (T2 − zI)−1 f, (T2 − ξI)−1 g . (6.12) & ' −1 Now deﬁne&the linear relation U from ' H1 = span (T1 − zI) N : |z| > 1 into −1 H2 = span (T2 − zI) N : |z| > 1 by the formula , + n n −1 −1 U= (T1 − zk I) fk , (T2 − zk I) fk . k=1

k=1

Then the identity (6.12) implies that U is a unitary operator from H1 onto H2 . In addition, U f = f for all f ∈ N, and n n n −1 −1 U T1 (T1 − zk I) fk = fk + U zk (T1 − zk I) fk k=1

=

n k=1

fk +

k=1 n k=1

−1

zk (T2 − zk I)

fk = T 2 U

k=1 n

−1

(T1 − zk I)

fk

.

k=1

Therefore, the simple parts of T1 and T2 are unitarily equivalent.

7. Characteristic properties of Q-functions of quasi-selfadjoint contractions The deﬁnition of the class Q(N) given in Section 6 can be seen as an analytical characterization for Q-functions of qsc-operators T as deﬁned in (4.1). Another characterization is established in the next theorem.

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Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

Theorem 7.1. Let N be a Hilbert space. The following conditions are equivalent: (i) the function Q(z) belongs to the class Q(N); (ii) (a) Q(z) ∈ L(N) is holomorphic in the domain |z| > 1 and with F ∈ L(N) it has the asymptotic expansion 1 1 1 Q(z) = − I + 2 F + o , z → ∞; z z z2 (b) the function −Q−1 (z) − zI − F is not constant, it has a holomorphic continuation onto Ext [−1, 1] as a bounded Nevanlinna function, and the strong limits Q−1 (±1) exist; (c) Q−1 (−1) − Q−1 (1) ≥ 0 and for all f, g ∈ N the following inequality holds: % −1 % % Q (−1) + Q−1 (1) f, g %2 −1 ≤ Q−1 (−1) − Q−1 (1) f, f Q (−1) − Q−1 (1) g, g . Proof. (i) ⇒ (ii) Let the function Q(z) belong to the class Q(N). Then (a) holds by deﬁnition, see (6.1). By Theorem 6.1 the function Q(z) has the operator representation Q(z) = PN (T − zI)−1 N, where T is a qsc-operator in a Hilbert space H ⊃ N, such that ran (T − T ∗ ) ⊂ N. Now (b) follows from parts (ii) and (v) of Proposition 4.1 and Proposition 4.3, see also the identity (4.19). The inequality in (c) is obtained from part (iv) of Proposition 4.1. (ii) ⇒ (i) Now assume that the function Q(z) has the properties (a)–(c). It follows from (a) and (b) that Q−1 (z) = −zI − F − G(z),

G(z) = o(1),

z → ∞.

Here G(z) ∈ NN [−1, 1] and G(∞) = 0. Now it follows from Theorem 2.3 that G(z) has the representation G(z) = K0 (A0 − zI)−1 (I − A20 )K0∗ , where A0 is a selfadjoint contraction in some Hilbert space H0 and K0 ∈ L(H0 , N). Moreover, according to (2.14) G(−1) = −Q−1 (−1) + I − F = K0 (I − A0 )K0∗ , G(1) = −Q−1 (1) − I − F = −K0 (I + A0 )K0∗ . This gives

⎧ −1 −1 ⎪ ⎨ Q (−1) − Q (1) = I − K0 K0∗ , 2 −1 −1 ⎪ ⎩ Q (−1) + Q (1) = K0 A0 K ∗ − F. 0 2 ∗ Now the assumption (c) implies that I − K0 K0 ≥ 0 and |((K0 A0 K0∗ − F )f, g)| ≤ DK0∗ f DK0∗ g,

f, g ∈ N.

(7.1)

Q-functions of Quasi-selfadjoint Contractions

49

By (2.3) there exists a contraction X in DK0∗ such that −F = −K0 A0 K0∗ + DK0∗ XDK0∗ .

(7.2)

Consider the Hilbert space H = H0 ⊕ N and let the operator T in H be given by the block form (3.2). Then T is a contraction and, in fact, a qsc-extension of the closed symmetric contraction A = A0 + K0 DA0 deﬁned on H0 . According to Schur-Frobenius formula (see (2.9), (2.10)) PN (T − zI)−1 N = −(G(z) + zI + F )−1 = Q(z),

|z| > 1,

i.e., Q(z) is the Q-function of T . Therefore, Q(z) belongs to the class Q(N).

The model established in Theorem 6.1 yields the following simple characterizations of Q-functions corresponding to the extreme selfadjoint contractive extensions Aµ and AM of A within the class Q(N). Recall, that all sc-extensions T = T ∗ of A can be described as the operator interval [Aµ , AM ], see [27]. Proposition 7.2. Let Q(z) belong to the class Q(N) and suppose that lim inf |(Q(x)f, f )| = ∞,

for all f ∈ N \ {0},

(7.3)

lim inf |(Q(x)f, f )| = ∞,

for all f ∈ N \ {0}.

(7.4)

x↑−1

or x↓1

Then Q(z) is a Nevanlinna function in NN [−1, 1] and it can be represented in the form Q(z) = PN (Aµ − zI)−1 N or Q(z) = PN (AM − zI)−1 N, z ∈ Ext [−1, 1], respectively, where Aµ and AM are the left and right extreme sc-extension of some symmetric contraction A. Proof. According to Theorem 6.1 the function Q(z) has the operator representation Q(z) = PN (T − zI)−1 N, where T is a qsc-operator in a Hilbert space H ⊃ N, such that ran (T − T ∗ ) ⊂ N. Moreover, T is a qsc-extension of the closed symmetric contraction A deﬁned by A = T dom A with dom A = H N. Let TR = (T + T ∗ )/2 and TI = (T − T ∗ )/2 be the real and the imaginary part of T , respectively, so that T = TR + iTI . Then for |x| > 1, −1

(T − xI)−1 = (TR − xI)−1/2 (I + iB)

(TR − xI)−1/2 ,

where

B = (TR − xI)−1/2 TI (TR − xI)−1/2 is a bounded selfadjoint operator. This shows that for all f ∈ N (Q(x)f, f ) = (I + iB)−1 (TR − xI)−1/2 f, (TR − xI)−1/2 f . Since (I + iB)−1 ≤ 1, one obtains

-2 |(Q(x)f, f )| ≤ -(TR − xI)−1/2 f - .

Now the assumption (7.3) implies that -2 lim inf -(TR − xI)−1/2 f - = ∞, x↑−1

for all f ∈ N \ {0}.

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Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

This means that ran (I + TR )1/2 ∩ N = {0}, cf., e.g., [7]. Since TR is a sc-extension of A, one concludes from the characterizations in (3.15) that TR = Aµ , cf. [27], [7], [22]. Now, in view of (3.14) TI = 0 and T = Aµ . The proof of the other statement is similar. Some further characteristic properties of Q-functions in the selfadjoint case, in particular, of Qµ - and QM -functions corresponding to the sc-extensions Aµ and AM have been established in [7], including some corrections to the results stated in [32].

8. Linear fractional transformations of the class Q(N) The Kre˘ın formula (5.2) and the discussion following it concerning the formulas in (5.8) give rise to a linear fractional transformation of Q-functions. Theorem 8.1. Let Q(z) belong to the class Q(N). Then the function −1

Q(z) (I + BQ(z))

,

|z| > 1,

belongs to the class Q(N) if and only if Q−1 (−1) + Q−1 (1) Q−1 (−1) − Q−1 (1) , . B∈B − 2 2

(8.1)

−1

Moreover, Q(z) (I + BQ(z)) is a Nevanlinna function of the class NN [−1, 1] if and only if B satisﬁes the conditions B + Q−1 (1) ≤ 0,

B + Q−1 (−1) ≥ 0.

(8.2)

Proof. First observe that, if B ∈ L(N) and (I + BQ(z))−1 ∈ L(N) for all |z| > 1, then it follows from (6.1) that 1 1 1 −1

, z → ∞, Q(z) := Q(z) (I + BQ(z)) = − I + 2 (F − B) + o z z z2

−1 (z) = Q−1 (z) + B. and clearly Q

Now assume that Q(z) ∈ Q(N). Then Q(z) ∈ L(N), |z| > 1, and since by −1 −1 Theorem 7.1 Q(z) , Q(z) ∈ L(N), |z| > 1, one has B, (I + BQ(z))−1 ∈ L(N)

−1 (±1) exist and satisfy for all |z| > 1. Moreover, the limit values Q

−1 (−1) = Q−1 (−1) + B, Q

−1 (1) = Q−1 (1) + B. Q

Now part (c) of Theorem 7.1 implies that % %2 −1 −1 % % % B + Q (−1) + Q (1) f, g % % % 2 −1 −1 Q (−1) − Q−1 (1) Q (−1) − Q−1 (1) f, f g, g ≤ 2 2 holds for all f, g ∈ N. Therefore, the condition (8.1) is satisﬁed.

(8.3)

Q-functions of Quasi-selfadjoint Contractions

51

Conversely, let the operator B ∈ L(N) satisfy the condition (8.1). By assumption Q(z) belongs to Q(N) and Theorem 6.1 shows that Q(z) = PN (T − z)−1 N, where T is qsc-operator in some Hilbert space H ⊃ N. Moreover, T is a qscextension of the symmetric contraction A = T H0 , H0 = H N. Now by Theorem 5.1 the assumption (8.1) means that B deﬁnes a qsc-extension T of A whose

= B. According to (5.8) the Q-function Q (z) resolvent is given by (5.2) with B T is of the form QT (z) = Q(z)(I + BQ(z))−1 , |z| > 1, and as a Q-function belongs to the class Q(N); see the discussion preceding Theorem 6.1. To prove the second part of the theorem, observe that in view of (4.5) −1 Q−1 (−1) = DK0∗ (Y + I)DK0∗ ,

(−1) = B + Q T

and

−1 (1) = DK0∗ (Y − I)DK0∗ , Q−1

(1) = B + Q T

where Y is a contraction in the subspace DK0∗ = ran DK0∗ . By Theorem 3.1 T is a selfadjoint contraction if and only if Y is a selfadjoint contraction in DK0∗ , or equivalently, B satisﬁes the conditions (8.2). Now, if (8.2) holds then T is selfadjoint and Q(z)(I + BQ(z))−1 = QT (z) ∈ NN [−1, 1]. Conversely, if QT (z) ∈ NN [−1, 1] then by part (vi) of Proposition 4.1 one

has F = F ∗ and consequently T = T ∗ , i.e., the conditions (8.2) are satisﬁed.

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[29] M.G. Kre˘ın and H. Langer, The defect subspaces and generalized resolvents of a Hermitian operator in the space Πκ , Funkcional. Anal. i Priloˇzen, 5 (1971), No. 2, 59–71 (English translation: Funct. Anal. Appl., 5 (1971), 136–146). [30] M.G. Kre˘ın and H. Langer, The defect subspaces and generalized resolvents of a Hermitian operator in the space Πκ , Funkcional. Anal. i Priloˇzen, 5 (1971), No. 3, 54–69 (English translation: Funct. Anal. Appl., 5 (1971), 217–228). ¨ [31] M.G. Kre˘ın and H. Langer, Uber die Q-function eines π-hermiteschen Operators in Raume Πκ , Acta Sci. Math. (Szeged), 34 (1973), 191–230. [32] M.G. Kre˘ın and I.E. Ovcharenko, On the Q-functions and sc-extensions of a Hermitian contraction with nondense domain, Sibirsk. Mat. J., 18 (1977), 1032–1056. [33] M.G. Kre˘ın and Sh.N. Saakyan, On some new results in the theory of the resolvent of Hermitian operators, Doklady Acad. Sci. SSSR, 169 (1966), 1269–1271. [34] H. Langer and B. Textorius, Generalized resolvents of contractions, Acta Sci. Math. (Szeged), 44 (1982), 125–131. [35] H. Langer and B. Textorius, Generalized resolvents of dual pairs of contractions, Operator Theory: Adv. Appl., 6 (1982), 103–118. [36] H. Langer and B. Textorius, On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space, Paciﬁc J. Math., 72 (1977), 135–165. [37] V.E. Lyantse and O.G. Storozh, Methods of the theory of unbounded operators, Naukova Dumka, Kiev, 1983. [38] M.M. Malamud, On extensions of sectorial operators and dual pairs of contractions, Sov. Math. Dokl., 39 (1989), 252–259. [39] M.M. Malamud and V.I. Mogilevski˘ı, On extensions of dual pairs of operators, Reports of National Academy of Sciences of Ukraine, (1997), 30–37. [40] M.M. Malamud and V.I. Mogilevski˘ı, Kre˘ın type formula for canonical resolvents of dual pairs of linear relations, Methods of Functional Analysis and Topology, 8 (2002), 72–100. [41] M.A. Naimark, On spectral functions of a symmetric operator, Izv. Akad. Nauk SSSR, Ser. Matem., 7 (1943), 285–296. [42] S. Saitoh, Theory of reproducing kernels and its applications, Pitman Research Notes in Mathematics Series, 189. Longman Scientiﬁc & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. [43] Yu.L. Shmul’yan, Two-sided division in the ring of operators, Mat. Zametki, 1 (1967), 605–610. [44] Yu.L. Shmul’yan, Operator balls, Teor. Funkci˘ı Funkcional. Anal. i Priloˇzen, 6 (1968), 68–81 (English translation: Integral Equations Operator Theory, 13 (1990), 864–882). [45] Yu.L. Shmul’yan and R.N. Yanovskaya, Blocks of a contractive operator matrix, Izv. Vuzov, Mat., 7 (1981), 72–75. [46] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, NorthHolland, New York, 1970. [47] B. Textorius, On generalized resolvents of nondensely deﬁned symmetric contractions, Acta Sci. Math. (Szeged), 49 (1985), 329–338.

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Yury Arlinski˘ı Department of Mathematical Analysis East Ukrainian National University Kvartal Molodyozhny 20-A Lugansk 91034 Ukraine e-mail: [email protected] Seppo Hassi Department of Mathematics and Statistics University of Vaasa P.O. Box 700 65101 Vaasa Finland e-mail: [email protected] Henk de Snoo Department of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen Nederland e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 55–73 c 2005 Birkh¨ auser Verlag Basel/Switzerland

A Class of Abstract Boundary Value Problems with Locally Deﬁnitizable Functions in the Boundary Condition Jussi Behrndt Dedicated to Professor Heinz Langer

Abstract. For a class of boundary value problems where the spectral parameter appears in the boundary condition in the form of a locally deﬁnitizable function linearizations are constructed and their local spectral properties are investigated. Mathematics Subject Classiﬁcation (2000). Primary: 47B50, 34B07; Secondary: 46C20, 47A06, 47B40. Keywords. Boundary value problems, locally deﬁnitizable operators and relations in Krein spaces, locally deﬁnitizable functions, boundary value spaces.

1. Introduction In this paper we study a class of boundary value problems with eigenvalue dependent boundary conditions. Let A be a closed symmetric operator or relation of defect one in a separable Krein space K, let {C, Γ0 , Γ1 } be a boundary value space for the adjoint A+ and let τ be a function locally holomorphic in some open subset of the extended complex plane which is symmetric with respect to the real line such that τ (λ) = τ (λ) holds. We investigate boundary value problems of the f following form: For a given k ∈ K ﬁnd a vector fˆ = f ∈ A+ such that f − λf = k

and

τ (λ)Γ0 fˆ + Γ1 fˆ = 0

(1.1)

holds. Under additional assumptions on τ and A a solution of this problem can be obtained with the help of the compressed resolvent of a selfadjoint extension

of A which acts in a larger Krein space. Making use of the coupling method A

and we study its local spectral from [6] we construct this so-called linearization A properties, which are closely connected with the solvability of (1.1).

56

J. Behrndt

More precisely, let Ω be some domain in C symmetric with respect to the real line such that Ω ∩ R = ∅ and the intersections of Ω with the upper and lower open half-planes are simply connected. We will assume that the selfadjoint extension A0 := ker Γ0 of A is deﬁnitizable over Ω, i.e., for every subdomain Ω of Ω with the same properties as Ω, Ω ⊂ Ω, there exists a selfadjoint projection E which reduces A0 such that A0 ∩ (EK)2 is deﬁnitizable in the Krein space EK and Ω belongs to the resolvent set of A0 ∩ ((1 − E)K)2 . With the help of approximative eigensequences or the local spectral function of A0 the spectral points of A0 in Ω ∩ R can be classiﬁed in points of positive and negative type and critical points (cf. [13], [16]). Further we assume that τ is a function which is deﬁnitizable in Ω, that is, for every domain Ω as Ω, Ω ⊂ Ω, the function τ can be written as the sum of a deﬁnitizable function (cf. [14], [15]) and a function holomorphic on Ω . Similarly to selfadjoint operators and relations deﬁnitizable over Ω the points in Ω ∩ R can be classiﬁed in points of positive and negative type and critical points. It was shown in [17] that τ can be represented with a selfadjoint relation T0 deﬁnitizable over Ω in some Krein space H such that the sign types of τ and T0 coincide in Ω ∩ R. If, in addition, the sign types of A0 and τ are “compatible” in Ω ∩ R (see Deﬁnition 2.8), then the selfadjoint relation A0 × T0 in the Krein space K × H

of the boundary value problem (1.1) is deﬁnitizable over Ω . The linearization A turns out to be a two-dimensional perturbation in resolvent sense of A0 ×T0 . Under some additional minimality assumptions on the selfadjoint relations A0 and T0 and with the help of a recent result of T.Ya. Azizov and P. Jonas which states that the inverse of a matrix-valued locally deﬁnitizable function is again locally deﬁnitizable

is deﬁnitizable over Ω . we prove in Theorem 3.6 that A The paper is organized as follows. In Section 2 we introduce the necessary notations and we recall the deﬁnitions of locally deﬁnitizable operators and relations and locally deﬁnitizable functions which can be found in, e.g., [13], [16] and [17]. Section 3 deals with boundary value problems of the form (1.1). After some preparatory work in Section 3.1 and Section 3.2 we formulate and prove the main

of the eigenvalue dependent boundary result in Section 3.3: The linearization A value problem (1.1) is locally deﬁnitizable. I thank P. Jonas for encouragement and critical help in the preparation of the manuscript.

2. Locally deﬁnitizable selfadjoint relations and locally deﬁnitizable functions 2.1. Notations and deﬁnitions Let (K, [·, ·]) be a separable Krein space. We study linear relations in K, that is, linear subspaces of K2 . The set of all closed linear relations in K is denoted by

C(K). Linear operators in K are viewed as linear relations via their graphs. For the usual deﬁnitions of the linear operations with relations, the inverse etc., we refer

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57 .

to [9]. We denote the sum (direct sum) of subspaces in K2 by (resp. ). The linear space of bounded linear operators deﬁned on a Krein space K1 with values in a Krein space K2 is denoted by L(K1 , K2 ). In the case K := K1 = K2 we simply write L(K). If (H, [·, ·]H ) is another separable Krein space the elements of K × H will be written in the form {k, h}, k ∈ K, h ∈ H. K × H equipped with the inner product [·, ·]K×H deﬁned by [{k, h}, {k , h }]K×H := [k, k ] + [h, h ]H ,

k, k ∈ K,

h, h ∈ H,

is also a Krein space. If S is a relation in K and T is a relation in H we shall write S × T for the direct product of S and T which is a relation in K × H, % . {s, t} % s t S×T = (2.1) % ∈ S, ∈ T . s {s , t } t {s,t} s, tˆ}, where For the pair {s ,t } on the right-hand side of (2.1) we shall also write {ˆ s t sˆ = s , tˆ = t . Let S be a closed linear relation in K. The resolvent set ρ(S) of S is the set of all λ ∈ C such that (S − λ)−1 ∈ L(K), the spectrum σ(S) of S is the complement of ρ(S) in C. The extended spectrum σ

(S) of S is deﬁned by σ

(S) = σ(S) if S ∈ L(K) and σ

(S) = σ(S) ∪ {∞} otherwise. We say that λ ∈ C belongs to the approximate point spectrum of S, denoted by σap (S), if there exists a sequence xn ∈ S, n = 1, 2, . . . , such that xn = 1 and limn→∞ yn − λxn = 0. The yn extended approximate point spectrum σ

ap (S) of S is deﬁned by + σap (S) ∪ {∞} if 0 ∈ σap (S −1 ) σ

ap (S) := . σap (S) if 0 ∈ σap (S −1 )

ap (S). We remark, that the boundary points of σ

(S) in C belong to σ Next we recall the deﬁnitions of the spectra of positive and negative type of a closed linear relation (see [16], [19]). For equivalent descriptions of the spectra of positive and negative type we refer to [16, Theorem 3.18]. Deﬁnition 2.1. Let S be a closed linear relation in K. A point λ ∈ σap (S) is said to be of positive type (negative type) with respect to S, if for every sequence xynn ∈ S, n = 1, 2 . . . , with xn = 1, limn→∞ yn − λxn = 0 we have lim inf [xn , xn ] > 0 resp. lim sup [xn , xn ] < 0 . n→∞

n→∞

If ∞ ∈ σ

ap (S), ∞ is said to be of positive type (negative type) with respect to S if 0 is of positive (resp. negative) type with respect to S −1 . An open subset ∆ of R is said to be of positive type (negative type) with respect to S if each point λ ∈ ∆ ∩ σ

(S) is of positive (resp. negative) type with respect to S. ∆ is called of deﬁnite type with respect to S if ∆ is either of positive or negative type with respect to S.

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Let A be a linear relation in K. The adjoint relation A+ ∈ C(K) is deﬁned as . % h % g ∈A . A+ := % [g , h] = [g, h ] for all h g

A is said to be symmetric (selfadjoint) if A ⊂ A+ (resp. A = A+ ). For a selfadjoint relation A in K the points of deﬁnite type introduced in Deﬁnition 2.1 to R. In fact, if, e.g., λ = ∞ is of positive type with respect belong to A, and xynn ∈ A is a sequence with xn = 1 and limn→∞ yn − λxn = 0, then |Im λ| lim inf [xn , xn ] = lim inf |Im [yn − λxn , xn ]| ≤ lim yn − λxn = 0 n→∞

n→∞

n→∞

implies λ ∈ R. 2.2. Locally deﬁnitizable selfadjoint relations in Krein spaces Let Ω be a domain in C symmetric with respect to the real axis such that Ω∩R = ∅ and the intersections of Ω with the upper and lower open half-planes are simply connected. Let A0 be a selfadjoint relation in the Krein space K such that σ(A0 ) ∩ (Ω\R) consists of isolated points which are poles of the resolvent of A0 , and no point of Ω ∩ R is an accumulation point of the non-real spectrum of A0 . Let ∆ be an open subset of Ω ∩ R. We say that A0 belongs to the class S ∞ (∆), if for every ﬁnite union ∆ of open connected subsets, ∆ ⊂ ∆, there exists m ≥ 1, M > 0 and an open neighborhood U of ∆ in C such that (A0 − λ)−1 ≤ M (1 + |λ|)2m−2 |Im λ|−m

(2.2)

holds for all λ ∈ U\R. We remark, that for an open subset ∆ of Ω ∩ R which is of positive type with respect to A0 the estimate (2.2) holds with m = 1 (see [16, Theorem 3.18] and [19] for the case of a bounded operator A0 ). Deﬁnition 2.2. Let Ω be a domain as above and let A0 be a selfadjoint relation in K such that σ(A0 ) ∩ (Ω\R) consists of isolated points which are poles of the resolvent of A0 and no point of Ω ∩ R is an accumulation point of the non-real spectrum of A0 in Ω. The relation A0 is said to be deﬁnitizable over Ω, if A0 ∈ S ∞ (Ω ∩ R) and every point µ ∈ Ω ∩ R has an open connected neighborhood Iµ in R such that both components of Iµ \{µ} are of deﬁnite type with respect to A0 . The next theorem is a variant of [16, Theorem 4.8]. The simple modiﬁcation of the proof is left to the reader. Theorem 2.3. Let A0 be a selfadjoint relation in K and let Ω be a domain as above. A0 is deﬁnitizable over Ω if and only if for every domain Ω with the same properties as Ω, Ω ⊂ Ω, there exists a selfadjoint projection E in K such that A0 can be decomposed in . A0 ∩ ((1 − E)K)2 A0 = A0 ∩ (EK)2 and the following holds:

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59

(i) A0 ∩ (EK)2 is a deﬁnitizable relation in the Krein space EK. (ii) σ

A0 ∩ ((1 − E)K)2 ∩ Ω = ∅. Let A0 be a selfadjoint relation in K which is deﬁnitizable over Ω, let Ω be a domain with the same properties as Ω, Ω ⊂ Ω, and let E be a selfadjoint projection with the properties as in Theorem 2.3. If E is the spectral function of the deﬁnitizable selfadjoint relation A0 ∩ (EK)2 in the Krein space EK (cf. [15, page 71], [10] and [18]), then the mapping δ → E (δ)E =: EA0 (δ)

(2.3)

deﬁned for all ﬁnite unions δ of connected subsets of Ω ∩ R the endpoints of which belong to Ω ∩R and are of deﬁnite type with respect to A0 ∩(EK)2 , is the spectral function of A0 on Ω ∩ R (see [16, Section 3.4 and Remark 4.9]). 2.3. Locally deﬁnitizable functions Let Ω be a domain as in the beginning of Section 2.2 and let τ be an L(Cn )valued piecewise meromorphic function in Ω\R which is symmetric with respect to the real line, that is τ (λ) = τ (λ)∗ for all points λ of holomorphy of τ . If, in addition, no point of Ω∩ R is an accumulation point of non-real poles of τ we write τ ∈ M n×n (Ω). The set of the points of holomorphy of τ in Ω\R and all points µ ∈ Ω ∩ R such that τ can be analytically continued to µ and the continuations from Ω ∩ C+ and Ω ∩ C− coincide, is denoted by h(τ ). In the next deﬁnition we introduce the sign type of open subsets in Ω ∩ R with respect to functions from the class M n×n (Ω) (see [17]). Deﬁnition 2.4. Let τ ∈ M n×n (Ω). An open subset ∆ ⊂ Ω ∩ R is said to be of positive type with respect to τ if for every x ∈ Cn and every sequence (µk ) of points in Ω ∩ C+ ∩ h(τ ) which converges in C to a point of ∆ we have lim inf Im (τ (µk )x, x) ≥ 0. k→∞

An open subset ∆ ⊂ Ω ∩ R is said to be of negative type with respect to τ if ∆ is of positive type with respect to −τ . ∆ is said to be of deﬁnite type with respect to τ if ∆ is of positive or negative type with respect to τ . Deﬁnition 2.5. A function τ ∈ M n×n (Ω) is called deﬁnitizable in Ω if the following holds. (i) Every point µ ∈ Ω ∩ R has an open connected neighborhood Iµ in R such that both components of Iµ \{µ} are of deﬁnite type with respect to τ . (ii) For every open subset ∆ in R, ∆ ⊂ Ω ∩ R, there exists m ≥ 1, M > 0 and an open neighborhood U of ∆ in C such that τ (λ) ≤ M (1 + |λ|)2m |Im λ|−m holds for all λ ∈ U\R.

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In [17] it is shown that a function τ ∈ M n×n (Ω) is deﬁnitizable in Ω if and only if for every ﬁnite union ∆ of open connected subsets of R such that ∆ ⊂ Ω∩R, τ can be written as the sum of an L(Cn )-valued deﬁnitizable function (see [14], [15]) and an L(Cn )-valued function which is locally holomorphic on ∆. The following theorem will be used in Section 3.2 and Section 3.3. It states that a locally deﬁnitizable function can be represented with a locally deﬁnitizable selfadjoint relation. A proof can be found in [17]. Theorem 2.6. Let τ be an L(Cn )-valued locally deﬁnitizable function in Ω and let Ω be a domain with the same properties as Ω, Ω ⊂ Ω. Then there exists a Krein space H, a selfadjoint relation T0 in H deﬁnitizable over Ω and a mapping γ ∈ L(Cn , H) with the following properties. (a) ρ(T0 ) ∩ Ω = h(τ ) ∩ Ω . (b) For a ﬁxed λ0 ∈ ρ(T0 ) ∩ Ω and all λ ∈ ρ(T0 ) ∩ Ω τ (λ) = Re τ (λ0 ) + γ + (λ − Re λ0 ) + (λ − λ0 )(λ − λ0 )(T0 − λ)−1 γ holds. (c) For any ﬁnite union ∆ of open connected subsets of R, ∆ ⊂ Ω ∩ R, such that the boundary points of ∆ are of deﬁnite type with respect to τ the spectral projection ET0 (∆) is deﬁned. If Ω is a domain with the same properties as Ω and Ω , Ω ⊂ Ω , and if we set E := ET0 (∆) + ET0 (Ω \R), then the minimality condition & ' EH = clsp 1 + (λ − λ0 )(T0 − λ)−1 Eγ x | λ ∈ ρ(T0 ) ∩ Ω , x ∈ Cn is fulﬁlled. (d) Any ﬁnite union ∆ of open connected subsets of R, ∆ ⊂ Ω ∩ R, is of positive (negative) type with respect to τ if and only if ∆ is of positive (resp. negative) type with respect to T0 . If τ and T0 are as in Theorem 2.6 we shall say that T0 is an Ω -minimal representing relation for τ . Remark 2.7. Let τ be an L(Cn )-valued locally deﬁnitizable function in Ω and let Ω be a domain with the same properties as Ω, Ω ⊂ Ω. If, in addition, τ is the restriction of a deﬁnitizable function (see [14], [15]) or if, in addition, the boundary of Ω is contained in h(τ ), then the selfadjoint relation T0 in Theorem 2.6 can be chosen such that the minimality condition & ' H = clsp 1 + (λ − λ0 )(T0 − λ)−1 γ x | λ ∈ ρ(T0 ) ∩ Ω , x ∈ Cn holds. The next deﬁnition connects sign types of locally deﬁnitizable functions and sign types of spectral points of locally deﬁnitizable relations. Deﬁnition 2.8. Let τ be an L(Cn )-valued locally deﬁnitizable function in Ω and let A0 be a selfadjoint relation in the Krein space K which is deﬁnitizable over Ω.

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61

We say that the sign types of τ and A0 are d-compatible in Ω if for every point µ ∈ Ω ∩ R there exists an open connected neighborhood Iµ ⊂ Ω ∩ R of µ such that each component of Iµ \{µ} is either of positive type with respect to τ and A0 or of negative type with respect to τ and A0 . If τ is a function which is deﬁnitizable in Ω, Ω is a domain as Ω, Ω ⊂ Ω, and T0 is an Ω -minimal representing relation for τ (see Theorem 2.6), then the sign types of τ and T0 are d-compatible in Ω .

3. Boundary value problems with locally deﬁnitizable functions in the boundary condition 3.1. Boundary value spaces and Weyl functions associated with symmetric relations in Krein spaces Let (K, [·, ·]) be a separable Krein space, let J be a corresponding fundamental

symmetry and let A ∈ C(K) be a closed symmetric relation in K. We say that A is of defect m ∈ N ∪ {∞}, if both deﬁciency indices n± (JA) = dim ker((JA)∗ − λ),

λ ∈ C± ,

of the symmetric relation JA in the Hilbert space (K, [J·, ·]) are equal to m. Here ∗ denotes the Hilbert space adjoint. We remark, that this is equivalent to the fact that there exists a selfadjoint extension of A in K and that each selfadˆ joint extension Aˆ of A in K satisﬁes dim A/A = m. We shall use the so-called boundary value spaces for the description of the selfadjoint extensions of closed symmetric relations in Krein spaces. The following deﬁnition is taken from [5]. Deﬁnition 3.1. Let A be a closed symmetric relation in the Krein space (K, [·, ·]). We say that {G, Γ0 , Γ1 } is a boundary value space for A+ if (G, (·, ·)) is a Hilbert space and there exist mappings Γ0 , Γ1 : A+ → G such that Γ = ΓΓ01 : A+ → G × G is surjective, and the relation [f , g] − [f, g ] = (Γ1 fˆ, Γ0 gˆ) − (Γ0 fˆ, Γ1 gˆ) f g holds for all fˆ = f , gˆ = g ∈ A+ . In the following we recall some basic facts on boundary value spaces which can be found in, e.g., [4] and [5]. For the Hilbert space case we refer to [11], [7] and [8]. Let A be a closed symmetric relation in K and let λ be a point of regular type of A. We denote by Nλ,A+ := ker(A+ − λ) = ran (A − λ)[⊥] the defect subspace of A at the point λ and we set ' & fλ % % ˆλ,A+ = N λfλ fλ ∈ Nλ,A+ .

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When no confusion can arise we write Nλ and Nˆλ instead of Nλ,A+ and Nˆλ,A+ . If there exists a selfadjoint extension A of A such that ρ(A ) = ∅ then we have A+ = A

.

Nˆλ

for all λ ∈ ρ(A ).

In this case there exists a boundary value space {G, Γ0 , Γ1 } for A+ such that ker Γ0 = A (cf. [5]). Let in the following A, {G, Γ0 , Γ1 } and Γ be as in Deﬁnition 3.1. It follows that the mappings Γ0 and Γ1 are continuous. The selfadjoint extensions A0 := ker Γ0

and

A1 := ker Γ1

of A are transversal, i.e., A0 ∩ A1 = A and A0 A1 = A+ . The mapping Γ induces, via & '

AΘ := Γ−1 Θ = fˆ ∈ A+ | Γfˆ ∈ Θ , Θ ∈ C(G), (3.1) a bijective correspondence Θ → AΘ between the set of all closed linear relations

C(G) in G and the set of closed extensions AΘ ⊂ A+ of A. In particular (3.1) gives a one-to-one correspondence between the closed symmetric (selfadjoint) extensions of A and the closed symmetric (resp. selfadjoint) relations in G. If Θ is a closed operator in G, then the corresponding extension AΘ of A is determined by AΘ = ker Γ1 − ΘΓ0 . (3.2) Assume that ρ(A0 ) = ∅ and denote by π1 the orthogonal projection onto the ﬁrst component of K × K. For every λ ∈ ρ(A0 ) we deﬁne the operators ˆλ )−1 ∈ L(G, K) and M (λ) = Γ1 (Γ0 |Nˆλ )−1 ∈ L(G). γ(λ) = π1 (Γ0 |N The functions λ → γ(λ) and λ → M (λ) are called the γ-ﬁeld and Weyl function corresponding to A and {G, Γ0 , Γ1 }. γ and M are holomorphic on ρ(A0 ) and the relations γ(ζ) = (1 + (ζ − λ)(A0 − ζ)−1 )γ(λ)

(3.3)

M (λ) − M (ζ)∗ = (λ − ζ)γ(ζ)+ γ(λ)

(3.4)

and

hold for all λ, ζ ∈ ρ(A0 ) (cf. [5]).

If Θ ∈ C(G) and AΘ is the corresponding extension of A (see (3.1)), then a point λ ∈ ρ(A0 ) belongs to ρ(AΘ ) if and only if 0 belongs to ρ(Θ − M (λ)). For λ ∈ ρ(AΘ ) ∩ ρ(A0 ) the well-known resolvent formula −1 γ(λ)+ (AΘ − λ)−1 = (A0 − λ)−1 + γ(λ) Θ − M (λ) holds (for a proof see, e.g., [5]).

(3.5)

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63

3.2. Locally deﬁnitizable functions as Weyl functions of symmetric relations Let, as in Section 2.2, Ω be a domain in C symmetric with respect to the real axis such that Ω ∩ R = ∅ and the intersections of Ω with the upper and lower open half-planes are simply connected. In the next proposition we consider boundary value spaces and Weyl functions associated with symmetric relations in Krein spaces which have the additional property that there exists a locally deﬁnitizable selfadjoint extension. We restrict ourselves to the case of defect one. Proposition 3.2. Let A be a closed symmetric relation of defect one in the Krein space K and assume that there exists a selfadjoint extension A0 of A which is deﬁnitizable over Ω. Let {C, Γ0 , Γ1 } be a boundary value space for A+ such that A0 = ker Γ0 . Then the corresponding Weyl function M is deﬁnitizable in Ω. If ∆ is an open subset of Ω ∩ R which is of positive (negative) type with respect to A0 , then ∆ is of positive (resp. negative) type with respect to M . Proof. As A0 is a selfadjoint relation which is deﬁnitizable over Ω it follows that the Weyl function M corresponding to A and {C, Γ0 , Γ1 } is piecewise meromorphic in Ω\R and no point of Ω ∩ R is an accumulation point of the non-real poles of M . Let λ0 ∈ ρ(A0 ). Making use of (3.3) and (3.4) we obtain that M is symmetric with respect to the real axis, Im M (λ0 ) = (Im λ0 )γ(λ0 )+ γ(λ0 ) and M (λ) = M (λ0 ) + (λ − λ0 )γ(λ0 )+ γ(λ) = Re M (λ0 ) − i (Im λ0 )γ(λ0 )+ γ(λ0 ) (3.6) + γ(λ0 )+ (λ − λ0 ) + (λ − λ0 )(λ − λ0 )(A0 − λ)−1 γ(λ0 ) = Re M (λ0 ) + γ(λ0 )+ (λ−Reλ0 ) + (λ− λ0 )(λ− λ0 )(A0 − λ)−1 γ(λ0 ) holds for all λ ∈ ρ(A0 ). Since A0 belongs to the class S ∞ (Ω ∩ R) it follows that M fulﬁls the second condition in Deﬁnition 2.5. Let µ ∈ Ω∩R and let Iµ ⊂ Ω∩R be an open connected neighborhood of µ in R such that both components of Iµ \{µ} are of deﬁnite type with respect to A0 . By [16, Theorem 3.18] a component of Iµ \{µ} is of positive (negative) type with respect to A0 if and only if it is of positive (resp. negative) type with respect to the function λ → (λ − Re λ0 ) + (λ − λ0 )(λ − λ0 )(A0 − λ)−1 . Now it follows from (3.6) that both components of Iµ \{µ} are of the same sign type with respect to A0 and M and therefore the Weyl function M is deﬁnitizable in Ω. The same argument shows that an open subset ∆ ⊂ Ω∩R which is of positive (negative) type with respect to A0 is also of positive (negative) type with respect to M . The next theorem is a variant of [3, Theorem 3.3]. For the convenience of the reader we sketch the proof. Theorem 3.3. Let τ be a complex-valued locally deﬁnitizable function in Ω and assume that τ is not identically equal to a constant. Let Ω be a domain with the

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same properties as Ω and Ω ⊂ Ω. Then there exists a Krein space H, a closed symmetric relation T of defect one in H and a boundary value space {C, Γ0 , Γ1 } for T + such that τ coincides with the corresponding Weyl function on Ω and T0 := ker Γ0 is an Ω -minimal representing relation for τ . Sketch of the proof of Theorem 3.3. Let τ be represented with an Ω -minimal selfadjoint relation T0 in a Krein space H as in Theorem 2.6. Let γ ∈ L(C, H) be as in Theorem 2.6 and ﬁx some λ0 ∈ h(τ ) ∩ Ω . For all λ ∈ h(τ ) ∩ Ω we deﬁne γ (λ) := 1 + (λ − λ0 )(T0 − λ)−1 γ ∈ L(C, H). The linear functional γ (λ)c → c deﬁned on ran γ (λ) is denoted by γ (λ)(−1) . The closed symmetric relation . % f % ∈ T0 % [g − µf, γ (µ) 1] = 0 T := g has defect one and does not depend on the choice of µ ∈ h(τ ) ∩ Ω . For some ﬁxed f fλ λ ∈ h(τ ) ∩ Ω we write the elements fˆ ∈ T + in the form fˆ = f0 + λf , where λ 0 f0 ∈ T0 and fλ ∈ ran γ (λ) = Nλ,T + . As in the proof of [3, Theorem 3.3] (see f0 also [7, Theorem 1]) one veriﬁes that {C, Γ0 , Γ1 }, where Γ0 fˆ := γ (λ)(−1) fλ , Γ1 fˆ := γ (λ)+ (f0 − λf0 ) + τ (λ)γ (λ)(−1) fλ , is a boundary value space for T + and the corresponding Weyl function coincides with τ on Ω . Remark 3.4. Let the function τ be deﬁnitizable in Ω and assume that τ is not identically equal to a constant. Let Ω be a domain with the same properties as Ω, Ω ⊂ Ω, and assume that τ is the restriction of a deﬁnitizable function or that the boundary of Ω is contained in h(τ ). If we choose T0 as in Remark 2.7 and T ⊂ T0 ⊂ T + as in Theorem 3.3, then the condition H = clsp {ran γ (λ) | λ ∈ ρ(T0 ) ∩ Ω } = clsp { Nλ,T + | λ ∈ ρ(T0 ) ∩ Ω } is fulﬁlled. In this case T is an operator. In the following proposition we use Deﬁnition 2.8. The statements will be useful in the proof of our main result in Section 3.3. Proposition 3.5. Let A be a closed symmetric relation of defect one in the Krein space K, let {C, Γ0 , Γ1 } be a boundary value space for A+ and denote by M the corresponding Weyl function. Assume that the selfadjoint relation A0 = ker Γ0 is deﬁnitizable over Ω and let τ be a complex-valued function which is deﬁnitizable in Ω such that the sign types of τ and A0 are d-compatible in Ω. Let Ω be a domain with the same properties as Ω, Ω ⊂ Ω, and let T0 be an Ω -minimal representing relation for τ in some Krein space H (see Theorem 2.6). Then the following holds:

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65

× H) is deﬁnitizable over Ω and the (i) The selfadjoint relation A0 × T0 ∈ C(K sign types of A0 × T0 and the functions τ and M are d-compatible in Ω . (ii) The function M + τ is deﬁnitizable in Ω. (iii) If τ (η) = 0 and (M + τ )(η ) = 0 for some η, η ∈ Ω, then the functions −1 M (λ) −1 M (λ) 0 and λ → − (3.7) λ → −1 −τ (λ)−1 0 −τ (λ)−1 are deﬁnitizable in Ω and their sign types are d-compatible with the sign types of the selfadjoint relation A0 × T0 in Ω . Proof. (i) Since A0 and T0 belong to S ∞ (Ω ∩ R) and S ∞ (Ω ∩ R), respectively, we conclude that A0 × T0 belongs to S ∞ (Ω ∩R). Let µ ∈ Ω ∩R and let Iµ ⊂ Ω ∩R be an open connected neighborhood of µ in R such that each component of Iµ \{µ} is of the same sign type with respect to A0 and τ . As T0 is an Ω -minimal representing relation for τ both components of Iµ \{µ} are of deﬁnite type with respect to A0 ×T0 and it follows that A0 × T0 is deﬁnitizable over Ω . The assumption that the sign types of τ and A0 are d-compatible in Ω implies that the sign types of τ and A0 × T0 as well as the sign types of M and A0 × T0 are d-compatible in Ω. (ii) For µ ∈ Ω ∩ R we choose an open connected neighborhood Iµ ⊂ Ω ∩ R of µ such that both components of Iµ \{µ} are of the same sign type with respect to A0 and τ . By Proposition 3.2 the sign types of M are the same as of A0 and therefore both components of Iµ \{µ} are of the same sign type with respect to M + τ . The growth properties of M and τ imply that M + τ fulﬁls the second condition in Deﬁnition 2.5 and therefore M + τ is deﬁnitizable in Ω. (iii) By [1, Theorem 2.5] the function −τ −1 is deﬁnitizable in Ω and it follows from the proof of [1, Theorem 2.5] that each point µ ∈ Ω ∩ R has an open connected neighborhood Iµ ⊂ Ω ∩ R such that both components of Iµ \{µ} are of the same sign type with respect to −τ −1 and τ . Therefore the sign types of −τ −1 and A0 are d-compatible in Ω. Now it is easy to see that the ﬁrst function in (3.7) is deﬁnitizable in Ω and its sign types are d-compatible with the sign types of A0 ×T0 in Ω . As the function M (λ) −1 λ → −1 −τ (λ)−1 is also deﬁnitizable in Ω another application of [1, Theorem 2.5] shows that the second function in (3.7) is deﬁnitizable in Ω and its sign types are d-compatible with the sign types of A0 × T0 in Ω . 3.3. The main result In this section we investigate the spectral properties of linearizations of a class of abstract eigenvalue dependent boundary value problems with locally deﬁnitizable functions in the boundary condition. Similar problems with a local variant of

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generalized Nevanlinna functions in the boundary condition have been considered in [3]. The main feature in Theorem 3.6 below is that the linearization turns out to be locally deﬁnitizable. Theorem 3.6. Let Ω be a domain as in the beginning of Section 3.2 and let A be a closed symmetric operator of defect one in the Krein space K such that K = clsp { Nλ,A+ | λ ∈ Ω} holds. Assume that there exists a selfadjoint extension A0 of A which is deﬁnitizable over Ω. Let {C, Γ0 , Γ1 } be a boundary value space for A+ , A0 = ker Γ0 , and denote by γ and M the corresponding γ-ﬁeld and Weyl function, respectively. Let τ be a nonconstant function which is deﬁnitizable in Ω, let Ω be a domain as Ω, Ω ⊂ Ω, choose H, T ⊂ T0 ⊂ T + and {C, Γ0 , Γ1 } as in Theorem 3.3 and assume that the condition H = clsp { Nλ,T + | λ ∈ Ω } is fulﬁlled. Let the sign types of τ and A0 be d-compatible in Ω, assume that the function M + τ is not identically equal to zero and deﬁne h0 := h(M ) ∩ h(τ ) ∩ h τ −1 ∩ h (M + τ )−1 . Then the relation (& ) % '

= A fˆ1 , fˆ2 ∈ A+ × T + % Γ1 fˆ1 − Γ1 fˆ2 = Γ0 fˆ1 + Γ0 fˆ2 = 0

(3.8)

is a selfadjoint extension of A in K × H which is deﬁnitizable over Ω and the

are d-compatible with the sign types of τ and M in Ω . The set sign types of A Ω \(R ∪ h0 ) is ﬁnite. For every k ∈ K and every λ ∈ h0 ∩ Ω the unique solution of the eigenvalue dependent boundary value problem f1 ˆ ˆ ˆ ∈ A+ , f1 − λf1 = k, τ (λ)Γ0 f1 + Γ1 f1 = 0, f1 = (3.9) f1 is given by

− λ)−1 {k, 0} = (A0 − λ)−1 k − γ(λ) M (λ) + τ (λ) −1 γ(λ)+ k, f1 = PK (A

f1 = λf1 + k.

(3.10)

in K × H with the Proof. 1. In this step we construct the selfadjoint relation A

help of the coupling method from [6, §5.2], we show that h0 ∩ Ω belongs to ρ(A) and that (3.10) is the unique solution of the boundary value problem (3.9). We follow the lines of [3, Proof of Theorem 4.1]. As the functions M , τ and M + τ are deﬁnitizable in Ω (see Proposition 3.5) [1, Theorem 2.3] implies that −τ −1 and −(M + τ )−1 are also deﬁnitizable in Ω. Let Ω and h0 be as in the assumptions of the theorem. As the non-real poles of the functions M , τ , τ −1 and (M + τ )−1 in Ω do not accumulate to Ω ∩ R we conclude from Ω ⊂ Ω that the set Ω \(R ∪ h0 ) is ﬁnite. Let H, T ⊂ T + and {C, Γ0 , Γ1 } be as in Theorem 3.3. Then τ is the corresponding Weyl function and the selfadjoint relation T0 = ker Γ0 is deﬁnitizable over Ω . We denote the γ-ﬁeld corresponding to {C, Γ0 , Γ1 } by γ and we set

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67

T1 := ker Γ1 . As {C, Γ1 , −Γ0 } is a boundary value space for T + with corresponding γ-ﬁeld and Weyl function λ → γ (λ)τ (λ)−1 and λ → −τ (λ)−1 , λ ∈ h(τ ) ∩ h τ −1 ∩ Ω , (3.11)

0 , Γ

1 }, where Γ

0 and Γ

1 are respectively, it follows without diﬃculty that {C2 , Γ + + 2 mappings from A × T into C deﬁned by & ' & ' Γ0 fˆ1 Γ1 fˆ1 ˆ ˆ ˆ ˆ

and Γ1 f1 , f2 := , Γ0 f1 , f2 := Γ1 fˆ2 −Γ0 fˆ2 {fˆ1 , fˆ2 } ∈ A+ × T + , is a boundary value space for A+ × T + with corresponding γ-ﬁeld γ(λ) 0 λ → γ

(λ) = , λ ∈ h(M ) ∩ h(τ ) ∩ h(τ −1 ) ∩ Ω , (3.12) 0 γ (λ)τ (λ)−1 and Weyl function /(λ) = λ → M

M (λ) 0 , 0 −τ (λ)−1

λ ∈ h(M ) ∩ h(τ ) ∩ h(τ −1 ) ∩ Ω .

in K × H corresponding to Θ = The selfadjoint relation A and (3.2) is given by

0 1 10

∈ L(C2 ) via (3.1)

= ker(Γ

1 − ΘΓ

0 ) A (& ) % ' = fˆ1 , fˆ2 ∈ A+ × T + % Γ1 fˆ1 − Γ1 fˆ2 = Γ0 fˆ1 + Γ0 fˆ2 = 0 .

can be written as For λ ∈ h0 ∩ Ω the resolvent of A 0 (A0 − λ)−1 −1 /(λ) −1 γ

+ γ (λ) Θ − M (A − λ) =

(λ)+ , 0 (T1 − λ)−1

(3.13)

(3.14)

/(λ))−1 one veriﬁes that the compressed resolvent (see (3.5)). Calculating (Θ − M

of A onto K is given by

− λ)−1 |K = (A0 − λ)−1 − γ(λ) M (λ) + τ (λ) −1 γ(λ)+ , λ ∈ h0 ∩ Ω . PK (A

− λ)−1 {k, 0} and f2 := PH (A

− λ)−1 {k, 0}. Then For k ∈ K we set f1 := PK (A {f1 , f2 }

⊂ A+ × T + ∈A {λf1 + k, λf2 } f2 ˆλ,T + . From (3.13) and since τ is the ∈N and fˆ1 := λff11+k ∈ A+ , fˆ2 := λf 2 Weyl function corresponding to {C, Γ0 , Γ1 } we get Γ1 fˆ1 = Γ1 fˆ2 = τ (λ)Γ0 fˆ2 = −τ (λ)Γ0 fˆ1 , and it follows that fˆ1 ∈ A+ is a solution of (3.9).

λ ∈ h0 ∩ Ω ,

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J. Behrndt

+ ˆ Let that the vector verify +that this solution f1 ∈ A is unique. Assume gus 1 gˆ1 = λg1 +k ∈ A is also a solution of (3.9), λ ∈ h0 ∩ Ω . Then fˆ1 − gˆ1 belongs to Nˆλ,A+ and 0 = τ (λ)Γ0 (fˆ1 − gˆ1 ) + Γ1 (fˆ1 − gˆ1 ) = τ (λ) + M (λ) Γ0 (fˆ1 − gˆ1 )

implies fˆ1 −ˆ g1 ∈ A0 ∩ Nˆλ,A+ as τ (λ)+M (λ) = 0. Therefore fˆ1 = gˆ1 since λ ∈ h(M ).

is deﬁnitizable over Ω and that the sign types 2. It remains to prove that A

of A are d-compatible with the sign types of the functions τ and M in Ω . In this step we show that for every point µ ∈ Ω ∩ R there exists an open connected neighborhood Iµ of µ in Ω ∩R such that both components of Iµ \{µ} are of deﬁnite

type with respect to A. As the sign types of τ and A0 are d-compatible in Ω, Proposition 3.5 implies that the selfadjoint relation A0 × T0 is deﬁnitizable over Ω . It is straightforward 0 , Γ 1 }, where to check that {C2 , Γ

1 − ΘΓ

0 , 0 := Γ Γ

1 := −Γ

0, Γ

0 = A.

The corresponding γ-ﬁeld is a boundary value space for A+ × T + with ker Γ * are deﬁned on ρ(A)

and for λ ∈ h0 ∩ Ω they are given by γ and Weyl function M /(λ) − Θ −1 λ → γ (λ) = γ (λ) M (3.15) and

−1 −1 *(λ) = −(M /(λ) − Θ)−1 = − M (λ) λ → M , −1 −τ (λ)−1 respectively. In particular *(λ) = Re M *(λ0 ) + γ (λ0 )+ (λ − Re λ0 ) M

− λ)−1 γ (λ0 ) + (λ − λ0 )(λ − λ0 )(A

(3.16)

holds for a ﬁxed λ0 ∈ h0 ∩ Ω and all λ ∈ h0 ∩ Ω (see the proof of Proposition 3.2). * is deﬁnitizable in Ω and the sign types of M * By Proposition 3.5 the function M and A0 × T0 are d-compatible in Ω . Let µ ∈ Ω ∩ R and assume, e.g., that a one-sided open connected neighborhood ∆+ of µ in R, ∆+ ⊂ Ω ∩ R, is of positive type with respect to A0 × T0 . * and A0 × T0 are d-compatible in Ω , it is no restriction to As the sign types of M *. Since A0 × T0 and A

assume that ∆+ is also of positive type with respect to M are both selfadjoint extensions of the symmetric relation A × T in K × H we have

− λ)−1 − ((A0 × T0 ) − λ)−1 ≤ 2 dim ran (A for all λ ∈ h0 ∩Ω . Let Ω∆+ be a domain with the same properties as Ω, Ω∆+ ⊂ Ω ,

is deﬁnitizable such that Ω∆+ ∩ R = ∆+ . It follows from [2, Corollary 2.5] that A over Ω∆+ and that for every ﬁnite union δ of open connected subsets in ∆+ , δ ⊂ ∆+ , such that the spectral projection EA (δ) is deﬁned the space EA (δ)(K×H)

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69

equipped with the inner product [·, ·]K×H is a Pontryagin space with ﬁnite rank of negativity. Let Ω be a domain with the same properties as Ω such that Ω ⊂ Ω and ∆+ ⊂ Ω ∩ R. By Theorem 2.6 there exists an Ω -minimal representing relation S *, that is, S is a selfadjoint relation in some Krein space G which is deﬁnitizable for M *) ∩ Ω , and with a suitable Λ ∈ L(C2 , G) we have over Ω , ρ(S) ∩ Ω = h(M *(λ) = Re M *(λ0 ) + Λ+ (λ − Re λ0 ) + (λ − λ0 )(λ − λ0 )(S − λ)−1 Λ (3.17) M for a ﬁxed λ0 ∈ ρ(S) ∩ Ω and all λ ∈ ρ(S) ∩ Ω . The spectral function of S on Ω ∩ R will be denoted by ES (comp. (2.3)). In the following we will assume that the point λ0 in (3.16) and (3.17) belongs

∩ ρ(S) ∩ Ω . This is no restriction. From (3.16) and (3.17) we obtain to ρ(A)

− λ)−1 (λ0 ) = Λ+ Λ and γ (λ0 )+ (A γ (λ0 ) = Λ+ (S − λ)−1 Λ γ (λ0 )+ γ

∩ ρ(S) ∩ Ω . Therefore the relation for all λ ∈ ρ(A) . % #n −1

∩ Ω )Λxk % λk ∈ ρ(S) ∩ ρ(A) k=1 (1 + (λk − λ0 )(S − λk ) # V := % n −1

) γ (λ0 )xk xk ∈ C2 , k = 1, . . . , n k=1 (1 + (λk − λ0 )(A − λk ) is isometric. The assumptions K = clsp { Nλ,A+ | λ ∈ Ω} imply K = clsp and H = clsp

( (

and H = clsp { Nλ,T + | λ ∈ Ω }

∩ Ω ran γ(λ) | λ ∈ ρ(S) ∩ ρ(A)

)

)

∩ Ω , ran γ (λ) | λ ∈ ρ(S) ∩ ρ(A)

respectively. From (3.12) and (3.15) we obtain ( )

∩ Ω , x ∈ C2 K × H = clsp γ (λ)x | λ ∈ ρ(S) ∩ ρ(A) and therefore ran V is dense in K × H. This implies that V is an isometric operator and the same holds for its closure V . Let δ be a ﬁnite union of open connected subsets in ∆+ , δ ⊂ ∆+ , such

Then that the boundary points of δ in R are of deﬁnite type with respect to A. (EA (δ)(K × H), [·, ·]K×H ) is a Pontryagin space with ﬁnite rank of negativity. As ∆+ is of positive type with respect to S the spectral projection ES (δ) is deﬁned and ES (δ)G equipped with the inner product from G is a Hilbert space. Writing

respectively, one ES (δ) and EA (δ) as strong limits of the resolvents of S and A, veriﬁes that V is reduced by ES (δ)G × EA (δ)(K × H). Then V δ := V ∩ ES (δ)G × EA (δ)(K × H) is a closed isometric operator from the Hilbert space ES (δ)G with dense range in the Pontryagin space EA (δ)(K×H). As in the proof of [12, Theorem 6.2] one veriﬁes that V δ is bounded and from this we conclude dom V δ = ES (δ)G and ran V δ = EA (δ)(K × H). The isometry of V δ implies that EA (δ)(K × H) is a Hilbert space.

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and choose a sequence uv n ∈ A

with un = 1 and Let ξ ∈ δ ∩ σ(A) n vn − ξun → 0 for n → ∞. From

∩ (I − E (δ))(K × H) 2 − ξ −1 ∈ L (I − E (δ))(K × H) A A A and lim (I − EA (δ))(vn − ξun ) = 0

n→∞

we obtain (I − EA (δ))un → 0 and EA (δ)un → 1 for n → ∞. As EA (δ)(K × H) is a Hilbert space we have lim inf [un , un ]K×H = lim inf [EA (δ)un , EA (δ)un ]K×H > 0, n→∞

n→∞

If ∞ belongs to δ∩

a similar reathat is, ξ is of positive type with respect to A. σ (A)

soning shows that ∞ is of positive type with respect to A. Therefore δ is of positive

As this is true for every ﬁnite union δ of open connected type with respect to A. subsets in ∆+ , δ ⊂ ∆+ , such that the boundary points of δ in R are of deﬁnite type

we conclude that ∆+ is also of positive type with respect to A. with respect to A Analogously one veriﬁes that a one-sided open connected neighborhood ∆− * is of µ in R, ∆− ⊂ Ω ∩ R, which is of negative type with respect to A0 × T0 and M

of negative type with respect to A. We have shown that for every point µ ∈ Ω ∩ R there is an open connected neighborhood Iµ in R such that both components of

Iµ \{µ} are of the same sign type with respect to A0 × T0 and A.

belongs to S ∞ (Ω ∩ R). For this we use the 3. It remains to verify that A relation (3.14). We show ﬁrst that the selfadjoint relation T1 = ker Γ1 in H is deﬁnitizable over Ω . As the function −τ (λ)−1 = Re −τ (λ0 )−1 + τ (λ0 )−1 γ (λ0 )+ (λ − Re λ0 ) + (λ − λ0 )(λ − λ0 )(T1 − λ)−1 γ (λ0 )τ (λ0 )−1 (see (3.11) and the proof of Proposition 3.2) is deﬁnitizable in Ω and the selfadjoint relation T0 is deﬁnitizable over Ω the same considerations as in step 2 of the *, A0 × T0 and A

show that every proof applied to −τ −1 , T0 and T1 instead of M point µ ∈ Ω ∩ R has an open connected neighborhood Iµ in R such that both components of Iµ \{µ} are of deﬁnite type with respect to T1 . By (3.5) we have (T1 − λ)−1 = (T0 − λ)−1 + γ (λ) −τ (λ)−1 γ (λ)+ for all λ ∈ h(τ ) ∩ h(τ −1 ) ∩ Ω . Since −τ −1 is deﬁnitizable in Ω the non-real spectrum of T1 in Ω does not accumulate to points in Ω ∩ R. The growth properties of −τ −1 (see Deﬁnition 2.5) and the resolvent of T0 imply T1 ∈ S ∞ (Ω ∩ R) and therefore T1 is deﬁnitizable over Ω . As the sign types of −τ −1 and A0 are d-compatible in Ω (see the proof of Proposition 3.5 (iii)) it follows that A0 × T1 is deﬁnitizable over Ω . The relation

− λ)−1 = (A0 × T1 − λ)−1 + *(λ) (A γ (λ)M γ (λ)+ ,

λ ∈ h0 ∩ Ω ,

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71

* and the resolvent of A0 × T0 show (cf. (3.14)) and the growth properties of M ∞

A ∈ S (Ω ∩ R). This completes the proof of Theorem 3.6. Remark 3.7. Let Ω, Ω , A ⊂ A0 , τ and T0 be as in Theorem 3.6 and let ∆ be an open connected subset in R, ∆ ⊂ Ω ∩ R, which is of positive (negative) type with respect to A0 and τ . As T0 is an Ω -minimal representing relation for τ it follows that ∆ is of positive (negative) type with respect to the selfadjoint relation A0 × T0 in K × H. From [2, Corollary 2.5] we obtain that for every ﬁnite union δ of open connected subsets in R, δ ⊂ ∆, such that the spectral projection EA (δ)

in (3.8) and the set δ is deﬁned, (E (δ)(K × H), [·, ·]K×H ) is corresponding to A A a Pontryagin space with ﬁnite rank of negativity (positivity). This can also be deduced from [3, Theorem 4.1]. Remark 3.8. The assumption H = clsp { Nλ,T + | λ ∈ Ω } in Theorem 3.6 implies

in (3.8) satisﬁes the minimality condition that the selfadjoint extension A ' &

− λ)−1 ){k, 0} | k ∈ K, λ ∈ ρ(A)

∩ Ω (3.18) K × H = clsp (1 + (λ − λ0 )(A

∩ Ω . This can be veriﬁed as in [3, Proof of Theorem 4.1]. for some ﬁxed λ0 ∈ ρ(A)

is a selfadjoint extension Let A ∈ C(K) be as in Theorem 3.6 and assume that B of A in some Krein space K × H which is deﬁnitizable over Ω such that the

− λ)−1 |K

onto K yields a solution of (3.9). Then PK (B compressed resolvent of B −1

− λ) |K coincide. If B

fulﬁls the minimality condition (3.18) with and PK (A

∩ Ω , respectively, and we

K × H and ρ(A) ∩ Ω replaced by K × H and ρ(B)

choose λ0 ∈ ρ(A) ∩ ρ(B) ∩ Ω , then , + # n

− λi )−1 ){ki , 0} %% λi ∈ ρ(A)

∩ ρ(B)

∩ Ω , (1 + (λi − λ0 )(A i=1 #n W := % −1

ki ∈ K, i = 1, 2, . . . , n ){ki , 0} i=1 (1 + (λi − λ0 )(B − λi ) is a densely deﬁned isometric operator in K×H with dense range in K×H and the

and B

same holds for its closure W . We denote the local spectral functions of A by EA and EB , respectively. Let ∆ be an open connected subset in R, ∆ ⊂ Ω ∩ R, such that EA (∆) is deﬁned. Then also EB (∆) is deﬁned and W is reduced by EA (∆)(K × H) × EB (∆)(K × H ). The closed isometric operator W∆ := W ∩ EA (∆)(K × H) × EB (∆)(K × H ) intertwines the resolvents of

1 := A

∩ E (∆)(K × H) 2 A A

1 := B

∩ E (∆)(K × H ) 2 , and B B

1 ) ∩ ρ(B

1 ) ∩ Ω and x ∈ dom W∆ we have i.e., for λ ∈ ρ(A

1 − λ)−1 x = (B

1 − λ)−1 W∆ x. W∆ (A In particular, the ranks of positivity and negativity of the inner products on the subspaces EA (∆)(K × H) and EB (∆)(K × H ) coincide.

72

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If, in addition to the assumptions above, (EA (∆)(K × H), [·, ·]K×H ) is a Pontryagin space, then EB (∆)(K × H ) equipped with the inner product from K × H is also a Pontryagin space and by [12, Theorem 6.2] the operator W∆ is an iso 1 and B

1 are metric isomorphism of EA (∆)(K × H) onto EB (∆)(K × H ), i.e., A isometrically equivalent. The case that the function τ is a real constant is excluded in Theorem 3.6. In this case we have the following theorem. Theorem 3.9. Let Ω be a domain as in the beginning of Section 3.2 and let A be a closed symmetric operator of defect one in the Krein space K such that K = clsp { Nλ,A+ | λ ∈ Ω} holds. Assume that there exists a selfadjoint extension A0 of A which is deﬁnitizable over Ω. Let {C, Γ0 , Γ1 } be a boundary value space for A+ , A0 = ker Γ0 , denote by γ and M the corresponding γ-ﬁeld and Weyl function, respectively, and let τ be a real constant. Then the relation A−τ = ker Γ1 + τ Γ0 is a selfadjoint extension of A in K which is deﬁnitizable over Ω. The sign types of M and A−τ are d-compatible in Ω. For every k ∈ K and every λ ∈ h(M ) ∩ h((M + τ )−1 ) ∩ Ω a solution of the boundary value problem f (3.19) f1 − λf1 = k, τ Γ0 fˆ1 + Γ1 fˆ1 = 0, fˆ1 = 1 ∈ A+ , f1 is given by

−1 γ(λ)+ k, f1 = (A−τ − λ)−1 k = (A0 − λ)−1 k − γ(λ) M (λ) + τ

f1 = λf1 + k.

Proof. The proof of Theorem 3.9 is a modiﬁcation of the proof of Theorem 3.6. Note ﬁrst that the relation (3.4) and K = clsp {ran γ(λ) | λ ∈ Ω∩ρ(A0 )} imply that the Weyl function M is not identically equal to a constant. Here it is obvious that the resolvent of A−τ yields a solution of the boundary value problem (3.19) (compare (3.1), (3.2) and (3.5)). As in step 2 and step 3 of the proof of Theorem 3.6 the function λ → −(M (λ)+τ )−1 , which by [1, Theorem 2.3] is deﬁnitizable in Ω, can be ˆ 1} ˆ 0, Γ regarded as the Weyl function corresponding to a boundary value space {C, Γ + ˆ for A with A−τ = ker Γ0 . Now the same arguments as in the proof of Theorem 3.6 show that A−τ is deﬁnitizable over Ω. The details are left to the reader.

References [1] T.Ya. Azizov, P. Jonas, On Locally Deﬁnitizable Matrix Functions, Preprint 21-2005, Institute of Mathematics, Technische Universit¨at Berlin. [2] J. Behrndt, P. Jonas, On Compact Perturbations of Locally Deﬁnitizable Selfadjoint Relations in Krein Spaces, Integral Equations Operator Theory 52 (2005), 17–44. [3] J. Behrndt, P. Jonas, Boundary Value Problems with Local Generalized Nevanlinna Functions in the Boundary Condition, to appear in Integral Equations Operator Theory.

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[4] V.A. Derkach, On Weyl Function and Generalized Resolvents of a Hermitian Operator in a Krein Space. Integral Equations Operator Theory 23 (1995), 387–415. [5] V.A. Derkach, On Generalized Resolvents of Hermitian Relations in Krein Spaces. J. Math. Sci. (New York) 97 (1999), 4420–4460. [6] V.A. Derkach, S. Hassi, M.M. Malamud, H.S.V. de Snoo, Generalized Resolvents of Symmetric Operators and Admissibility. Methods Funct. Anal. Topology 6 (2000), 24–53. [7] V.A. Derkach, M.M. Malamud, Generalized Resolvents and the Boundary Value Problems for Hermitian Operators with Gaps. J. Funct. Anal. 95 (1991), 1–95. [8] V.A. Derkach, M.M. Malamud, The Extension Theory of Hermitian Operators and the Moment Problem. J. Math. Sci. (New York) 73 (1995), 141–242. [9] A. Dijksma, H.S.V. de Snoo, Symmetric and Selfadjoint Relations in Krein Spaces I. Oper. Theory Adv. Appl. 24, Birkh¨ auser Verlag Basel (1987), 145–166. [10] A. Dijksma, H.S.V. de Snoo, Symmetric and Selfadjoint Relations in Krein Spaces II, Ann. Acad. Sci. Fenn. Math. 12, 1987, 199–216. [11] V.I. Gorbachuk, M.L. Gorbachuk, Boundary Value Problems for Operator Diﬀerential Equations. Kluwer Academic Publishers, Dordrecht (1991). [12] I.S. Iohvidov, M.G. Krein, H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indeﬁnite Metric. Mathematical Research 9 (1982), AkademieVerlag Berlin. [13] P. Jonas, On a Class of Unitary Operators in Krein Space. Oper. Theory Adv. Appl. 17, Birkh¨ auser Verlag Basel (1986), 151–172. [14] P. Jonas, A Class of Operator-Valued Meromorphic Functions on the Unit Disc, Ann. Acad. Sci. Fenn. Math. 17 (1992), 257-284. [15] P. Jonas, Operator Representations of Deﬁnitizable Functions. Ann. Acad. Sci. Fenn. Math. 25 (2000), 41–72. [16] P. Jonas, On Locally Deﬁnite Operators in Krein Spaces. in: Spectral Theory and Applications, Theta Foundation, (2003). [17] P. Jonas, On Operator Representations of Locally Deﬁnitizable Functions, Oper. Theory Adv. Appl. 162, Birkh¨ auser Verlag, Basel (2005), 165–190. [18] H. Langer, Spectral Functions of Deﬁnitizable Operators in Krein Spaces. Functional Analysis Proceedings of a Conference held at Dubrovnik, Yugoslavia, November 2-14 (1981), Lecture Notes in Mathematics 948, Springer Verlag Berlin-Heidelberg-New York (1982), 1–46. [19] H. Langer, A. Markus, V. Matsaev, Locally Deﬁnite Operators in Indeﬁnite Inner Product Spaces, Math. Ann. 308 (1997), 405–424. Jussi Behrndt Institut f¨ ur Mathematik, MA 6-4 Technische Universit¨ at Berlin Straße des 17. Juni 136 D-10623 Berlin Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 75–95 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Riesz Bases of Root Vectors of Indeﬁnite Sturm-Liouville Problems with Eigenparameter Dependent Boundary Conditions, I ´ Paul Binding and Branko Curgus Abstract. We consider a regular indeﬁnite Sturm-Liouville problem with two self-adjoint boundary conditions, one being aﬃnely dependent on the eigenparameter. We give suﬃcient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space. Mathematics Subject Classiﬁcation (2000). Primary 34L10, 34B24, 34B09, 47B50. Keywords. Sturm-Liouville equations, indeﬁnite weight functions, Riesz bases.

1. Introduction We consider a regular indeﬁnite Sturm-Liouville boundary eigenvalue problem of the form −(p f ) + q f = λ r f on [−1, 1]. (1.1) The coeﬃcients 1/p, q, r in (1.1) are assumed to be real and integrable over [−1, 1], p(x) > 0, and x r(x) > 0 for almost all x ∈ [−1, 1]. We impose two boundary conditions on (1.1) (only one of which is λ-dependent): L b(f ) = 0,

Mb(f ) = λ Nb(f ).

(1.2)

where L, M and N are 1 × 4 non-zero (row) matrices and the boundary mapping b is deﬁned for all f in the domain of (1.1) by T b(f ) = f (−1) f (1) (pf )(−1) (pf )(1) . We shall utilize an operator theoretic framework developed in [3]. Under Condition 2.1 below, a self-adjoint operator A in the Krein space L2,r (−1, 1) ⊕ C∆ can be associated with the eigenvalue problem (1.1), (1.2). Here ∆ is a non-zero

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real number which is determined by M and N – see Section 2 for details. We remark that the topology of this Krein space is that of the corresponding Hilbert space L2,|r| (−1, 1) ⊕ C|∆| . (In the rest of the paper we abbreviate L2,r (−1, 1) to L2,r and L2,|r| (−1, 1) to L2,|r| .) Our main goal in this paper is to provide suﬃcient conditions on the coeﬃcients in (1.1), (1.2) under which there is a Riesz basis of the above Hilbert space consisting of the union of bases for all the root subspaces of the above operator A. This will be referred to for the remainder of this section as the Riesz-basis property of A. Completeness and expansion theorems with a stronger topology, but in a smaller space corresponding to the form domain of the operator A, have been considered by many authors – see [3] (and the references there) and [12]. Although the topology of the Krein space L2,r ⊕ C∆ is weaker than the topology of the form domain, which in our case is a Pontryagin space, the expansion question turns out to be much more challenging mathematically. Indeed, even for the case when the boundary conditions are λ-independent this problem is nontrivial. In our notation, this case corresponds to L being a nonsingular 2 × 4 matrix, with the second equation in (1.2) suppressed. The Rieszbasis property of the operator corresponding to A, now deﬁned in L2,r , has been discussed by several authors, e.g., in [2, 6, 9, 14, 15]. The ﬁrst general suﬃcient condition for this was given by Beals [2], who required the weight function r to behave like a power of the independent variable x in an open neighborhood of the turning point x = 0, although his method does allow more general weight functions. Reﬁnements of Beals’s method in [9] and [15] show that a “one-sided” condition on r (i.e., in only a half-neighborhood of x = 0 ) is enough to guarantee the Riesz-basis property. That some extra condition on r is indeed necessary follows from [15] where Volkmer showed that weight functions r exist for which the corresponding SturmLiouville problem (1.1), under the conditions used here, does not have the Rieszbasis property. Explicit examples of such weight functions were given in [1] and [10]. Recently, Parfyonov [13] has given an explicit necessary and suﬃcient condition for the Riesz basis property in the case p = 1, q = 0 with odd weight function r. Here, and in most of the above references, Dirichlet boundary conditions were imposed. General self-adjoint (perhaps non-separated, but still λ-independent) bound´ ary conditions were treated by Curgus and Langer [6]. They showed that if the essential boundary conditions, i.e., those not including derivatives, were separated, then a Beals-type condition in a neighborhood of x = 0 was suﬃcient for the Riesz-basis property. But if some of the non-separated boundary conditions were essential then [6] established the Riesz-basis property only by imposing extra restrictions on the weight function in (half-)neighborhoods of both endpoints of the interval [−1,1]. Again, some extra restriction is necessary, since in [4] we gave an explicit example of (1.1) under the conditions used here, satisfying a Beals-type condition at x = 0, but without the Riesz-basis property. Of course at least one (in fact one, in this antiperiodic case) boundary condition was essential and nonseparated. In some sense, then, the boundary ±1 behaves as a turning point under such boundary conditions.

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In summary, the Riesz-basis property is quite subtle, and depends signiﬁcantly on the nature of the boundary conditions even when they are independent of λ. In this paper and its sequel, we shall examine the analogous situation for the cases of one and two λ-dependent boundary conditions, where the possibilities for the (λ-dependent) boundary conditions are much greater. As in the λ-independent case, a condition on the weight function is needed near the turning point x = 0 to ensure the Riesz-basis property of A. We shall develop such a condition (which is implied by the ones discussed above) in Section 4. Depending on the nature of the boundary conditions (1.2), we may also need a condition near the boundary, and this is discussed in Section 5. It should be remarked that for the case of exactly one λ-dependent boundary condition treated here we need only one such condition, near either x = −1 or x = 1, and this can be viewed as a “one-sided” condition at ±1. In the case of two λ-dependent boundary conditions we shall also need a condition involving both boundary points x = −1 and x = 1. It turns out that all the above conditions have a common core. This is not immediately obvious, since there are diﬀerences between the “turning points” 0 and ±1. For example, when the boundary conditions are separated, the values of f and f are equal at 0 but are independent at −1 and 1. The common core, which will also be needed in Part II, involves the notion of smoothly connected half-neighborhoods, and this is deﬁned and studied in Section 3. In order to apply the above conditions, we use a criterion in Theorem 2.2, equivalent to the Riesz-basis property of A, involving a positive homeomorphism of L2,r ⊕ C∆ with the form domain of A as an invariant subspace. This, together with certain molliﬁcation arguments, is used for our main results, which are detailed in Section 6. To paraphrase these, we recall that a λ-independent boundary condition is essential if it does not include derivatives. Similarly, a λ-dependent boundary condition will be called essential if it does not include derivatives in the λ-terms. In Theorem 6.1 we discuss situations when a condition on r near x = 0 suﬃces for the Riesz-basis property of A. For example this holds when the ﬁrst (λ-independent) boundary condition in (1.2) is either non-essential, or essential and separated, and the second (λ-dependent) one is non-essential. If the latter condition is essential instead, then the same result holds if a sign condition is also satisﬁed, and this includes a result of Fleige [11], which is the only reference we know where the Riesz-basis property of A has been studied for λ-dependent boundary conditions. In Theorems 6.2 and 6.3 we consider those cases of (1.2) which are not covered by Theorem 6.1. Then we require a condition near just one of the boundary points ±1, not both as in [6]. The choice of the boundary point is arbitrary in Theorem 6.2 which deals with the case when the boundary conditions in (1.2) are, respectively, essential non-separated and non-essential. In Theorem 6.3, however, this choice is not arbitrary but depends on the sign of the number ∆ used in deﬁning the inner product on L2,r ⊕ C∆ .

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2. Operators associated with the eigenvalue problem The maximal operator Smax in L2,r (−1, 1) = L2,r associated with (1.1) is deﬁned by 1 Smax : f → (f ) := −(pf ) + qf , f ∈ D(Smax ), r where ' & D(Smax ) = Dmax = f ∈ L2,r : f, pf ∈ AC[0, 1], (f ) ∈ L2,r . We deﬁne the boundary mapping b by T b(f ) := f (−1) f (1) (pf )(−1) (pf )(1) , and the concomitant matrix Q corresponding to ⎡ 0 0 −1 ⎢0 0 0 Q := i ⎢ ⎣1 0 0 0 −1 0

f ∈ D(Smax ),

b by ⎤ 0 1⎥ ⎥. 0⎦ 0

We notice that Q = Q−1 . Integrating by parts we easily calculate that 1 1 Smax f g r − f Smax g r = i b(g)∗ Qb(f ), f, g ∈ D(Smax ). −1

−1

Throughout, we shall impose the following non-degeneracy and self-adjointness conditions on the boundary data. Condition 2.1. The row vectors L, M and N in (1.2) satisfy: ⎡ ⎤ L (1) the 3 × 4 matrix ⎣M⎦ has rank 3, N (2) LQL∗ = MQM∗ = NQN∗ = LQM∗ = LQN∗ = 0, (3) i MQ−1 N∗ is a non-zero real number and we deﬁne i . ∆=− MQ−1 N∗

(2.1)

Clearly the boundary value problem (1.1)–(1.2) will not change if row reduction is applied to the coeﬃcient matrix 7 6 L 0 . (2.2) M N In what follows we will assume that the 2 × 8 matrix in (2.2) is row reduced to row echelon form (starting the reduction at the bottom right corner). After the row reduction, we write the row vectors L and N as L = L e Ln , N = Ne Nn . (2.3) If either of the 1 × 2 matrices Ln , Nn is non-zero, the corresponding boundary condition is called “non-essential”. In any case these matrices do not appear in the representation of the form domain of A, discussed below, but they will play an

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important role in our conditions for Riesz bases in Section 6. The 1 × 2 matrices Le and Ne represent the “essential” boundary conditions if the non-essential parts Ln and Nn are zero matrices. Next we deﬁne a Krein space operator associated with the problem (1.1)– (1.2). We consider the linear space L2,|r| ⊕ C, equipped with the inner product 6 7 1 g f := , f gr + w∆z , f, g ∈ L2,|r| , z, w ∈ C. w z −1 Then L2,|r| ⊕ C, [ · , · ] is a Krein space, which we denote by L2,r ⊕ C∆ . A fundamental symmetry on this Krein space is given by 7 6 J0 0 , (2.4) J := 0 sgn ∆ where sgn ∆ ∈ {−1, 1} and J0 : L2,r → L2,r is deﬁned by x ∈ [−1, 1].

(J0 f )(x) := f (x) sgn(r(x)),

Then [J · , · ] is a positive deﬁnite inner product which turns L2,r ⊕ C∆ into a Hilbert space L2,|r| ⊕ C|∆| . The topology of L2,r ⊕ C∆ is deﬁned to be that of L2,|r| ⊕ C|∆| , and a Riesz basis of L2,r ⊕ C∆ is deﬁned as a homeomorphic image of an orthonormal basis of L2,|r| ⊕ C|∆| . We deﬁne the operator A in the Krein space L2,r ⊕ C∆ on the domain ⎧ ⎫ ⎨6 7 L2,r ⎬ f ∈ ⊕ : f ∈ D Smax , L b(f ) = 0, z = Nb(f ) D(A) = (2.5) ⎩ z ⎭ C∆ by

6 A

7 7 6 Smax f f , := Mb(f ) Nb(f )

6

7 f ∈ D(A). Nb(f )

(2.6)

Using [3, Theorems 3.3 and 4.1] we see that this operator is self-adjoint in L2,r ⊕ C∆ and in particular: (i) A is quasi-uniformly positive [7] (and therefore deﬁnitizable) in L2,r ⊕ C∆ . (ii) A has a discrete spectrum. (iii) The root subspaces corresponding to real distinct eigenvalues of A are mutually orthogonal in the Krein space L2,r ⊕ C∆ . (iv) All but ﬁnitely many eigenvalues of A are semisimple and real. For further properties of A, we refer the reader to [3, Theorem 3.3]. From (i), (ii) and the characterization of the regularity of the critical point inﬁnity for deﬁnitizable operators in Krein spaces given in [5, Theorem 3.2], we then obtain the following, which is our central tool. Theorem 2.2. Let F (A) denote the form domain of A. There exists a basis for each root subspace of A, so that the union of all these bases is a Riesz basis of L2,|r| ⊕ C|∆| if and only if there exists a bounded, boundedly invertible, positive operator W in L2,r ⊕ C∆ such that W F (A) ⊂ F(A).

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In order to apply this result, we need to characterize the form domain F (A). To this end, let Fmax be the set of all functions f in L2,r , absolutely continuous 1 on [−1, 1], such that −1 p |f |2 < +∞. On Fmax we deﬁne the essential boundary mapping be : Fmax → C2 by T be (f ) := f (−1) f (1) , f ∈ Fmax . Clearly be is surjective. By [3, Theorem 4.2], there are four possible cases for the form domain F (A) of A: If Ln = 0 and Nn = 0, then ⎫ ⎧ ⎬ ⎨6 7 L2,r f ∈ ⊕ : f ∈ Fmax , z ∈ C . (2.7) F(A) = ⎭ ⎩ z C∆ If Ln = 0 and Nn = 0, then ⎧ ⎫ ⎨6 7 L2,r ⎬ f ∈ ⊕ : f ∈ Fmax , Le be (f ) = 0, z ∈ C . F (A) = ⎩ z ⎭ C∆ If Ln = 0 and Nn = 0, then ⎧ ⎫ 7 L2,r ⎨6 ⎬ f ∈ ⊕ : f ∈ Fmax . F (A) = ⎩ Ne be (f ) ⎭ C∆ If Ln = 0 and Nn = 0, then ⎧ ⎫ 7 L2,r ⎨6 ⎬ f ∈ ⊕ : f ∈ Fmax , Le be (f ) = 0 . F (A) = ⎩ Ne be (f ) ⎭ C∆

(2.8)

(2.9)

(2.10)

To construct an operator W as in Theorem 2.2 we need to impose conditions on the coeﬃcients p and r in (1.1). In all cases we need Condition 4.1 in a neighborhood of 0, and in some cases we also need one of two conditions, 5.1 or 5.2, on r in neighborhoods of −1 or 1. These will be discussed in Sections 4 and 5 respectively.

3. Smooth connection and associated operator To prepare the ground for the conditions mentioned above (and in Part II), we develop the concept of smoothly connected half-neighborhoods. A closed interval of non-zero length is said to be a left half-neighborhood of its right endpoint and a right half-neighborhood of its left endpoint. Let ı be a closed subinterval of [−1, 1]. By Fmax (ı) we denote the set of all functions f in L2,r (ı) which are absolutely continuous on ı and such that ı p |f |2 <

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+∞. Note that Fmax [−1, 1] is the space Fmax deﬁned below Theorem 2.2. In the next deﬁnition aﬃne function α means α(t) = a + α t where a, α , t ∈ R. Deﬁnition 3.1. Let p and r be the coeﬃcients in (1.1). Let a, b ∈ [−1, 1] and let ha and hb , respectively, be half-neighborhoods of a and b which are contained in [−1, 1]. We say that the ordered pair (ha , hb ) is smoothly connected if there exist (a) (b) (c)

positive real numbers and τ , non-constant aﬃne functions α : [0, ] → ha and β : [0, ] → hb , non-negative real functions ρ and deﬁned on [0, ],

such that (i) (ii) (iii) (iv) (v)

α(0) = a and β(0) = b, p ◦ α and p ◦ β are locally integrable on the interval (0, ], ρ ◦ α−1 ∈ Fmax α([0, ]) , 1/τ < % < τ a.e. % on [0, ], %r β(t) % p β(t) % , and (t) = , for t ∈ (0, ]. ρ(t) = %% r α(t) % p α(t)

The numbers α , β , (the slopes of α, β, respectively) and ρ(0) are called the parameters of the smooth connection. Remark the function α in Deﬁnition 3.1 is aﬃne, the condition ρ ◦ α−1 ∈ 3.2. Since Fmax α([0, ]) in (iii) is equivalent to ρ ∈ AC[0, ] and |ρ (t)|2 p(α(t))dt < +∞. (3.1) 0

Under the assumption that 1/τ < < τ a.e. on [0, ], it follows that property (3.1) is equivalent to ρ ∈ AC[0, ] and |ρ (t)|2 p(β(t))dt < +∞. 0

To illustrate Deﬁnition 3.1, we make the following Deﬁnition 3.3. Let ν and a be real numbers and let ha be a half-neighborhood of a. Let g be a function deﬁned on ha . Then g is called of order ν on ha if there exists g1 ∈ C 1 (ha ) such that g(x) = (x − a)ν g1 (x)

and

g1 (x) = 0,

x ∈ ha .

Example 3.4. Let a, b ∈ [−1, 1]. Let ha and hb , respectively, be half-neighborhoods of a and b contained in [−1, 1]. Assume that the coeﬃcient r in (1.1) is of order ν on both half-neighborhoods ha and hb . Assume also that the functions p and 1/p are bounded on ha and hb (or, alternatively, that p is of order µ on both halfneighborhoods ha and hb .) Then lengthy, but straightforward, reasoning shows that the half-neighborhoods ha and hb are smoothly connected. Moreover the parameters of the smooth connection are non-zero numbers.

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Remark 3.5. Throughout the paper we use the following convention: A product of functions is deﬁned to have value 0 whenever one of its terms has value zero, even if some other terms are not deﬁned. & ' Theorem 3.6. Let ı and j be closed intervals, ı, j ∈ [−1, 0], [0, 1] . Let a be an endpoint of ı and let b be an endpoint of j. Denote by a1 and b1 , respectively, the remaining endpoints. Assume that the half-neighborhoods ı of a and j of b are smoothly connected with parameters α , β and ρ(0). Then there exists an operator S : L2,|r|(ı) → L2,|r|(j) such that:

(S-1) S ∈ L L2,|r|(ı), L2,|r| (j) , S ∗ ∈ L L2,|r|(j), L2,|r| (ı) . (S-2) (Sf )(x) = 0, |x − b1 | ≤ 12 for all f ∈ L2,|r|(ı), and (S ∗ g)(x) = 0, |x − a1 | ≤ 12 for all g ∈ L2,|r|(j). (S-3) SFmax (ı) ⊂ Fmax (j), S ∗ Fmax (j) ⊂ Fmax (ı). (S-4) For all f ∈ Fmax (ı) and all g ∈ Fmax (j) we have lim f (x), lim (Sf )(y) = |α | x→a

y→b y∈j

x∈ı

lim (S ∗ g)(x) = |β |ρ(0) lim g(y).

x→a x∈ı

y→b y∈j

Proof. Let > 0 be the real number and α and β the aﬃne functions introduced in Deﬁnition 3.1. Thus α(0) = a and β(0) = b. It is no loss of generality to assume that each of the intervals α [0, ] and β [0, ] has a length < 1/2. Let α1 : [0, 1] → ı and β1 : [0, 1] → j be strictly monotonic and continuously diﬀerentiable bijections such that α1 (x) = α(x) and β1 (x) = β(x) for all x ∈ [0, ]. Then α1 (1) = a1 and β1 (1) = b1 . Let φ : [0, 1] → [0, 1], φ ∈ C 1 [0, 1], be such that φ(t) = 1,

0 ≤ t ≤ /2,

φ(t) = 0,

≤ t ≤ 1.

(3.2)

Deﬁne the operator S : L2,|r|(ı) → L2,|r|(j) by (Sf ) β1 (t) := |α |f α1 (t) φ(t),

f ∈ L2,|r| (ı),

t ∈ [0, 1].

(3.3)

Clearly S is linear. In what follows we shall use the combination of property (3.2) and Remark 3.5 to simplify the notation and calculations. For example these imply that in the deﬁnition (3.3) of S we could use β and α instead of β1 and α1 without changing the substance of the deﬁnition. At various points of the proof we shall employ the monotonic (increasing or decreasing) substitutions x = β(t),

α(t) = ξ.

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To prove that S is bounded we let f ∈ L2,|r| (ı) and calculate 2 |(Sf )(x)| |r(x)|dx = sgn(β ) |(Sf )(β(t))|2 |r(β(t))| β dt 0

j

= |α |2 |β |

|f (α(t))|2 |φ(t)|2 ρ(t) |r(α(t))|dt

0 2

≤ |α | |β |R |f (α(t))|2 |φ(t)|2 |r(α(t))|dx 0 ≤ |α ||β |R |f (ξ)|2 |r(ξ)|dξ, ı

where R is an upper bound of the function ρ. The above calculation proves that S is bounded and S ≤ |α ||β |R. that To verify the ﬁrst claim in (S-2), let |x−b1 | < 1/2 and f ∈ L2,|r|(ı). Note the length of j is 1, the endpoints of j are b, b1 , and β(0) = b. Since β [0, ] has the length < 1/2 and since β1 is strictly monotonic we conclude that t = β1−1 (x) > . Therefore, by (3.2), (Sf )(x) = |α |f α(t) φ(t) = 0. This proves the ﬁrst claim in (S-2). To prove SFmax (ı) ⊂ Fmax (j), let f ∈ Fmax (ı). By deﬁnition (3.3), since f is absolutely continuous on ı and φ ∈ C 1 [0, 1], the function Sf is absolutely continuous on j and for almost all t ∈ [0, 1] we have β (Sf ) (β(t)) = |α | α f (α(t))φ(t) + f (α(t))φ (t) . (3.4) To prove that Sf ∈ Fmax (j) we need to show that (Sf ) ∈ L2,p (j), that is 2 |(Sf ) (x)| p(x)dx = |β | |(Sf ) (β(t))|2 p(β(t))dt < +∞.

(3.5)

0

j

We consider each summand in (3.4) separately. By (3.2), the second function in the sum in (3.4) is a continuous function which vanishes outside of the interval [/2, ]. Since by assumption p ◦ β is an integrable function on [/2, ], it follows that 1

|f (α(t))φ (t)|2 p(β(t))dt < +∞.

(3.6)

0

Using the notation and assumptions from Deﬁnition 3.1, for the ﬁrst function in the sum in (3.4) we have 1 2 2 |f (α(t))| |φ(t)| p(β(t))dt = |f (α(t))|2 |φ(t)|2 (t) p(α(t))dt 0 0 ≤τ |f (α(t))|2 |φ(t)|2 p(α(t))dt (3.7) 0 τ ≤ |f (ξ)|2 p(ξ)dξ. |α | ı

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Since f ∈ L2,p (ı) the last expression is ﬁnite. Based on (3.4), (3.5), (3.6) and (3.7) we conclude that (Sf ) ∈ L2,p (j) and consequently Sf ∈ Fmax (j). The next step in the proof is to calculate S ∗ : L2,|r| (j) → L2,|r| (ı). Note that S ∗ is calculated with respect to the Hilbert space inner products on the underlying spaces. Property (3.2) allows us to consider only aﬃne changes of variable in the integrals below. Let f ∈ L2,|r|(ı) and g ∈ L2,|r|(j). Then (Sf )(x) g(x) |r(x)|dx j (Sf )(β(t)) g(β(t)) |r(β(t))| dt = |β | 0 = |β ||α | f (α(t)) φ(t) g(β(t)) ρ(t) |r(α(t))| dt 0 = |β | f (ξ) φ(α−1 (ξ)) g(β(α−1 (ξ))) ρ(α−1 (ξ)) |r(ξ)| dξ. ı

Therefore for g ∈ L2,|r| (j) we have (S ∗ g) x := |β | ρ φ (g ◦ β) α−1 (x) , Thus

(S ∗ g) α(t) = |β |g β(t) ρ(t) φ(t),

x ∈ ı.

g ∈ L2,|r| (j),

t ∈ [0, 1].

(3.8)

∗

As the adjoint of a bounded operator, the operator S is bounded. To verify the second part of (S-2) let |x − a1 | < 1/2 and g ∈ L2,|r|(j). Note that the length of ı is 1 and α(0) = a. Since α [0, ] has length < 1/2 and since α1 is strictly monotonic we conclude that t = α−1 1 (x) > . Therefore, by (3.2), (S ∗ g)(x) = |β |g β(t) ρ(t)φ(t) = 0. To prove S ∗ Fmax (j) ⊂ Fmax (ı), let g ∈ Fmax (j). Since g and ρ are absolutely continuous and φ ∈ C 1 [0, 1], the function (S ∗ g) ◦ α is absolutely continuous on [0, 1]. Diﬀerentiation of (3.8) yields α (S ∗ g) α(t) = |β | ρ φ (g ◦ β) (t) + ρ φ (g ◦ β) (t) + β ρ φ (g ◦ β) (t) , (3.9) for almost all t ∈ [0, 1]. To prove that S ∗ f ∈ Fmax (ı) we need to show that (S ∗ f ) ∈ L2,p (ı), that is |(S ∗ f ) (ξ)|2 p(ξ)dξ = |α | |(S ∗ f ) (α(t))|2 p(α(t))dt < +∞. (3.10) ı

0

We prove that each summand on the right-hand side of (3.9) belongs to L2,p (ı). By (3.2), the second summand is a continuous function which vanishes outside of

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the interval [/2, ]. Since p ◦ α is an integrable function on [/2, ], it follows that 1 |g(β(t))φ (t)ρ(t)|2 p(α(t))dt < +∞. (3.11) 0

Next, we consider the third summand in (3.9). Since ρ is continuous on [0, 1] we can consider only φ (g ◦ β): 1 % % % % %g β(t) φ(t)%2 p(α(t))dt = %g β(t) φ(t)%2 1 p(β(t))dt (t) 0 0 % %2 %g β(t) φ(t)% p(β(t))dt ≤τ (3.12) 0 τ = |g (x)|2 p(x)dx < +∞. |β | j Finally, for the ﬁrst summand in (3.9), it is suﬃcient to consider ρ φ, since g ◦ β is absolutely continuous. By (3.1) 1 2 2 |ρ (t) φ(t)| p(α(t))dt ≤ |ρ (t)| p(α(t))dt < +∞. (3.13) 0

0

Based on (3.9), (3.10), (3.11), (3.12) and (3.13) we conclude that (S ∗ f ) ∈ L2,p (ı) and consequently S ∗ f ∈ Fmax (ı). Thus we have veriﬁed the properties (S-1), (S-2), (S-3). Since (S-4) is clear the theorem is proved.

4. Condition at 0 and associated operator Condition 4.1 (Condition at 0). Let p and r be coeﬃcients in (1.1). Denote by 0− a generic left and by 0+ a generic right half-neighborhood of 0. We assume that at least one of the four ordered pairs of half-neighborhoods (0− , 0− ),

(0− , 0+ ),

(0+ , 0− ),

is smoothly connected with connection parameters |α0 | = |β0 |ρ0 (0).

(0+ , 0+ ), α0 , β0

(4.1)

and ρ0 (0) such that

Theorem 4.2. Assume that the coeﬃcients p and r satisfy Condition 4.1. Then there exists an operator W0 : L2,r → L2,r such that (a) W0 is bounded on L2,|r| . (b) The operator J0 W0 −I is nonnegative on the Hilbert space L2,|r| . In particular W0−1 is bounded and W0 is positive on the Krein space L2,r . (c) (W0 f )(x) = (Jf )(x), 12 ≤ |x| ≤ 1, f ∈ L2,r . (d) W0 Fmax [−1, 1] ⊂ Fmax [−1, 1]. Proof. Let α0 , β0 , and ρ0 (0) be given by Condition 4.1. Recall that |α0 | = |β0 |ρ0 (0). Let φ0 : [−1, 1] → [0, 1], φ0 ∈ C 1 [−1, 1] be an even function such that φ0 (0) = 1

and

φ0 (x) = 0

for 1/2 ≤ |x| ≤ 1.

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86 Deﬁne the operators

P0,− : L2,|r| (−1, 0) → L2,|r|(−1, 0) and P0,+ : L2,|r|(0, 1) → L2,|r|(0, 1) by (P0,− f )(x) = f (x) φ0 (x),

f ∈ L2,|r|(−1, 0), x ∈ [−1, 0],

(P0,+ f )(x) = f (x) φ0 (x),

f ∈ L2,|r|(0, 1),

x ∈ [0, 1].

Then P0,− and P0,+ are self-adjoint operators with the following properties: (P0,− f )(x) = 0,

f ∈ L2,|r|(−1, 0), −1 ≤ x ≤ − 21 ,

(4.2)

(P0,+ f )(x) = 0,

f ∈ L2,|r|(0, 1),

(4.3)

P0,− Fmax [−1, 0] ⊂ Fmax [−1, 0],

1 2

≤ x ≤ 1,

P0,+ Fmax [0, 1] ⊂ Fmax [0, 1],

(4.4)

and (P0,− f )(0−) = f (0−),

f ∈ Fmax [−1, 0],

(4.5)

(P0,+ f )(0+) = f (0+),

f ∈ Fmax [0, 1].

(4.6)

Here, the value of a function at 0± represents its one sided limit. Condition 4.1 requires that one of the four ordered pairs of half neighborhoods is smoothly connected. For such a pair, Theorem 3.6 guarantees the existence of a speciﬁc operator which we denote by S0 . For each of the four pairs we shall use diﬀerent combinations of scaled operators P0,− , P0,+ , S0 and S0∗ to deﬁne a bounded block operator L2,|r|(−1, 0) L2,|r| (−1, 0) ⊕ ⊕ → L2,|r| (0, 1) L2,|r|(0, 1)

X0 : with the following properties

(X0∗ f )(x) = 0,

1/2 ≤ |x| ≤ 1,

X0 Fmax [−1, 1] ⊂ Fmax [−1, 1], (X0 f )(0) ∗ X0 Fmax [−1, 1] ∗ (X0 f )(0+) + (X0∗ f )(0−) These properties of X0 and

= f (0),

f ∈ Fmax [−1, 1],

(4.7) (4.8) (4.9)

⊂ Fmax [−1, 0] ⊕ Fmax [0, 1],

(4.10)

= −2f (0),

(4.11)

f ∈ Fmax [−1, 1].

X0∗

imply that the operator W0 = J X0∗ X0 + I

has all the properties stated in the theorem. Since we assume that |α0 | = |β0 |ρ0 (0), the system γ1 |α0 | + γ2 = 1,

γ1 |β0 |ρ0 (0) + γ2 = −3

has a nontrivial real solution γ1 , γ2 . We use this solution in the deﬁnitions below.

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Case 1. Assume that the half-neighborhoods 0− , 0− in (4.1) are smoothly connected. Then by Theorem 3.6 there exists an operator S0 : L2,|r|(−1, 0) → L2,|r|(−1, 0) which satisﬁes (S-1)–(S-4) in Theorem 3.6 with ı = j = [−1, 0], a = b = 0. In particular, for f ∈ Fmax [−1, 0], (S0 f )(0−) = |α0 | f (0−), We deﬁne X0 and calculate X0∗ as 7 6 0 γ1 S0 + γ2 P0,− , X0 = 0 P0,+

(S0∗ f )(0−) = |β0 | ρ0 (0) f (0−).

X0∗

6

γ S ∗ + γ2 P0,− = 1 0 0

0 P0,+

7 .

Case 2. Assume that the half-neighborhoods 0− , 0+ in (4.1) are smoothly connected. Then by Theorem 3.6 there exists an operator S0 : L2,|r| (−1, 0) → L2,|r|(0, 1) which satisﬁes (S-1)–(S-4) in Theorem 3.6 with ı = [−1, 0], j = [0, 1], a = b = 0. In particular, for f ∈ Fmax [−1, 0], (S0 f )(0+) = |α0 | f (0−), We deﬁne X0 and calculate X0∗ as 7 6 0 P , X0 = 0,− γ1 S0 γ2 P0,+

(S0∗ f )(0−) = |β0 | ρ0 (0) f (0+).

X0∗ =

6

P0,− 0

7 γ1 S0∗ . γ2 P0,+

Case 3. Assume that the half-neighborhoods 0+ , 0− in (4.1) are smoothly connected. Then by Theorem 3.6 there exists an operator S0 : L2,|r| (0, 1) → L2,|r| (−1, 0) which satisﬁes (S-1)–(S-4) in Theorem 3.6 with ı = [0, 1], j = [−1, 0], a = b = 0. In particular, for f ∈ Fmax [0, 1], (S0 f )(0−) = |α0 | f (0+), We deﬁne X0 and calculate X0∗ as 7 6 γ2 P0,− γ1 S0 , X0 = 0 P0,+

(S0∗ f )(0+) = |β0 | ρ0 (0) f (0−).

X0∗

6 γ P = 2 0,− γ1 S0∗

0 P0,+

7 .

Case 4. Assume that the half-neighborhoods 0+ , 0+ in (4.1) are smoothly connected. Then by Theorem 3.6 there exists an operator S0 : L2,|r| (0, 1) → L2,|r|(0, 1) which satisﬁes (S-1)–(S-4) in Theorem 3.6 with ı = [0, 1], j = [0, 1], a = b = 0. In particular, for f ∈ Fmax [0, 1], (S0 f )(0+) = |α0 | f (0+),

(S0∗ f )(0+) = |β0 | ρ0 (0) f (0+).

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88

We deﬁne X0 and calculate X0∗ as 7 6 0 P , X0 = 0,− 0 γ1 S0 + γ2 P0,+

X0∗ =

6

P0,− 0

7 0 . γ1 S0∗ + γ2 P0,+

First note that in each of the four cases above, the operator X0 is bounded since each of its components is bounded. In each of the four cases above, the property (4.7) follows from (S-2) in Theorem 3.6, and properties (4.2) and (4.3). Let f ∈ Fmax [−1, 1]. Since γ1 |α0 | + γ2 = 1, the function X0 f is continuous in each case and (X0 f )(0) = f (0). This, (4.4), (4.5), (4.6), (S-3) and (S-4) in Theorem 3.6 imply (4.8). Inclusion (4.10) follows similarly. In each of the above cases, equation (4.11) is a consequence of γ1 |β0 |ρ0 (0) + γ2 = −3, (4.5), (4.6) and (S-4) in Theorem 3.6. This proves the theorem.

Remark 4.3. Note the behavior of the operator W0 in Theorem 4.2 at the boundary of the interval [−1, 1]: 7 6 7 6 (W0 f )(−1) −f (−1) , f ∈ Fmax [−1, 1]. = f (1) (W0 f )(1) This property of W0 will be used in Section 6. In the next section, under additional assumptions on the coeﬃcients p and r in a neighborhood of −1 and 1, we shall construct operators W with speciﬁed behaviors at −1 and 1.

5. Conditions at –1 and 1, and associated operators In this section we show that under additional assumptions on the coeﬃcients p and r near −1 we can construct an operator W−1 with prescribed behavior at −1 and under additional assumptions near 1 we can construct an operator W+1 with prescribed behavior at 1. Condition 5.1 (Condition at −1). Let p and r be coeﬃcients in (1.1). We assume that a right half-neighborhood of −1 is smoothly connected to a right half-neigh borhood of −1 with the connection parameters α−1 , β−1 and ρ−1 (0) such that |α−1 | = |β−1 |ρ−1 (0). Condition 5.2 (Condition at 1). Let p and r be coeﬃcients in (1.1). We assume that a left half-neighborhood of 1 is smoothly connected to a left half neighborhood of 1 with the connection parameters α+1 , β+1 and ρ+1 (0) such that |α+1 | = |β+1 |ρ+1 (0). In the rest of this section we shall need two operators analogous to P0,− and P0,+ introduced in Section 4. Let φ1 : [−1, 1] → [0, 1] be a smooth even function such that φ1 (−1) = 1,

φ1 (x) = 0 for 0 ≤ |x| ≤ 1/2,

φ1 (1) = 1.

(5.1)

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Deﬁne the operators P1,− : L2,|r| (−1, 0) → L2,|r|(−1, 0) and P1,+ : L2,|r|(0, 1) → L2,|r|(0, 1) by

(P1,− f )(x) = f (x) φ1 (x),

f ∈ L2,|r|(−1, 0), x ∈ [−1, 0],

(5.2)

x ∈ [0, 1]. (5.3) (P1,+ f )(x) = f (x) φ1 (x), f ∈ L2,|r|(0, 1), Then P1,− and P1,+ are self-adjoint operators with the following properties: (P1,− f )(x) = 0,

f ∈ L2,|r| (−1, 0), − 21 ≤ x ≤ 0,

(5.4)

(P1,+ f )(x) = 0,

f ∈ L2,|r| (0, 1),

(5.5)

P1,− Fmax [−1, 0] ⊂ Fmax [−1, 0],

0≤x≤

1 2,

P1,+ Fmax [0, 1] ⊂ Fmax [0, 1],

(5.6)

and (P1,− f )(−1+) = f (−1+), (P1,+ f )(1−) = f (1−),

f ∈ Fmax [−1, 0],

(5.7)

f ∈ Fmax [0, 1].

(5.8)

Proposition 5.3. Assume that the coeﬃcients p and r satisfy Condition 5.1. Let µ be an arbitrary complex number. Then there exists an operator W−1 : L2,r → L2,r such that (a) W−1 is bounded on L2,|r| . (b) The operator J0 W−1 − I is nonnegative on the Hilbert space L2,|r|. In particular (W−1 )−1 is bounded and W−1 is positive on the Krein space L2,r . (c) (W−1 f )(x) = (Jf )(x), − 21 ≤ x ≤ 1, f ∈ L2,r . (d) W−1 Fmax [−1, 1] ⊂ Fmax [−1, 0] ⊕ Fmax [0, 1]. (e) (W−1 f )(−1) = µf (−1) for all f ∈ Fmax [−1, 1]. Proof. We use the notation introduced in Condition 5.1. By Theorem 3.6 there exists a bounded operator S−1 : L2,|r|(−1, 0) → L2,|r| (−1, 0) such that ∗ S−1 Fmax [−1, 0] ⊂ Fmax [−1, 0] and S−1 Fmax [−1, 0] ⊂ Fmax [−1, 0],

and, for all f ∈ Fmax [−1, 0], (S−1 f )(−1) = |α−1 | f (−1),

∗ (S−1 f )(−1) = |β−1 | ρ−1 (0)f (−1).

Let µ be an arbitrary complex number. Since we assume that |α−1 | = |β−1 | ρ−1 (0), the complex numbers γ1 and γ2 can be chosen such that

γ1 |α−1 | + γ2 = 1,

γ 1 |β−1 | ρ−1 (0) + γ 2 = −µ − 1.

∗ We deﬁne X−1 and calculate X−1 as 7 6 γ S + γ2 P1,− 0 , X−1 = 1 −1 0 0

∗ X−1 =

6

∗ γ 1 S−1 + γ 2 P1,− 0

7 0 . 0

Then for all f ∈ Fmax [−1, 1] we have ∗ (X−1 f )(−1) = f (−1) and (X−1 f )(−1) = (−µ − 1)f (−1).

Therefore

∗ W−1 = J X−1 X−1 + I has all the properties stated in the proposition.

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The proof of the next proposition is very similar to the preceding proof, and will be omitted. Proposition 5.4. Assume that the coeﬃcients p and r satisfy Condition 5.2. Let µ be an arbitrary complex number. Then there exists an operator W+1 : L2,r → L2,r such that (a) W+1 is bounded on L2,|r| . (b) The operator J0 W+1 − I is nonnegative on the Hilbert space L2,|r| . In particular (W+1 )−1 is bounded and W+1 is positive on the Krein space L2,r . (c) (W+1 f )(x) = (Jf )(x), −1 ≤ x ≤ 12 , f ∈ L2,r . (d) W+1 Fmax [−1, 1] ⊂ Fmax [−1, 0] ⊕ Fmax [0, 1]. (e) (W+1 f )(1) = µf (1) for all f ∈ Fmax [−1, 1].

6. Riesz basis of root vectors In this section we return to the eigenvalue problem (1.1)–(1.2) and the operator A associated with it. We start with cases when the conditions in Section 5 are not needed. We remark that the notation of Section 2 is used extensively in the rest of this section. Theorem 6.1. Assume that the following three conditions are satisﬁed. (a) The coeﬃcients p and r satisfy Condition 4.1. (b) One of the following is true: (i) Ln = 0, (ii) L = [1 0 0 0], (iii) L = [0 1 0 0]. (c) One of the following is true: (i) Nn = 0, (ii) N = [1 0 0 0] and ∆ < 0, (iii) N = [0 1 0 0] and ∆ > 0. Then there is a basis for each root subspace of A, so that the union of all these bases is a Riesz basis of L2,|r| ⊕ C|∆| . Proof. Assume ﬁrst that Nn = 0. By (2.7), the form domain of A when Ln = 0 is given by ⎧ ⎫ ⎨6 7 L2,r ⎬ f ∈ ⊕ : f ∈ Fmax , z ∈ C , F (A) = ⎩ z ⎭ C∆

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91

and in the other two cases in (b), (2.8) gives ⎧ ⎫ ⎨6 7 L2,r ⎬ f ∈ ⊕ : f ∈ Fmax , z ∈ C, Le be (f ) = 0 , F (A) = ⎩ z ⎭ C∆ where Le be (f ) = f (−1) in case (b-ii) and Le be (f ) = f (1) in case (b-iii). Next assume (c-ii). Then Ne be (f ) = f (−1) and (2.9) shows that ⎧ ⎫ 7 L2,r ⎨6 ⎬ f ∈ ⊕ : f ∈ Fmax F (A) = ⎩ f (−1) ⎭ C∆ when Ln = 0, and in the other cases in (b), (2.10) gives ⎧ ⎫ 7 L2,r ⎨6 ⎬ f ∈ ⊕ : f ∈ Fmax , Le be (f ) = 0 . F (A) = ⎩ f (−1) ⎭ C∆ Finally, assume (c-iii). Then Ne be (f ) = f (1) and (2.9) shows that ⎧ ⎫ 7 L2,r ⎨6 ⎬ f ∈ ⊕ : f ∈ Fmax F (A) = ⎩ f (1) ⎭ C∆ when Ln = 0, and in the other cases in (b), (2.10) gives ⎧ ⎫ 7 L2,r ⎨6 ⎬ f ∈ ⊕ : f ∈ Fmax , Le be (f ) = 0 . F (A) = ⎩ f (1) ⎭ C∆ Let W0 be the operator constructed in Theorem 4.2, and let 6

W0 W = 0

0 sgn(∆)

7 :

L2,r ⊕ C∆

→

L2,r ⊕ . C∆

A straightforward veriﬁcation shows that W is a bounded, boundedly invertible, positive operator in the Krein space L2,r ⊕ C∆ and W F (A) ⊂ F(A) in each of the above listed cases. Consequently, the theorem follows from Theorem 2.2. In the next result we shall assume that one of the conditions from Section 5 is satisﬁed. Theorem 6.2. Assume that the following three conditions are satisﬁed. (a) The coeﬃcients p and r satisfy Condition 4.1, and one of Conditions 5.1, 5.2. (b) Nn = 0. (c) L = [u v 0 0] with uv = 0. Then there is a basis for each root subspace of A, so that the union of all these bases is a Riesz basis of L2,|r| ⊕ C|∆| .

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´ P. Binding and B. Curgus

Proof. Under the assumptions of the theorem, (2.8) shows that the form domain of A is given by ⎧ ⎫ ⎨6 7 L2,r ⎬ f ∈ ⊕ : f ∈ Fmax , z ∈ C, uf (−1) + vf (1) = 0 . F (A) = ⎩ z ⎭ C∆ Deﬁne the following two Krein spaces: ˙ 2,r ( 1 , 1). K1 := L2,r (−1, − 21 )[+]L K0 := L2,r − 21 , 12 , 2

(6.1)

Extending the functions in K0 and K1 by 0 onto the rest of [−1, 1], we can consider the spaces K0 and K1 as subspaces of L2,r . Then ˙ 1. L2,r = K0 [+]K Assume that the functions p and r satisfy Conditions 4.1 and 5.1. Let W0 be the operator constructed in Theorem 4.2 and let W−1 be the operator constructed in Proposition 5.3 with µ = 1. Then properties (c) in Theorem 4.2 and Proposition 5.3, imply that K0 and K1 are invariant under W0 and W−1 . As we chose µ = 1, we have (W−1 f )(−1) = f (−1) and (W−1 f )(1) = f (1). Deﬁne ˙ −1 |K . W01 := W0 |K0 [+]W 1

(6.2)

Since W0 and W−1 are bounded, boundedly invertible and positive in the Krein space L2,r , so is the the operator W01 . Also, W01 Fmax [−1, 1] ⊂ Fmax [−1, 1] and 7 7 6 6 f (−1) (W01 f )(−1) . (6.3) = f (1) (W01 f )(1) If the functions p and r satisfy Conditions 4.1 and 5.2, then, instead of W−1 , we use the operator W+1 constructed in Proposition 5.4 with µ = −1. Redeﬁning the operator W01 as ˙ +1 |K (6.4) W01 := W0 |K0 [+]W 1 we see that it is again bounded, boundedly invertible, and positive in the Krein space L2,r , W01 Fmax [−1, 1] ⊂ Fmax [−1, 1] and (since we use µ = −1) 7 7 6 6 f (−1) (W01 f )(−1) . (6.5) =− f (1) (W01 f )(1) Now a simple inspection shows that, in both above cases, the operator 7 L2,r 6 L2,r W01 0 : ⊕ → ⊕ W = 0 ∆ C∆ C∆ is bounded, boundedly invertible and positive in the Krein space L2,r ⊕ C∆ and W F (A) ⊂ F(A). Thus the theorem again follows from Theorem 2.2. Our ﬁnal result covers the remaining cases, but in the interests of presentation we shall impose no conditions on L. Of course, there is some overlap with Theorem 6.1.

Riesz Bases of Root Vectors, I

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Theorem 6.3. Assume that the following two conditions are satisﬁed. (a) The coeﬃcients p and r satisfy Condition 4.1, and (i) Condition 5.1 if ∆ > 0, (ii) Condition 5.2 if ∆ < 0. (b) Nn = 0. Then there is a basis for each root subspace of A, so that the union of all these bases is a Riesz basis of L2,|r| ⊕ C|∆| . Proof. In this case (2.9) shows that the form domain of A is ⎧ ⎫ 7 L2,r ⎨6 ⎬ f ∈ ⊕ : f ∈ Fmax F (A) = ⎩ Ne be (f ) ⎭ C∆ if Ln = 0, and

⎫ L2,r ⎬ f ∈ ⊕ : f ∈ Fmax , Le be (f ) = 0 F (A) = ⎭ ⎩ Ne be (f ) C∆ ⎧ ⎨6

7

by (2.10) if Ln = 0. Let W01 : L2,r → L2,r be the operator constructed in the proof of Theorem 6.2, and deﬁne W : L2,r ⊕ C∆ → L2,r ⊕ C∆ by 7 6 0 W01 . W = 0 sgn(∆) As before, in both cases the properties of W01 imply that W is a bounded, boundedly invertible, positive operator in the Krein space L2,r ⊕ C∆ and W F (A) ⊂ F(A). Now the theorem follows from Theorem 2.2. Remark 6.4. It is instructive to look at the above results from the viewpoint of non-essential boundary conditions (Ln , Nn = 0) and essential ones (whose essential parts Le , Ne can be separated or not). Let us call a boundary condition essentially separated if it is either non-essential, or else its essential part is separated. Theorem 6.1 states that if both boundary conditions are essentially separated, then subject to the sign conditions in (c-ii) and (c-iii), Condition 4.1 suﬃces for the existence of a Riesz basis of root vectors. If any of these assumptions fail, then we impose conditions from Section 5. In particular, if the λ-dependent boundary condition is non-essential, then either of these conditions suﬃce, but in other cases the choice is governed by the sign of ∆. We conclude with a simple example. Example 6.5. We suppose that L = [d1 d2 d3 d4 ], M = [m1 m2 m3 m4 ], N = [0 γ 0 0], where (d3 , d4 ) = (0, 0) and γm4 > 0. Note that the only λ-dependent term in (1.2) involves f (1).

´ P. Binding and B. Curgus

94 We calculate

MQ−1 N∗ = −iγm4 , so by (2.1), ∆ > 0. It follows from Theorem 6.1 with condition (b-i) that Condition 4.1 suﬃces for the existence of a Riesz basis of root vectors. This example overlaps with [11, Corollary 3.8], where separated boundary conditions, also satisfying d2 = d4 = m1 = m3 = 0, d3 = m4 = 1, γ > 0, were considered by Fleige for a Krein-Feller equation instead of (1.1). Acknowledgment We thank a referee for a very careful reading of the submitted version of this article. This has led to the correction of a number of inaccuracies.

References [1] N.L. Abasheeva, S.G. Pyatkov, Counterexamples in indeﬁnite Sturm-Liouville problems. Siberian Advances in Mathematics. Siberian Adv. Math. 7 (1997), 1–8. [2] R. Beals, Indeﬁnite Sturm-Liouville problems and half-range completeness. J. Diﬀerential Equations 56 (1985), 391–407. ´ [3] P.A. Binding, B. Curgus, Form domains and eigenfunction expansions for diﬀerential equations with eigenparameter dependent boundary conditions. Canad. J. Math. 54 (2002), 1142–1164. ´ [4] P.A. Binding, B. Curgus, A counterexample in Sturm-Liouville completeness theory. Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 244–248. ´ [5] B. Curgus, On the regularity of the critical point inﬁnity of deﬁnitizable operators. Integral Equations Operator Theory 8 (1985), 462–488. ´ [6] B. Curgus, H. Langer, A Krein space approach to symmetric ordinary diﬀerential operators with an indeﬁnite weight function. J. Diﬀerential Equations 79 (1989), 31–61. ´ [7] B. Curgus, B. Najman, Quasi-uniformly positive operators in Krein space. Operator theory and boundary eigenvalue problems (Vienna, 1993), 90–99, Oper. Theory Adv. Appl., 80, Birkh¨ auser, 1995. [8] A. Dijksma, Eigenfunction expansions for a class of J-self-adjoint ordinary diﬀerential operators with boundary conditions containing the eigenvalue parameter. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 1–27. [9] A. Fleige, The “turning point condition” of Beals for indeﬁnite Sturm-Liouville problems. Math. Nachr. 172 (1995), 109–112. [10] A. Fleige, A counterexample to completeness properties for indeﬁnite Sturm-Liouville problems. Math. Nachr. 190 (1998), 123–128. [11] A. Fleige, Spectral theory of indeﬁnite Krein-Feller diﬀerential operators. Mathematical Research, 98 Akademie Verlag, 1996. [12] H. Langer, A. Schneider, On spectral properties of regular quasideﬁnite pencils F − λG. Results Math. 19 (1991), 89–109. [13] A.I. Parfyonov, On an embedding criterion for interpolation spaces and application to indeﬁnite spectral problems. Siberian Math. J. 44 (2003), 638–644.

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[14] S.G. Pyatkov, Interpolation of some function spaces and indeﬁnite Sturm-Liouville problems. Diﬀerential and integral operators (Regensburg, 1995), 179–200, Oper. Theory Adv. Appl. 102, Birkh¨ auser, (1998). [15] H. Volkmer, Sturm-Liouville problems with indeﬁnite weights and Everitt’s inequality. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 1097–1112. Paul Binding Department of Mathematics and Statistics University of Calgary Calgary Alberta, T2N 1N4 Canada e-mail: [email protected] ´ Branko Curgus Department of Mathematics Western Washington University Bellingham WA 98225 USA e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 97–134 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Minimal Models for Nκ∞ -functions Aad Dijksma, Annemarie Luger and Yuri Shondin To Heinz Langer, wishing him a happy retirement

Abstract. We present explicit realizations in terms of self-adjoint operators and linear relations for a non-zero scalar generalized Nevanlinna function (z) = −1/N (z) under the assumption that N (z) has N (z) and the function N exactly one generalized pole which is not of positive type namely at z = ∞. The key tool we use to obtain these models is reproducing kernel Pontryagin spaces. Mathematics Subject Classiﬁcation (2000). Primary 47B25, 47B50, 47B32; Secondary 47A06. Keywords. Generalized Nevanlinna function, generalized pole, realization, model, reproducing kernel spaces, Pontryagin spaces, self-adjoint operator, symmetric operator, linear relation, block operator matrix.

1. Introduction An n × n matrix function N is called a generalized Nevanlinna function with κ negative squares if (i) it is deﬁned and meromorphic on C \ R, (ii) it satisﬁes N (z) = N (z ∗ )∗ for all z ∈ D(N ), the domain of holomorphy of N , and (iii) the kernel N (ζ) − N (z)∗ KN (ζ, z) = , ζ, z ∈ D(N ), ζ − z∗ has κ negative squares. Here the expression on the right-hand side for ζ = z ∗ is to be understood as N (ζ). If κ = 0, the function N is called a Nevanlinna function; The authors gratefully acknowledge support from the “Fond zur F¨ orderung der wissenschaftlichen Forschung” (FWF, Austria, grant number P15540-N05), the Netherlands Organization for Scientiﬁc Research NWO (grant NWO 047-008-008), and the Research Training Network HPRNCT-2000-00116 of the European Union.

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A. Dijksma, A. Luger and Y. Shondin

in this case N is holomorphic on C \ R, satisﬁes N (z) = N (z ∗ )∗ there and the kernel condition is equivalent to the condition Im N (z) ≥ 0, Im z = 0. Im z The class of n × n matrix functions with κ negative squares is denoted by Nκn×n and by Nκ when the functions are scalar. A realization for a function N ∈ Nκn×n in some Pontryagin space P is a pair (A, Γz ) consisting of a self-adjoint relation A in P with a nonempty resolvent set ρ(A) and a corresponding Γ-ﬁeld Γz , that is, a family of mappings Γz : Cn → P, z ∈ ρ(A), which satisfy Γz = (IP + (z − ζ)(A − z)−1 )Γζ ,

ζ, z ∈ ρ(A),

and

N (ζ) − N (z)∗ = Γ∗z Γζ , ζ, z ∈ ρ(A), z = ζ ∗ . ζ − z∗ If a point z0 ∈ ρ(A) is ﬁxed this implies the following representation of N : N (z) = N (z0 )∗ + (z − z0∗ )Γ∗z0 IP + (z − z0 )(A − z)−1 Γz0 , z ∈ D(N ).

The function N is determined by the self-adjoint relation A in P and the Γ-ﬁeld Γz up to an additive constant hermitian n × n matrix. The space P is called the state space of the realization (A, Γz ). The realization (A, Γz ) can always be chosen minimal which means that span {Γz c | z ∈ (A), c ∈ Cn } = P. In that case the negative index of the state space P is equal to the number of negative squares of the kernel KN (ζ, z) and D(N ) = ρ(A); see [16, Theorem 1.1]. Two minimal realizations of N are unitarily equivalent. With a minimal realization (A, Γz ) often a symmetric restriction S of the relation A is associated and deﬁned by % S = {{f, g} ∈ A % Γ∗z0 (g − z0∗ f ) = 0}. This deﬁnition is independent of z0 ∈ D(N ), S is an operator, and Γz maps Cn onto the defect subspace ran (S − z ∗ )⊥ of S at z. The triplet (A, Γz , S) is called a model in P for the realization of N or, for short, a model for the function N in P. The model will be called minimal if the realization is minimal. If n = 1 the function ϕ(z) = Γz 1 = (IP + (z − z0 )(A − z)−1 )ϕ(z0 ), called a defect function for S and A, spans the defect subspace of S at z and the representation of N takes the form N (z) = N (z0 )∗ + (z − z0∗ )ϕ(z), ϕ(z0 )P . Every N ∈ Nκ admits a basic factorization of the form N (z) = r# (z)N1 (z)r(z),

(1.1)

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99

where N1 ∈ N0 and r is a rational function whose zeros (poles) are the generalized zeros (poles) of N in C+ ∪ R (C− ∪ R, respectively) which are not of positive type; for deﬁnitions and a proof of (1.1), see, for example, [10] and [9]. Here and in the sequel for a vector function f we denote by f # the function f # (z) = f (z ∗ )∗ . If κ1 is the number of zeros of r and κ2 is the number of poles of r (counted according to their multiplicities), then κ = max {κ1 , κ2 }. If τ = κ1 − κ2 is positive (negative) then z = ∞ is a generalized pole (zero) of N which is not of positive type and with degree of non-positivity |τ |. In particular, if r is a polynomial (necessarily of degree κ), then z = ∞ is the only generalized pole of N and not of positive type; if on the other hand κ1 = 0 (so that κ2 = κ), then z = ∞ is the only generalized zero of N and not of positive type. In this paper we are describing minimal models for functions 0 = N ∈ Nκ = −N −1 (which also belongs to Nκ ) under the assumption that the latter and N belongs to belongs to the class Nκ∞ considered in [12]. By deﬁnition, a function N the class Nκ∞ if and only if it belongs to Nκ and has a representation of the form N(z) = c# (z)N0 (z)c(z) + p(z),

(1.2)

where N0 (z) is a Nevanlinna function with the properties lim y Im N0 (iy) = +∞,

y→∞

lim y −1 N0 (iy) = 0,

y→∞

Re N0 (i) = 0,

(1.3)

), and p is some real polynomial. As c(z) = (z − z0 )m with m ∈ N0 and z0 ∈ D(N explained in [12], the representation (1.2) (with (1.3)) is irreducible and implies (z). that z = ∞ is the only generalized pole of non-positive type of the function N The ﬁrst two conditions in (1.3) are equivalent to the fact that in the minimal model for N0 the symmetric operator is densely deﬁned in the state space. The third condition is simply a normalization. In the deﬁnition of the class Nκ∞ given in [12] it was required that the point z0 belongs to the set C \ R, but in view of ) and the deﬁnition [12, Remark 1.3] z0 may belong to the possibly larger set D(N is independent of the choice of z0 ∈ D(N ). The minimal models, which we obtain and which are related to the irreducible representation (1.2) of N , for N and N n m m have a state space of the form K = H0 ⊕ C ⊕ C ⊕ C , n = max {deg p − 2m, 0}, equipped with the indeﬁnite inner product G · , · K , where H0 is the state space for a minimal model of the function N0 and the Gram matrix G is the 4 × 4 block matrix given by (6.1) with blocks determined by the polynomials p and q from (z). In [5], [6], [10], and [25] the minimal models for N the realization (4.1) of N related to the basic factorization (1.1) are studied. The model considered in [25] has a state space which is a subspace with ﬁnite co-dimension of L = H1 ⊕ Cκ ⊕ Cκ equipped with the indeﬁnite inner product GL · , · L , where H1 is the state space for a minimal model of the function N1 and the Gram matrix is given by ⎛ ⎞ IH1 0 0 0 ICκ ⎠ . GL = ⎝ 0 0 ICκ 0

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The model in the present paper is more detailed than this model because we consider a more special class of generalized Nevanlinna functions. To motivate our study of the model problem we list some applications where functions of the form (1.2) play a role. First we note that Nevanlinna functions N0 (z) satisfying the asymptotic conditions in (1.3) (in the following we disregard the normalization condition) appear naturally (i) as a Q-function of the minimal operator associated with a self-adjoint boundary value problem for a formally symmetric ordinary diﬀerential expression (Titchmarsh-Weyl coeﬃcient) and (ii) as the main ingredient in the formula of the resolvent for the singular perturbation A0 + α−1 χ · , χ0

(1.4)

of an unbounded self-adjoint operator A0 in a Hilbert space H0 with inner product · , · 0 , generated by a generalized element χ ∈ H−1 \ H0 : N0 (z) = (z − z0∗ )ϕ0 (z), ϕ0 (z0 )0 − iIm z0 ϕ0 (z0 ), ϕ0 (z0 )0 + c, where ϕ(z) = (A0 − z)−1 χ and c is a real number; see [1, 2]. Here, in the scale of Hilbert spaces associated with H0 and A0 , the space H−m is the dual of the space Hm = dom |A0 |m equipped with the inner product (|A0 | + 1)m · , (|A0 | + 1)m · 0 , m = 1, 2 . . .. As explained in [13], generalized Nevanlinna functions of the form (1.2) with deg p ≤ 2m play a similar role as in (ii) but now for the strongly singular perturbation (1.4) with χ ∈ H−m−1 \H−m . Furthermore, in [19] and [24] point-like perturbations of the Laplacian in R3 were constructed to describe the low energy asymptotic behavior k cot δ0 (k) =

n

aj k 2j + o(k 2n )

j=1

of the quantum mechanical scattering data at zero orbital momenta, where E = k 2 is the energy of scattering particle and δ0 (k) is the scattering phase. This construction amounts to building a model of the generalized Nevanlinna function of the √ form (1.2) with c(z) = 1, deq p > 0, and N0 (z) = − −z. In the two papers just mentioned two diﬀerent models in Pontryagin spaces were given. To describe a given truncated series of low energy scattering with non-zero angular momentum models for generalized Nevanlinna functions (1.2) with arbitrary deg c(z) and deg p(z) are needed. Some models of this kind where considered in [8]. As a further motivation for the models in this paper, we discuss in Section 8 an approximation problem where generalized Nevanlinna functions of the form (1.2) with various values of deg c(z) and deg p(z) appear. We summarize the contents of the seven sections which come after this introduction. In Section 2 we recall the main theorem from [12] which characterizes = −1/N under the assumption that N realizations of the functions N and N ∞ belongs to the class Nκ . The self-adjoint operator A and the self-adjoint relation are related via inﬁnite coupling. This notion from [20] in the models for N and N is explained after Theorem 2.1. In the sequel we make it a point to indicate this

Minimal Models for Nκ∞ -functions

101

connection between various versions of the two models. To do this, we also consider minimal models for certain one-dimensional perturbations A α of A = A 0 , where α is a real number. The key tool in the further analysis of the realizations in Section 2 is the theory of reproducing kernel Pontryagin spaces and in Section 3 we collect some theorems from this theory which will be used in the sequel. In particular, we recall the so-called canonical models. The irreducible representation (1.2) induces a canonical model for the generalized Nevanlinna matrix functions ⎛ ⎞ N0 (z) 0 0 0 ⎜ 0 q(z) 0 0 ⎟ p0 (z) c# (z)

⎜ ⎟ N (z) = ⎝ , , M (z) = 0 0 p0 (z) c# (z)⎠ c(z) 0 0 0 c(z) 0 where the real polynomials q and p0 are uniquely determined by the polynomial p in (1.2) via the equality p(z) = c# (z)q(z)c(z) + p0 (z) and the requirement that in which the deg p0 ≤ 2m − 1. In Section 4 we present models for N and N

) is the state space. See Theorem 4.1, where, as in reproducing kernel space L(N all our theorems (unless stated otherwise), the case n = deg q > 0 and m > 0 is considered. The resolvents of the corresponding self-adjoint operators/relation are given in Corollary 4.5. The cases where n = 0 or m = 0 are considered separately in Theorem 4.6 and Theorem 4.7; these cases are important in our examples. The

) admits the decomposition L(N

) = L(N0 ) ⊕ L(q) ⊕ L(M ), where the space L(N direct summands are the reproducing kernel spaces associated with the functions N0 , q and M . In Section 5 we study special bases for the last two summands and the associated Gram matrices (see Lemma 5.1). These bases allow us to identify

) with L = L(N0 ) ⊕ Cn ⊕ Cm ⊕ Cm . The corresponding matrix representations L(N of the models in Theorems 4.1, 4.6, and 4.7 are given in Theorems 6.1, 6.3, and 6.4 in Section 6. In that section we also determine formulas for the compressions of the resolvents of the self-adjoint operators/relation in the models and the compressions of the operators/relation themselves to the subspaces L(N0 ) and L(N0 ) ⊕ Cn ; see Theorem 6.5 and Theorem 6.6. By changing the bases slightly, the self-adjoint operator in the model for N can be given in a block operator matrix form, and this result is shown in Section 7. In Section 8, the last section of this paper, we give some examples and discuss an approximation problem associated with the Bessel diﬀerential expression. We thank the referees for their useful comments.

2. Characterization of the class Nκ∞ In [12] the following characterization of the class Nκ∞ was established. We recall that if A is a self-adjoint operator or a self-adjoint relation in some Pontryagin space P and w an element in P, then w is called cyclic for A if span {w, (A − z)−1 w | z ∈ ρ(A)} = P,

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or, equivalently, if for some (and then for every) z0 ∈ ρ(A), the function ϕ(z) = w + (z − z0 )(A − z)−1 w generates the space P, that is, span {ϕ(z) | z ∈ ρ(A)} = P. If A is an operator then w is cyclic for A if and only if span {(A − z)−1 w | z ∈ ρ(A)} = P. (z) = −N (z)−1 , the following four Theorem 2.1. For the functions N (z) and N assertions are equivalent. (i) N (z) has a representation : 8 9 N (z) = (A − z)−1 w, w P , z ∈ D(N ), (2.1) where A is a self-adjoint operator in a Pontryagin space P with negative index κ, w ∈ P is a cyclic element for A with the property w ∈ dom Am+n−1 \ dom Am+n for some integers m, n ∈ N0 , m+ n > 0, the subspace L = span {w, Aw, . . . , Am−1 w, Am w, . . . , Am+n−1 w} has index of non-positivity κ, and Aj w, Ak w = 0, Aj w, Ak w = 0,

0 ≤ j, k ≤ m + n − 1, j + k ≤ 2m + n − 2, 0 ≤ j, k ≤ m + n − 1, j + k = 2m + n − 1.

(ii) N (z) ∈ Nκ , z = ∞ is the only generalized zero of non-positive type of N (z), and N (z) has a representation N (z) = −

2m+2n−1 j=2m+n

sj−1 1 + 2m+2n−1 M (z) zj z

(2.2)

with m, n ∈ N0 , m+ n > 0, real numbers sj , j = 2m + n − 1, . . . , 2m+2n−2, s2m+n−1 = 0 if n > 0, and a function M (z) with the properties limy→∞ M (iy) = 0,

limy→∞ y 2 Re M (iy) = +∞.

(z) has a representation (iii) N − z)−1 )u, u , (z) = N (z0∗ ) + (z − z0∗ )(I + (z − z0 )(A N P P

), z ∈ D(N

), A is a self-adjoint relation in a Pontryagin space P with where z0 ∈ D(N = ∅, u ∈ P is a cyclic element for A, the root negative index κ, ρ(A) space L of A at z = ∞ is spanned by m + n vectors w1 , w2 , . . . , wm+n , at ∞, L has index of non-positivity κ and which form a Jordan chain of A span {w1 , w2 , . . . , wm } is its isotropic subspace. If m = 0 and P0 denotes the L, which is a uniformly positive subspace orthogonal projection onto H0 = P then P0 u ∈ / dom A. of P,

Minimal Models for Nκ∞ -functions

103

(z) ∈ N ∞ , the irreducible representation of N(z) (iv) N being κ (z) = c(z)# N0 (z)c(z) + p (z), N

c(z) = (z − z0 )m ,

where N0 ∈ N0 has the properties lim y −1 N0 (iy) = 0,

lim y Im N0 (iy) = ∞,

y→∞

y→∞

Re N0 (i) = 0,

), p (z) = # ak z k is a real polynomial of degree , and m ∈ N0 , z0 ∈ D(N k=0 we set n = max { − 2m, 0}. The Pontryagin spaces in (i) and (iii) can be chosen the same and then the element w1 in (iii) can be chosen to coincide with w in (i) and to satisfy w1 , u = 1; in this With A and w from (i) and the case wj = Aj−1 w, j = 1, . . . , m + n, and L = L. coeﬃcients sj , 2m+ n− 1 ≤ j ≤ 2m+ 2n− 2 in (2.2), sj = 0 if 0 ≤ j ≤ 2m+ n− 2, it holds sj = Ar w, As w if r + s = j, 0 ≤ r, s ≤ m + n − 1, and, if n > 0, then s2m+n−1 = 1/a2m+n = 1/a , where a is the leading coeﬃcient of the polynomial p in (iv). The relation between the negative index κ of the Pontryagin spaces in (i) and (iii) and the integers m, and n is given by ⎧ if n = 0, ⎨ m m + n+1 if n > 0, n odd , a < 0, κ= 2 ⎩ m + n2 otherwise . at ∞ in part (iii) is a regular Note that in case m = 0, the root space L of A subspace, whereas if m > 0 it is degenerate with an m-dimensional isotropic part. In the ﬁrst case ∞ is called a critical singular point and in the second case it is called a singular critical point of A. = A∞ , where A∞ is To the last part of the theorem can be added that A deﬁned through inﬁnite coupling of A and w. This means that it is obtained as the limit in the resolvent sense of the self-adjoint operator A α = A + α · , ww

(2.3)

by letting α → ∞: Since for α ∈ R \ {0}, (A α − z)−1 = (A − z)−1 −

· , ϕ(z ∗ )P ϕ(z), N (z) + 1/α

we have (A∞ − z)−1 = (A − z)−1 −

ϕ(z) = (A − z)−1 w,

(2.4)

· , ϕ(z ∗ )P ϕ(z). N (z)

(2.5)

N (z) , 1 + αN (z)

(2.6)

For later reference we note that (2.4) implies (A α − z)−1 w, w =

which for α = 0 is consistent with (2.1) and for α = ∞ with (A∞ − z)−1 w = 0, which follows from (2.5).

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Formula (2.2) is related to the moment problem for generalized Nevanlinna functions, the numbers sj being the moments; see, for example, [23] and [3]. The purpose of this paper is to provide some explicit minimal models for the operator = A∞ . To derive these models we use the theory of A = A 0 and the relation A reproducing kernels.

3. Reproducing kernel Pontryagin spaces and canonical models A by now well-known model for N ∈ Nκn×n is described in the following theorem. Here the state space is the reproducing kernel space L(N ) associated with the kernel KN (ζ, z). Recall that the elements of this space are n-vector functions deﬁned and holomorphic on D(N ), that the functions KN ( · , z)c, where z runs through D(N ) and c runs through Cn , are dense in L(N ), and that the kernel has the reproducing property: f, KN ( · , z)cL(N ) = c∗ f (z),

f ∈ L(N ), c ∈ Cn .

Whenever deﬁned we denote by Rz the diﬀerence-quotient operator and, for later use, by Eζ the operator of evaluation at the point ζ, that is, Rz f (ζ) =

f (ζ) − f (z) , ζ −z

Eζ f = f (ζ),

(3.1)

where f is a vector function. Theorem 3.1. Let N ∈ Nκn×n be given. Then: & ' (i) A = {f, g} ∈ L(N )2 | ∃ c ∈ Cn : g(ζ) − ζf (ζ) ≡ c is a self-adjoint relation in L(N ) with ρ(A) = ∅, and N (ζ) − N (z) Γz c (ζ) = KN (ζ, z ∗ )c = c, c ∈ Cn , ζ −z is a corresponding Γ-ﬁeld. The pair (A, Γz ) is a minimal realization of N . (ii) The resolvent of A is the diﬀerence-quotient operator in L(N ): (A − z)−1 = Rz ,

z ∈ ρ(A). ' (iii) S = {f, g} ∈ L(N ) | g(ζ) − ζf (ζ) ≡ 0 is a symmetric operator in the space L(N ) with equal defect indices n − d, where d = dim ker Γz . Moreover, σp (S) = ∅ and the adjoint of S is given by & ' S ∗ = span {Γz h, zΓz h} | h ∈ Cn , z ∈ D(N ) & ' = {f, g} ∈ L(N )2 | ∃ c, d ∈ Cn : g(ζ) − ζf (ζ) ≡ c − N (ζ)d . &

2

For the proof of this theorem and remarks concerning its origin we refer to [10, Theorem 2.1]. The minimal realization of N described here is called the canonical realization of N and the triplet (A, Γz , S) is called the canonical model. For these canonical models only, to denote the dependence on N we often write AN , ΓN z , SN etc. instead of A, Γz , S, etc.

Minimal Models for Nκ∞ -functions

105

By N we denote the operator of multiplication by the function N : (Nf )(ζ) = N (ζ)f (ζ).

(3.2)

Theorem 3.2. Let N ∈ Nκn×n , assume that N (z) is invertible for some point ∈ N n×n and the following statements = −N −1 . Then N z ∈ D(N ), and set N κ hold. ) to L(N ) is unitary; its inverse is the operator of (i) N as a mapping from L(N multiplication by N −1 . (ii) We have ( ) % )2 % ∃ d ∈ Cn : g(ζ) − ζf (ζ) ≡ N (ζ)d = N−1 AN N = {f, g} ∈ L(N A N

) = ∅. and hence ρ(A N (iii) For 0 = c ∈ Cn and j = 0, 1, . . ., we have ζ j N (ζ)c ∈ L(N ) if and only if ). ζ j c ∈ L(N The theorem coincides in part with [10, Corollary 2.3]. Part (i) follows from the kernel identity N (ζ)KN (ζ, z)N (z)∗ = KN (ζ, z), part (ii) from (i) and Theorem 3.1 (i), and part (iii) follows from (i). The inclusions in (iii) hold for j = 0 if and only if z = ∞ is a generalized zero of N or, equiv . That we use the notation A for the alently, z = ∞ is a generalized pole of N N operator/relation N−1 AN N in part (ii) of Theorem 3.2 comes from applying our convention that if A is the self-adjoint operator/relation in a model for N then we for the corresponding operator/relation associated with N : If N ∈ Nκn×n write A is invertible at some point in D(N ), then the triplet (AN , KN ( · , z ∗ ), SN ) is the and the triplet canonical model for N , K ( · , z ∗ )N (z), S ) (A N N N

(3.3)

), because it is isomorphic is a minimal model for the function N = N in L(N under N with the canonical model (AN , ΓN z , SN ) for N in L(N ). For use in the next section we recall the following theorem (see [10, Theorem 2.4]). A function N ∈ Nκn×n is called strict if for some non-real point z0 ∈ D(N ) it holds : ker KN (ζ, z0 ) = {0}. ζ∈D(N )

Theorem 3.3. Suppose that N ∈ Nκn×n is strict and let (A, Γz , S) be the canonical model for N . Then: (i) A relation is a canonical self-adjoint extension of S if and only if it is of the form % & ' AA,B = {f, g} ∈ L(N )2 % ∃ h ∈ Cn : g(ζ) − ζf (ζ) ≡ (A + N (ζ)B)h

106

A. Dijksma, A. Luger and Y. Shondin with n × n matrices A and B satisfying the relations A = n, A∗ B − B ∗ A = 0. rank B

If AA,B and AA ,B are two such canonical self-adjoint extensions of S then AA ,B = AA,B if and only if A = A C and B = B C for some invertible n × n matrix C. (ii) ρ (AA,B ) = ∅ if and only if for some non-real point z0 ∈ D(N ) the matrices A + N (z0 )B and A + N (z0 )∗ B are invertible. In this case for z ∈ ρ (AA,B ) ∩ ρ(A): −1 −1 (AA,B − z) = (A−z)−1−Γz B (A + N (z)B) Γ∗z∗ . (3.4) We specialize to case n = 1 and assume 0 = N ∈ L(N ). Then on account of Theorem 3.1 (i) and (ii), w = N is a cyclic element of L(N ) for AN and N (z) = (AN − z)−1 w, wL(N ) . The operator

α

AN = AN + α · , wL(N ) w,

α ∈ R,

can also be written as

α

AN = {{f, g} ∈ L(N )2 |∃c ∈ C : g(ζ) − ζf (ζ) = (1 + αN (ζ))c}.

(3.5)

This can be seen by comparing the resolvents (2.4) and (3.4) applied to this situ α ation. If we let α → ∞, then in the resolvent sense AN converges to A∞ N

=

{{f, g} ∈ L(N )2 |∃c ∈ C : g(ζ) − ζf (ζ) = N (ζ)c}

=

SN + {{0, cw} | c ∈ C} = SN + {0} × (dom SN )⊥ .

The set on the right-hand side after the ﬁrst equality can at least formally be obtained from the set on the right-hand side of (3.5) by replacing {f, g} by {αf, αg} and letting α → ∞. Now we apply the unitary map N and ﬁnd that (the constant ), 1 is a cyclic element for A , and function) 1 ∈ L(N N + α · , 1 ) α = N−1 A N = A (A N N N L(N)

α

(3.6)

)2 |∃c ∈ C : g(ζ) − ζf (ζ) = (α − N (ζ))c}. = {{f, g} ∈ L(N and 1 we have Hence in the inﬁnite coupling of A N )∞ = A = N−1 A∞ (A N N. N N

4. Minimal models in the space L(N) From now on we assume that (1) N is a non-zero scalar generalized Nevanlinna function in Nκ , = −1/N ∈ Nκ∞ , (2) N

(3.7)

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107

has representation (1.2), and (3) N ) belongs to the possibly smaller set D(N0 ). (4) z0 ∈ D(N in the form We rewrite the irreducible representation (1.2) of N (z) = c# (z)(N0 (z) + q(z))c(z) + p0 (z), N

c(z) = (z − z0 )m ,

(4.1)

where q and p0 are real polynomials such that c# (z)q(z)c(z) + p0 (z) = p(z), 0 = deg p0 < 2m, and n = deg q is the number appearing in Theorem 2.1. With the decomposition (4.1) we associate the generalized Nevanlinna matrix functions ⎛ ⎞ N0 (z) 0 0 0 ⎜ q(z) 0 0 ⎟ p0 (z) c# (z)

(z) = ⎜ 0 ⎟ . N , M (z) = ⎝ 0 0 p0 (z) c# (z)⎠ c(z) 0 0 0 c(z) 0

) with kernel K ( · , · ) can be It follows that the reproducing kernel space L(N N

) = L(N0 ) ⊕ L(q) ⊕ L(M ). If n > 0 and decomposed as the orthogonal sum L(N m > 0, then the elements of L(q) are the polynomials of degree < n and the elements of L(M ) are 2-vector functions with polynomial entries. Unless stated otherwise we assume n > 0 and m > 0. If n = 0 or m = 0, then L(q) = {0} or L(M ) = {0} and the formulas simplify; we consider these cases separately. in the space L(N

). For In this section we give minimal models for N and N this we introduce the vector function v(z) = c(z) c(z) 1 c(z)(N0 (z) + q(z)) and the following ⎛ 1 0 ⎜0 1 Aα = ⎜ ⎝0 0 0 0

4 × 4 matrices ⎞ ⎛ 0 0 0 ⎜0 0 0⎟ ⎟, B = −⎜ ⎝0 α 0⎠ 1 0 1

0 0 0 1

0 0 1 0

⎞ 1 1⎟ ⎟, 0⎠ 0

⎛ 0 ⎜0 B = − ⎜ ⎝0 1

0 0 0 1

0 0 0 0

⎞ 1 1⎟ ⎟. 0⎠ 0

Theorem 4.1. Assume the conditions (1)–(4) hold. in L(N

) are given by the triplets (i) The minimal models of N and N

(B, KN ( · , z ∗ )v(z)N (z), S)

and

K ( · , z ∗ )v(z), S),

(B, N

where & '

) | ∃h ∈ C4 : g (ζ) − ζ f (ζ) = (A0 + N

(ζ)B)h , B = {f , g } ∈ L(N

(4.2)

' &

(ζ)B)h = {f , g } ∈ L(N

) | ∃h ∈ C4 : g(ζ) − ζ f (ζ) = (IC4 + N , B

(4.3)

and & '

) | ∃h ∈ C4 with h3 = 0 :

(ζ)B)h S = {f , g } ∈ L(N g(ζ) − ζ f (ζ) = (A0 + N .

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A. Dijksma, A. Luger and Y. Shondin

) is (ii) S has defect (1, 1) and the family of all its self-adjoint extensions in L(N

α given by B and B , α ∈ R, where B α

= =

with

B + α · , w

L(N ) w

& '

) | ∃h ∈ C4 :

(ζ)B)h (4.4) {f , g} ∈ L(N g(ζ) − ζ f (ζ) = (Aα + N w

= 0 0

1

). 0 ∈ L(N

= B ∞ , the limit of B α in the resolvent sense Moreover, B = B 0 and B by letting α → ∞. Note that S has defect (1, 1) shows that it does not coincide with SN in the

in L(N

), which has defect (4, 4) if n = 0 and (3, 3) canonical representation of N if n = 0. On account of (1)–(4) the four equivalent statements in Theorem 2.1 hold For the Pontryagin space P we take the reproducing kernel Pontryafor N and N. ). For P we take L(N ) but we identify it with L(N ) via the unitary gin space L(N map N deﬁned by (3.2). Since ∞ is a generalized zero of N , we have N ∈ L(N ) (see [10, Corollary 2.3(iii)]). It follows that (2.1) holds with w = N and A = AN ) we have w = w1 = 1 and A = A in P and in the identiﬁcation of P with L(N N and so N (z) = (AN − z)−1 1, 1L(N) . The expansion (2.2) for N implies that the functions w1 , w2 , . . . , wm+n (or, what amounts to the same, w, Aw, . . . , Am+n−1 w) are given by 1, ζ, . . . , ζ m+n−1 (see [3, Lemma 5.2]). Here we use that the moments sj are zero for 0 ≤ j ≤ 2m + n − 2. in statement (iii) of Theorem 2.1 holds with A = A The representation for N N and u = KN ( · , z0∗ ). (4.5) Notice that u, w1 L(N ) = 1 by the reproducing property of the kernel. Since (AN − z)−1 1 = Rz 1 = 0, we see directly that AN is a relation with a nontrivial multi-valued part: 1 ∈ AN (0). In Section 3 we showed that the triplets , K ( · , z ∗ )N (z), S ) (A N N N

and (AN , KN ( · , z ∗ ), SN )

in L(N ) and that A is the limit in the resolvent are minimal models of N and N N

α , and (3.6) and (3.7). The main ; see (3.3), Theorem 3.1 applied to N sense of A N are isomorphic copies of idea of the proof of the theorem is that B α and B

α A and A . The isomorphism is given in Lemma 4.4 below. We begin with two N

N

technical lemmas. h1 ∈ L(M ) and h2 = 0, then deg h1 < m. Lemma 4.2. If h2

Minimal Models for Nκ∞ -functions

109

Proof. Since deg p0 < 2m, the space L(M ) is spanned by the 2m linearly independent 2-vector functions # j p0 j c R0 , R0 , j = 1, 2, . . . , m, c 0 where R0 is the diﬀerence-quotient operator given by (3.1) with z = 0. If h2 = 0, then # h1 j c | j = 1, 2, . . . , m}, ∈ span {R0 h2 0 which implies deg h1 < m. Lemma 4.3. Assume N0 ∈ N0 satisﬁes the conditions in (1.3). If h1 and h2 are polynomials and N0 h1 + h2 ∈ L(N0 ), then h1 = h2 = 0. Proof. Whenever deﬁned we have Rw0 (f g)(ζ) = Rw0 (f )(ζ)g(w0 ) + f (ζ)Rw0 (g)(ζ). Let h1 and h2 be polynomials such that N0 h1 + h2 ∈ L(N0 ). Assume that h1 and h2 are not both identically equal to 0. If we apply Rw0 with w0 ∈ D(N0 ) a number of times to the function N0 h1 +h2 ∈ L(N0 ) and use the above formula and that Rw0 (N0 )(ζ)c ∈ L(N0 ), c ∈ C, we ﬁnd that there is pair of complex numbers (c1 , c2 ) = (0, 0) such that N0 c1 + c2 ∈ L(N0 ). If N0 c1 + c2 = 0, then N0 is a real constant and therefore, on account of the last equality in (1.3) equal to zero. But this is in contradiction with the ﬁrst equality in (1.3). If N0 c1 + c2 = 0, then, by Theorem 3.1(iii) applied to N0 , ∗ {0, 0} = {0, N0 c1 + c2 } ∈ SN , 0

which implies that the minimal operator SN0 in L(N0 ) is not densely deﬁned. This is in contradiction with the ﬁrst two equalities in (1.3). These contradictions imply that h1 and h2 are identically equal to 0.

) → L(N ) deﬁned by (Vf )(ζ) = v # (ζ) f (ζ) is Lemma 4.4. The mapping V : L(N unitary. The corresponding mapping in [10, Lemma 3.1], where realizations of N related to its basic factorization (1.1) are considered, is a partial isometry but not necessarily injective. Proof of Lemma 4.4. A straightforward calculation shows v # (ζ)KN (ζ, z)v(z ∗ ) = KN (ζ, z).

(4.6)

We claim that the number of negative squares of the kernels KN and KN coincide. Indeed, the ﬁrst number is the sum of the number of negative squares of the scalar function q and the matrix function M . The ﬁrst of which is equal to (n + 1)/2 if n is odd and the leading coeﬃcient of the polynomial q is negative, and otherwise it is n/2 . The kernel of M has m negative squares since z0 is the only zero of M in the closed upper half-plane and its multiplicity is m. By Theorem 2.1, the sum of these numbers equals κ, which is the number of negative squares of KN . Hence

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[4, Theorem 1.5.7.] implies that V is a surjective partial isometry. We show that it is in fact a unitary mapping, that is, ker V is trivial. Assume there is an element f (ζ) = f (ζ) a(ζ) b(ζ) d(ζ) ∈ ker V. Then c# (ζ)f (ζ) + c# (ζ)a(ζ) + b(ζ) + c# (ζ)(N0 (ζ) + q(ζ))d(ζ) ≡ 0.

(4.7)

Since N0 is holomorphic at the point z0∗ , we have that f ∈ L(N0 ) is holomorphic at z0∗ also and the equality (4.7) implies that the polynomial b has a zero of order at least m at z0∗ . So we have b(ζ) = (ζ − z0∗ )m b1 (ζ) for some polynomial b1 . Equality (4.7) implies N0 h1 + h2 = −f ∈ L(N0 ) with polynomials h1 = d and h2 = a + b1 + qd. By Lemma 4.3, h1 = h2 = 0, that is, d = 0 and a + b1 = 0. Now we use Lemma 4.2: Since b d ∈ L(M ) and d = 0, the lemma yields that deg b < m, which implies b1 = 0 and hence b = 0 and a = 0. Finally, on account of (4.7), we have f = 0. We conclude that f = 0, that is, ker V = {0}. and S by the formulas Proof of Theorem 4.1. We ﬁrst deﬁne B, B, V, B = V−1 A N

= V−1 A V, B N

= V−1 S V S = B ∩ B N

and claim that they coincide with the relations in part (i) of the theorem. Assuming the claim is true, we have, according to (4.6), that under the unitary mapping V the

) is the isomorphic copy of the element K ( · , z ∗ ) in element KN ( · , z ∗ )v(z) in L(N N ). Hence the triplets in (i) are isomorphic copies of the minimal models (3.3) L(N in L(N

). It remains to prove the claim. It and (AN , KN ( · , z ∗ ), SN ) for N and N is easy to see that & '

)2 | ∃d ∈ C : v # (ζ)( (ζ)d , B = {f , g} ∈ L(N g (ζ) − ζ f (ζ)) = N & ' =

)2 | ∃c ∈ C : v # (ζ)( B {f , g} ∈ L(N g (ζ) − ζ f (ζ)) = c . First we prove formula (4.2). Denote by BA0 ,B the relation deﬁned by the right hand side of (4.2). If {f , g} ∈ BA0 ,B and h = h1 h2 h3 h4 then

(ζ)B)h = −N (ζ)h3 , g(ζ) − ζ f (ζ) = v # (ζ)(A0 + N (4.8) v # (ζ) hence BA0 ,B ⊂ B. Since BA0 ,B and B are self-adjoint operators, see Theorem 3.3, equality holds. In the same way, if {f , g} ∈ BI 4 ,B , the operator deﬁned by the C right-hand side of (4.3), then

(ζ)B)h = h3 , v # (ζ)( g (ζ) − ζ f (ζ)) = v # (ζ)(IC4 + N

(4.9)

and equality prevails because both self-adjoint relations have ⊂ B nonempty resolvent sets. The formula for S follows from (4.8) and (4.9). This completes the proof of part (i). As to (ii) we ﬁrst deﬁne the operators B α by hence BI

C4 , B

) α V, B α = V−1 (A N

α ∈ R.

Minimal Models for Nκ∞ -functions

111

On account of (3.6), we have

L(N ) w,

B α = B + α · , w where

w

= V−1 1 = 0 0

0 .

1

(4.10)

We now show (4.4). From (2.4), applied to this situation, we have (B α − z)−1 = (B − z)−1 −

L(N ) · , (B − z ∗ )−1 w N (z) + 1/α

(B − z)−1 w

and, by Theorem 3.3, −1

(BAα ,B − z)−1 = (AN − z)−1 − ΓNz Ez ,

B(Aα + N (ζ)B)

where Ez = Γ∗N z∗ is the operator of evaluation at the point z. For α = 0, the last equality yields

(ζ)B)−1 Ez (B − z)−1 = (AN − z)−1 − ΓN z B(A0 + N and, on account of (4.10), (B − z)−1 w

= N (z)ΓNz

v(z). Combining these relations we ﬁnd (B α − z)−1 − (BAα ,B − z)−1 6 −1

= ΓN z −B(A0 + N(ζ)B) −

7 N (z)2 # −1

Ez . v(z)v (z) + B(Aα + N (ζ)B) N (z) + α1

A straightforward calculation shows that the expression in square brackets vanishes, which implies (4.4) for all α ∈ R. = B ∞ , where the relation on the Clearly, B = B 0 and, because of (3.7), B right-hand side is obtained via inﬁnite coupling of B and w,

that is, by taking the limit of B α in the resolvent sense by letting α → ∞. Finally, the statement concerning S and its extensions follow from the corresponding results for SN and the unitarity of V. The following corollary is an immediate consequence of Theorem 3.3. We set ⎛ ⎞ c(z)c# (z) c(z)c# (z) c(z) α − p0 (z) ⎜c(z)c# (z) c(z)c# (z) ⎟ c(z) α − p0 (z) ⎟ Kα (z) = ⎜ # # ⎝ c# (z) c (z) 1 (N0 (z) + q(z))c (z) ⎠ α − p0 (z) α − p0 (z) (N0 (z) + q(z))c(z) (N0 (z) + q(z))(α − p0 (z)) and

⎛ 0 ⎜0 1 K(z) = limα→∞ Kα (z) = ⎜ ⎝0 α 1

0 0 0 1

0 0 0 0

⎞ 1 ⎟ 1 ⎟. ⎠ 0 N0 (z) + q(z)

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A. Dijksma, A. Luger and Y. Shondin

are given by Corollary 4.5. The resolvents of B α and B N (z) Γ Kα (z)Ez , (B α − z)−1 = (AN − z)−1 + 1 + αN (z) N z − z)−1 = (A − z)−1 + Γ K(z)E (B z, N

Nz

(4.11) (4.12)

), where (AN − z)−1 is the diﬀerence-quotient operator in the space L(N ∗ ∗ ∗ ΓNz

= diag {KN0 ( · , z ), Kq ( · , z ), KM ( · , z )},

and Ez = (ΓN z∗ )∗ is the evaluation operator at the point z. From (4.11) it readily follows that

w

= (B α − z)−1 w,

N (z) , 1 + αN (z)

α ∈ R ∪ {∞},

which is consistent with (2.6). It remains to discuss the simpliﬁcations if n = 0 or m = 0. The case n = 0 and m > 0: Then q(z) = q0 with q0 ∈ R, hence Kq (ζ, z) = 0 and L(q) = {0}; as to the 2 × 2 matrix function M (z): deg p0 (z) < 2m and the space

(z) becomes the 3 × 3 matrix function L(M ) is nontrivial. Now N

) = L(N0 ) ⊕ L(M ).

(z) = diag {N0 (z) + q0 , M (z)} and L(N N

(z) + q0 ) the operator V With v(z) = c(z) 1 c(z)(N 0 : L(N ) → L(N ) of

in multiplication by v # (z) = c# (z) 1 c# (z)(N0 (z) + q0 ) is unitary and w −1 (4.10) becomes w

:= V 1 = 0 1 0 . Theorem 4.6. Assume n = 0 and m > 0. Then Theorem 4.1 and Corollary 4.5 remain true provided in all 4 × 4 matrices the 2-nd row and the 2-nd column are S,

and B α the space C4 and the entry h3 deleted and in the formulas for B, B, 3 are replaced by C and h2 . The case n > 0 and m = 0: Now c(z) = 1, q(z) is a nonconstant real (z) = N0 (z)+q(z). polynomial, and the irreducible representation (1.2) becomes N

From N (z) = diag {N0 (z), q(z)} it follows that L(N ) = L(N0 ) ⊕ L(q). With the

) → L(N ) of multiplication 1 1 the mapping V : L(N vector function v(z) = # by v (z) = 1 1 is unitary. Theorem 4.7. Assume n > 0 and m = 0 and for α ∈ R deﬁne the operators & '

) | ∃h ∈ C2 : g (ζ) − ζ f (ζ) = (Aα + N

(ζ)B)h , g } ∈ L(N B α = {f , 1 0 , −1 α

where Aα =

B=−

0 0

1 . 1

= A = B ∞ , and S = B ∩ B, (i) Theorem 4.1 holds provided B = B 0 , B N which takes the form & S = {f , g } ∈ AN | ∃h ∈ C : ( g (ζ) − ζ f (ζ)) = h 1 −1 }.

Minimal Models for Nκ∞ -functions

113

(ii) Corollary 4.5 becomes: For α ∈ R, (B

α

−1

− z)

1 1 1 Ez , Γ − N0 (z) + q(z) − α N z 1 1

−1

= (AN − z)

), where (AN − z)−1 is the diﬀerence-quotient operator in the space L(N ΓN z = diag {KN0 ( · , z ∗ ), Kq ( · , z ∗ )}, and Ez = (ΓN z∗ )∗ is point evaluation at z.

5. A decomposition of L(N) In this section we choose a basis in L(q) ⊕ L(M ) and determine the associated

see Lemma 5.1 below. In the next section we identify L(q)⊕L(M ) Gram matrix G; n m

and exhibit with C ⊕ C ⊕ Cm equipped with an inner product determined by G

α and the relation B. As in [12] we the matrix representations of the operators B

) the linearly independent elements deﬁne in L(N

vj = (B − z0∗ )j−1 w,

j = 1, . . . , m, m + 1, . . . , m + n, ∗

and with ϕ(z) = ϕ( · , z) = KN ( · , z )v(z) 1 d j−1 − z0 )−j+1 ϕ(z0 ) = ϕ(z) |z=z0 , uj = (B (j − 1)! d z

j = 1, . . . , m.

Here, on account of (4.6), u1 = V−1 u, where u is given by (4.5). Moreover, we introduce the three subspaces L0 = span {v1 , . . . , vm }, L = span {vm+1 , . . . , vm+n }, M = span {u1 , . . . , um }. at ∞ is the direct sum L +L ˙ 0 and According to Theorem 2.1, the root space of B 0 L is its isotropic part. ˙ Lemma 5.1. We have L(q) = L and L(M ) = L0 +M. The basis elements for L(q) and L(M ) can be written more explicitly as vm+j (ζ) = 0 (ζ − z0∗ )j−1 0 0 , j = 1, . . . , n, and vj (ζ)

=

uj (ζ)

=

0 0 0 0

(ζ − z0∗ )j−1 Rzj 0 p0 (ζ)

0 , (ζ − z0 )m−j ,

j = 1, . . . , m, j = 1, . . . , m,

where Rz stands for the diﬀerence-quotient operator at z. The Gram matrix asso˙ ciated with this basis for the space L(q) ⊕ (L0 +M) is given by ⎛ ⎞ Gq 0 0

=⎝0 0 ICm ⎠ , G 0 ICm Gp0

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A. Dijksma, A. Luger and Y. Shondin

in which Gq = (qi,j )ni,j=1 has entries qi,j = vm+j , vm+i L(N ) =

m+j−1 m+i−1 k=0

l=0

m+i−1 m+j −1 (−z0∗ )m+j−1−k (−z0 )m+i−1−l sk+l , l k

where sk+l are the moments of N in (2.2), and Gp0 = (pi,j )m i,j=1 has entries pi,j = uj , ui L(N ) =

(5.1)

1 d j−1 d i−1 p0 (z) − p0 (w∗ ) %% 1 . % (j − 1)! (i − 1)! dz dw∗ z − w∗ z=w=z0

The formulas for the basis element uj (ζ) and the entry pi,j of the Gram matrix Gp0 in this lemma are independent of the way the polynomial p0 (z) is # 0 k written. If we write p0 (ζ) = k=0 pk (ζ − z0 ) and set pk = 0 if k > 0 , then 0 # Rzj 0 p0 (ζ) = pk (ζ − z0 )k−j and straightforward calculations yield k=j

pi,j =

2m−j k=i

k−1 pk+j−1 (z0∗ − z0 )k−i ; i−1

so, in particular, if z0 ∈ R then pi,j = pi+j−1 . After the proof of the lemma we give some other formulas for the Gram matrix Gq as well. Proof of Lemma 5.1. Since v1 = w,

the element v1 is of the given form. We calculate vj = (B − z0∗ )vj−1 for j = 2, . . . , m + n. Write Bvj−1 as Bvj−1 (ζ) = f (ζ) a(ζ) b(ζ) d(ζ) ∈ L(N0 ) ⊕ L(q) ⊕ L(M ). Then, by (4.4), there exists a vector h = h1 h2 h3 h4 ∈ C4 such that ⎛ ⎞ ⎞ ⎛ h1 − N0 (ζ)h4 f (ζ) ⎜ ⎟ ⎟ ⎜ a(ζ) h2 − q(ζ)h4 ⎜ ⎟ ⎟ ⎜ (5.2) = ⎜ ⎜ ⎟. ⎟ ⎝b(ζ) − ζ(ζ − z0∗ )j−2 ⎠ ⎝−c# (ζ)(h1 + h2 ) − p0 (ζ)h3 ⎠ d(ζ)

−c(ζ)h3 + h4

By Lemma 4.3, h1 = h4 = 0. Since d is a polynomial of degree less than m also h3 = 0. Thus b is the ﬁrst component of an element in L(M ) whose second component d = 0, therefore, see Lemma 4.2, the degree of b is less than m. The equality between the third components of the vectors in (5.2) now reads as b(ζ) − ζ(ζ − z0∗ )j−2 = −c# (ζ)h2 .

(5.3)

For 2 ≤ j ≤ m, a comparison of the degrees of the polynomials on both sides, yields h2 = 0. Thus b(ζ) = (ζ − z0∗ )j−1 + z0∗ (ζ − z0∗ )j−2 and hence we have vj (ζ) = 0 0 (ζ − z0∗ )j−1 0 , j = 1, . . . , m.

Minimal Models for Nκ∞ -functions

115

If j = m + 1 then (5.3) implies h2 = 1 and hence we ﬁnd vm+1 (ζ) = 0 1 0 0 . Now the formula for vj , j = m+ 2, . . . , n, can be checked in a similar way as above. It is easy to see that the element u1 = KN ( · , z0∗ )v(z0 ) has the stated form. By (4.12), we have for 2 ≤ j ≤ m, ⎛ ⎞ ⎛ ⎞ 0 0 ⎜ ⎟ ⎜ ⎟ 0 0 − z0 )−1 uj−1 (ζ) = (A − z0 )−1 ⎜ ⎟=⎜ j ⎟. uj (ζ) = (B N ⎝ Rzj−1 ⎠ ⎝ ⎠ R p (ζ) p (ζ) 0 z0 0 0 (ζ − z0 )m−j+1

(ζ − z0 )m−j

the zeros come from the facts that L(q) ⊥ L(M ) As to the Gram matrix G, 0 and L is neutral. The formula for Gq follows from expanding

(B − z0∗ )m+i−1 w

L(N ) qi,j = (B − z0∗ )m+j−1 w, in terms of B l w,

B k w

L(N ) N )k 1 = AlN N, AkN N L(N ) = Al w, Ak wP = sk+l . = (AN )l 1 , (A L(N)

are obtained from the reproducing kernel property of The entries ICm in G KN (ζ, z): % 1 d j−1 % KN ( · , z ∗ )v(z), vi L(N ) % uj , vi L(N ) = (j − 1)! dz z=z0 % j−1 d 1 % = (z − z0 )i−1 % = δij , 1 ≤ i, j ≤ m. (j − 1)! dz z=z0 Finally, the formula for pi,j in the lemma readily follows from 1 1 (j − 1)! (i − 1)! % d j−1 d i−1 % ∗ ∗ K ( · , z )v(z), K ( · , w )v(w) . × %

N N L( N ) dz d w∗ z=w=z0

uj , ui L(N ) =

We claim that the Gram matrix Gq = (vm+j , vm+i L(N ) )ni,j=1 in Lemma 5.1 is lower diagonal with respect to the second diagonal. To see this we use the equality vm+j , vm+i L(N ) = vm+j+1 , vm+i−1 L(N ) + (z0∗ − z0 )vm+j , vm+i−1 L(N ) , (5.4) which readily follows from the relation vm+i = (B − z0∗ )vm+i−1 . Since L(q) is orthogonal to L0 the recurrence relation (5.4) implies vm+j , vm+1 L(N ) = 0,

j = 1, . . . , n − 1,

and hence, again with (5.4), also the lower triangular form of Gq . Furthermore, (5.4) also implies that the entries on the second diagonal: vm+d , vm+n+1−d L(N )

116

A. Dijksma, A. Luger and Y. Shondin

are independent of d = 1, . . . , n. Note that Gq is not a Hankel matrix in general, however, it is if z0 = z0∗ . In the following two propositions we present two other formulas for Gq . The ﬁrst one is in terms of the real coeﬃcients τj of q: q(z) = τn z n + τn−1 z n−1 + · · · + τ1 z + τ0 ,

τn = 0.

The Gram matrix Sq associated with the standard basis {1, ζ, . . . , ζ n−1 } in L(q) is equal to ⎞−1 ⎛ ⎛ ⎞ τ1 τ2 · · · τn−1 τn 0 0 ··· 0 σn−1 ⎜ τ2 τ3 · · · ⎜ 0 τn 0⎟ 0 · · · σn−1 σn ⎟ ⎟ ⎜ ⎜ ⎟ Sq = ⎜ . =⎜ . ⎟ . . . . .. ⎟ . .. .. .. .. .. ⎠ ⎝ .. ⎝ .. . ⎠ 0 0 σn−1 σn · · · σ2n−3 σ2n−2 τn 0 · · · where the real numbers σj are deﬁned by the expansion 1 σn−1 σ2n−2 1 = n + · · · + 2n−1 + O( 2n ), q(z) z z z

z = iy, y ↑ ∞

(5.5)

(see, for example, [10, Theorem 3.4]). Notice that τ0 does not play a role. Since Gq is the Gram matrix of the basis vm+1 , vm+2 , . . . , vm+n , we express this basis in terms of the standard basis via the n × n matrix H: 1 ζ − z0∗ (ζ − z0∗ )2 . . . (ζ − z0∗ )n−1 = 1 ζ ζ 2 . . . ζ n−1 H. and then we have Gq = H∗ Sq H. If H = (hi,j )ni,j=1 then the entries in the jth column are given by ⎧ ⎨ j−1 (−z0∗ )j−i , i = 1, . . . j, i−1 hi,j = ⎩ 0, i = j + 1, . . . n. The connection with the moments sj for N given by (2.2) can be obtained from the fact that the asymptotics of N (z) = −1/N(z) in Theorem 2.1(ii) is the same as the asymptotics of the function −1/(c# (z)q(z)c(z)): Write for |z| > |z0 |, ∞ 1 tm+k−1 k m+k−1 (5.6) = , t = z m+k−1 0 m−1 c(z) z m+k k=0

(note that tm−1 = 1) and set ⎛ tm−1 tm ⎜ 0 t m−1 ⎜ T=⎜ . .. ⎝ .. . 0 0

··· ···

tm+n−3 tm+n−4 .. .

···

0

⎞ tm+n−2 tm+n−3 ⎟ ⎟ .. ⎟ . . ⎠ tm−1

Minimal Models for Nκ∞ -functions

117

Using (5.5) and (5.6) to calculate the asymptotics of −1/(c#(z)q(z)c(z)) and comparing it with (2.2) we ﬁnd that ⎛ ⎞ 0 0 ··· 0 s2m+n−1 ⎜ 0 0 · · · s2m+n−1 s2m+n ⎟ ⎜ ⎟ MN = ⎜ ⎟ = T∗ Sq T. .. .. .. .. ⎝ ⎠ . . . . s2m+n−1

s2m+n

···

s2m+2n−3

s2m+2n−2

Hence we have proved the following proposition. Proposition 5.2. With H, Sq , T and MN as deﬁned above we have Gq = H∗ T−∗ MN T−1 H = H∗ Sq H. The ﬁrst equality follows from the formula for Gq given in Lemma 5.1. The triangular forms of the matrices H with 1 on the diagonal and Sq with σn−1 = 1/τn on the second diagonal yield the triangular form of Gq with 1/τn on the second diagonal.

) To derive yet another formula for Gq , we identify the elements vm+j ∈ L(N ∗ j−1 with the functions (ζ − z0 ) ∈ L(q), j = 1, . . . , n. Also for later use, we introduce the vector polynomial sq (z) = (si (z))ni=1 ,

si (z) = Rzi 0∗ q(z),

(5.7)

where Rz is the diﬀerence-quotient operator at z. The kernel Kq ( · , z) can be expressed in the basis {vm+1 , . . . , vm+n } as n Kq ( · , z) = vm+i si (z ∗ ). (5.8) i=1

For this, write q as q(z) =

n #

qk (z−z0∗ )k , then si (z) =

#n

k=0

Kq (ζ, z) = =

n k=1 n

and hence

n k (ζ − z0∗ )k − (z ∗ − z0∗ )k qk = qk (ζ − z0∗ )i−1 (z ∗ − z0∗ )k−i ζ − z∗ i=1

(ζ − z0∗ )i−1

i=1

∗ k−i k=i qk (z−z0 )

k=1

n k=i

qk (z ∗ − z0∗ )k−i =

n

vm+i (ζ) si (z ∗ ).

i=1

For the next proposition, we choose n distinct points z1 , . . . , zn ∈ C, denote by V the n × n Vandermonde matrix ⎛ ⎞ 1 z1 − z0∗ . . . (z1 − z0∗ )n−1 ⎜1 z2 − z0∗ . . . (z2 − z0∗ )n−1 ⎟ ⎜ ⎟ V = ⎜. ⎟, .. .. ⎝ .. ⎠ . . ∗ ∗ n−1 1 zn − z0 . . . (zn − z0 ) and deﬁne the n × n matrix S = (si,j )ni,j=1 by si,j = si (zj∗ ). Proposition 5.3. With the above notation Gq = S−∗ V.

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# Proof. If S−1 = (ti,j )ni,j=1 , then on account of (5.8), vm+i = nk=1 Kq ( · , zk )tk,i , i, j = 1, . . . , n, and by the reproducing kernel property, n n vm+j , vm+i L(q) = t∗k,i vm+j , Kq ( · , zk )L(q) = t∗k,i vm+j (zk ) =

k=1 n

k=1

(S−∗ )i,k (zk − z0∗ )j−1 = (S−∗ V)i,j .

k=1

6. Minimal models in the space (K; G) in the In this section we construct minimal models for the functions N and N n m m orthogonal sum K = L(N0 )⊕C ⊕C ⊕C , where L(N0 ) is the reproducing kernel Hilbert space associated with the Nevanlinna function N0 in the representation . The inner product on Cm will be denoted by (x, y)m = y∗ x, x, y ∈ Cm ; (4.1) of N the index m in the inner product will be omitted when it is clear from the context. We denote by (K; G) the linear space K equipped with the indeﬁnite inner product G · , · K deﬁned by the Gram matrix ⎛ ⎞ IL(N0 ) 0 0 0 ⎜ 0 Gq 0 0 ⎟ ⎟, G=⎜ (6.1) ⎝ 0 0 0 ICm ⎠ 0 0 ICm Gp0 where Gq and Gp0 are given in Lemma 5.1. Because of Lemma 4.3 we have N0 ∈ L(N0 ). But since the element Rw0 N0 = KN0 ( · , w0∗ ),

w0 ∈ D(N0 ),

belongs to L(N0 ), we see that N0 is a generalized element belonging to the space L(N0 )−1 deﬁned in the Introduction. Thus the pairing f0 , N0 between an element f0 ∈ dom AN0 and N0 is well deﬁned: f0 , N0 = (AN0 − w0 )f0 , KN0 ( · , w0 )L(N0 ) = g0 (w0 ) − w0 f0 (w0 ),

(6.2)

where g0 = AN0 f0 and the right-hand side is independent of w0 ∈ D(N0 ). In this connection we write χ−1 for N0 ∈ L(N0 )−1 and we deﬁne ϕ0 (z) = (AN0 − z)−1 χ−1 = KN0 ( · , z ∗ ),

z ∈ D(N0 ).

By Theorem 3.1, the triplet (AN0 , ϕ0 (z), SN0 ) is a minimal model for N0 in L(N0 ). To keep the formulation of the next theorem short, we introduce the following notation. We write 0 n q(z) = qj (z − z0∗ )j , p0 (z) = pk (z − z0 )j , (6.3) j=0

j=0

set pk = 0 for k > 0 , and deﬁne the column vectors ∗ q = q0 · · · qn−1 ∈ Cn , p = p0 · · · pm−1 ∈ Cm .

Minimal Models for Nκ∞ -functions

119

We use the column vector polynomials m

m

sq (z) = (sj (z))nj=1 , tp0 (z) = (tj (z))m j=1 , r1p0 (z) = (r1j (z))j=1 , r2p0 (z) = (r1j (z))j=1 . Here sq (z) is as in (5.7), tj (z) = Rzj 0 p0 (z), and the entries of the last two vectors are the coeﬃcients in the expansions Rz Rzm0 p0 (ζ) =

m

r1j (z)(ζ − z0∗ )j−1 ,

j=1

and 0

Rzj 0 p0 (ζ)(z

− z0 )

j−1

=

j=m+1

m

r2j (z)(ζ − z0∗ )j−1 .

j=1

Furthermore, we write b(z) = (z − z0∗ )m−1 (z − z0∗ )m−2 · · · 1 , d(z) = 1 (z − z0 ) · · · (z − z0 )m−1 . We denote by Jm (z0 ) the m × m Jordan block matrix at z0 with Jm (z0 )em,1 = z0 em,1 , where for j = 1, 2, . . . , m, em,j stands for the jth element in the standard orthogonal basis of Cm . Theorem 6.1. Assume the conditions (1)–(4). (i) For α ∈ R, let C α be the set of all pairs of the form

⎧⎛ ⎞ ⎛ ⎞⎫ ⎪ ⎪ g0 + w0 λχ0 ⎪ ⎪ ⎪ ⎜ f0 +λχ0 ⎟ ⎜ ⎟⎪ ⎪ ⎪ ⎪ ∗ ⎟ ⎜−f , χ ⎟⎪ ⎬ ⎨⎜ a e − λ(N (w )e + q) + J (z ) a + (b, e )e ⎜ ⎟ ⎜ 0 −1 n,1 0 0 n,1 n 0 m,m n,1 ⎟ ⎜ ⎟ ,⎜ ⎟ , ∗ ⎟ ⎜ ⎟⎪ ⎪⎜ b −λG e + J (z ) b + (d, αe − p)e ⎪ p0 m,m m 0 m,1 m,1 ⎪ ⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ d λem,m + Jm (z0 )d

with χ0 = ϕ0 (w0 ), {f0 , g0 } ∈ AN0 , λ ∈ C, a ∈ Cn such that an = λqn , and b, d ∈ Cm , where w0 is a ﬁxed point in D(N0 ). Then C α is the graph of a self-adjoint operator (also denoted by C α ) in the space (K; G). be the set of all pairs of the form (ii) Let C ⎧⎛ ⎞ ⎛ ⎪ f0 +λχ0 ⎪ ⎪ ⎪⎜ ⎟ ⎜ ⎪ ⎨⎜ ⎜ a ⎟ ⎜ ⎟ ⎜−f0 , χ−1 en,1 ⎜ ⎟ ,⎜ ⎜ ⎟ ⎜ ⎪ b ⎪ ⎪ ⎝ ⎠ ⎝ ⎪ ⎪ ⎩ d

⎞⎫

⎪ g0 + w0 λχ0 ⎪ ⎪ ⎟⎪ ⎪ ⎬ − λ(N0 (w0 )en,1 + q) + Jn (z0 )∗ a + (b, em,m )en,1 ⎟ ⎟ , ⎟ ⎟⎪ −λGp0 em,m + Jm (z0 )∗ b + µem,1 ⎪ ⎠⎪ ⎪ ⎪ ⎭ λem,m + Jm (z0 )d

with χ0 = ϕ0 (w0 ), {f0 , g0 } ∈ AN0 , λ, µ ∈ C, a ∈ Cn such that an = λqn , b ∈ Cm , and d ∈ Cm such that d1 = 0, where w0 is a ﬁxed point in D(N0 ). is a self-adjoint relation in (K; G). Then C in the space (K; G) are given by the triplets (iii) The minimal models of N and N (C, N (z)Γz , S)

and

Γz , S), (C,

120

A. Dijksma, A. Luger and Y. Shondin and where C = C 0 , S = C ∩ C, ⎛

⎞ ϕ0 (z)c(z) ⎜ ⎟ sq (z)c(z) ⎟ Γz = ⎜ ⎝r2p0 (z) + b(z)c(z)(N0 (z) + q(z))⎠ . d(z)

(iv) The family of all self-adjoint extensions of S in (K; G) is given by C α , Moreover, C = C ∞ , the limit in the resolvent sense of C α α ∈ R, and C. as α → ∞. Note that C α is of the form (2.3): ⎛ ⎞ ⎛ ⎞ 0 0 < ; ⎜ 0 ⎟ ⎜ 0 ⎟ ⎟ ⎜ ⎟ C α = C + α · , ⎜ ⎝em,1 ⎠ ⎝em,1 ⎠ . 0 0 is multi-valued: 0 0 em,1 Note also C

0 ∈ C(0), S can also be written as

⎧⎧⎛ ⎞ ⎛ ⎞⎫⎫∗ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨⎨⎜ ⎟ ⎜ ⎟⎬⎬ 0 0

α ⎜ ⎟,⎜ ⎟ ∩ S=C ⎝0⎠ ⎝em,1 ⎠⎪⎪ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩⎩ ⎭⎪ 0 0 and that

C α

⎧⎧⎛ ⎞ ⎛ ⎞⎫⎫ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨⎜ ⎨⎪ ⎟ ⎜ ⎟⎬⎬ 0 0 ⎜ ⎟,⎜ ⎟ = S + span 0 (z0 ))em,1 ⎠⎪⎪ , ⎪⎝ 0 ⎠ ⎝(α − N ⎪ ⎪ ⎪ ⎩ ⎭ ⎩⎪ ⎭⎪ em,1 z0 em,1

in particular, the domain of C α is dense and independent of α ∈ R. In the proof of Theorem 6.1 we identify – according to the basis discussed in

) with K = L(N0 )⊕Cn ⊕Cm ⊕Cm and the relations B α , Section 5 – the space L(N and S deﬁned in the theorem. To explain the identiﬁcation, and S with C α , C, B,

) be given as let the vector function f ∈ L(N f (ζ) = f (ζ) a(ζ)

b(ζ)

d(ζ)

,

where L(q) a(ζ) =

n j=1

n aj vm+j (ζ) 2 = aj (ζ − z0∗ )j−1 j=1

(6.4)

Minimal Models for Nκ∞ -functions and

˙ L0 +M

=

m

b(ζ) d(ζ)

121

bj vj (ζ) 3,4 + dj uj (ζ) 3,4

j=1

⎛# ⎞ m djRzj 0 p0 (ζ)+bj (ζ − z0∗ )j−1 ⎜j=1 ⎟ ⎟. =⎜ m # ⎝ ⎠ dj (ζ − z0 )m−j j=1

Then f will be identiﬁed with the element ⎛ ⎞ f ⎜a⎟ ⎜ ⎟ ∈ K, ⎝b⎠ d where a = a1 . . . an , etc. We write f f w

in (4.10) we have w

0 0 em,1 0 .

b d . For example, for

a

) given by Proof of Theorem 6.1. Identify the vector functions f , g ∈ L(N

= f (ζ) a1 (ζ) b1 (ζ) d1 (ζ) , g (ζ) = g(ζ) a2 (ζ) b2 (ζ) d2 (ζ) f(ζ) with the elements

f f

b2 d2 in K. Here for i = 1, 2, the entries of the vectors ai = ai,1 . . . ai,n , etc., appear as coeﬃcients in the representations ⎛# ⎞ m j ∗ j−1 d R p (ζ)+bi,j (ζ − z0 ) n ⎜j=1 i,j z0 0 ⎟ bi (ζ) ⎟. =⎜ ai (ζ) = ai,j (ζ − z0∗ )j−1 , m # ⎝ ⎠ di (ζ) m−j di,j (ζ − z0 ) j=1 a1

b1

d1

,

g g

a2

j=1

According to Theorem 4.1 we have

g (ζ)} ∈ B α ⇐⇒ ∃ h1 {f(ζ), ⎛

h2

h3

h4

∈ C4 :

⎞ h1 − N0 (ζ)h4 ⎜ ⎟ h2 − q(ζ)h4 ⎟ g (ζ) − ζ f (ζ) = ⎜ ⎝−c# (ζ)(h1 + h2 ) + (α − p0 (ζ))h3 ⎠ . −c(ζ)h3 + h4

(6.5)

Comparison of the fourth components on both sides of this equality yields h3 = d1,1 and d2,j = z0 d1,j + d1,j+1 , j = 1, . . . , m − 1;

d2,m = z0 d1,m + h4 .

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A. Dijksma, A. Luger and Y. Shondin

In the same way the second components in (6.5) yield h4 = a1,n /qn and a1,n , a2,1 = z0∗ a1,1 + h2 − q0 qn a1,n , j = 2, . . . , n. a2,j = z0∗ a1,j + a1,j−1 − qj−1 qn We now consider the third components in (6.5). Inserting the expressions for d2,j already obtained we ﬁnd that m

d2,j

j=1

reduces to

0

pk (ζ − z0 )k−j −

0 m

d1,j pk (ζ − z0 )k−j+1 + z0 (ζ − z0 )k−j

j=1 k=j

k=j m

d1,j pj−1 − h3 p0 (ζ) + h4

j=1

0

pk (ζ − z0 )k−m .

k=m

In the last term we may replace 0 by 2m − 1, because 0 < 2m and pk = 0 if k > 0 . Using the relation (5.1) the last term can now be rewritten as m # h4 um , uj L(N ) (ζ − z0∗ )j−1 . Then in the third components there only remain j=1

powers of ζ − z0∗ and comparing these we ﬁnd h1 + h2 = b1,m and b2,1 = z0∗ b1,1 + αd1,1 −

m

pj−1 d1,j − h4 um , u1 L(N ) ,

j=1

b2,j = z0∗ b1,j + b1,j−1 − h4 um , uj L(N ) ,

j = 2, . . . , m.

If we rewrite N0 (ζ) in the ﬁrst component of (6.5) as (ζ − w0 )KN0 (ζ, w0∗ ) + N0 (w0 ) we ﬁnd g(ζ) − w0 h4 KN0 (ζ, w0∗ ) − ζ f (ζ) − h4 KN0 (ζ, w0∗ ) = h1 − h4 N0 (w0 ). = = >? @ >? @ =:g0 (ζ)

=:f0 (ζ)

So {f0 , g0 } ∈ AN0 and with λ = a1,n /qn = h4 we have f = f0 + λKN0 ( · , w0∗ ), g = g0 + w0 λKN0 ( · , w0∗ ).

(6.6)

Because N0 ∈ L(N0 ), we have that KN0 ( · , w0∗ ) ∈ dom AN0 , hence the decomposition (6.6) is unique. Furthermore, we have that h1 = f0 , N0 + λN0 (w0 ). Indeed, since {f0 , g0 } ∈ AN0 , the diﬀerence g0 (ζ) − ζf0 (ζ) is identically equal to a constant and hence, on account of (6.2), h1 − λN0 (w0 ) = h1 − h4 N0 (w0 ) = g0 (ζ) − ζf0 (ζ) = g0 (w0 ) − w0 f0 (w0 ) = f0 , N0 . Hence h2 = b1,m − h1 = b1,m − f0 , N0 − λN0 (w0 ). Together these formulas show that {f (ζ), g (ζ)} ∈ B α can be identiﬁed with a pair of elements in K of the form described in the theorem. Hence under the identiﬁcation B α coincides with C α .

Minimal Models for Nκ∞ -functions

123

can be identiﬁed with B can be proved in a similar way and therefore (ii) That C the details are omitted.

) and K, the Γ-ﬁeld K ( · , z ∗ )v(z) in Theo(iii) In the identiﬁcation between L(N N rem 4.1 coincides with the Γ-ﬁeld Γz in (iii). (z) = (z 2 + 1)N0 (z) + γ3 z 3 + γ2 z 2 + γ1 z + γ0 , where N0 is Example. Consider N a Nevanlinna function satisfying (1.3) and the γj ’s are real numbers with γ3 = 0. in the form (4.1) and (6.3): We rewrite N (z) = (z + i){N0 (z) + q1 (z + i) + q0 }(z − i) + p1 (z − i) + p0 N (i) = γ0 − γ2 + i(γ1 − γ3 ). with q1 = γ3 , q0 = γ2 − iγ3 , p1 = γ1 − γ3 and p0 = N ∈ N ∞ , where κ = 2 if γ3 < 0 and κ = 1 if γ3 > 0. The Then m = 1, n = 1, N κ is K = L(N0 ) ⊕ C ⊕ C2 equipped with the indeﬁnite inner state space for N and N product G · , · K in which the Gram matrix is given by 1 0 1 . , G = diag IL(N0 ) , 1 γ1 − γ3 γ3 We take w0 = i, and recall χ0 (z) = KN0 ( · , z ∗ ) ∈ L(N0 ) and χ−1 = N0 ∈ L(N0 )−1 . We ﬁnd that the graph of the self-adjoint operator C α is the set of all elements of the form ⎧⎛ ⎞ ⎛ ⎞⎫ ⎪ ⎪ g0 + iλχ0 ⎪ ⎪ ⎪ ⎜ f0 + λχ0 ⎟ ⎜ ⎪ ⎪ ⎟⎪ ⎪ ⎜ ⎬ ⎨⎜ ⎟⎪ λγ3 ⎟ ⎜ ⎟ ⎜ −f0 , χ−1 − λ(N0 (i) + γ2 ) + b ⎟ ⎜ ⎟ ,⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎪ ⎪ ⎪ b ⎪ ⎪ ⎠ ⎝ −λ(γ1 − γ3 ) − ib + (α − N (i))d⎠ ⎪ ⎪ ⎪⎝ ⎪ ⎪ ⎭ ⎩ d λ + id is the set of all elements with {f0 , g0 } ∈ AN0 , λ, b, d ∈ C. The self-adjoint relation C of the form ⎧⎛ ⎞ ⎛ ⎞⎫ ⎪ ⎪ f0 + λχ0 g0 + iλχ0 ⎪ ⎪ ⎪ ⎜ ⎪ ⎪ ⎟ ⎜ ⎟⎪ ⎪ ⎬ ⎨⎜ ⎟ ⎜ −f , χ ⎟⎪ λγ − λ(N (i) + γ ) + b 3 0 −1 0 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ,⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎪ ⎪ ⎪ µ b ⎪ ⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ λ 0 with {f0 , g0 } ∈ AN0 , λ, b, µ ∈ C. From Corollary 4.5 we obtain the following theorem. We set S1 (z; z0 ) = 0 and if n ≥ 2, ⎞ ⎛ 0 1 (z − z0 ) . . . (z − z0 )n−2 ⎟ ⎜ .. .. .. ⎟ ⎜ . . . ⎟ ⎜ ⎟ ⎜ . . .. .. Sn (z; z0 ) = ⎜ (z − z0 ) ⎟ ⎟ ⎜ ⎟ ⎜ .. ⎠ ⎝ . 1 0 and we recall that the deﬁnitions of the matrix functions Kα (z) and K(z) are given just before Corollary 4.5.

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A. Dijksma, A. Luger and Y. Shondin

in Theorem 6.2. The resolvents of the operators C α , α ∈ R, and the relation C K are given by Sm (z; z0∗ ) r1p0 (z)b# (z) (C α −z)−1 = diag (AN0 − z)−1 , Sn (z; z0∗ ), # 0 Sm (z; z0∗) N (z) r2p0 (z) b(z) Kα (z) diag ϕ0 (z), sq (z), + d(z) 0 1 + αN (z) # d (z) tp0 (z) , × diag · , ϕ0 (z ∗ )L(N0 ) , d# (z), 0 b# (z) where ϕ0 (z) = KN0 ( · , z ∗ ), and Sm (z; z0∗ ) r1p0 (z)b# (z) −1 −1 ∗ (C−z) = diag (AN0 − z) , Sn (z; z0 ), # 0 Sm (z; z0∗ ) r2p0 (z) b(z) K(z) +diag ϕ0 (z), sq (z), d(z) 0 # d (z) tp0 (z) ∗ # × diag · , ϕ0 (z )L(N0 ) , d (z), . 0 b# (z) As to the proof of the theorem we only mention the following identiﬁcations between the elements and operators in L(q) and Cn : −1 Kq ( · , z ∗ ) sq (z) = (Jn (z0 )∗ − z (q − q(z)en,1 ), · , Kq ( · , z) Gq · , sq (z ∗ ) = d# (z), (Aq − z)−1 Sn (z; z0∗ ) = (I + (z0∗ − z)Jn (0))−1 Jn (0). The ﬁrst identiﬁcation follows from (5.7) and (5.8) which show that sq (z) is the vector representation of the element Kq ( · , z ∗ ) ∈ L(q) relative to the basis {vm+i }ni=1 . The linear functional · , Kq ( · , z ∗ )L(q) on L(q) can be identiﬁed with (Gq · , sq (z ∗ )) = d# (z) viewed as a mapping from Cn to C. This also follows from (6.4). In the theorem r1p0 (z)b# (z) is an m × m matrix polynomial of rank 1. Also, the matrix r2p0 (z) b(z) d(z) 0 whose entries are column m-vectors should be viewed as a mapping from C2 to C2m , whereas the matrix # d (z) tp0 (z) 0 b# (z) whose entries are row m-vectors should be seen as a mapping from C2m to C2 . The matrices are related via the formula # # r2p0 (z) b(z) 0 ICm d (z) tp0 (z) , = d(z) 0 ICm Gp0 0 b# (z) which corresponds to a part of the identity Γ∗N z∗ = Ez .

Minimal Models for Nκ∞ -functions

125

We now consider the analogs of Theorem 6.1 in the cases n = 0 and m = 0. The case n = 0 and m > 0: Here K = L(N0 ) ⊕ Cm ⊕ Cm and the Gram matrix takes the form ⎛ ⎞ IL(N0 ) 0 0 0 ICm ⎠ . G=⎝ 0 0 ICm Gp0 Theorem 6.3. Assume n = 0 and m > 0. Then Theorem 6.1 holds if we delete the second component in all 4-vectors in the formulas, omit in (i) and (ii) the statement “a ∈ Cn such that an = λqn ”, add in (i) and (ii) the statement “bm = f0 , χ−1 + λ(N0 (w0 ) + q0 )”, and set q(z) = q0 in Γz in (iii). The case n > 0 and m = 0: Now K = L(N0 ) ⊕ Cn and G = diag {IL(N0 ) , Gq }. Theorem 6.4. Assume n > 0 and m = 0. Then Theorem 6.1 holds if C α is the set of all pairs of the form ⎧⎛ ⎨ f ⎝ 0 ⎩

⎞⎫

⎞ ⎛

⎬ + λχ0 ⎠ ⎝ g0 + w0 λχ0 ⎠ , ∗ a −f0 , χ−1 en,1 − λ(N0 (w0 )en,1 + q + αen,1 ) + Jn (z0 ) a ⎭ (6.7) with χ0 = ϕ0 (w0 ), {f0 , g0 } ∈ AN0 , λ ∈ C, and a ∈ Cn such that an = λqn , where w0 is a ﬁxed point in D(N0 ), if = AN0 ⊕ {{a, Jn (z0 )∗ a + µen,1 } | a ∈ Cn , an = 0, µ ∈ C}, C and if Γz = ϕ0 (z), sq (z) .

Next we give the formulas for the compressions of the resolvents (C α −z)−1 , − z)−1 to the subspaces L(N0 ) and L(N0 ) ⊕ Cn of (K; G) in the α ∈ R, and (C case n > 0 and m > 0; similar formulas can be obtained in the other two cases. We denote by P0 and P1 the orthogonal projections in (K; G) onto L(N0 ) and L(N0 ) ⊕ Cn . Theorem 6.5. (i) For α ∈ R, P0 (C α − z)−1 |L(N0 ) = (AN0 − z)−1 − with parameter Tα (z) = q(z) +

1 · , ϕ0 (z ∗ )L(N0 ) ϕ0 (z) N0 (z) + Tα (z)

p0 (z) − α , and for α = ∞, c# (z)c(z)

− z)−1 |L(N ) = (AN0 − z)−1 . P0 (C 0 (ii) For α ∈ R, P1 (C

α

−1

− z)

|L(N0 )⊕Cn

(AN0 − z)−1 = 0

1 − N0 (z) + Tα (z)

0 Sn (z; z0∗ )

· , ϕ0 (z ∗ )L(N0 ) ϕ0 (z) ϕ0 (z)d# (z) · , ϕ0 (z ∗ )L(N0 ) sq (z)

sq (z)d# (z)

,

126

A. Dijksma, A. Luger and Y. Shondin and for α = ∞, P1 (C

∞

−1

− z)

|L(N0 )⊕Cn =

(AN0 − z)−1 0

0

Sn (z; z0∗ )

.

The resolvent formula (Aτ − z)−1 = (AN0 − z)−1 −

1 · , ϕ0 (z ∗ )ϕ0 (z) N0 (z) + τ

describes the family of all self-adjoint extensions Aτ of SN0 in L(N0 ) in terms of the parameter τ ∈ R ∪ {∞}. If the number τ is replaced by a generalized Nevanlinna function τ (z) the formula describes the minimal self-adjoint extensions which act in spaces containing L(N0 ). This is called Krein’s resolvent formula. In part (i) the parameter describing C α with α ∈ R is explicitly given by τ (z) = Tα (z) ∈ Nκ . Related to Krein’s formula here are the references [21, Theorem 4.7], [13, Theorem 4.2], and [7, Theorem 3.5]. with It is of interest to compare the compressed resolvents of C α and C the compressions of these operators/relation themselves. Recall Stenger’s lemma (see [15, Theorem 3.3 and a remark after the theorem]) that if A is a self-adjoint

and H is a Hilbert or Pontryagin relation with ρ(A) = ∅ in a Pontryagin space H

such that dim H

H < ∞, then the compression of A to H, that subspace of H, is, the linear relation PH A |H = {{f, PH g} | {f, g} ∈ A, f ∈ H},

onto H, is self-adjoint in H. In our where PH is the orthogonal projection in H case L(N0 ) is a Hilbert subspace and L(N0 ) ⊕ Cn is a Pontryagin subspace of the Pontryagin space K and both have a ﬁnite codimension. Therefore the compressions just mentioned are self-adjoint. The following theorem follows directly from Theorem 6.1 and its versions for the special cases n = 0 or m = 0. Theorem 6.6. (i) If n > 0 and m ≥ 0, then L(N ) = AN0 , P0 C α |L(N0 ) = P0 C| 0 and if n = 0 and m > 0, then (in graph notation) L(N ) P0 C α |L(N0 ) = P0 C| 0 = {{f0 + λχ0 , g0 + w0 λχ0 }|{f0 , g0 } ∈ AN0 , f0 , χ−1 + λ(N0 (w0 ) + q0 ) = 0}. (ii) If n > 0 and m > 0, then L(N )⊕Cn P1 C α |L(N0 )⊕Cn = P1 C| 0 and their graphs coincide with the set of all pairs of the form (6.7) with α = 0.

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7. Block operator matrix models in the space (K; H) Changing the basis we have considered in the previous sections we can write C α in Theorem 6.1 with w0 = z0∗ in a block operator matrix form, which is not and C possible with the basis used so far. Set ⎛ ⎞ 1 IL(N0 ) ( · , en,n )χ0 0 0 ⎜ qn ⎟ ⎜ ICn 0 0 ⎟ T =⎜ 0 ⎟, ⎝ 0 0 I m 0 ⎠ C

0

0

0

ICm

where χ0 = ϕ0 (z0∗ ), and deﬁne the operators D α = T −1 C α T , , and the Gram matrix = T −1 CT relation D ⎛ 1 IL(N0 ) ( · , en,n )χ0 ⎜ qn ⎜1 h0 ⎜ ( · , en,n )en,n + Gq H = T ∗ GT = ⎜ · , χ0 L(N0 ) en,n qn2 ⎜ qn ⎝ 0 0 0

0

α ∈ R, the linear ⎞ 0 0 0 ICm

0

⎟ ⎟ 0 ⎟ ⎟, ⎟ ICm ⎠ Gp0

where h0 = χ0 , χ0 L(N0 ) = KN0 (z0 , z0 ). Since T −1 w

= w,

we have

K w,

D α = D + αH · , w

D = T −1 CT.

The space K equipped with the indeﬁnite inner product H · , · K will be are self-adjoint in (K; H). The relation denoted by (K; H). Clearly, D α and D can be obtained via inﬁnite coupling of D and w, D

that is, as limit of D α in the resolvent sense by letting α → ∞. The following theorem shows that D α can be expressed by means of block operator matrices. We use the notation and D explained directly above Theorem 6.1. be as deﬁned above. Then: Theorem 7.1. Let D α , α ∈ R, and D (i) dom D α = (dom AN0 ) ⊕ Cn ⊕ Cm ⊕ Cm and on this domain D α has the block matrix form ⎞ ⎛ AN0 D12 0 0 ⎟ ⎜ ⎟ ⎜ D21 D22 ( · , em,m )en,1 0 ⎟ ⎜ ⎟ ⎜

α 1 D =⎜ ⎟, − ( · , en,n )Gq em,m Jm (z0 )∗ ( · , αem,1 − p)em,1 ⎟ ⎜ 0 ⎟ ⎜ qn ⎠ ⎝ 1 0 ( · , en,n )em,m 0 Jm (z0 ) qn where with χ0 = ϕ0 (z0 ) qn−1 1 ( · , en,n ) − ( · , en,n−1 ) χ0 , D12 = 2 qn qn

D21 = − · , χ−1 en,1 ,

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1 ( · , en,n ) N0 (z0∗ )en,1 + q . qn

= (dom AN0 ) ⊕ Cn ⊕ Cm ⊕ (Cm {em,1 }) and (ii) dom D = {{f , ∃µ ∈ C : D g}|f ∈ dom D, g = Df + 0 0 µem,1

0

},

where D = D 0 .

z , S),

(D, Γ

z , S)

are minimal models for N and N in (iii) The triplets (D, N (z)Γ

(K; H), where S = D ∩ D and ⎛ ⎞ (ϕ0 (z) − ϕ0 (z0 ))c(z) ⎜ ⎟ sq (z)c(z)

z = ⎜ ⎟ Γ ⎝r2p0 (z) + b(z)c(z)(N0 (z) + q(z))⎠ . d(z) The theorem follows directly from Theorem 6.1 with w0 = z0∗ , the deﬁnitions and Γ

z = T −1 Γz . Note that Γ

z0 = 0 0 0 em,1 . of D α and D, From the theorems in Section 6 one can easily obtain formulas for the resol We vents, the compressions of the resolvent and the compressions of D α and D. leave the details to the reader.

8. Examples We give two examples and discuss an approximation problem taken from [11], [17], and [14] to which we refer for details and proofs. They are related to the Bessel diﬀerential expression ν 2 − 1/4 y(x) x2 on (0, 1] with a self-adjoint boundary condition at the regular endpoint x = 1 and on (0, ∞), which is limit point at x = ∞. We recall the series expansion of the Bessel function ∞ z 2k z ν (−1)k Jν (z) = , (8.1) 2 k!Γ(k + ν + 1) 2 ν y(x) = −y (x) +

k=0

where the series on the right converges absolutely, and uniformly in any bounded domain of z and ν. We denote by 1 Yν (z) (Jν (z)cos (νπ) − J−ν (z)), = sin (νπ) 1 (1) (J−ν (z) − Jν (z)e−iνπ ), Hν (z) = isin (νπ) π (1) Kν (z) = i π2 ei 2 ν Hν (iz) the Neumann function of order ν, the ﬁrst Hankel function of order ν, and the Basset (or MacDonald) function of order ν, respectively; see, for example, [18].

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Example. See [11]. We consider ν on the interval (0, 1] and impose the boundary condition y(1) = 0 at the regular endpoint x = 1 on all functions y. (i) First assume 0 < ν < 1. Then the minimal realization S of ν in H0 = L2 (0, 1) is a symmetric operator with defect indices (1, 1). We denote by A0 the self-adjoint operator extension of S with graph A0 = {{y, ν y} ∈ S ∗ |limx↓0 xν−1/2 y(x) = 0}. The function π z ν/2 x1/2 ϕ(x, z) = − 2 sin πν

√ √ √ J−ν ( z) √ Jν (x z) − J−ν (x z) Jν ( z)

(8.2)

(8.3)

belongs to ker (S ∗ − z) and is a defect function for S and A0 with corresponding Q-function √ π ν J−ν ( z) √ . N (z) = − z (8.4) 2 sin πν Jν ( z) Thus the following relations hold: ϕ(z) = (I + (z − z0 )(A0 − z)−1 )ϕ(z 0 ),

(z) − N (w)∗ N = ϕ(z), ϕ(w) 0 . (8.5) z − w∗

(ii) Now assume ν > 1, ν = 2, 3, . . .. Then the results are quite diﬀerent from those in (i): The minimal realization of ν in H0 is self-adjoint, the function ϕ( · , z) in in (8.4) is now (8.3) is well deﬁned but it does not belong to H0 , and the function N a generalized Nevanlinna function with κ = [(ν + 1)/2] negative squares. Thus the model for this function involves a self-adjoint operator or relation in a Pontryagin ∈ N ∞ : Let zn , n = 1, 2 . . ., space with κ negative squares. In [11] we show that N √κ −ν/2 Jν ( z) in increasing order. be the enumeration of the zeros of the function z admits (The zeros are positive and, since the function is entire, countable.) Then N the decomposition (z) = z 2κ (N0 (z) + q0 ) + p0 (z), N where N0 (z) =

∞ n=1

q0 =

zn 1 − 2 zn − z zn + 1

2znν−2κ , Jν (zn )2

∞ 1 (2κ) 2znν−2κ−1 (0) − , N (2κ)! (zn2 + 1)Jν (zn )2 n=1

and p0 (z) =

2κ−1 j=0

pj z j ,

pj =

1 (j) N (0). j!

∈ N ∞. The Nevanlinna function N0 satisﬁes the relations (1.3) and hence N κ κ Theorem 6.3 with m = κ > 0, z0 = 0, and Gp0 = (pi+j−1 )i,j=1 yields the descrip and N = −1/N. tion of the models for N

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Example. See [17]. We now consider ν on (0, ∞). The endpoint x = ∞ is limit point so we do not need to impose a condition at this endpoint. Where possible we use the same notation as in the previous example. (i) First assume 0 < ν < 1. Then the minimal realization S of v in H0 = L2 (0, ∞) is a symmetric operator with defect indices (1, 1). The self-adjoint extension A0 of S deﬁned by formula (8.2) is uniquely determined by the facts that its spectrum σ(A0 ) = √ [0, ∞) is absolutely continuous and that the functions y(x, λ) = c(λ)x1/2 Jν (x λ), λ ∈ [0, ∞), form a complete set of generalized eigenfunctions of A0 , where c(λ) is some normalizing factor. The function √ √ ν ϕ(x, z) = x(−z) 2 Kν (x −z) (8.6) belongs to ker (S ∗ − z) and is a defect function for S and A0 with corresponding Q-function (z) = − π N (−z)ν . (8.7) 2 sin πν is Thus the relations (8.5) are also valid in this case. It follows from (8.7) that N a Nevanlinna function, which satisﬁes the limit conditions in (1.3). (ii) Now assume ν > 1 and ν = 2, 3, . . .. Then, as in the previous example, the minimal realization of ν in H0 is self-adjoint, the function ϕ( · , z) in (8.6) does in (8.7) is a generalized Nevanlinna function not belong to H0 , and the function N with κ = [(ν + 1)/2] negative squares. Here the branch of (−z)ν is chosen so that (−z)ν = rν e iν(θ−π) if z = re iθ , 0 < θ < 2π. For any z0 ∈ (−∞, 0) the function N admits the decomposition (z) = (z − z0 )2κ (N0 (z) + q0 ) + p0 (z), N where

N0 (z) = 0

∞

t 1 − 2 t−z t +1

tν dt, 2(t − z0 )2κ

q0 = −

and p0 (z) =

2κ−1 j=0

pj (z − z0 ) , j

1 (j) π(−1)j+1 pj = N (z0 ) = j! 2 sin πν

π , 4 sin πν 2

ν (−z0 )ν−j . j

Since N0 is a Nevanlinna function which satisﬁes the relations (1.3), we have that ∈ N ∞ . Hence Theorem 6.3 with m = κ > 0 applies and provides the description N κ (z) and N (z) = −1/N (z). Since z0 is real, the Gram matrix of the models for N κ Gp0 is given by Gp0 = (pi+j−1 )i,j=1 . Inspired by [29] and [28], we discuss an approximation problem in Nκ∞ ; for details we refer to the paper [14] in preparation. In the context of the discussion around (1.4), the problem is to approximate strongly singular perturbations by ∈ N ∞ (κ > 0) with irreducible smoother perturbations. Consider a function N κ representation (4.1): (z) = (z − z0 )2κ (N0 (z) + q0 ) + p0 (z), N

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131

j ∈ N ∞ with irreducible representation (4.1): and a sequence of functions N κ j (z) = N0j (z) + qj (z), N

j = 1, 2, . . . ,

where N0 and all N0j are Nevanlinna functions satisfying (1.3), z0 is a real number belonging to their common domain D of holomorphy, q0 ∈ R, p0 (z) and qj (z) are real polynomials with deg p0 ≤ 2κ − 1, and n = deg qj is either 2κ or 2κ ± 1: if n is odd and the leading coeﬃcient of qj (z) is negative, then n = 2κ − 1; if n is odd and the leading coeﬃcient of qj (z) is positive, then n = 2κ + 1. Assume that, as j converges to N uniformly on compact subsets of D. The approximation j → ∞, N problem with variable spaces then is to describe this convergence in terms of the j and N and of the corresponding state spaces. We rewrite N j in the models of N following form j (z) = (z − z0 )2κ (M0j (z) + q0j ) + p0j (z) + qj,2κ+1 (z − z0 )2κ+1 , N

(8.8)

where M0j (z) is a Nevanlinna function, p0j (z) is a real polynomial with deg p0j ≤ 2κ − 1, and q0j , qj,2κ+1 ∈ R with qj,2κ+1 ≥ 0 (if qj,2κ+1 > 0 then it is the leading coeﬃcient of qj (z)). The convergence assumption is equivalent to the convergence of M0j (z) + q0j to N0 (z) + q0 uniformly on compact subsets of D, the pointwise convergence of the polynomials p0j (z) to p0 (z), and the convergence qj,2κ+1 → 0, j (z) need not be irreducible, and so as j → ∞. The representation (8.8) of N models will have to be constructed, which fall outside the scope of this paper. Approximation of operators with variation of the space in which they act has been considered in [22, pp. 512, 513]; for such approximations in an indeﬁnite setting, see [27] and [26]. The application we have in mind is related to ν and the last example. In [14] in (8.7) with ν > 1, ν = 2, 3, . . . , can be approximated we show that the function N by functions of the form δ (z) = N δ (z) + q δ (z) N (8.9) 0

by letting δ ↓ 0. Here q δ (z) is some real polynomial of degree [ν] with coeﬃcients depending on δ, which we will not further specify here, and the function N0δ is obtained as follows. Consider the family of regularized diﬀerential expressions lν,δ y(x) = −y (x) +

ν 2 − 1/4 y(x) (x+δ)2

on (0, ∞), where the parameter δ varies over some interval (0, δ0 ), δ0 > 0. Let Sδ be the minimal operator associated with lν,δ in the Hilbert space H0 = L2 (0, ∞); it is symmetric and its defect indices are (1, 1). Each self-adjoint extension of Sδ can be obtained as the restriction of the maximal operator Sδ ∗ by the boundary condition y (0) = αy(0) with α ∈ R ∪ {∞}. We denote by Aδ the extension corresponding to α = ∞. The function √ 1/2 Kν ((x+δ) −z) √ , γ = 2ν−1 Γ(ν), ϕδ (x, z) = γ(x+δ) δ ν Kν (δ −z)

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is a defect function for Sδ and Aδ . The function considered in (8.9) is by deﬁnition the function √ √ Kν (δ −z) δ 2 1−2ν √ N0 (z) = γ δ . (8.10) −z Kν (δ −z) It satisﬁes the relation N0δ (z) − N0δ (w)∗ = ϕδ (z), ϕδ (w)0 z − w∗ and hence is a Q-function for Sδ and Aδ . It follows that N0δ is a Nevanlinna function with integral representation ∞ t 1 δ − 2 dσδ (t) + Re N0δ (i), N0 (z) = t−z t +1 0 where, for t ≥ 0, dσδ (t) =

1 2γ 2 1 √ dt. Im N0δ (t + i0) dt = 2 2ν 2 √ π π δ Jν (δ t) + Yν2 (δ t)

(8.11)

δ in (8.9) belongs to N ∞ with It will be shown (in [14]) that the function N κ in (8.7) uniformly on compact subsets κ = [(ν + 1)/2] and, if δ ↓ 0, converges to N ). Note that the representation (8.9) of N δ is irreducible and corresponds of D(N whose irreducible to (4.1) with m = 0. This in contrast with the limit function N representation corresponds to (4.1) with m = κ > 0. We conclude the paper with a ﬁnal remark. Remark 8.1. From the beginning up to and including Section 7 we may replace the factor c(z) = (z − z0 )m by c(z) = (z − z1 ) · · · (z − zm ) with zj ∈ D(N0 ) to obtain similar but more general models as in [13, Sections 6 and 7].

References [1] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable models in quantum mechanics, Springer, 1988. [2] S. Albeverio and P. Kurasov, Singular perturbations of diﬀerential operators, London Mathematical Society Lecture Notes 271, Cambridge Univ. Press, 2000. [3] D. Alpay, A. Dijksma, and H. Langer, Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions, Linear Algebra Appl. 387C (2004), 313–342. [4] D. Alpay, A. Dijksma, J. Rovnyak, and H. de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Operator Theory: Adv. Appl. 96, Birkh¨ auser Verlag, Basel, 1997. [5] V.A. Derkach and S. Hassi, A reproducing kernel space model for Nκ -functions, Proc. Amer. Math. Soc. 131 (12) (2003), 3795–3806. [6] V. Derkach, S. Hassi, and H. de Snoo, Operator models associated with singular perturbations, Methods Funct. Anal. Topology 7 (3) (2001), 1–21.

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[7] V. Derkach, S. Hassi, and H. de Snoo, Singular perturbations of self-adjoint operators, Mathematical Physics, Analysis and Geometry 6 (2003), 349–384. [8] J.F. van Diejen and A. Tip, Scattering from generalized point interaction using selfadjoint extensions in Pontryagin spaces, J. Math. Phys. 32 (3) (1991), 630–641. [9] A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, A factorization result for generalized Nevanlinna functions of the class Nκ , Integral Equations Operator Theory 36 (2000), 121–125. [10] A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, Minimal realizations of scalar generalized Nevanlinna functions related to their basic factorization, Operator Theory: Adv. Appl., 154, Birkh¨ auser Verlag, Basel, 2004, 69–90. [11] A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, Singular and regular point-like perturbations of the Bessel operator on (0, 1] in a Pontryagin space, in preparation. [12] A. Dijksma, H. Langer, and Yu. Shondin, Rank one perturbations at inﬁnite coupling in Pontryagin spaces, J. Funct. Anal. 209 (2004), 206–246. [13] A. Dijksma, H. Langer, Y. Shondin, and C. Zeinstra, Self-adjoint operators with inner singularities and Pontryagin spaces, Operator Theory: Adv., Appl., 118, Birkh¨ auser Verlag, Basel, 2000, 105–175. [14] A. Dijksma, A. Luger, and Yu. Shondin, Approximation in Nκ∞ with variable state spaces (tentative title), in preparation. [15] A. Dijksma, H. Langer, and H.S.V. de Snoo, Unitary colligations in Πκ -spaces, characteristic functions and Straus extensions, Paciﬁc J. Math. 125 (2) (1986), 347–362. [16] A. Dijksma, H. Langer, and H.S.V. de Snoo, Eigenvalues and pole functions of Hamiltonian systems with eigenvalue depending boundary conditions, Math. Nachr. 161 (1993), 107–154. [17] A. Dijksma and Yu. Shondin, Singular point-like perturbations of the Bessel operator in a Pontryagin space, J. Diﬀerential Equations 164 (2000), 49–91. [18] A. Erd´elyi, Higher transcendental functions, vol. ii, Mcgraw-Hill, New York, 1953. [19] C.J. Fewster, Generalized point interactions for the radial Schrodinger equation via unitary dilation, J. Phys. A: 28 (1995), 1107–1127. [20] F. Gesztesy and B. Simon, Rank one perturbation at inﬁnite coupling, J. Funct. Anal. 128 (1995), 245–252. [21] S. Hassi, M. Kaltenb¨ack, and H.S.V. de Snoo, The sum of matrix Nevanlinna functions and self-adjoint extensions in exit spaces, Operator Theory: Adv., Appl., 103, Birkh¨ auser Verlag, Basel, 1998, 137–154. [22] T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag, Heidelberg, 1966. ¨ [23] M.G. Krein and H. Langer, Uber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Πκ zusammenh¨ angen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187–236. [24] V.I. Kruglov and B.S. Pavlov, Zero-range potentials with inner structure: ﬁtting parameters for resonance scattering, Preprint arXiv.org: quant-ph/0306150. [25] M. Langer and A. Luger, Scalar generalized Nevanlinna functions: realizations with block operator matrices, Operator Theory: Adv. Appl. 162, Birkh¨ auser Verlag, Basel, 2005, 253–267.

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[26] H. Langer and B. Najman, Perturbation theory for deﬁnizable operators in Krein spaces, J. Operator Theory 9(1983), 297–317. [27] B. Najman, Perturbation theory for selfadjoint operators in Pontrjagin spaces, Glasnik Mat. 15(35) (1980), 351–371. [28] Yu. Shondin, On approximation of high order singular perturbations, J. Phys. A: Math. Gen. 38(2005), 5023–5039. [29] O.Yu. Shvedov, Approximations for strongly singular evolution equations, J. Funct. Anal. 210(2) (2004), 259–294. Aad Dijksma Department of Mathematics University of Groningen P.O. Box 800 9700 AV Groningen The Netherlands e-mail: [email protected] Annemarie Luger Institute for Analysis and Scientiﬁc Computing Vienna University of Technology Wiedner Hauptstrasse 8–10 A-1040 Vienna Austria e-mail: [email protected] Yuri Shondin Department of theoretical Physics State Pedagogical University Str. Ulyanova 1 Nizhny Novgorod 603950 Russia e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 135–145 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Generalized Friedrichs Extensions Associated with Interface Conditions for Sturm-Liouville Operators Andreas Fleige, Seppo Hassi, Henk de Snoo and Henrik Winkler Dedicated to Heinz Langer on the occasion of his retirement

Abstract. For a class of Sturm-Liouville operators with an interface condition at an interior point all selfadjoint realizations are determined. This result is obtained via a description of the selfadjoint extensions of the coupling of two symmetric operators. The (generalized) Friedrichs extension, when it exists, is determined. Suﬃcient conditions for the (generalized) Friedrichs extension to exist are given. Mathematics Subject Classiﬁcation (2000). Primary 47A10, 47B25; Secondary 34B05, 34B24. Keywords. Symmetric operator, selfadjoint extension, (generalized) Friedrichs extension, boundary triplet, Weyl function, interface condition.

1. Introduction Let −DpD + q be a Sturm-Liouville expression with real coeﬃcients on the subset [−b, 0)∪(0, b] of R. It is assumed that there are ﬁxed separated boundary conditions at −b and b and that there is a selfadjoint interface condition at 0 of the form (pu )(0+) = τ (u(0+) − u(0−)),

(pu )(0+) = (pu )(0−),

when τ ∈ R, and of the form u(0+) = u(0−),

(pu )(0+) = (pu )(0−),

when τ = ∞. For τ ∈ R ∪ {∞} these interface conditions describe all selfadjoint extensions of the symmetric operator which is given by the interface condition u(0+) = u(0−),

(pu )(0+) = (pu )(0−) = 0,

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together with the ﬁxed separated boundary conditions at −b and b. The purpose of this note is to indicate conditions which show the existence of a generalized Friedrichs extension (cf. [5], [6]) and to determine the corresponding value of τ ∈ R ∪ {∞} under mild conditions on the singularity at 0. Note that the conditions on the coeﬃcients p and q at 0 can be further relaxed along the lines of, e.g., [10]. It is clear that interface conditions at an interior point can be easily described via the orthogonal coupling of symmetric operators. The selfadjoint extensions for such orthogonal couplings are best expressed in terms of the corresponding boundary triplets; see the extension results in [1]. The description of generalized Friedrichs extensions can also be given in terms of sesquilinear forms along the lines of [3], which will be done in [4].

2. Preliminaries 2.1. Boundary triplets Let S be a closed symmetric operator (or relation) with equal defect numbers in a Hilbert space H with inner product (·, ·). Recall that a relation in H is a linear subspace of the Cartesian product H × H, and its elements are denoted as ordered pairs. Let Nλ = ker (S ∗ − λ) be the defect subspace of S, and denote λ := { fλ = {fλ , λfλ } : fλ ∈ Nλ }, N

λ ∈ C,

λ ⊂ S ∗ . A boundary triplet Π = {H, Γ0 , Γ1 } of S ∗ consists of a Hilbert so that N space H and the boundary mappings Γj , j = 0, 1, from S ∗ to H such that Γ : f → { Γ0 f, Γ1 f} from S ∗ into H × H is surjective and such that the identity (f , g) − (f, g ) = Γ1 f, Γ0 g − Γ0 f, Γ1 g (2.1) H

H

holds for all f = {f, f }, g = {g, g } ∈ S ∗ . The mappings Γi deﬁne two selfadjoint extensions Ai of S via Ai = ker Γi , i = 0, 1. The mapping Γ : f → {Γ0 f, Γ1 f} induces a one-to-one correspondence between the closed extensions H of S which are intermediate (i.e., which satisfy S ⊂ H ⊂ S ∗ ) and the closed linear relations τ in H, via H := { f ∈ S ∗ : Γf ∈ −τ −1 }. (2.2) In particular, this correspondence (2.2) is one-to-one between all selfadjoint extensions H of S and all selfadjoint relations τ in H, cf. [1], [2]. When S is densely deﬁned, then often the mappings Γ0 and Γ1 are interpreted as being deﬁned on dom S ∗ instead of on S ∗ , i.e., in that case one speaks of Γ0 f and Γ1 f when f = {f, f } ∈ S ∗ . This convention will be followed in the present note. Associated to the boundary triplet Π are two operator functions: the Weyl function M (λ), deﬁned by λ } M (λ) = { {Γ0 fλ , Γ1 fλ } : fλ ∈ N

λ ∈ ρ(A0 ),

(2.3)

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137

the graph of a bounded linear operator in H, and the γ-ﬁeld γ(λ) deﬁned by λ }, γ(λ) = { {Γ0 fλ , fλ } : fλ ∈ N

λ ∈ ρ(A0 ),

(2.4)

the graph of a bounded linear operator from H to Nλ . Both functions are holomorphic on ρ(A0 ). The relation between the Weyl function M (λ) and the γ-ﬁeld γ(λ) is given by M (λ) − M (µ)∗ = γ(µ)∗ γ(λ). (2.5) λ−µ ¯ The identity (2.5) implies that M (λ) is the Q-function of the pair {S, A0 }. Each Weyl function belongs to the class N of Nevanlinna functions, i.e., is holomorphic ¯ and Im M (λ) ≥ 0 for λ ∈ C+ . Moreover, on C \ R, and satisﬁes M (λ)∗ = M (λ), M (λ) is strict, i.e., 0 ∈ ρ(Im M (λ)) for all λ ∈ C \ R. In general, the Weyl function M (λ) determines up to unitary isomorphisms, a model for the symmetric operator S and its selfadjoint extension A0 . 2.2. Generalized Friedrichs extensions For the purposes of this paper it is suﬃcient to consider the case of a symmetric operator S with defect numbers (1, 1), in which case the Weyl function M (λ) is a scalar function. The Weyl function M (λ) is said to belong to the Kac class N1 if ∞ Im M (iy) dy < ∞. y 1 In this case there is a real limit γ = lim M (iy) ∈ R.

(2.6)

y→∞

It follows from (2.2) (identifying the selfadjoint relations in R with R ∪ {∞}) that there is a one-to-one correspondence between all selfadjoint extensions of S in H and all numbers in R ∪ {∞} via A(τ ) = { f ∈ S ∗ : Γ0 f = −τ Γ1 f},

τ ∈ R ∪ {∞}.

(2.7)

Recall that the selfadjoint extension A(τ ) is determined by τ ∈ R∪{∞} via Kre˘ın’s formula: (A(τ ) − z)−1 = (A0 − z)−1 − γ(z)(M (z) + 1/τ )−1 (·, γ(¯ z )),

z ∈ C \ R.

Note that A(0) corresponds with A0 = ker Γ0 and that A(∞) corresponds with A1 = ker Γ1 . The Weyl functions Mτ (λ) of the selfadjoint extensions A(τ ) of S in (2.7) are related by Mτ (λ) =

M (λ) − τ , 1 + τ M (λ)

τ ∈ R ∪ {∞}.

(2.8)

If M (λ) belongs to the Kac class N1 then all Mτ (λ), τ ∈ R ∪ {∞}, belong to N1 , except for τ = −1/γ where γ is given by (2.6). The value τ = −1/γ gives rise to an exceptional selfadjoint extension of S, the generalized Friedrichs extension, cf. [5], [6].

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Closely related to the Kac class N1 is the class S of Stieltjes functions. A Nevanlinna function M (λ) is said to belong to the class S if both M (λ) and λM (λ) are Nevanlinna functions. Equivalently, M (λ) belongs to S if and only if M (λ) belongs to N1 and is analytic on C \ [0, ∞), such that lim M (x) ≥ 0,

x→−∞

cf. [8]. Analogously, a Nevanlinna function M (λ) is said to belong to the class S− if both M (λ) and λM (−λ) are Nevanlinna functions. Now M (λ) belongs to S− if and only if M (λ) belongs to N1 and is analytic on C \ (−∞, 0], such that lim M (x) ≤ 0.

x→∞

2.3. Coupling of symmetric operators with defect numbers (1, 1) Consider two closed symmetric operators S+ and S− in the Hilbert spaces H+ and H− and assume that their defect numbers are (1, 1). Form the orthogonal sum H = H+ ⊕H− and deﬁne in H the closed symmetric operator S = S+ ⊕S− . Clearly, + ∗ ∗ S ∗ = S+ ⊕ S− and the defect numbers of S are (2, 2). Let Π+ = {H, Γ+ 0 , Γ1 } and − − ∗ ∗ Π− = {H, Γ0 , Γ1 } be boundary triplets for S+ and S− with γ-ﬁelds γ+ (λ) and γ− (λ). Then − − Γ 0 = Γ+ Γ 1 = Γ+ (2.9) 0 ⊕ Γ0 , 1 ⊕ Γ1 , forms a boundary triplet for the orthogonal sum S ∗ . Observe that the corresponding γ-ﬁeld and Weyl function are of the form γ+ (λ) ⊕ γ− (λ),

M+ (λ) ⊕ M− (λ),

(2.10) A+ 0

where γ± (λ) and M± (λ) correspond to the selfadjoint extensions = ker Γ+ 0 − − and A0 = ker Γ0 of S+ and S− , respectively. It is also of interest to consider one-dimensional symmetric extensions S of S, cf. [1], [5]. + − − Proposition 2.1. Let Π+ = {H, Γ+ 0 , Γ1 } and Π− = {H, Γ0 , Γ1 } be boundary ∗ ∗ triplets for S+ and S− with γ-ﬁelds γ+ (λ), γ− (λ) and Weyl functions M+ (λ), M− (λ), respectively. Then the linear relation S deﬁned by S = { f = f+ ⊕ f− ∈ S ∗ ⊕ S ∗ : Γ+ f+ = Γ− f− = Γ+ f+ + Γ− f− = 0 }, +

−

0

0

1

1

is closed and symmetric in H = H+ ⊕ H− and has defect numbers (1, 1). Its adjoint S ∗ is given by S ∗ = { f = f+ ⊕ f− ∈ S ∗ ⊕ S ∗ : Γ+ f+ = Γ− f− , }, +

−

0

0

and the defect spaces of S are of the form Nλ (S ∗ ) = ker (S ∗ − λ) = { γ+ (λ) ⊕ γ− (λ) h : h ∈ H }. A boundary triplet for S ∗ is given by + −

∗

∗ Π = { H, Γ+ 0 S , (Γ1 + Γ1 ) S },

and the corresponding Weyl function is given by M+ (λ) + M− (λ).

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All selfadjoint extensions A(τ ) of S in H are in one-to-one correspondence with τ ∈ R ∪ {∞} via A(τ ) = { f = f+ ⊕ f− ∈ S ∗ ⊕ S ∗ : Γ+ f+ = Γ− f− , Γ+ f+ = −τ (Γ+ f+ + Γ− f− ) }. +

−

0

0

0

1

1

In particular, − + − ∗ ∗ A(∞) = { f = f+ ⊕ f− ∈ S+ ⊕ S− : Γ+ 0 f+ = Γ0 f− , Γ1 f+ + Γ1 f− = 0 }.

Now assume that the Weyl functions M+ (λ) and M− (λ) corresponding to the + − − selfadjoint extensions A+ 0 = ker Γ0 and A0 = ker Γ0 , respectively, each belong to N1 . Clearly, the sum M+ (λ)+M− (λ) then also belongs to the class N1 . Therefore, the symmetric extension S of S+ ⊕ S− has a generalized Friedrichs extension. In fact, if limy→∞ M+ (iy) = 0 and limy→∞ M− (iy) = 0, then the value τ = ∞ corresponds to the generalized Friedrichs extension.

3. Sturm-Liouville expressions and interface conditions 3.1. Sturm-Liouville operators Let p and q be real-valued functions on an interval (0, b], b > 0, such that 1/p and q are integrable on (0, b], and consider the Sturm-Liouville expression L+ := −DpD + q on (0, b]. The following lemma can be checked directly. Lemma 3.1. Let L+,max be the maximal diﬀerential operator in L2 (0, b) associated with L+ on (0, b], b > 0. Then: (i) the restriction S+ of L+,max deﬁned by S+ = { u ∈ dom L+,max : u(0) = (pu )(0) = u(b) = 0 } is a closed, densely deﬁned, and symmetric operator with defect numbers (1, 1); (ii) the adjoint of S+ is given by ∗ S+ = { u ∈ dom L+,max : u(b) = 0 }; ∗ (iii) a boundary triplet for S+ is deﬁned by the boundary mappings Γ+ 0 u = −(pu )(0),

Γ+ 1 u = u(0).

Now let p and q be real-valued functions on an interval [−b, 0), b > 0, such that 1/p and q are integrable on [−b, 0), and consider the Sturm-Liouville expression L− = −DpD + q on the interval [−b, 0). The following analog of Lemma 3.1 is immediate. Lemma 3.2. Let L−,max be the maximal diﬀerential operator in L2 (−b, 0) associated with (3.1) on [−b, 0), b > 0. Then: (i) the restriction S− of L−,max deﬁned by S− = { u ∈ dom L−,max : u(0) = (pu )(0) = u(−b) = 0 } is a closed, densely deﬁned, and symmetric operator with defect numbers (1, 1);

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(ii) the adjoint of S− is given by ∗ S− = { u ∈ dom L−,max : u(−b) = 0 }; ∗ (iii) a boundary triplet for S− is deﬁned by the boundary mappings Γ− 0 u = −(pu )(0),

Γ− 1 u = −u(0),

The corresponding Weyl functions can be calculated by ﬁxing a fundamental system for the equation −(pu ) + qu = λu,

λ ∈ C.

(3.1)

Let ϕ(·, λ) and ψ(·, λ) denote the fundamental solutions of (3.1) on (0, b] which satisfy the initial conditions ϕ(0, λ) = 1, (pϕ )(0, λ) = 0, (3.2) ψ(0, λ) = 0, (pψ )(0, λ) = −1. Introduce the function M+ (λ) by M+ (λ) = −

ψ(b, λ) , ϕ(b, λ)

(3.3)

and deﬁne a solution of L+ u = λu by χ+ (·, λ) = ψ(·, λ) + M+ (λ)ϕ(·, λ). Then ∗ χ+ (x, λ) satisﬁes χ+ (b, λ) = 0 and thus it spans ker (S+ − λ). Furthermore, M+ (λ) is the Weyl function corresponding to the boundary triplet in (iii) of Lemma 3.1, + since Γ+ 0 χ+ (·, λ) = 1, Γ1 χ+ (·, λ) = M+ (λ), cf. (2.3). Likewise it can be shown that χ+ (·, λ) is the corresponding γ-ﬁeld, cf. (2.4). Observe that the functions ϕ(x, λ) and ψ(x, λ) in (3.2) can also be seen as a fundamental system for the equation (3.1) on the interval [−b, 0). Introduce the function M− (λ) by ψ(−b, λ) M− (λ) = , (3.4) ϕ(−b, λ) and deﬁne a solution of L− u = λu by χ− (·, λ) = ψ(·, λ) − M− (λ)ϕ(·, λ). Then ∗ χ− (x, λ) satisﬁes χ− (−b, λ) = 0 and thus it spans ker (S− − λ). Completely analogous to the previous case, M− (λ) is the Weyl function corresponding to the bound− ary triplet in (iii) of Lemma 3.2, since Γ− 0 χ− (·, λ) = 1, Γ1 χ− (·, λ) = M− (λ). Furthermore, χ− (·, λ) is the corresponding γ-ﬁeld. 3.2. Coupling of Sturm-Liouville operators ∗ ∗ in L2 [0, b] and S− in L2 [−b, 0] are used to deﬁne a The two diﬀerential operators S+ 2 diﬀerential operator in L [−b, b] by means of an interface conditions at the origin. ∗ ∗ ∗ ∗ For this purpose deﬁne the orthogonal sum S ∗ = S+ ⊕ S− of S+ and S− in 2 L [−b, b], so that S = S+ ⊕ S− has defect numbers (2, 2). A boundary triplet for S ∗ is obtained from Lemmas 3.1 and 3.2 and the construction in (2.9), which means for the present situation −(pu )(0+) u(0+) , u ∈ dom S ∗ . Γ0 u = , Γ1 u = (3.5) −(pu )(0−) −u(0−)

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Here the domain dom S ∗ is given by dom S ∗ = { u ∈ dom L+,max ⊕ L−,max : u(−b) = u(b) = 0 }, i.e., by the set of all u ∈ L2 [−b, b] which satisfy u, pu ∈ AC([−b, b] \ {0}), (pu ) ∈ L2 [−b, b], u(−b) = u(b) = 0.

(3.6)

The notation u(0±) = limt→±0 u(t) is used to indicate a reference to the underlying ∗ operator S± . Observe that the corresponding γ-ﬁeld and Weyl function are of the form (2.10), where γ± (λ) and M± (λ) correspond to the selfadjoint extensions A+ 0 and A− 0 determined by the boundary condition (pu )(0+) = 0 and (pu )(0−) = 0, respectively. The following proposition is an immediate consequence of Proposition 2.1 and Lemmas 3.1 and 3.2. ∗ ∗ Proposition 3.3. Let S ∗ = S+ ⊕ S+ be the diﬀerential operator in L2 [−b, b], where ∗ ∗ S+ and S− are as in Lemma 3.1 and Lemma 3.2, respectively, and let {C2 , Γ0 , Γ1 } be a boundary triplet for S ∗ determined by the boundary mappings Γ0 , Γ1 in (3.5), so that the γ-ﬁeld and the Weyl function are of the form (2.10). Then: (i) the linear relation S deﬁned by

S = { u ∈ dom S ∗ : u(0+) = u(0−),

(pu )(0+) = (pu )(0−) = 0 }

(3.7)

is a closed symmetric extension of S in L2 [−b, b] with defect numbers (1, 1); (ii) the adjoint of S is given by S ∗ = { u ∈ dom S ∗ : (pu )(0+) = (pu )(0−) := (pu )(0) };

(3.8)

0, Γ

1 } for S ∗ is determined by (iii) a boundary triplet {C, Γ

0 u = −(pu )(0), Γ

1 u = u(0+) − u(0−); Γ

(iv) the corresponding Weyl function is equal to M+ (λ) + M− (λ). Moreover, the selfadjoint extensions A(τ ) of S in L2 [−b, b] are given for τ ∈ R by dom A(τ ) = { u ∈ dom S ∗ : (pu )(0+) = τ (u(0+) − u(0−)), (pu )(0+) = (pu )(0−) }, and for τ = ∞ by dom A(∞) = { u ∈ dom S ∗ : u(0+) = u(0−), (pu )(0+) = (pu )(0−) }.

4. Generalized Friedrichs extensions and interface conditions The Sturm-Liouville expressions in Section 3 satisfy conditions which guarantee the interpretation as diﬀerential operators in the spaces L2 [0, b] and L2 [−b, 0] with Weyl functions belonging to the general Nevanlinna class N. Under additional conditions it can be shown that the Weyl functions belong to the Kac class N1 .

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Lemma 4.1. Assume that the coeﬃcient p in the Sturm-Liouville expression L+ = −DpD + q satisﬁes p(x) > 0 for 0 < x ≤ b and b dx < ∞, (4.1) p(x) 0 and that the coeﬃcient q(x) ≥ c, c ∈ R, is bounded from below and integrable on (0, b]. Then the Weyl function M+ (λ) in (3.3) belongs to the Kac class N1 , it is analytic on C \ [c, ∞), and lim M+ (λ) = 0.

(4.2)

λ→−∞

In particular, if q(x) ≥ 0 on (0, b], then M+ (λ) belongs to the class S of Stieltjes functions. Proof. It suﬃces to show that the spectral measure σ of M+ (λ) satisﬁes the relation supp σ ⊂ [c, ∞), and that (4.2) holds. It follows from the relation (3.1) that x x ϕ(x, λ) = 1 + ϕ (t, λ)dt, p(x)ϕ (x, λ) = (q(x) − λ)ϕ(t, z)dt, (4.3) 0

0

and these relations imply for 0 < x ≤ b and λ ≤ c that ϕ (x, λ) ≥ 0, and further that the functions ϕ(x, λ) and p(x)ϕ (x, λ) are nondecreasing with respect to x. In particular, ϕ(b, λ) has no zeros oﬀ [c, ∞), and according to (3.3) the relation supp σ ⊂ [c, ∞) is shown. The Wronskian identity and the initial conditions (3.2) imply that ψ(x, λ) 1 − = , ϕ(x, λ) p(x)ϕ(x, λ)2 and hence, with the relation (3.3), that b M+ (λ) = 0

1 dt . ϕ(t, λ)2 p(t)

(4.4)

As ϕ(x, λ) ≥ 1 for 0 < x ≤ b and λ ≤ c, it follows from the relations (4.3) that lim p(x)ϕ (x, λ) = ∞,

λ→−∞

lim ϕ(x, λ) = ∞.

λ→−∞

By dominated convergence, the relation (4.4) implies the relation (4.2). If q(x) ≥ 0 then supp σ ⊂ [0, ∞) and it follows from [8] that M+ (λ) ∈ S. The following result can be established in a completely analogous fashion. Lemma 4.2. Assume that the coeﬃcient p in the Sturm-Liouville expression L− = −DpD + q on [−b, 0) satisﬁes p(x) < 0 for −b ≤ x < 0 and 0 dx , (4.5) −∞ < −b p(x)

Generalized Friedrichs Extensions

143

and that the coeﬃcient q(x) ≤ c, c ∈ R, is bounded from above and integrable on [−b, 0). Then the Weyl function M− (λ) in (3.3) belongs to the Kac class N1 , it is analytic on C \ (−∞, c], and lim M− (λ) = 0.

(4.6)

λ→∞

In particular, if q(x) ≤ 0 on [−b, 0), then M− (λ) belongs to the class S− . A combination of Lemmas 4.1 and 4.2 in conjunction with the remarks following Proposition 2.1 now leads to a description of the generalized Friedrichs extension in the situation of Proposition 3.3. Proposition 4.3. Assume that the coeﬃcients p and q of the Sturm-Liouville expression −DpD + q on [−b, b] satisfy the assumptions of Lemma 4.1 on (0, b], and the assumptions of Lemma 4.2 on [−b, 0). Then the symmetric operator S in (3.7) has a generalized Friedrichs extension and it corresponds to τ = ∞. It is also possible to obtain a similar result for the case when the sign of the coeﬃcient p is not ﬁxed on (0, b] or on [−b, 0). Deﬁne for this purpose the following functions x 1 P0 (x) = dt, 0 |p(t)| x |t| P1 (x) = dt, 0 |p(t)| x 1 P01 (x) = dt. 0 P0 (t) The following result is inspired by [7] (and could be stated in a slightly more general fashion, similar to the formulations in [7]). Proposition 4.4. Assume that one of the following functions P01 (δ) P1 (δ)|p(δ)|

or

P0 (δ) δ

(4.7)

is locally integrable in a neighborhood of 0. Then the symmetric operator S in (3.7) has a generalized Friedrichs extension corresponding to τ = ∞. Proof. First consider the case of the interval (0, b]. It suﬃces to recall the inequality [7, Theorem 4.1 and (4.8)]: δ(y) 1 + y 0 P0 (t) dt |M+ (iy)| ≤ C , (4.8) yδ(y) where δ(y) is a monotonically decreasing function with δ(y) → 0,

yδ(y) → ∞,

y → ∞,

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which satisﬁes further conditions, cf. [7, Lemma 3.2]. Hence, if for some y0 > 0, the following inequality δ(y) ∞ 1 + y 0 P0 (t) dt dy < ∞ (4.9) y 2 δ(y) y0 is satisﬁed, then the function M+ (λ) ∈ N1 and limy→∞ M+ (iy) = 0. Now if one of the functions in (4.7) is locally integrable near 0 then there exists a monotonically decreasing, absolutely continuous function δ(y) : [y0 , ∞) → (0, 1) with the required properties such that the inequality (4.9) holds, cf. [7]. Completely similar to the case of the interval (0, b], the case of the interval [−b, 0) can be treated.

References [1] V.A. Derkach, S. Hassi, M.M. Malamud, and H.S.V. de Snoo, Generalized resolvents of symmetric operators and admissibility, Methods of Functional Analysis and Topology, 6 (2000), 24–55. [2] V.A. Derkach, S. Hassi, M.M. Malamud, and H.S.V. de Snoo, Boundary relations and their Weyl families, Trans. Amer. Math. Soc., to appear. [3] A. Fleige, S. Hassi, and H.S.V. de Snoo, A Kre˘ın space approach to representation theorems and generalized Friedrichs extensions, Acta Sci. Math. (Szeged), 66 (2000), 633–650. [4] A. Fleige, S. Hassi, H.S.V. de Snoo, and H. Winkler, Sesquilinear forms corresponding to a non-semibounded Sturm-Liouville operator, in preparation. [5] S. Hassi, M. Kaltenb¨ ack, and H.S.V. de Snoo, Triplets of Hilbert spaces and Friedrichs extensions associated with the subclass N1 of Nevanlinna functions, J. Operator Theory, 37 (1997), 155–181. [6] S. Hassi, H. Langer, and H.S.V. de Snoo, Selfadjoint extensions for a class of symmetric operators with defect numbers (1, 1), 15th OT Conference Proceedings, (1995), 115–145. [7] S. Hassi, M. M¨ oller, and H.S.V. de Snoo, Sturm-Liouville operators and their spectral functions, J. Math. Anal. Appl., 282 (2003), 584–602. [8] I.S. Kac and M.G. Kre˘ın, R-functions-analytic functions mapping the upper halfplane into itself, Supplement II to the Russian edition of F.V. Atkinson, Discrete and continuous boundary problems, Mir, Moscow, 1968 (Russian) (English translation: Amer. Math. Soc. Transl., (2) 103 (1974), 1–18. [9] I.S. Kac and M.G. Kre˘ın, On the spectral functions of the string, Supplement II to the Russian edition of F.V. Atkinson, Discrete and continuous boundary problems, Mir, Moscow, 1968 (Russian) (English translation: Amer. Math. Soc. Transl., (2) 103 (1974), 19–102). [10] H.-D. Niessen and A. Zettl, Singular Sturm-Liouville problems: The Friedrichs extension and comparison of eigenvalues, Proc. London Math. Soc., 64 (1992), 545–578.

Generalized Friedrichs Extensions Andreas Fleige Am S¨ udwestfriedhof 27 44137 Dortmund Deutschland e-mail: [email protected] Seppo Hassi Department of Mathematics and Statistics University of Vaasa P.O. Box 700 65101 Vaasa Finland e-mail: [email protected] Henk de Snoo Department of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen Nederland e-mail: [email protected] Henrik Winkler Department of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen Nederland e-mail: [email protected]

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Operator Theory: Advances and Applications, Vol. 163, 147–162 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Spectral Properties of Operator Polynomials with Nonnegative Coeﬃcients Karl-Heinz F¨orster and B´ela Nagy Dedicated to Professor Dr. Heinz Langer

Abstract. We study properties of the polynomial Q(λ) = λm I − S(λ) where S(λ) = λl Al +· · ·+λA1 +A0 and 1 ≤ m < l. The coeﬃcients Al , . . . , A0 are in the positive cone of an ordered Banach algebra or are positive operators on a complex Banach lattice E and I is the identity. We study the properties of the spectral radius of S(λ) if λ is a nonnegative real number, and its connection with the existence of spectral divisors with nonnegative coeﬃcients in the considered sense. We prove factorization results for nonnegative elements in an ordered decomposing Banach algebra with closed normal algebra cone and in the Wiener algebra. Earlier results on monic (nonnegative) operator polynomials are applied to the operator polynomial class studied here. Mathematics Subject Classiﬁcation (2000). MSC(2000): Primary 47A56; Secondary 46H99, 47B65 . Keywords. Operator polynomials, positive coeﬃcients, factorization, ordered Banach algebras, spectral radius.

1. Introduction In the present paper we consider the operator polynomial Q(λ) = λm I − λl Al − · · · − λA1 − A0 , where Al , . . . , A0 are nonnegative (= positive) operators on a complex Banach lattice E, Al = 0 and I is the identity operator on E. Most of our results will be valid and proved for polynomials of the type above with coeﬃcients in the cone of an ordered Banach algebra. This work was completed with partial support of the Hungarian National Science Grants OTKA Nos T-030042 and T- 047276 and partial support of the DAAD and the Technical University of Berlin.

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Concerning the used concepts of Banach algebras we refer to the monographs T.W. Palmer[18] and I. Gohberg, S. Goldberg and M. A. Kaashoek [10], whereas concerning the spectral theory of operator polynomials we refer to the monographs A.S. Markus [16] and L. Rodman [19], concerning the theory of nonnegative operators on Banach lattices we refer to the monographs P. Meyer-Nieberg [17] and H.H. Schaefer [23]. We will adopt several results on operator polynomials for our more general case, if no essential diﬀerences in their proofs appear. Monic operator polynomials (i.e., m > l) with nonnegative operator coeﬃcients Al , . . . , A0 have been considered in [4], [13], [15] and [19]. Here we investigate mainly the case 1 ≤ m < l. We change the notation and consider in the following polynomials q(λ) = λm e − λl al − · · · − λa1 − a0 ,

λ ∈ C,

(1.1)

where e is the unit element of a complex Banach algebra A and the coeﬃcients aj belong to a cone C in A for j = 0, 1, . . . , l; we say also, they are nonnegative (with respect to this cone). We deﬁne the polynomial s(λ) = λl al + · · · + λa1 + a0 .

(1.2)

By (s(λ)) we denote the spectral radius of s(λ). One of our main results is the following (see Section 5) : Let 1 ≤ m < l and (s(r0 )) < r0m for some r0 > 0. Then q(·) has a monic (right) spectral divisor of degree m with nonnegative coeﬃcients and a comonic (left) spectral divisor with nonnegative coeﬃcients. In Section 4 we prove factorization results for nonnegative elements in an abstract ordered decomposing Banach algebra with closed normal algebra cone. As corollaries we obtain results on factorizations in the Wiener algebra. In Section 2 we recall the notation of an ordered Banach algebra with closed (normal) cone, [20], and collect some spectral properties of elements in such cones. In Section 3 we study the function s : [0, ∞[−→ R+ with s (r) = (s(r)).

(1.3)

An important property of this function is its log-log convexity (for this notion see the text below). In the last section we consider operator polynomials and apply known results on monic operator polynomials to our case here.

2. Ordered Banach algebras In this section we consider ordered Banach algebras in the sense of [20]. We assume the reader to be familiar with the deﬁnition and elementary properties of Banach algebras. Throughout this section A will denote a (real or complex) Banach algebra with unit e and zero element 0. As in [20, §3] we call a subset C ⊂ A an algebra cone if C satisﬁes the following conditions:

Spectral Properties of Operator Polynomials 1. C + C ⊂ C, 3. C · C ⊂ C,

149

2. λ C ⊂ C for all λ ≥ 0, 4. e ∈ C.

C is called a proper cone, if −C ∩C = {0}. Any cone in A induces an ordering “≤” on A in the following way: a≤b if and only if b−a∈C (for every a, b ∈ A). It is well known that this is a partial ordering on A, i.e., ≤ is reﬂexive and transitive; ≤ is antisymmetric if and only if C is proper. If C is an algebra cone in A, then the induced partial ordering ≤ satisﬁes for a, b ∈ A and a scalar λ : 1 . 0 ≤ a, 0 ≤ b =⇒ 0 ≤ a + b, 3 . 0 ≤ a, 0 ≤ b =⇒ 0 ≤ a · b,

2 . 0 ≤ a, 0 ≤ λ =⇒ 0 ≤ λa 4 . 0 ≤ e.

Conversely, if ≤ is a partial ordering on A such that 1 .–4 . hold, then C = {a ∈ A : 0 ≤ a} is an algebra cone which induces ≤. If A is ordered by an algebra cone, then we call A an ordered Banach algebra. An algebra cone C of A is said to be normal if there exists a constant γ > 0 such that 0 ≤ a ≤ b in A implies a ≤ γ b. It is easy to see that a normal algebra cone is proper. In applications the Banach algebra L(E) of all linear bounded operators in a (complex) Banach lattice E with cone L(E)+ is an important example for an ordered Banach algebras with a closed normal algebra cone. Especially the algebra Cn×n of all complex n × n matrices with the cone of all entrywise nonnegative matrices is such an algebra. In the next proposition we collect some properties of an ordered Banach algebra with a normal cone. Proposition 2.1. Let A be an ordered Banach algebra with a closed normal algebra cone C. Then 1. The spectral radius is a monotone function on C; i.e., if 0 ≤ a ≤ b, then (a) ≤ (b). 2. For all a ∈ C its spectral radius (a) belongs to its spectrum σ(a); i.e., (a) ∈ σ(a) for a ∈ C. 3. Let a ∈ C and let λ ∈ C. Then λ ∈ / σ(a) and (λe − a)−1 ∈ C if and only if λ is real and (a) < λ. Proof. The ﬁrst two assertion are proved in [20, Theorem 4.1.1 and Proposition 5.1] We will prove the third assertion. It is clear that (a) < λ is suﬃcient for λ ∈ / σ(a); (λe − a)−1 ∈ C follows from the expansion of (λe − a)−1 at ∞ (C. Neumann’s series). Suppose that (λe − a)−1 ∈ C for some λ ∈ / σ(a). For n = 0, 1, 2 . . . set xn = (λe − a)−n , where x0 = (λe − a)0 = e. Clearly xn ∈ C, xn = 0 and λxn = axn + xn−1 for n = 1, 2, . . .. By induction on n it follows from the last equality that λn xn ∈ C, λn−1 xn ∈ C and λn xn ≥ λn−1 xn−1 ≥ x0 = e. Now we can proceed as in the proof of [23, Appendix 2.3, p. 264] to obtain that (a) < λ.

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3. The spectral radius of s(·) as a function on the nonnegative axis In this section we investigate the function s : [0, r0 [−→ R+ with s (r) = (s(r)). where s(·) is holomorphic on the open disc Dr0 = {z ∈ C | |z| < r0 }, so s(λ) =

∞

λj aj

for |λ| < r0 ,

j=0

and that the coeﬃcients aj are in the cone C of an ordered complex Banach algebra A. The next lemma may be known. For sake of completeness we include a proof; it is an appropriate modiﬁcation of the proof of [2, Lemma 3] Lemma 3.1. Let C be a normal closed cone in a complex Banach algebra. Then there exists a constant β > 0 such that s(λ) ≤ βs(|λ|),

(s(λ)) ≤ (s(|λ|))

for all functions s(·) which are holomorphic in Dr0 with coeﬃcients in C and for all λ ∈ Dr0 . Proof. C is normal; i.e., there exists a positive constant γ such that x, y ∈ C x ≤ y imply x ≤ γ y . Then u ∈ C, v ∈ A and

and

−u ≤ v ≤ u imply v ≤ 2γ u . Indeed, from u − v ≥ 0 and u + v ≥ 0 there follows 2γ u = γ u + v + (u − v) ≥ max{ u + v , u − v } ≥ 12 ( u + v + u − v ) ≥ v . Let λ = reiφ and θ ∈ [0, 2π]. Then s(λ) = eiθ s(λ) = ≤ 2 sup 0≤ω≤2π

∞

∞

rk (cos(θ + kφ) + i sin(θ + kφ))ak

k=0

r cos(ω + kφ)ak . k

k=0

Now −rk ak ≤ rk cos(ω + kφ)ak ≤ rk ak , −

∞

r k ak ≤

k=0

Therefore s(λ) ≤ 4γ

∞ k=0

∞ # k=0 k

for k = 0, 1, 2, . . .. Adding, we have

rk cos(ω + kφ)ak ≤

∞

rk ak ∈ C.

k=0

rk ak = s(|λ|). The inequality of the spectral

radii follows now from (s(λ)) ≤ β(s(|λ))k for k = 0, 1, 2, . . . (note that sk (·) is also holomorphic in Dr0 and has nonnegative coeﬃcients) and the well-known formula for the spectral radius.

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151

By a result of E. Vesentini (see [1, p. 52]) the function λ −→ log((s(λ))) (and then also the function λ −→ (s(λ)) ) is subharmonic on Dr0 . Since the coeﬃcients aj are in C we obtain from the lemma above s (r) = max (s(λ)) for |λ|=r

all r ∈ [0, r0 [. From the theory of subharmonic functions (see [14], Theorem 2.13) we obtain Proposition 3.2. Under the conditions above the function s is continuous and not decreasing, and log(s (·)) is convex in log(r) on ]0, r0 [. log(s (·)) is convex in log(r) on ]0, r0 [ means that the function ηs : ] − ∞, log(r0 )[→ R

with

t −→ log(s (et ))

for

r1 , r2 ∈]0, r0 [,

(3.1)

is convex, or equivalently s (r1 τ r2 1−τ ) ≤ s (r1 )τ S (r2 )1−τ

τ ∈ [0, 1].

We call a function which satisﬁes the last functional inequality log-log convex. Fundamental properties of log-log convex functions give Proposition 3.3. Assume that the function s(·) satisﬁes the conditions above. Let k ∈ N0 . 1. Let 0 < r1 < r2 < r3 < r0 be such that s (rj ) = rjk for j = 1, 2, 3. Then s (r) = rk for all r ∈ [r1 , r3 ]. 2. Let r1 , r2 > 0 be such that r1 = r2 and s (rj ) = rjk for j = 1, 2, let s be diﬀerentiable in r1 , and let s (r1 ) = kr1k−1 . Then s (r) = rk for all r in the closed interval with the endpoints r1 and r2 . Proof. The function ηs is convex. Conditions 1 imply that the graphs of ηs and of the line R → R with t −→ kt have three diﬀerent points in common. Therefore they coincide on the corresponding interval, this implies the assertion. Conditions 2 imply that the graphs of ηs and of the line have two diﬀerent points in common and in one of the points the line is the tangent of ηs in this point. Again, they coincide on the corresponding interval, and we obtain the assertion. Proposition 3.4. Assume that the function s(·) satisﬁes the conditions above. Let q(λ) = λm e−s(λ) for all λ ∈ Dr0 and some m ∈ N. Assume that there exist positive numbers r1 and r2 such that 0 < r1 < r2 < r0 and s (r) < rm for r ∈]r1 , r2 [. Then σ(q(·)) ∩ {λ ∈ C | r1 < |λ| < r2 } = ∅. Proof. For λ ∈ C with r1 < |λ| < r2 we obtain from Lemma 3.1 that (s(λ)) ≤ (s(|λ|)) = s (|λ|) < |λm |. Therefore q(λ) = λm (e − λ−m s(λ)) is invertible.

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Example 3.5. For n ∈ N and kj ∈ n × n (weighted cyclic) matrix ⎛ 0 λk1 ⎜ ⎜ 0 0 ⎜ .. .. S(λ) = ⎜ ⎜ . . ⎜ ⎝ 0 0 λkn 0

N0 = N ∪ {0} for j = 1, 2, . . . n consider the 0

···

λk2 .. .

..

··· ···

. 0 0

0 .. . .. . λkn−1 0

⎞ ⎟ ⎟ ⎟ ⎟ , λ ∈ C. ⎟ ⎟ ⎠

We assume that l := max kj ≥ 1. Then S(·) is a matrix polynomial of 1≤j≤n

degree l with entrywise nonnegative matrices as coeﬃcients. These matrices are nonnegative operators on the Banach lattice Cn and elements in the normal cone R+ n×n of the nonnegative n × n square matrices in the Banach algebra Cn×n of the complex n × n square matrices with the spectral norm. Let λ = 0. Then σ(S(λ)) = {z ∈ C : z n = λk } where k = k1 + · · · + kn . The eigenvalues of S(λ) have simple (geometric and algebraic) multiplicities, the corresponding eigenvectors of S(λ) to z ∈ σ(S(λ)) are multiples of (1, λ−k1 z, λ−k1 −k2 z 2 , . . . , λ−k1 −···−kn−1 z n−1 ). For r > 0 the matrix S(r) is nonnegative and irreducible. S(1) is row-stochastic; i.e., S(1)1 = 1, where 1 is the vector in Cn which has all comk ponents equal to 1. We have S (r) = r n . Therefore S is not convex if k < n. For k = n we obtain S (r) = r for r > 0, therefore S (1) = S (1) = 1 and S (1) = 0 in this case. For L(λ) = λm − S(λ) we have σ(L) = {z ∈ C : z nm = z k }, especially σ(L) = C if and only if mn = k.

4. Factorization in ordered decomposing Banach algebras In this section we introduce the concept of an ordered decomposing Banach algebra and prove a factorization result on elements in the algebra cone which will be applied in the next section to polynomials with nonnegative coeﬃcients. A Banach algebra A is called a decomposing Banach algebra if A is the direct sum of two closed subalgebras A+ and A− ; see [3], [10], [11]. Let P+ denote the bounded linear projection of A onto A+ annihilating A− . Then P− := I − P+ is the bounded linear projection of A onto A− annihilating A+ . Let A be at the same time an ordered Banach algebra with algebra cone C and a decomposing Banach algebra (with the projections P+ and P− ). We call A an ordered decomposing Banach algebra, if C is invariant under P+ and under P− ; i.e., P± C ⊂ C. The following theorem is an order theoretic version of Lemma 5.1 in Chapter 1 of [8]; cf. also Theorem 23.3 in [16]. Theorem 4.1. Let A be an ordered decomposing Banach algebra with closed normal algebra cone C. If a ∈ C and (a) < 1,

Spectral Properties of Operator Polynomials

153

then e − a admits a factorization e − a = b+ (e − b− ) with b+ ∈ A+ , e − b+ ∈ A+ ∩ C, b− ∈ A− ∩ C, the elements b+ and e − b− are −1 invertible, and b−1 − e ∈ A− ∩ C and (e − b− )−1 ∈ C. + ∈ A+ ∩ C, (e − b− ) Proof. Let a ∈ C and (a) < 1. We deﬁne the bounded linear operator T+ : A → A by T+ x = P+ (xa) for all x ∈ A. Then T+ C ⊂ A+ ∩ C and 0 ≤ T+ e = P+ a ≤ a. Therefore 0 ≤ T+ 2 e ≤ T+ a = P+ a2 ≤ a2 , and then 0 ≤ T+ n e ≤ an for n = 1, 2, . . .. The normality of the cone C implies T+ n e ≤ γan for n = 1, 2, . . . and some ∞ # constant γ. Therefore from (a) < 1 it follows that x ˆ = T+ n e converges, belongs to A+ ∩ C and is a solution of the equation

n=0

x − P+ (xa) = x − T+ x = e. Set b− = e − x ˆ+x ˆa, then b− = P− (ˆ xa) ∈ A− ∩ C. Next we consider the operator T− : A → A with T− y = P− (ay) for all y ∈ A. As above, T− C ⊂ A− ∩ C and ∞ # T− n e ≤ γan for n = 1, 2, . . . . Therefore yˆ = T− n e converges, belongs to n=0

C, and is a solution of the equation y − P− (ay) = y − T− y = e. Set b+ = e − P+ (aˆ y). Then b+ ∈ A+ (note that we assume e ∈ A+ ), e − b+ = P+ (aˆ y ) ∈ A+ ∩ C and b+ = (e − a)(e + P− (aˆ y )). The last equation implies x ˆ b+ = (e − b− )(e + P− (aˆ y )), and this is equivalent to xˆb+ − e = −b− + P− (aˆ y) − b− P− (aˆ y ). In the last equation the left-hand side belongs to A+ , while the right-hand side belongs to A− . Therefore both sides are equal to zero, consequently x ˆb+ = e = (e − b− )(e + P− (aˆ y )). Thus b+ is left invertible, and e − b− is right invertible. We employ now a standard argument to prove that these elements are invertible. We replace in the argument above a by τ a, where τ ∈ [0, 1]. Then τ a ∈ C and (τ a) < 1, and the formulae for x ˆ, yˆ, b+ and b− show that we have to replace them by ∞ ∞ # # x ˆ(τ ) = (τ T+ )m e, yˆ(τ ) = (τ T− )m e, n=0

b+ (τ ) = e − τ P+ (aˆ y (τ )),

n=0

b− (τ ) = τ P− (ˆ x(τ )a),

respectively. These functions are continuous on [0, 1], b+(τ ) is left invertible and e − b− (τ ) is right invertible for all τ ∈ [0, 1] , and b+ (0) = e − b− (0) = e is invertible. By Lemma 23.2 in [16], b+ = b+ (1) and b− = b− (1) are invertible and ˆ ∈ A+ ∩ C, (e − b− )−1 = e + P− (aˆ y ) ∈ C and (e − b− )−1 − e = P− (aˆ y) ∈ b−1 + = x A− ∩ C. We obtained above the equation b+ = (e − a)(e + P− (aˆ y)). Multiplying on the right by e − b− , we obtain b+ (e − b− ) = e − a.

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We will apply Theorem 4.1 to the Wiener algebra W (A; T), see [11], where A is an ordered Banach algebra and T = T1 denotes the unit circle, i.e., ∞

W (A; T) = {a(·) : a(λ) =

λk ak , λ ∈ T, (an ) ∈ lZ1 (A)}

k=−∞

where lZ1 (A) = {(ak )k∈Z ⊂ A|

∞ #

ak is convergent}. Clearly, ak is the kth

k=−∞

Fourier coeﬃcient of a(·) ∈ W (A; T). Let A# be a Banach algebra. Then W (A; T) is a Banach algebra with norm ∞ a(·)W = k=−∞ ak . W (A; T) is the direct sum of the closed subalgebras W+ (A; T) = W−,0 (A; T) =

{ a(·) { a(·)

: :

ak = 0 for k < 0}, ak = 0 for k ≥ 0}.

Obviously, W+ (A; T) is the set of those functions in W (A; T) that admit continuous extensions onto {λ ∈ C : |λ| ≤ 1} and are analytic on D1 , while W−,0 (A; T) is the set of those functions in W (A; T) that admit continuous extensions onto {λ ∈ C | |λ| ≥ 1}, are analytic on {λ ∈ C; |λ| > 1} and vanish at inﬁnity. Clearly, the operator P+ deﬁned on W (A; T) by (P+ a(·))(λ) =

∞

λk ak ,

λ ∈ T,

k=0

is the projection of W (A; T) onto W+ (A; T) annihilating W−,0 (A; T). eW (·) with eW (λ) ≡ e for all λ ∈ T is the identity in W (A; T). For a(·) ∈ W (A; T) we denote by W (a(·)) the spectral radius of a(·) in the Banach algebra W (A; T). Let A be an ordered Banach algebra with algebra cone C. Then W (A; T) is an ordered Banach algebra with algebra cone CW = {a(·) ∈ W (A; T) : ak ∈ C for all k ∈ Z}. If C is normal (or closed), then CW is normal (or closed, respectively). The following corollary is the reformulation of Theorem 4.1 for the Wiener algebra W (A; T); see also Theorem 23.4 in [16]. Corollary 4.2. Let A be an ordered Banach algebra with closed normal algebra cone C. Suppose a(·) ∈ W (A; T) such that a(·) ∈ CW

and W (a(·)) < 1

Then eW (·) − a(·) admits a canonical factorization with respect to the unit circle; i.e., there exist b+ (·) ∈ W+ (A; T) and b− (·) ∈ W−,0 (A; T) such that e − a(λ) = b+ (λ)(e − b− (λ))

for all λ ∈ T,

and b+ (·) ∈ W+ (A; T), b− (·) ∈ W− (A; T) ∩ CW , eW (·) − b+ (·) ∈ W+ (A; T) ∩ CW , b+ (·)−1 ∈ W+ (A; T) ∩ CW , (eW (·) − b− (·))−1 − eW (·) ∈ W−,0 (A; T) ∩ CW and (eW (·) − b− (·))−1 ∈ CW .

Spectral Properties of Operator Polynomials

155

For functions in W (A; T) with only ﬁnitely many non-zero Fourier coeﬃcients we obtain stronger results. Note that W (A; T) is a subalgebra of C(A; T) := {f | f : T → A is continuous} and a(·) ∞ := max a(λ) A ≤ a(·) W for all a(·) ∈ |λ|=1

W (A; T). In the following we sometimes write for a ∈ A for clarity aA and A (a) instead of a and (a), respectively. Lemma 4.3. Let A be a Banach algebra, and let a(·) ∈ W (A; T) be such that ak = 0 for all k ∈ Z with |k| ≥ m for some m ∈ N. Then a(·) W ≤ (2m + 1) a(·) ∞

and

W (a(·)) = ∞ (a(·)).

Proof. The inequality follows from Cauchy’s formula for the coeﬃcients ak of a(·), see [16, p.127]. Applying this inequality to an (·), we obtain an (·) W ≤ (2nm + 1) an (·) ∞ for n = 1, 2 . . .. Using the known formula for the spectral radius, we have that W (a(·)) ≤ ∞ (a(·)). The reverse inequality follows from an (·) ∞ ≤ an (·) W for n = 1, 2 . . .. The next lemma is very similar to Lemma 3.1, and the proof of Lemma 3.1 needs only minor technical changes. Lemma 4.4. Let A be an ordered Banach algebra with closed normal algebra cone C. Then there exists a β > 0 such that a(·) ∞ ≤ β a(1) A

and

∞ (a(·)) = A (a(1))

for all a(·) ∈ CW . Theorem 4.5. Let A be an ordered Banach algebra with closed normal algebra cone C. Suppose that a(·) ∈ W (A; T) and m ∈ N is such that ak = 0 for |k| > m,

a(·) ∈ CW

and

A (a(1)) < 1.

Then eW (·) − a(·) admits a canonical factorization with respect to the unit circle and all assertions of Corollary 4.2 hold. Proof. Lemmata 4.3 and 4.4 show that W (a(·)) = ∞ (a(·)) = A (a(1)) < 1. Therefore, the assumptions of Corollary 3.4 are satisﬁed. The results of this section can be used to study matrix functions having values which are (in a certain sense) entrywise nonnegative matrices; for the factorization theory of general matrix functions relative to a curve we refer to [11]. In the next section we will apply the last theorem to polynomials with coeﬃcients in the cone of an ordered Banach algebra.

5. Factorization of polynomials with nonnegative coeﬃcients In this section we apply the results of the preceding section to obtain a factorization of polynomials with coeﬃcients in a normal cone of an ordered Banach algebra; the next theorem is the main result of this paper.

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Theorem 5.1. Let A be an ordered Banach algebra with closed normal algebra cone l # C. Let q(·) be a polynomial q(λ) = λm e − λj aj with 1 ≤ m < l and aj ∈ C for j=0

j = 0, 1 . . . , l, and al = 0. Suppose there exists a r0 > 0 such that (s(r0 )) < r0 m , where s(λ) =

l #

λj aj .

j=0

Then 1. The A-valued function e − a(·) with a(λ) =

l #

λj−m aj = λ−m s(λ) for λ = 0

j=0

admits a (right ) canonical factorization with respect to the circle Tr0 = {λ ∈ C | |λ| = r0 }; 2. q(·) has a monic spectral (right ) divisor c(·) of degree m with nonnegative coeﬃcients, i.e., c(λ) = λm e −

m−1

cj λj ,

cj ∈ C

(5.1)

j=0

for j = 0, 1, 2 . . . , m − 1, and σ(c(·)) = σ(q(·)) ∩ {λ ∈ C | |λ| < r0 }. 3. Let b(·) be the (uniquely deﬁned ) polynomial of degree l − m such that q(λ) = b(λ)c(λ) for all λ ∈ C. Then b(λ) = b0 (e −

l−m

λj bj ), b0 is invertible, b−1 0 ∈ C and bj ∈ C

(5.2)

j=1

for j = 1, 2, . . . , l − m, σ(b(·)) = σ(q(·)) ∩ {λ ∈ C : |λ| > r0 }. −1

4. c(t)

∈ C for r0 ≤ t; b(t)−1 ∈ C for 0 ≤ t ≤ r0 .

Proof. 1. The function ar0 (·) with ar0 (λ) = a(r0 λ) for λ ∈ T1 satisﬁes the assumptions of Theorem 4.5. Therefore we have e − ar0 (λ) = b+ (λ)(e − b− (λ)) for λ ∈ T1 with some b+ (·) ∈ W+ (A; T) and some b− (·) ∈ W−,0 (A; T). 2. Then q(r0 λ) = r0m b+ (λ)(λm e − λm b− (λ)) for λ ∈ T1 . Now b+ (·) and b−1 + (·) have analytic extensions onto D1 , and b− (·) has an analytic extension onto {z ∈ C : |z| > 1} which vanishes at inﬁnity. Then it follows (cf. 22.11 in [16]) that c(λ) = λm e − λm b− (r0−1 λ) is a monic polynomial of degree m and c(·) is a spectral divisor of q(·) with σ(c(·)) = σ(q(·)) ∩ {z ∈ C : |z| < r0 }. Now b− (·) ∈ CW , therefore λm b− (r0−1 λ) = λm−1 cm−1 + · · · + c0 with cj ∈ C. 3. That b(·) is uniquely deﬁned follows from Lemma 22.8 in [16]; this Lemma is proved for an algebra of operators but its proof works in our more general situation. From the proof of part 2 it follows that b(λ) = b+ (r0−1 λ) for λ ∈ C . b+ (·)−1 ∈

Spectral Properties of Operator Polynomials CW ∩ W+ (A; T) implies σ(b(·)) ⊂ C\Dr0 and b(λ)−1 =

∞ #

157

λk fk for |λ| ≤ r0 with

k=0

fk ∈ C for k = 0, 1, 2 . . . . Therefore b(t)−1 ∈ C for 0 ≤ t ≤ r0 ; especially, b0 is ∈ C. From Corollary 4.2 we know that eW (·) − b+ (·) ∈ CW . invertible and b−1 0 Then e − b0 ∈ C and bj ∈ C for j = 1, . . . , l − m. 4. From t > (c(·)) we obtain ⎛ ⎞−1 m−1 m−1 ∞ c(t)−1 = ⎝tm e − tj cj ⎠ = t−m cˆk ∈ C where cˆ = tj−m cj ∈ C. j=0

j=0

k=0

The proof for b(t)−1 ∈ C is similar.

Corollary 5.2. Let A be an ordered Banach algebra with closed normal cone C and #l let q(·) be the polynomial q(λ) = λm e − j=0 λi aj = λm e − s(λ) where 1 ≤ m < l and aj ∈ C, al = 0. Then there exist polynomials b(·) and c(·) as in (5.1) and (5.2) with q(λ) = b(λ)c(λ) for all λ ∈ C and an r0 > 0 such that part 4 of Theorem 5.1 holds if and only if (s(r0 )) < r0m . Any factorization with these properties is unique. Proof. We have to prove the “only if” part. Let r0 m e − s(r0 ) = q(r0 ) = b(r0 )c(r0 ) be such that b(r0 )−1 and c(r0 )−1 exist and belong to C. Then r0 m e − s(r0 ) has an inverse in C. It follows from Proposition 2.1.3 that (s(r0 )) < r0 m . The last assertion follows from [16, Lemma 22.8]. Theorem 5.3. Let A be an ordered complex Banach algebra with closed normal cone C and let the polynomials q(·) and s(·) satisfy the assumptions of the last theorem; hence S (r0 ) < r0m for some r0 > 0. Then 1. Either s (t) < tm for all t ∈]0, r0 ], or there exists exactly one r1 ∈]0, r0 [ such that S (r1 ) = r1 m . In the second case we have s (t) s (t)

< tm > tm

for for

r1 < t ≤ r0 , 0 < t < r1 ,

(5.3) (5.4)

# j r1 = (c(·)) ∈ σ(c(·)) and r1 m = (c+ (r1 )), here c+ (λ) = m−1 j=0 cj λ , see (5.1). 2. Either s (t) < tm for all t ∈ [r0 , ∞[, or there exists exactly one r2 ∈]r0 , ∞[ such that s (r2 ) = r2 m . In the second case we have s (t) s (t)

< tm > tm

for for

r0 ≤ t < r2 , r2 < t,

(5.5) (5.6)

r2 ∈ σ(b(·)), σ(b(·)) ⊂ {λ ∈ C : r2 ≤ |λ|} and (b+ (r2 )) = 1, here b+ (λ)) = l−m # j λ bj , see (5.2). j=1

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Proof. 1. Assume that s (t) ≥ tm for some t ∈]0, r0 [. The function s (·) is continuous, by the intermediate-value theorem there exists an r1 ∈ [t, r0 [ such that s (r1 ) = r1m . For any such r1 we obtain from the convexity of ηs (·) (see Proposition ηs (u0 ) − mu1 (u − u1 ) + mu1 for u1 ≤ u ≤ u0 . Here uj = log rj 3.2) that ηs (u) ≤ u0 − u1 for j = 0, 1. Clearly ηs (u0 ) < mu0 implies ηs (u) < mu for u1 < u ≤ u0 . This is equivalent to s (t) < tm for t ∈]r1 , r0 ]. Now it is clear there is exactly one r ∈]0, r0 ] such that s (r) = rm . The inequalities (5.3) and (5.4) are now clear. For this r1 we have r1 m ∈ σ(s(r1 )) by Proposition 2.1.1. Therefore r1 ∈ σ(q(·)) ∩ Dr0 = σ(c(·)) by Theorem 5.1.2, thus r1 ≤ (c(·)). Set r = (c(·)). Then r ∈ [r1 , r0 ]. Further r ∈ boundary σ(c(·)) ⊂ σ(q(·)), see [16, Lemma 22.3, p.112], therefore rm ∈ σ(s(r)), and then rm ≤ (s(r)). From (5.3) we obtain r1 = r = (c(·)). For monic polynomials of degree m with coeﬃcients in C we have (c+ (·)) = (c(·))m = rm , see [19, Proposition 2.1]. 2. Assume that s (t) ≥ tm for some t ∈]r0 , ∞[. A similar argument as in the proof of part 1 shows that there exists exactly one r2 ∈]r0 , ∞[ with s (r2 ) = r2 m and s (t) < tm for t ∈ [r0 , r[. The inequalities (5.5) and (5.6) are now clear. For a proof of the other assertions we deﬁne ˜b(·) by ˜b(λ) = λl−m b−1 0 b(1/λ) for λ = 0 and ˜b(0) = bl−m , here we use (5.2). Then ˜b(·) is the monic polynomial with nonnegative coeﬃcients ˜b(λ) = λl−m e − (λl−m−1 b1 + · · · + bl−m ) = λl−m e − ˜b+ (λ). Note that σ(˜b(·))\{0} = { λ1 : λ ∈ σ(b(·))} = { λ1 : λ ∈ σ(q(·)), r0 ≤ |λ|} by Theorem 5.1.3. We will show that r12 = (˜b(·)). Now q(r2 ) = r2 m−l b0˜b( r12 )c(r2 ) is singular but b0 and c(r2 ) are invertible, thus ˜b( r12 ) is singular. Then r12 ∈ σ(˜b(·)), and r12 ≤ (˜b(·)). Set r˜ = (˜b(·)). Then r˜ ∈ boundary σ(˜b(·)), and b( r1˜ ) is singular. Therefore q( 1r˜ ) is not invertible, see [19, Proposition 2.1], and r0 < r1˜ ≤ r2 . Thus 1 1 1 1 1 r˜m ∈ σ(s( r˜ )), and then r˜m ≤ (s( r˜ )). From (5.5) we obtain r˜ = r2 . Now r2 ∈ { λ1 : λ ∈ σ(˜b(·)),λ = 0} = σ(b(·)) ⊂ {λ ∈ C : r2 ≤ |λ|}. By [19, Proposition 2.1] we have r2m−l = (˜b( r12 ), and then (b+ (r2 )) = (r2 l−m˜b+ ( r12 ) = 1.

6. Operator polynomials with nonnegative coeﬃcients In this section we consider the operator polynomial Q(λ) = λm I − λl Al − · · · − λA1 − A0 = λm I − S(λ)

(6.1)

where A0 , . . . , Al are nonnegative (= positive) operators on a complex Banach lattice E with Al = 0. We assume that 1 ≤ m < l and (S(r0 )) < r0m for some positive number r0 . The results of the preceding section (Theorem 5.1) show that Q(·) can be factorized in a special way, namely Q(λ) = B0 B(λ)C(λ)

for all λ ∈ C.

(6.2)

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Here B0 is invertible with B0−1 ∈ L(E)+ , B(·) is a comonic operator polynomial of degree l − m with nonnegative coeﬃcients, i.e., B(λ) = I − λB1 − · · · − λl−m Bl−m = I − B+ (λ),

(6.3)

B0−1 Al

where Bj ∈ L(E)+ for j = 1, . . . , l − m and Bl−m = = 0, and C(·) is a monic operator polynomial of degree m with nonnegative coeﬃcients, i.e., C(λ) = λm I − λm−1 Cm−1 − . . . − λ1 C1 − C0 = λm I − C+ (λ)

(6.4)

where Cj ∈ L(E)+ for j = 0, 1, . . . , m − 1. We will use results on monic (and comonic) operator polynomials with nonnegative coeﬃcients from [4], [5] and [19] to describe spectral properties of Q(·) on the circles with radius r such that S (r) = rm . Theorem 6.1. Let E, Q(·), S(·), B(·), C(·), B+ (·), C+ (·), B0 and r0 be as above, especially (S(r0 )) < r0 m . Then for all r ∈ [0, ∞[ with S (r) = rm the following statements hold: 1. rm ∈ σ(S(r)) and r ∈ σ(Q(·)). 2. Let 0 < r < r0 . Then r is a pole of order k of Q−1 (·) and the space of all Jordan chains of Q(·) corresponding to r has a ﬁnite dimension h if and only if r is a pole of R(·, r−m+1 C+ (r)) with residue of ﬁnite rank h. 3. Let r0 < r. Then r is a pole of order k of Q−1 (·) and the space of all Jordan chains of Q(·) corresponding to r has a ﬁnite dimension h if and only if 1 is a pole of R(·, B+ (r)) with residue of ﬁnite rank h. 4. If r is a pole of Q−1 (·) and the space of all Jordan chains of Q(·) corresponding to r is ﬁnite-dimensional, then {λ ∈ σ(Q(·)) | | λ | = r} = r · G. here G is the union of ﬁnitely many (ﬁnite) groups of roots of unity and consists entirely of poles of Q−1 (·) 5. Let r and λ0 be poles of Q−1 (·) with | λ0 | = r. Then the following hold: (a) The order of the pole λ0 of Q−1 (·) is not greater than the order of the pole r of Q−1 (·). (b) Every nontrivial Jordan chain of Q(·) corresponding to λ0 is a linearly independent set. Proof. Let 0 < r < r0 be such that S (r) = rm . Then B(λ0 ) is invertible for all λ0 ∈ Tr by Theorem 5.1.3. Therefore Q−1 (·) has a pole of order k at λ0 if and only C −1 (·) has a pole of order k at λ0 . By [16, Lemma 22.5, p.123] the Jordan chains of Q(·) and C(·) corresponding to λ0 coincide. Now the assertions of the theorem concerning the case 0 < r < r0 follow from the corresponding results of [19]; namely, assertion 2 follows from [19, Corollary 5.5], assertion 4 follows from [19, Corollary 4.7] and assertion 5(b) follows from [19, Corollary 5.4]. Assertion 5(a) is a consequence of the Pringsheim Theorem, see [22, p.262]. Let r0 < r be such that S (r) = rm . Then C(λ0 ) is invertible for all λ0 ∈ Tr by Theorem 5.1.3. Therefore Q−1 (·) has a pole of order k at λ0 if and only if B −1 (·)

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has a pole of order k at λ0 . Direct computation shows: xj for j = 0, 1, . . . , p form a Jordan chain of Q(·) corresponding to λ0 if and only if the vectors j j yj = C (j−i) (λ0 )xi for j = 0, 1, . . . , p i i=0 form a Jordan chain of B(·) corresponding to λ0 . Therefore Q−1 (·) has a pole of order k at λ0 if and only B −1 (·) has a pole of order k at λ0 ; recall that the order of a pole is the length of the longest nontrivial Jordan chain corresponding to it and that a Jordan chain is called nontrivial if its ﬁrst vector is non-zero. Further, it follows that the space of Jordan chains of Q(·) corresponding to λ0 has a ﬁnite dimension h if and only if the space of Jordan chains of B(·) corresponding to λ0 ˜ has a ﬁnite dimension h. We deﬁne now the monic operator polynomial B(·) by l−m ˜ B(1/λ) for λ = 0. Note that (see the proof of Theorem 5.3.2) B(λ) = λ 1 1 ˜ : λ ∈ σ(B(·))} and r˜ = = (B(·)). λ r ˜ is the comonic companion operator Then the companion operator CB˜ of B(·) of B(·), see [12, p. 187]; note that the results formulated there for the matrix case hold also for bounded linear operators in Banach spaces. Note also that in [16, §12] slightly diﬀerent companion operators are deﬁned, but the two variants are cogredient, i.e., they coincide after a permutation of some operator rows and corresponding operator columns. Therefore the results of [16, §12 and §13] hold for the companion operators deﬁned above and the following sequence of equivalences holds. Q−1 (·) has a pole of order k at λ0 (and the space of Jordan chains of Q(·) corresponding to λ0 has ﬁnite dimension h) if and only if B −1 (·) has a pole of order k at λ0 (and the space of Jordan chains of B(·) corresponding to λ0 has a ﬁnite dimension h, respectively) if and only if (I −·CB˜ )−1 has a pole of order k at λ0 (and the space of Jordan chains of (I −·CB˜ ) corresponding to λ0 has ﬁnite a dimension h, respectively), see [16, Lemma 12.5 and 12.6 and the remarks on p. 63] if and only if the resolvent R(·, CB˜ ) = (·I − CB˜ )−1 has a pole at λ˜0 = λ10 of order k (and the residuum of this resolvent at λ0 has ﬁnite rank h, respectively), see [16, Lemma 12.8] if and only if ˜ −1 (·) has a pole of order k at λ˜0 and the space of Jordan chains of B(·) ˜ correB ˜ sponding to λ0 has ﬁnite a dimension h, respectively), see [15, 2.1, Hilfssatz 1]. Now assertion 4 follows from [19, Corollary 4.7]. Assertion 5(a) is a consequence of the Pringsheim Theorem, see [22, p. 262]. From the connection between the Jordan chains of Q(·) and B(·) given above it follows directly that the nontrivial Jordan chains of Q(·) corresponding to λ0 = 0 are linearly independent if and only if the nontrivial Jordan chains of B(·) corresponding to λ0 = 0 are linearly independent if and only if the nontrivial ˜ corresponding to λ˜0 = 1 are linearly independent, see [16, Jordan chains of B(·) λ0 ˜ σ(B(·))\{0} ={

Spectral Properties of Operator Polynomials

161

Lemma 12.3 and 12.7 and remarks on p.63]. Assertion 5(b) follows now from [19, Corollary 5.4]. If λ0 = r, we can continue the equivalences above by [19, Corollary 5.5], and obtain: Q−1 (·) has a pole of order k at r and the space of Jordan chains of Q(·) corresponding to r has a ﬁnite dimension h if and only if ˜+ (˜ r˜ is a pole of the resolvent R(·, r˜−l+m+1 )B r ) and the residuum of this resolvent at r˜ has ﬁnite rank h, and this assertion is equivalent to 1 is a pole of the resolvent R(·, B+ (r)) and the residuum of this resolvent at 1 has ˜+ (˜ r ) = r˜B+ (r)). ﬁnite rank h (since r˜−l+m+1 B Acknowledgement The authors wish to thank two referees for their careful work and constructive remarks.

References [1] B. Aupetit, A Primer on Spectral Theory. Springer-Verlag, New York – Berlin – Heidelberg, 1991. [2] F.F. Bonsall, Endomorphismus of a Partially Ordered Vector Space without Order Unit. J. London Math. Soc. 30 (1955), 144–153. [3] K.F. Clancey and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators. Birkh¨ auser, Basel and Boston, 1981. [4] K.-H. F¨ orster and B. Nagy, Some Properties of the Spectral Radius of a Monic Operator Polynomial with Nonnegative Compact Coeﬃcients. Integral Equations Operator Theory, 14 (1991), 794–805. [5] K.H. F¨ orster and B. Nagy, On the Linear Independence of Jordan Chains. Operator Theory: Advances and Applications, 122 (2001), 229–245. [6] H.R. Gail, S.L. Hantler and B.A. Taylor. Spectral Analysis of M/G/1 and G/M/1 Type Markov Chains. Adv. Appl. Prob. 28 (1996), 114–165. [7] H.R. Gail, S.L. Hantler and B.A. Taylor. Matrix-Geometric Invariant Measures for G/M/1 Type Markov Chains. Comm. Statist. Stochastic Models 14 (3) (1998), 537– 569. [8] I. Gohberg and I. Feldman. Convolution Equations and Projection Methods for their Solution. Amer. Math. Soc., Providence, R. I., 1974. [9] I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators. I.OT49. Birkh¨ auser Verlag, 1990. [10] I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, Vol. II. Birkh¨ auser Verlag, Basel and Boston, 1993. [11] I. Gohberg, M.A. Kaashoek and I.M. Spitkovsky, An Overview of Matrix Factorization Theory and Operator Applications, in: Operator Theory: Advances and Applications, vol. 141, pp. 1–102, Birkh¨ auser Verlag, Basel and Boston, 2003. [12] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials. Academic Press, New York, 1982.

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[13] K.P. Hadeler, Eigenwerte von Operatorpolynomen. Archive Rat. Mech. Appl. 20 (1965), 72–80. [14] W.K. Hayman, P.B. Kennedy, Subharmonic Functions. Academic Press, London – New York – San Francisco, 1976. ¨ [15] G. Maibaum, Uber Scharen positiver Operatoren. Math. Ann. 184 (1970), 238–256. [16] A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils. Transl. of Math. Monographs, vol. 71. Amer. Math. Soc., Providence, 1988. [17] P. Meyer-Nieberg, Banach Lattices. Springer-Verlag, New York, 1991. [18] T.W. Palmer, Banach Algebras and The General Theory of *-Algebras. Cambridge University Press, Cambridge, 1994. [19] R.T. Rau, On the Peripheral Spectrum of a Monic Operator Polynomial with Positive Coeﬃcients. Integral Equations Operator Theory 15 (1992), 479–495. [20] H. Raubenheimer and S. Rode, Cones in Banach Algebras. Indag. Math. (N.S.) 7 (4) (1996), 489–502. [21] L. Rodman, An Introduction to Operator Polynomials. Operator Theory: Advances and Applications, vol. 38, Birkh¨ auser: Basel-Boston-Berlin, 1989. [22] H.H. Schaefer, Banach Lattices and Positive Operators. Springer Verlag, New York, 1980. [23] H.H. Schaefer, Topological Vector Spaces. Springer Verlag, New York, 1971. Karl-Heinz F¨ orster Technische Universit¨ at Berlin Fakult¨ at II Institut f¨ ur Mathematik, MA 6-4 Straße des 17. Juni 136 D-10623 Berlin Germany e-mail: [email protected] B´ela Nagy Department of Analysis Institute of Mathematics Technical University of Budapest H-1521 Budapest Hungary e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 163–189 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Szeg˝ o Pairs of Orthogonal Rational Matrixvalued Functions on the Unit Circle Bernd Fritzsche, Bernd Kirstein and Andreas Lasarow Dedicated to Professor Heinz Langer

Abstract. We study distinguished pairs of orthonormal systems of rational matrix-valued functions on the unit circle, namely the so-called Szeg˝o pairs. These pairs are determined by an initial condition and a sequence of strictly contractive q × q matrices, which is called the sequence of Szeg˝o parameters. The Szeg˝ o parameters contain essential information on the underlying q × q nonnegative Hermitian Borel measure on the unit circle. Mathematics Subject Classiﬁcation (2000). Primary 42C05, 47A56. Keywords. Orthogonal rational matrix-valued functions, Recurrence relations of Szeg˝ o-type, Szeg˝ o parameters, Favard-type theorem.

0. Introduction This paper continues the line of investigations started in [FKL1]–[FKL3] where we realized ﬁrst steps towards a Szeg˝o theory of orthogonal rational matrix-valued functions on the unit circle T. The main feature of our conception of Szeg˝ o theory is the distinguished role of the Christoﬀel-Darboux formulas (see [FKL2], [FKL3]). As a cornerstone of our approach we introduced in [FKL2, Section 6] the notions of left and right Christoﬀel-Darboux pairs for rational matrix-valued functions. The recursion formulas for left and right Christoﬀel-Darboux pairs which were obtained in Section 2 of [FKL3] realize a crucial step in constructing our Szeg˝ o theory. Namely, these recursion formulas indicated that there is a one-to-one correspondence between Christoﬀel-Darboux pairs and sequences of matrices which are connected to particular signature matrices. The work of the third author of the present paper was supported by the German Academy of Natural Scientists Leopoldina by means of the Federal Ministry of Education and Research on badge BMBF-LPD 9901/8-88.

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The main topic of this paper is to discuss those pairs of orthogonal systems of rational matrix-valued functions which are direct generalizations of the Szeg˝ o pairs of orthonormal systems of matrix polynomials which were discussed by Delsarte, Genin, and Kamp in [DGK] (cf. [DFK, Section 3.6]), who developed the theory along the classical case of orthogonal polynomials of Szeg˝ o [S] (see also [GS], [G1], and [G2]). The present paper is also guided by the work of Bultheel, Gonz´alez-Vera, Hendriksen, and Nj˚ astad on scalar orthogonal rational functions (see, e.g., [BGHN1], [BGHN2], and the probably ﬁrst work on this topic by Djrbashian [Dj]). Note that, in the case of matrix polynomials and in the scalar case of rational functions, Szeg˝ o recursions for orthogonal functions were derived in [DGK, Section V] and in [BGHN2, Chapter 4], respectively, using a diﬀerent way as demonstrated below. Szeg˝o pairs of rational matrix-valued functions on the unit circle T are special pairs of orthogonal systems which are determined by an initial condition and a sequence of strictly contractive q × q matrices which are called the Szeg˝o parameters. In [FKL3] it is already shown that, in the generic case, each pair of orthonormal systems of rational matrix-valued functions is characterized by an initial condition and a sequence of jqq -unitary matrices. Roughly speaking, Szeg˝ o pairs of orthonormal systems of rational matrix-valued functions correspond to sequences of positive Hermitian jqq -unitary matrices. Since there is a one-to-one correspondence between jqq -unitary matrices and strictly contractive q × q matrices (see, e.g., [D, Theorem 1.2]), Szeg˝o pairs of rational matrix-valued functions on the unit circle T can be described by an initial condition and a sequence of strictly contractive q × q matrices. A brief synopsis is as follows. In addition to several notations we will use, we explain in Section 1 some basics on rational matrix-valued functions and, especially, we recall the Christoﬀel-Darboux formulas which every pair of orthonormal rational matrix-valued functions satisﬁes. In Section 2, we introduce the notion Szeg˝o pair of orthonormal systems in the context of rational matrix-valued functions and, by an application of the Christoﬀel-Darboux formulas, we will see that such pairs of orthonormal systems are determined by an initial condition and a sequence of strictly contractive q × q matrices via certain recurrence relations. Then, in Section 3, we study an inverse question about this and we prove a result which is often called Favard-type theorem. Our Favard-type theorem says that if we have a pair of rational matrix-valued functions which satisﬁes the initial condition and which is recursively connected by some strictly contractive q × q matrices based on the recurrence relations in Section 2, then there is a (not necessarily unique) nonnegative Hermitian q × q matrix measure on the Borelian σ-algebra on the unit circle with respect to which the sequences of rational matrix-valued functions form left and right orthonormal systems, respectively. Finally, we will see in Section 4 that the Szeg˝ o parameter En , which determines the elements Xn and Yn of a Szeg˝ o pair of rational matrix-valued functions by the recurrence relations assuming the knowledge of the previous elements Xn−1 and Yn−1 , can be computed by some integral formulas and, in particular, in terms of Xn−1 and Yn−1 .

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165

1. Notation and preliminaries Throughout this paper, let p and q belong to the set N of all positive integers. Let us use C, Z, and N0 to denote the sets of all complex numbers, of all integers, and of all nonnegative integers, respectively. If m ∈ Z and if n ∈ Z or n = +∞, then we will write Nm,n for the set of all integers k which satisfy m ≤ k ≤ n. If X is a nonempty set, then let Xp×q be the set of p × q matrices each entry of which belongs to X. The notation 0p×q stands for the null matrix that belongs to the set Cp×q of all p × q complex matrices, and the identity matrix which belongs to Cq×q will be denoted by Iq . If the size of a null matrix or a identity matrix is obvious, we will omit the indices. If A ∈ Cq×q , then rank A and det A indicate the rank of A and the determinant of A, √ respectively. Furthermore, if A is a nonnegative Hermitian q × q matrix, then A stands for the (unique) nonnegative Hermitian square root of A. The symbol T stands for the unit circle, D for its interior, and E for its exterior with respect to the extended complex plane C0 := C ∪ {∞}, i.e., let T := {z ∈ C : |z| = 1}, D := {z ∈ C : |z| < 1}, and E := C0 \ (D ∪ T). We will use the notation T1 to designate the set of all sequences (αj )∞ j=1 of complex numbers which satisfy αj αk = 1 for all positive integers j and k. Obviously, if (αj )∞ j=1 ∈ T1 , then αj ∈ T for all j ∈ N. For technical reasons, throughout this paper, we set α0 := 0. Let Bp×q (respectively, B1 ) be the σ-algebra of all Borel subsets of Cp×q (respectively, C), and let BT := B1 ∩ T. The linear Lebesgue measure on (T, BT ) is designated by λ. If f is a matrix-valued function deﬁned on a subset M of C0 with T ⊆ M, then we will write f for the restriction of f to T. If G is a nonempty ∗ subset 0 and if f is a matrix-valued function deﬁned on G, then f (z) is short ofC ∗ for f (z) , z ∈ G. & ' If α ∈ C \ (T ∪ {0}), then bα denotes the function bα : C0 \ α1 → C given by & ' + α α−w for w ∈ C \ α1 |α| 1−αw (1.1) bα (w) := 1 for w = ∞ . |α| Further, let b0 : C → C be deﬁned by b0 (w) := w for each w ∈ C. Clearly, if α ∈ D, then the function bα is exactly the elementary Blaschke factor corresponding to α. Now we turn our attention to left and right Cq×q -modules of rational matrixvalued functions with prescribed pole structure. Let τ ∈ N or τ = +∞, and let (αj )τj=1 be a sequence of complex numbers. Further, let πα,0 : C0 → C be the constant function with value 1 and let Rα,0 denote the set of all constant complexvalued functions deﬁned on C0 . For each n ∈ N1,τ , let πα,n: C → C be deﬁned by πα,n (w) :=

n A j=1

(1 − αj w)

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and let Rα,n designate the set of all complex-valued functions f which are rational 1 P with some complex polynomial P of and which admit a representation f = πα,n degree not greater than n. Further, for each n ∈ N0,τ , let . n n $ $ 1 Pα,n := and Zα,n := {αj }, αj j=1 j=1 B & ' 1 := 0, and 0j=1 α1j := ∅. For where we use the conventions 10 := ∞, ∞ each n ∈ N0,τ , every function which belongs to Rα,n is holomorphic in C0 \ Pα,n . Obviously, if 0 ≤ n < τ , then Rα,n does not depend on the numbers αj , j ∈ Nn+1,τ . Identifying a constant complex-valued function deﬁned on C0 with its restriction to C, one can easily see that in the case αj = 0 for all j ∈ N1,n the class Rα,n coincides with the set Pn of all complex-valued polynomials of degree not greater than n. If n1 and n2 are integers with 0 ≤ n1 ≤ n2 ≤ τ , then Rα,n1 ⊆ Rα,n2 . If a sequence (αj )∞ j=1 of complex numbers is given, then let Rα,∞ :=

∞ $

Rα,n .

n=0

Every function f ∈ Rα,∞ is holomorphic in C0 \

B ∞ ( j=1

1 αj

) .

q×q -submodule For each n ∈ N0 , the class Rp×q α,n can be considered as a right C q×q p×q of the right C -module Rα,∞ . On the other hand, for each n ∈ N0 , the class p×p -submodule of the left Cp×p -module Rp×q Rp×q α,n is also a left C α,∞ . q×q to denote the constant Now let (αj )∞ j=1 ∈ T1 . We will use Bα,0 : C0 → C (q)

(q)

matrix-valued function with value Iq and, for each n ∈ N, let the function Bα,n : C0 \ Pα,n → Cq×q be deﬁned by ⎛ ⎞ n A (q) (w) := ⎝ bαj (w)⎠ Iq . Bα,n j=1

) ( (q) If τ ∈ N0 or τ = +∞, then Bα,k : k ∈ N0,τ is both a basis of the right q×q -module Rq×q Cq×q -module Rq×q α,τ and a basis of the left C α,τ . (For a more detailed discussion of these modules we refer to [FKL1].) In particular, if n ∈ N0 and if q×q X ∈ Rq×q such that α,n , then there are unique matrices A0 , A1 , . . . , An ∈ C #n (q) X = j=0 Aj Bα,j and the reciprocal rational (matrix-valued ) function X [α,n] of #n (q) [α,n] := j=0 A∗n−j Bβ,j where the X with respect to (αj )∞ j=1 and n is given by X sequence (βj )∞ j=1 is deﬁned by βk := αn+1−k for each k ∈ N1,n and βj := αj for each integer j with j ≥ n + 1. In particular,

and

X [α,n] ∈ Rq×q α,n

(1.2)

X [α,n] (αn ) = A∗n .

(1.3)

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If n ∈ N0 and if αj = 0 for each j ∈ N1,n , then every X ∈ Rq×q α,n is a q × q matrix polynomial of degree not greater than n and X [α,n] is exactly the reciprocal matrix ˜ [n] of X with respect to the unit circle T and the formal degree n polynomial X (cf. [DGK] or also [DFK, Section 1.2]). A mapping F whose domain is the σ-algebra BT of all Borelian subsets of T and whose values are nonnegative Hermitian complex q × q matrices is called nonnegative Hermitian ∞ q × q∞Borel measure on T if it is countably additive, i.e., if B # F satisﬁes F Ak = F (Ak ) for every inﬁnite sequence (Ak )∞ k=1 of pairk=1

k=1

wise disjoint sets which belong to BT . We will use Mq≥ (T, BT ) to denote the set of all nonnegative Hermitian q × q Borel measures on T. Let F ∈ Mq≥ (T, BT ). For each j ∈ N1,q and each k ∈ N1,q , the entry function Fjk of F in the jth row and kth column is a complex-valued measure. One can easily #q see that F is absolutely continuous with respect to the trace measure τ F := j=1 Fjj of F . An ordered pair [Φ, Ψ] consisting of Borel measurable matrix-valued functions Φ : T → Cp×q and Ψ : T → Cs×q is called left-integrable with respect to F if each entry of the matrix-valued function ΦFτ Ψ∗ is integrable with respect to τ F and the corresponding integral is deﬁned by ΦdF Ψ∗ := ΦFτ Ψ∗ dτ F. (1.4) T T We will also write T Φ(z)F (dz)Ψ∗ (z) for the integral given in (1.4). An ordered pair [Φ, Ψ] consisting of Borel measurable matrix-valued functions Φ : T → Cq×p and Ψ : T → Cq×s is said to be right-integrable with respect to F if [Φ∗ , Ψ∗ ] to F. It is known that the set p × q-L2l (T, BT , F ) is left-integrable with respect 2 respectively, q × p − Lr (T, BT , F ) of all matrix-valued functions Φ : T → Cp×q (respectively, Φ : T → Cq×p ) for which [Φ, Φ] is left-integrable (respectively, rightintegrable) with respect to F is a left (respectively, right) Cp×p -semi Hilbert module, where the Gramian structure is given by (Φ, Ψ)F,l := respectively, (Φ, Ψ)F,r := ΦdF Ψ∗ Φ∗ dF Ψ . (1.5) T

T

For more details on the integration theory with respect to nonnegative Hermitian q × q measures, we refer to Kats [Kt] and Rosenberg [R]. Now let a sequence (αj )∞ j=1 of numbers which belong to C \ T and a nonnegative Hermitian q × q Borel measure F on T be given. Then one can immediately 2 q×p see that {X : X ∈ Rp×q α,∞ } ⊆ p × q − Ll (T, BT , F ) and {X : X ∈ Rα,∞ } ⊆ 2 p×q q×p q × p − Lr (T, BT , F ) hold. If X and Y belong to Rα,∞ (respectively, Rα,∞ ), then (X, Y )F,l (respectively, (X, Y )F,r ) is short for (X, Y )F,l (respectively, (X, Y )F,r ). q×q For each n ∈ N0 , it is readily checked that Rq×q -semi Hilbert α,n is both a left C 2 q×q submodule of q × q − Ll (T, BT , F ) and a right C -semi Hilbert submodule of q × q − L2r (T, BT , F ). Under a certain assumption on the measure F the space q×q Rq×q -Hilbert module. This is based on a non-degeneracy concept which α,n is a C is studied in [FKL1, Section 5]. If n ∈ N0 , then a nonnegative Hermitian q × q

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Borel measure F on T is called non-degenerate of order n if the block Toeplitz (F ) (F ) matrix Tn := (Γj−k )nj,k=0 is nonsingular where (F ) Γj := z −j F (dz), j ∈ Z, T

are the Fourier coeﬃcients of F . For each n ∈ N0 , we will use Mq,n ≥ (T, BT ) to denote the set of all F ∈ Mq≥ (T, BT ) which are non-degenerate of order n. Clearly, (T, BT ) ⊆ Mq,n Mq,n+1 ≥ ≥ (T, BT ) holds for all n ∈ N0 . A nonnegative Hermitian q×q Borel measure F on T is said to be non-degenerate of order ∞ if F belongs to Mq,∞ ≥ (T, BT ) :=

∞ :

Mq,n ≥ (T, BT ).

n=0

Observe that in [FKL1, Section 5] several characterizations of the set Mq,n ≥ (T, BT ) q,n are given. If n ∈ N0 and if F ∈ M≥ (T, BT ), in view of [FKL1, Theorem 5.8] one can easily check that Rq×q α,n (with the Gramian structure given in (1.5)) is both a left and a right Cq×q -Hilbert module. If F belongs to Mq,∞ ≥ (T, BT ), the space q×q turns out to be both a left and a right C -pre-Hilbert module. Rq×q α,∞ q ∞ Let (αj )j=1 ∈ T1 and let F ∈ M≥ (T, BT ). Further, let τ ∈ N0 or let τ = +∞. A sequence (Xk )τk=0 of matrix-valued functions which belong to Rq×q α,∞ is called a left (respectively, right) orthonormal system corresponding to (αj )∞ j=1 and F if the following two conditions are satisﬁed: (i) For each k ∈ N0,τ , the function Xk belongs to Rq×q α,k . (ii) For each j ∈ N0,τ and each k ∈ N0,τ , respectively, (Xj , Xk )F,r = δjk I . (Xj , Xk )F,l = δjk I We call a pair [(Xk )τk=0 , (Yk )τk=0 ] consisting of a left orthonormal system τ corresponding to (αj )∞ j=1 and F and a right orthonormal system (Yk )k=0 corresponding to (αj )∞ j=1 and F a pair of orthonormal systems corresponding to ∞ (αj )j=1 and F . In [FKL2, Corollary 4.4] it is veriﬁed that if (αj )∞ j=1 ∈ T1 , if τ ∈ N0 or τ = +∞, and if F ∈ Mq≥ (T, BT ), then there exists a pair [(Xk )τk=0 , (Yk )τk=0 ] of orthonormal systems corresponding to (αj )∞ j=1 and F if and only if the underlying measure F is non-degenerate of order τ . Pairs of orthonormal systems satisfy the following Christoﬀel-Darboux formulas. (Xk )τk=0

q,τ Theorem 1.1. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Further, let [(Xk )τk=0 , (Yk )τk=0 ] be a pair of orthonormal systems corresponding to (αj )∞ j=1 and F . For all n ∈ N1,τ and every choice of v and w in C0 \ Pα,n , n−1 ∗ 1 − bαn (v)bαn (w) Xk∗ (v)Xk (w) = Yn[α,n] (v) Yn[α,n] (w) − Xn∗ (v)Xn (w), k=0

n−1 ∗ 1 − bαn (v)bαn (w) Yk (v)Yk∗ (w) = Xn[α,n] (v) Xn[α,n] (w) − Yn (v)Yn∗ (w), k=0

Szeg˝o Pairs of Orthogonal Rational Matrix-valued Functions

n Xk∗ (v)Xk (w) 1 − bαn (v)bαn (w)

= and

169

∗ Yn[α,n] (v)

k=0 Yn[α,n] (w)

− bαn (v)bαn (w)Xn∗ (v)Xn (w),

n Yk (v)Yk∗ (w) 1 − bαn (v)bαn (w) k=0

∗ = Xn[α,n] (v) Xn[α,n] (w) − bαn (v)bαn (w)Yn (v)Yn∗ (w). A proof of Theorem 1.1 is given in [FKL2, Theorem 5.4 and Corollary 5.5]. Note that, conversely, systems of rational matrix-valued functions satisfying identities of the type given in Theorem 1.1 are necessarily orthonormal systems corresponding to some nonnegative Hermitian measure (see [FKL3, Theorems 3.10 and 4.12]). q,τ Remark 1.2. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). τ τ Further, let [(Xk )k=0 , (Yk )k=0 ] be a pair of orthonormal systems corresponding to (αj )∞ j=1 and F . For each n ∈ N1,τ , the following statements are satisﬁed (see [FKL2, Remark 6.2, Theorems 6.7, 6.9, and 6.10]):

(a) There is a number zn ∈ T such that zn · det Xn (w) = det Yn (w)

and det Xn[α,n] (w) = zn · det Yn[α,n] (w)

are satisﬁed for every choice of w ∈ C0 \ Pα,n . [α,n] (b) If |αn | < 1, then det Xn vanishes nowhere in E \ Pα,n and det Xn vanishes nowhere in D \ Pα,n−1 . [α,n] (c) If |αn | > 1, then det Xn vanishes nowhere in D \ Pα,n and det Xn vanishes nowhere in E \ Pα,n−1 .

2. Szeg˝ o pairs of orthonormal systems Inspired by the case of matrix polynomials we turn our attention to distinguished pairs of orthonormal systems of rational matrix-valued functions. q,τ Deﬁnition 2.1. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). For each k ∈ N0,τ , let + α k if αk = 0 |αk | (2.1) ηk := −1 if αk = 0 .

Let [(Xk )τk=0 , (Yk )τk=0 ] be a pair of orthonormal systems corresponding to (αj )∞ j=1 and F . Then [(Xk )τk=0 , (Yk )τk=0 ] is called a Szeg˝ o pair of orthonormal systems corresponding to (αj )∞ j=1 and F if for each n ∈ N1,τ the following statements holds: (i) If (1 − |αn |)(1 − |αn−1 |) > 0, then the matrices −1 ηn ηn−1 (1 − |αn−1 |2 ) [α,n−1] Yn−1 (αn−1 ) Yn[α,n] (αn−1 ) 1 − αn αn−1

(2.2)

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B. Fritzsche, B. Kirstein and A. Lasarow and

−1 [α,n−1] ηn ηn−1 (1 − |αn−1 |2 ) [α,n] Xn (αn−1 ) Xn−1 (αn−1 ) 1 − αn αn−1

(2.3)

are both positive Hermitian. (ii) If (1 − |αn |)(1 − |αn−1 |) < 0, then the matrices

and

−1 [α,n−1] |αn−1 |2 − 1 Yn (αn−1 ) Xn−1 (αn−1 ) 1 − αn αn−1

(2.4)

−1 |αn−1 |2 − 1 [α,n−1] Yn−1 (αn−1 ) Xn (αn−1 ) 1 − αn αn−1

(2.5)

are both positive Hermitian. Observe that, in view of Remark 1.2, the inverse matrices occurring in the formulae (2.2)–(2.5) are well deﬁned. The notion introduced in Deﬁnition 2.1 is a generalization of the corresponding notion in the matrix polynomial case, i.e., in the case that αj = 0 for all j ∈ N (see [DGK] and [DFK, Deﬁnition 3.6.5]). q,τ Remark 2.2. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). From [FKL2, Proposition 3.7 and Corollary 4.4] and the polar decomposition of matrices one can immediately see that there exists a Szeg˝o pair of orthonormal systems corresponding to (αj )∞ j=1 and F . q,τ Remark 2.3. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). From [FKL2, Proposition 3.7] and the polar decomposition of matrices one can also see that the following statements hold:

(a) If [(Xk )τk=0 , (Yk )τk=0 ] is a Szeg˝o pair of orthonormal systems corresponding to (αj )∞ j=1 and F , then, for all unitary complex q × q matrices U and V, the ˜ pair [(Xk )τk=0 , (Y˜k )τk=0 ] given for all k ∈ N0,τ by + + UXk if αk ∈ D Yk V if αk ∈ D ˜ ˜ and Yk := Xk := ∗ V Xk if αk ∈ E Yk U∗ if αk ∈ E is also a Szeg˝o pair of orthonormal systems corresponding to (αj )∞ j=1 and F . ˜ k )τ , (Y˜k )τ ] are Szeg˝ (b) If [(Xk )τk=0 , (Yk )τk=0 ] and [(X o pairs of orthonormal k=0 k=0 ˜ 0 (0) −1 and systems corresponding to (αj )∞ and F , then U := X0 (0) X j=1 −1 Y0 (0) are unitary complex q × q matrices and for all k ∈ N0,τ V := Y˜0 (0) the identities + + UXk if αk ∈ D Yk V if αk ∈ D ˜k = and Y˜k = X ∗ V Xk if αk ∈ E Yk U∗ if αk ∈ E are satisﬁed.

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171

q,τ Remark 2.4. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Furτ τ ther, let [(Xk )k=0 , (Yk )k=0 ] be a Szeg˝o pair of orthonormal systems corresponding to (αj )∞ j=1 and F . In view of Remark 1.2 and Deﬁnition 2.1, one can get that there exists a z ∈ T such that for all n ∈ N1,τ and all w ∈ C0 \ Pα,n the equalities

z · det Xn (w) = det Yn (w)

and

det Xn[α,n] (w) = z · det Yn[α,n] (w)

are satisﬁed. q,τ Deﬁnition 2.5. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). For each k ∈ N0,τ , let ηk be given by (2.1). Further, let [(Xk )τk=0 , (Yk )τk=0 ] be a τ Szeg˝o pair of orthonormal systems corresponding to (αj )∞ j=1 and F . Then (En )n=1 given by ⎧ [α,n] −1 ⎨ ηn ηn−1 Xn (αn−1 ) Yn (αn−1 ) if (1−|αn |)(1−|αn−1 |) > 0 ∗ (2.6) En:= −1 [α,n] ⎩η η (αn−1 ) Xn (αn−1 ) if (1−|αn |)(1−|αn−1 |) < 0 . n n−1 Yn

is said to be the sequence of Szeg˝ o parameters corresponding to [(Xk )τk=0 , (Yk )τk=0 ]. Note that, in view of Remark 1.2, the inverses occurring in (2.6) are well deﬁned. The notion introduced in Deﬁnition 2.5 is a generalization of the corresponding notion in the matrix polynomials case (see [DGK] and [DFK, Deﬁnition 3.6.6]). This can be seen from [DFK, Lemma 3.6.14 and Proposition 3.6.5]. Moreover, under the assumptions of Deﬁnition 2.5 from [FKL2, Remark 6.2 and Lemma 6.5] and Remark 1.2, for each n ∈ N1,τ , it follows that ⎧ −1 ⎨ ηn ηn−1 Xn[α,n] (αn−1 ) Yn (αn−1 ) if (1−|αn |)(1−|αn−1 |) > 0 ∗ En = (2.7) −1 [α,n] ⎩η η Y (α ) X (α ) if (1−|αn |)(1−|αn−1 |) < 0 . n n n−1 n n−1 n−1 In the matrix polynomial case the Szeg˝ o parameters can be used to obtain recursion formulas for Szeg˝o pairs of orthonormal systems (of matrix polynomials). We will see that these formulas can be generalized to the rational case. q,τ Remark 2.6. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Let [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝o pair of orthonormal systems corresponding to τ (αj )∞ o parameters. Then j=1 and F , and let (En )n=1 be the sequence of its Szeg˝ q,τ T F belongs to M≥ (T, BT ). Moreover, [(YkT )τk=0 , (XkT )τk=0 ] is a Szeg˝o pair of orT T τ thonormal systems corresponding to (αj )∞ j=1 and F , and (En )n=1 is the sequence of its Szeg˝ o parameters (see [FKL1, Remark 5.11], [FKL2, Remark 2.10 and Remark 3.5], and (2.7)). q,τ Remark 2.7. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). If A is a positive Hermitian q×q matrix, then [FKL1, Remarks 5.3, 3.10, and 5.13] yield √ −1 −1 √ A belongs that H : BT → Cq×q given by H(B) := A F (T) F (B) F (T) τ τ also to Mq,τ (T, B ) and satisﬁes H(T) = A. Moreover, if [(X ) , (Y T k k=0 k )k=0 ] is a ≥ ∞ Szeg˝o pair of orthonormal systems corresponding to (αj )j=1 and F , and if (En )τn=1

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is the sequence of its Szeg˝o parameters, then, in view of [FKL2, Remark 2.8], it √ −1 τ τ √ −1 is not hard to see that Xk F (T) A , A F (T)Yk k=0 is a Szeg˝o k=0 τ pair of orthonormal systems corresponding to (αj )∞ j=1 and H, and (En )n=1 is also the sequence of the Szeg˝o parameters corresponding to this Szeg˝ o pair. q,τ Lemma 2.8. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Let τ τ [(Xk )k=0 , (Yk )k=0 ] be a pair of orthonormal systems corresponding to (αj )∞ j=1 and F . For each n ∈ N1,τ and each w ∈ C \ Pα,n , then ∗ ∗ [α,n] Yn (αn−1 ) Yn[α,n] (w) − Xn (αn−1 ) Xn (w) [α,n−1] ∗ [α,n−1] = 1 − bαn (αn−1 )bαn (w) Yn−1 (αn−1 ) Yn−1 (w)

and

∗ ∗ Xn[α,n] (w) Xn[α,n] (αn−1 ) − Yn (w) Yn (αn−1 ) [α,n−1] [α,n−1] ∗ = 1 − bαn (w)bαn (αn−1 ) Xn−1 (w) Xn−1 (αn−1 ) .

Proof. Apply Theorem 1.1 and use (1.1). Proposition 2.9. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ For each k ∈ N0,τ , let ηk be given by (2.1). For each n ∈ N1,τ , let ⎧ C 1−|αn |2 ⎨ if (1 − |αn |)(1 − |αn−1 |) > 0 1−|αn−1 |2 C ρn := ⎩ − |αn |2 −1 if (1 − |αn |)(1 − |αn−1 |) < 0 . 1−|αn−1 |2

Mq,τ ≥ (T, BT ).

(2.8)

o pair of orthonormal systems corresponding to Let [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝ τ (αj )∞ and F , and let (E ) be the sequence of its Szeg˝ o parameters. For each n n=1 j=1 n ∈ N1,τ , the following statements hold: (a) If (1 − |αn |)(1 − |αn−1 |) > 0, then −1 2 ηn ηn−1 (1 − |αn−1 |2 ) [α,n−1] I − E∗n En = ρ2n Yn−1 (αn−1 ) Yn[α,n] (αn−1 ) 1 − αn αn−1 and I−

En E∗n

=

ρ2n

2 −1 [α,n−1] ηn ηn−1 (1 − |αn−1 |2 ) [α,n] Xn (αn−1 ) Xn−1 (αn−1 ) . 1 − αn αn−1

(b) If (1 − |αn |)(1 − |αn−1 |) < 0, then 2 −1 [α,n−1] |αn−1 |2 − 1 ∗ 2 I − E n E n = ρn Yn (αn−1 ) Xn−1 (αn−1 ) 1 − αn αn−1 and I − En E∗n = ρ2n

−1 |αn−1 |2 − 1 [α,n−1] Yn−1 (αn−1 ) Xn (αn−1 ) 1 − αn αn−1

(c) The Szeg˝ o parameter En is a strictly contractive q × q matrix.

2 .

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173

Proof. Let n ∈ N1,τ . First we consider the case (1 − |αn |)(1 − |αn−1 |) > 0. Hence, Remark 1.2 implies det Yn[α,n] (αn−1 ) = 0. (2.9) Using (2.9), (2.6), and Lemma 2.8 we obtain [α,n] ∗ Yn (αn−1 ) I − E∗n En Yn[α,n] (αn−1 ) ∗ ∗ = Yn[α,n] (αn−1 ) Yn[α,n] (αn−1 ) − Xn (αn−1 ) Xn (αn−1 ) [α,n−1] ∗ [α,n−1] (2.10) = 1 − |bαn (αn−1 )|2 Yn−1 (αn−1 ) Yn−1 (αn−1 ). From (1.1) we see that 1 − |bαn (αn−1 )|2 =

(1 − |αn |2 )(1 − |αn−1 |2 ) . (1 − αn αn−1 )(1 − αn αn−1 )

(2.11)

Since ηn and ηn−1 are unimodular complex numbers from (2.8), (2.10), (2.9), and (2.11), we can conclude that, in the considered case, the ﬁrst equation in (a) is fulﬁlled. In view of (2.7), in the case (1 − |αn |)(1 − |αn−1 |) < 0, the ﬁrst formula in (b) can be veriﬁed similarly. Using Remark 2.6 the second equation in (a) and (b) follows from the ﬁrst one, respectively. Thus, parts (a) and (b) are proved. In view of Deﬁnition 2.1 and (2.8), part (c) is a consequence of parts (a) and (b). In the following, if a sequence (αj )∞ j=1 ∈ T1 is given, we will use the notations ηk and ρn which are deﬁned by (2.1) for all k ∈ N0 and by (2.8) for all n ∈ N, respectively. q,τ Proposition 2.10. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). τ τ o pair of orthonormal systems corresponding to Let [(Xk )k=0 , (Yk )k=0 ] be a Szeg˝ τ (αj )∞ and F , and let (E ) be the sequence of its Szeg˝ o parameters. For each n n=1 j=1 n ∈ N1,τ and each w ∈ C \ Pα,n , then the following statements hold:

(a) If (1 − |αn |)(1 − |αn−1 |) > 0, then Yn (w) − ηn ηn−1 Xn[α,n] (w)En = ρn

1 − αn−1 w bαn−1 (w)Yn−1 (w) I − E∗n En 1 − αn w

and Xn (w) − ηn ηn−1 En Yn[α,n] (w) = ρn

1 − αn−1 w bαn−1 (w) I − En E∗n Xn−1 (w). 1 − αn w

(b) If (1 − |αn |)(1 − |αn−1 |) < 0, then Yn (w) − ηn ηn−1 Xn[α,n] (w)En = ρn

1 − αn−1 w [α,n−1] Xn−1 (w) I − E∗n En 1 − αn w

and Xn (w) − ηn ηn−1 En Yn[α,n] (w) = ρn

1 − αn−1 w [α,n−1] I − En E∗n Yn−1 (w). 1 − αn w

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Proof. Let n ∈ N1,τ and w ∈ C \ Pα,n . At ﬁrst we shall prove the ﬁrst equation in (a) and hence we consider in the following the case (1 − |αn |)(1 − |αn−1 |) > 0. Using (2.6) and Lemma 2.8 we get Yn[α,n] (w) − ηn ηn−1 E∗n Xn (w) −∗ [α,n] ∗ ∗ Yn (αn−1 ) Yn[α,n] (w) − Xn (αn−1 ) Xn (w) = Yn[α,n] (αn−1 ) −∗ [α,n−1] ∗ [α,n−1] Yn−1 (αn−1 ) Yn−1 (w) = 1 − bαn (αn−1 )bαn (w) Yn[α,n] (αn−1 ) and therefore, in view of [FKL2, Remarks 2.4 and 2.8] and Proposition 2.9, Yn (w) − ηn ηn−1 Xn[α,n] (w)En −1 = bαn (w) − bαn (αn−1 ) Yn−1 (w)Yn[α,n−1] (αn−1 ) Yn[α,n] (αn−1 ) bαn (w) − bαn (αn−1 ) ηn ηn−1 (1 − αn αn−1 ) = Yn−1 (w) I − E∗n En . 2 (1 − |αn−1 | )ρn Taking into account the identity bαn (w) − bαn (αn−1 ) ηn ηn−1 (1 − αn αn−1 ) 1 − αn−1 w bαn−1 (w) = 1 − αn w 1 − |αn |2 it follows the ﬁrst identity in (a). Now let (1 − |αn |)(1 − |αn−1 |) < 0. Using (2.7), Lemma 2.8, (1.1), (2.8), and Proposition 2.9 we can conclude Yn (w) − ηn ηn−1 Xn[α,n] (w)En ∗ ∗ −∗ Yn (αn−1 ) = Yn (w) Yn (αn−1 ) − Xn[α,n] (w) Xn[α,n] (αn−1 ) [α,n−1] [α,n−1] ∗ −∗ = bαn (w)bαn (αn−1 ) − 1 Xn−1 (w) Xn−1 (αn−1 ) Yn (αn−1 ) ∗ −1 [α,n−1] (|αn |2 − 1)(1 − αn−1 w) [α,n−1] Xn−1 (w) Yn (αn−1 ) = Xn−1 (αn−1 ) (1 − αn αn−1 )(1 − αn w) 1 − αn−1 w [α,n−1] Xn−1 (w) I − E∗n En . = ρn 1 − αn w Hence, the ﬁrst identity in (b) is also proved. The second equation in (a) and (b) can be obtained using Remark 2.6, [FKL2, Remark 2.11], and the ﬁrst one, respectively. For each strictly contractive complex q × q matrix E, let H(E) be the so-called Halmos extension of E, i.e., let √ √ −1 −1 I − EE∗ E I − E∗ E H(E) := . √ √ −1 −1 E∗ I − EE∗ I − E∗ E Furthermore, in view of Deﬁnition 2.1, Deﬁnition 2.5, and part (c) of Proposition τ 2.9, if (αj )∞ j=1 ∈ T1 and if (En )n=1 is a sequence of strictly contractive complex q × q matrices where τ ∈ N or τ = ∞, then for n ∈ N1,τ and w ∈ C \ Pα,n we set 0 (w)Iq 0 I b H(En ) αn−1 Hα,E;n (w) := q 0 ηn ηn−1 Iq 0 Iq

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0 bαn−1 (w)Iq 0 I H(E∗n ) q 0 Iq 0 ηn ηn−1 Iq if (1 − |αn |)(1 − |αn−1 |) > 0 and in the case (1 − |αn |)(1 − |αn−1 |) < 0 we put bαn−1 (w)Iq 0 0 Iq 0 Iq Hα,E;n (w) := H(En ) 0 ηn ηn−1 Iq 0 Iq Iq 0 and 0 Iq 0 bαn−1 (w)Iq 0 Iq ∗ H(En ) Gα,E;n (w) := . 0 Iq 0 ηn ηn−1 Iq Iq 0

and

Gα,E;n (w) :=

q,τ Theorem 2.11. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). τ τ o pair of orthonormal systems corresponding to Let [(Xk )k=0 , (Yk )k=0 ] be a Szeg˝ τ (αj )∞ and F , and let (E ) o parameters. For all n n=1 be the sequence of its Szeg˝ j=1 n ∈ N1,τ and all w ∈ C \ Pα,n , the following recurrence relations are satisﬁed: Xn−1 (w) Xn (w) 1 − αn−1 w Hα,E;n (w) = ρn [α,n] [α,n−1] 1 − αn w Yn (w) Yn−1 (w) and 1 − αn−1 w [α,n−1] Yn−1 (w), Xn−1 (w) Gα,E;n (w). Yn (w), Xn[α,n] (w) = ρn 1 − αn w

Proof. Let n ∈ N1,τ and w ∈ C \ Pα,n . Suppose (1−|αn |)(1−|αn−1 |) > 0. We have Xn (w) 0 I H(−En ) q [α,n] 0 ηn ηn−1 Iq Yn (w) −1 [α,n] −1 I − En E∗n Xn (w) − ηn ηn−1 En I − E∗n En Yn (w) = −1 [α,n] −1 −E∗n I − En E∗n Xn (w) + ηn ηn−1 I − E∗n En Yn (w) ⎛ ⎞ −1 [α,n] Xn (w) − ηn ηn−1 En Yn (w) I − En E∗n ⎠ . = ⎝ (2.12) −1 [α,n] −E∗n Xn (w) + ηn ηn−1 Yn (w) I − E∗n En From Proposition 2.10, [FKL2, Remark 2.8 and Proposition 2.13], and the identity ηn−1 (αn−1 − w) 1 − αn−1 w bαn−1 (w) = 1 − αn w 1 − αn w we get Yn[α,n] (w) − ηn ηn−1 E∗n Xn (w) −ηn (ηn−1 αn−1 w − ηn−1 ) [α,n−1] I − E∗n En Yn−1 (w) = ρn 1 − αn w 1 − αn−1 w [α,n−1] I − E∗n En Yn−1 (w). (2.13) = ρn ηn ηn−1 1 − αn w Using (2.13) and again Proposition 2.10 we can see that the right-hand side of (2.12) is equal to 1 − αn−1 w bαn−1 (w)Xn−1 (w) ρn . [α,n−1] 1 − αn w Yn−1 (w)

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Taking into account H(En )H(−En ) = I thus the ﬁrst identity for the considered case (1−|αn |)(1−|αn−1 |) > 0 follows. Analogously, one can prove that this identity is satisﬁed if (1 − |αn |)(1 − |αn−1 |) < 0. In view of [FKL2, Remark 2.8 and Proposition 2.13], the second recurrence formula is a consequence of the ﬁrst one. q,τ Corollary 2.12. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). τ τ o pair of orthonormal systems corresponding to Let [(Xk )k=0 , (Yk )k=0 ] be a Szeg˝ τ (αj )∞ and F , and let (E ) be the sequence of its Szeg˝ o parameters. For every n n=1 j=1 choice of n ∈ N1,τ and w ∈ C \ Pα,n , the following statements hold: (a) If (1 − |αn |)(1 − |αn−1 |) > 0, then −1 1 − αn−1 w [α,n−1] bαn−1 (w)Xn−1 (w) + En Yn−1 (w) I − En E∗n Xn (w) = ρn 1 − αn w

and Yn (w) = ρn

−1 1 − αn−1 w [α,n−1] bαn−1 (w)Yn−1 (w) + Xn−1 (w)En I − E∗n En . 1 − αn w

(b) If (1 − |αn |)(1 − |αn−1 |) < 0, then −1 1 − αn−1 w [α,n−1] bαn−1 (w)En Xn−1 (w) + Yn−1 (w) I − En E∗n Xn (w) = ρn 1 − αn w and Yn (w) = ρn

−1 1 − αn−1 w [α,n−1] bαn−1 (w)Yn−1 (w)En + Xn−1 (w) I − E∗n En . 1 − αn w

Proof. Apply Theorem 2.11.

At the end of this section we note again that the particular case of q × q matrix polynomials is considered in [DGK] and [DFK, Section 3.6]. In this special case the recurrence relations stated in Corollary 2.12 coincide with those given in [DFK, Lemma 3.6.12].

3. Szeg˝ o pairs of rational matrix-valued functions Motivated by the matrix polynomial case and Corollary 2.12 we introduce the following notion. τ Deﬁnition 3.1. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let (En )n=1 be a sequence of strictly contractive q × q matrices. Further, let X0 and Y0 be nonsingular complex q×q matrices such that X∗0 X0 = Y0 Y0∗ , and let X0 and Y0 be the constant matrix-valued functions (deﬁned on C0 ) with values X0 and Y0 , respectively. For all n ∈ N1,τ such that (1−|αn |)(1−|αn−1 |) > 0, let Xn and Yn be the matrix-valued functions which are given for all w ∈ C \ Pα,n by −1 1 − αn−1 w [α,n−1] bαn−1 (w)Xn−1 (w) + En Yn−1 (w) I − En E∗n Xn (w) := ρn 1 − αn w

Szeg˝o Pairs of Orthogonal Rational Matrix-valued Functions and Yn (w) := ρn

177

−1 1 − αn−1 w [α,n−1] bαn−1 (w)Yn−1 (w) + Xn−1 (w)En I − E∗n En . 1 − αn w

For all n ∈ N1,τ such that (1 − |αn |)(1 − |αn−1 |) < 0, let Xn and Yn be the matrix-valued functions which are deﬁned for all w ∈ C \ Pα,n by −1 1 − αn−1 w [α,n−1] bαn−1 (w)En Xn−1 (w) + Yn−1 (w) I − En E∗n Xn (w) := ρn 1 − αn w and Yn (w) := ρn

−1 1 − αn−1 w [α,n−1] bαn−1 (w)Yn−1 (w)En + Xn−1 (w) I − E∗n En . 1 − αn w

o pair of rational matrix-valued Then the pair [(Xk )τk=0 , (Yk )τk=0 ] is called the Szeg˝ τ functions generated by [(αj )∞ j=1 ; (En )n=1 ; X0 , Y0 ]. q,τ Remark 3.2. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). τ τ Let [(Xk )k=0 , (Yk )k=0 ] be a Szeg˝o pair of orthonormal systems corresponding to τ (αj )∞ o parameters. Further, let j=1 and F , and let (En )n=1 be a sequence of Szeg˝ X0 := X0 (0) and Y0 := Y0 (0). From [FKL2, Remark 5.3] we know that X0 and Y0 are nonsingular matrices such that X∗0 X0 = Y0 Y0∗ holds. Hence, Corollary 2.12 shows that [(Xk )τk=0 , (Yk )τk=0 ] is the Szeg˝o pair of rational matrix-valued functions τ generated by [(αj )∞ j=1 ; (En )n=1 ; X0 , Y0 ]. τ Remark 3.3. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let (En )n=1 be a sequence of strictly contractive q × q matrices. Further, let X0 and Y0 be nonsingular complex q × q matrices such that X∗0 X0 = Y0 Y0∗ . Let [(Xk )τk=0 , (Yk )τk=0 ] be the Szeg˝ o τ pair of rational matrix-valued functions generated by [(αj )∞ ; (E ) ; X , Y ]. n 0 0 n=1 j=1 For each k ∈ N0,τ , from Deﬁnition 3.1, (1.1), and (1.2) one can easily see that the matrix-valued functions Xk and Yk both belong to Rq×q α,k . τ Remark 3.4. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let (En )n=1 be a sequence of strictly contractive complex q × q matrices. Furthermore, let X0 and Y0 be nonsingular complex q × q matrices such that X∗0 X0 = Y0 Y0∗ , let X0 and Y0 be the constant matrix-valued functions with values X0 and Y0 respectively, and for each n ∈ N1,τ let Xn and Yn be a matrix-valued function belonging to Rq×q α,n . By using the same arguments as in the proof of Theorem 2.11, one can verify that [(Xk )τk=0 , (Yk )τk=0 ] is the Szeg˝o pair of rational matrix-valued functions generated τ by [(αj )∞ j=1 ; (En )n=1 ; X0 , Y0 ] if and only if for each n ∈ N1,τ and each w ∈ C\Pα,n the recurrence relations stated in Proposition 2.10 are fulﬁlled.

As already in the introduction announced, we are going now to prove some results with respect to the pairs of sequences of rational matrix-valued functions deﬁned by Deﬁnition 3.1, which are often called Favard-type theorems (cf. [BGHN2, Chapter 8]). At ﬁrst, we study the case that a ﬁnite sequence of strictly contractive complex q × q matrices forms the basis of the pairs.

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m Theorem 3.5. Let (αj )∞ j=1 ∈ T1 and let m ∈ N. Further, let (En )n=1 be a sequence of strictly contractive complex q×q matrices, let X0 and Y0 be nonsingular complex m q × q matrices such that X∗0 X0 = Y0 Y0∗ , and let [(Xk )m o k=0 , (Yk )k=0 ] be the Szeg˝ ∞ pair of rational matrix-valued functions generated by [(αj )j=1 ; (En )m n=1 ; X0 , Y0 ]. Then Fm : BT → Cq×q deﬁned by % %% −1 −∗ 1 − |αm |2 % 1 Fm (B) := Xm (z) Xm (z) λ(dz) (3.1) 2 2π B |z − αm |

is a (well-deﬁned ) nonnegative Hermitian q × q measure which belongs to the set m Mq,∞ (T, BT ) and [(Xk )m o pair of orthonormal systems cork=0 , (Yk )k=0 ] is a Szeg˝ ≥ ∞ o responding to (αj )j=1 and Fm . Moreover, (En )m n=1 is exactly the sequence of Szeg˝ m , (Y ) ]. parameters corresponding to [(Xk )m k k=0 k=0 Proof. From Remark 3.3 we know that, for each k ∈ N0,m , the matrix-valued functions Xk and Yk both belong to Rq×q α,k . The matrix 0 I jqq := q 0 −Iq is obviously a 2q × 2q signature matrix, i.e., j2qq = jqq and j∗qq = jqq hold. We are m q×q going to verify that [(Xk )m k=0 , (Yk )k=0 ] is a pair of Rα,∞ -sequences which is jqq recursively connected, i.e., that X0 and Y0 are constant matrix-valued functions which satisfy X0∗ X0 = Y0 Y0∗ , that there is a sequence (Un )m n=1 of complex 2q × 2q matrices such that + jqq if (1 − |αn |)(1 − |αn−1 |) > 0 U∗n jqq Un = (3.2) −jqq if (1 − |αn |)(1 − |αn−1 |) < 0 for all n ∈ N1,m and that Xn (w) 1 − αn−1 w (w)Iq b = ρn Un αn−1 [α,n] 0 Yn (w) 1 − αn

0 Iq

Xn−1 (w) [α,n−1] Yn−1 (w)

(3.3)

for all n ∈ N1,m and all w ∈ C \ Pα,n . Obviously, X0 and Y0 are constant matrixvalued functions which satisfy X0∗ X0 = Y0 Y0∗ . For each n ∈ N1,m , the matrix ⎧ Iq 0 ⎪ ⎪ H(En ) if (1 − |αn |)(1 − |αn−1 |) > 0 ⎪ ⎨ 0 ηn ηn−1 Iq Un := ⎪ 0 Iq 0 Iq ⎪ ⎪ if (1 − |αn |)(1 − |αn−1 |) < 0 H(E ) ⎩ n 0 ηn ηn−1 Iq Iq 0 fulﬁlls (3.2). Let n ∈ N1,m and let w ∈ C \ Pα,n . In view of Deﬁnition 3.1 and [FKL2, Remark 2.8 and Proposition 2.13], if (1 − |αn |)(1 − |αn−1 |) > 0 then Yn[α,n] (w) = ρn ηn ηn−1

−1 1−αn−1w [α,n−1] Yn−1 (w)+bαn−1 (w)E∗n Xn−1 (w) . (3.4) I−E∗n En 1−αn w

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Analogously, in the case (1 − |αn |)(1 − |αn−1 |) < 0, we have Yn[α,n] (w) = ρn ηn ηn−1

−1 1−αn−1w [α,n−1] E∗n Yn−1 (w)+bαn−1 (w)Xn−1 (w) . (3.5) I−E∗n En 1−αn w

Consequently, (3.3) is valid. By virtue of [FKL3, Lemma 3.11], for all z ∈ T, the matrix Xm (z) is nonsingular. From [FKL1, Example 5.4] we know then that Fm is a well-deﬁned nonnegative Hermitian measure which belongs to Mq,∞ ≥ (T, BT ). m , (Y ) ] is an orthonorApplying [FKL3, Theorem 4.4] we obtain that [(Xk )m k k=0 k=0 mal system corresponding to (αj )∞ and F . Let n ∈ N . We consider the case m 1,m j=1 that the inequality (1 − |αn |)(1 − |αn−1 |) > 0 holds. From (3.4) we see that

I − E∗n En Yn[α,n] (αn−1 ) = ρn ηn ηn−1

1 − |αn−1 |2 [α,n−1] Y (αn−1 ) 1 − αn αn−1 n−1

and, in view of [FKL3, Lemma 3.11], −1 1 − |αn−1 |2 [α,n−1] I − E∗n En = ρn ηn ηn−1 Yn−1 (αn−1 ) Yn[α,n] (αn−1 ) . 1 − αn αn−1

(3.6)

In particular, the matrix stated in (2.2) is positive Hermitian. For each w ∈ C\Pα,n , from Deﬁnition 3.1 we get Xn[α,n] (w) I − En E∗n 1 − αn−1 w [α,n−1] Xn−1 (w) + bαn−1 (w)Yn−1 (w)E∗n = ρn ηn ηn−1 1 − αn w and consequently, in view of [FKL3, Lemma 3.12], −1 [α,n−1] 1 − |αn−1 |2 [α,n] X I − En E∗n = ρn ηn ηn−1 (αn−1 ) Xn−1 (αn−1 ). 1 − αn αn−1 n Therefore the matrix stated in (2.3) is also positive Hermitian. In the case that (1 − |αn |)(1 − |αn−1 |) < 0 holds one can analogously check that the matrices stated in (2.4) and (2.5) are both positive Hermitian. If (1 − |αn |)(1 − |αn−1 |) > 0, then Deﬁnition 3.1 and (3.6) provide us −1 ηn ηn−1 Xn (αn−1 ) Yn[α,n] (αn−1 ) −1 −1 1−|αn−1 |2 [α,n−1] = ηn ηn−1 ρn I−En E∗n En Yn−1 (αn−1 ) Yn[α,n] (αn−1 ) = En . 1−αn αn−1 If (1 − |αn |)(1 − |αn−1 |) < 0, from (3.5) and Deﬁnition 3.1 we obtain similarly −1 ∗ ηn ηn−1 Yn[α,n] (αn−1 ) Xn (αn−1 ) −1 ∗ −1 ∗ [α,n−1] 1 − |αn−1 |2 ∗ = ρn I − En En En Yn−1 (αn−1 ) Xn (αn−1 ) = En . 1 − αn αn−1 The proof is complete.

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τ Corollary 3.6. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N, let (En )n=1 be a sequence of strictly contractive complex q × q matrices, let X0 and Y0 be nonsingular complex q × q matrices such that X∗0 X0 = Y0 Y0∗ , and let [(Xk )τk=0 , (Yk )τk=0 ] be the Szeg˝ o pair of τ rational matrix-valued functions generated by [(αj )∞ j=1 ; (En )n=1 ; X0 , Y0 ]. Further, ∞ let the sequences (Xm )∞ m=τ +1 and (Ym )m=τ +1 of rational matrix-valued functions be deﬁned for all m ∈ Nτ +1,∞ and w ∈ C0 \ Pα,m by C ⎧ 1−|αm |2 αm−1 −w ⎨ ηm−1 1−|α if (1−|αm |)(1−|αm−1 |) > 0 2 1−α w Xm−1 (w) m−1 | m C Xm (w) := ⎩ − |αm |2 −1 1−αm−1 w [α,m−1] (w) if (1−|αm |)(1−|αm−1 |) < 0 1−|αm−1 |2 1−αm w Ym−1

and

C ⎧ 1−|αm |2 αm−1 −w ⎨ ηm−1 1−|α 2 1−α w Ym−1 (w) m−1 | m C Ym (w) := ⎩ − |αm |2 −1 1−αm−1 w [α,m−1] (w) 1−|αm−1 |2 1−αm w Xm−1

if (1−|αm |)(1−|αm−1 |) > 0 if (1−|αm |)(1−|αm−1 |) < 0 ,

∞ o and let Em := 0q×q for all m ∈ Nτ +1,∞ . Then [(Xk )∞ k=0 , (Yk )k=0 ] is the Szeg˝ ∞ pair of rational matrix-valued functions generated by [(αj )j=1 ; (En )∞ ; X , Y ]. 0 0 n=1 Moreover, for each m ∈ Nτ,∞ , if the nonnegative Hermitian q × q Borel measure ∞ Fm on T is given as in (3.1) then Fm = Fτ , [(Xk )∞ o pair k=0 , (Yk )k=0 ] is the Szeg˝ ∞ of orthonormal systems corresponding to (αj )j=1 and Fτ , and (En )∞ is exactly n=1 ∞ the sequence of Szeg˝ o parameters corresponding to [(Xk )∞ k=0 , (Yk )k=0 ].

Proof. Since 0q×q is a strictly contractive q × q matrix, from Deﬁnition 3.1, (2.8), ∞ (1.1), and (2.1) it follows immediately that [(Xk )∞ o pair k=0 , (Yk )k=0 ] is the Szeg˝ ∞ of rational matrix-valued functions generated by [(αj )j=1 ; (En )∞ n=1 ; X0 , Y0 ]. Furthermore, an application of Theorem 3.5 and [FKL2, Remark 2.5, Remark 6.2, and Lemma 6.5] implies for all m ∈ Nτ +1,∞ and z ∈ T the identity % % % 1 − |αm |2 % |z − ατ |2 ∗ ∗ % Xm (z) Xm (z) = %% Xτ (z) Xτ (z). % 2 2 1 − |ατ | |z − αm | Therefore, we obtain for each m ∈ Nτ +1,∞ the equality Fm = Fτ . Thus, we ∞ can ﬁnally conclude from Theorem 3.5 that [(Xk )∞ o pair of k=0 , (Yk )k=0 ] is the Szeg˝ ∞ orthonormal systems corresponding to (αj )j=1 and Fτ , and that (En )∞ n=1 is exactly ∞ the sequence of Szeg˝o parameters corresponding to [(Xk )∞ k=0 , (Yk )k=0 ]. ∞ Theorem 3.7. Let (αj )∞ j=1 ∈ T1 , let (En )n=1 be a sequence of strictly contractive complex q × q matrices, and let X0 and Y0 be nonsingular complex q × q matri∞ ces such that X∗0 X0 = Y0 Y0∗ . Let [(Xk )∞ o pair of rational k=0 , (Yk )k=0 ] be a Szeg˝ ∞ matrix-valued functions generated by [(αj )j=1 ; (En )∞ ; X , Y ]. 0 0 Then there is an n=1 ∞ ∞ (T, B ) such that [(X ) , (Y ) ] is a Szeg˝ o pair of orthonormal F ∈ Mq,∞ T k k=0 k k=0 ≥ ∞ ∞ systems corresponding to (αj )j=1 and F . Moreover, (En )n=1 is the sequence of ∞ Szeg˝ o parameters corresponding to [(Xk )∞ k=0 , (Yk )k=0 ].

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m Proof. According to Theorem 3.5, for each m ∈ N, [(Xk )m o k=0 , (Yk )k=0 ] is a Szeg˝ ∞ pair of orthonormal systems corresponding to (αj )j=1 and the nonnegative Hermitian q × q measure Fm which is given by (3.1). From [FKL2, Remark 5.3] we know that Fm (T) = (X∗0 X0 )−1 for all m ∈ N. By virtue of [FKL3, Lemma 4.7], ∞ there are an F ∈ Mq≥ (T, BT ) and a subsequence (Fmn )∞ n=1 of (Fm )m=1 which converges weakly to F (cf. [FK] or [FKL3, Section 4]). Applying [FKL3, Lemma 4.6] (see also [FK, Satz 3]), for all nonnegative integers j and k, we obtain ∗ Xj dF Xk = lim Xj dFmn Xk∗ (Xj , Xk )F,l = T

n→∞

T

= lim (Xj , Xk )Fmn ,l = δj,k Iq n→∞

and analogously (Yj , Yk )F,r = δjk Iq . ∞ Hence, in view of Remark 3.3, [(Xk )∞ k=0 , (Yk )k=0 ] is a pair of orthonormal systems ∞ corresponding to (αj )j=1 and F . The rest of the assertion follows immediately from [FKL2, Corollary 4.4] and Theorem 3.5.

4. Integral representations of Szeg˝ o parameters In the following, for each w ∈ C, let fw : C → C and gw : C → C be deﬁned by fw (z) := z − w

and gw (z) := 1 − wz,

(4.1)

respectively. If a sequence (αj )∞ j=1 ∈ T1 is given, we will again use the notation ηk given by (2.2). q Remark 4.1. Let (αj )∞ j=1 ∈ T1 , let n ∈ N, and let F ∈ M≥ (T, BT ). Let X and Y belong to Rq×q α,n−1 . Further, let P be a polynomial over C of degree one. In view of [FKL2, Remark 2.5], one obtains that gα1 P X [α,n−1] and gα1 P Y belong to Rq×q α,n n n and that 1 1 P X [α,n−1], Y [α,n−1] = X, PY . gαn gαn F,l F,r q,τ Lemma 4.2. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). o pair of orthonormal systems corresponding to Let [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝ τ (αj )∞ o parameters. For each j=1 and F , and let (En )n=1 be the sequence of its Szeg˝ n ∈ N1,τ , the following statements hold:

(a) If (1 − |αn |)(1 − |αn−1 |) > 0, then for all Z ∈ Rq×q α,n−1 , we have gαn−1 [α,n−1] fαn−1 En Yn−1 , Z = ηn−1 Xn−1 , Z , gαn gαn F,l F,l gα fα [α,n−1] Z, n−1 Xn−1 En = ηn−1 Z, n−1 Yn−1 , gαn gαn F,r F,r

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and

gαn−1 Z, Xn−1 gαn

En = ηn−1

F,l

fαn−1 [α,n−1] Z, Yn−1 gαn

, F,l

gα [α,n−1] fα = ηn−1 Xn−1 , n−1 Z . En Yn−1 , n−1 Z gαn gαn F,r F,r

(b) If (1 − |αn |)(1 − |αn−1 |) < 0, then fαn−1 gαn−1 [α,n−1] En Xn−1 , Z = ηn−1 Y ,Z , gαn gαn n−1 F,l F,l gα fα [α,n−1] Z, n−1 Yn−1 En = ηn−1 Z, n−1 Xn−1 , gαn gαn F,r F,r fαn−1 gαn−1 [α,n−1] Z, Yn−1 En = ηn−1 Z, Xn−1 , gαn gαn F,l F,l and

gαn−1 [α,n−1] fαn−1 Z = ηn−1 Yn−1 , Z . En Xn−1 , gαn gαn F,r F,r

Proof. Let n ∈ N1,τ and Z ∈ Rq×q α,n−1 . From [FKL2, Remark 2.1 and Lemma 3.6] we get that (Xn , Z)F,l = 0 and (Z, Yn )F,r = 0. Application of Corollary 2.12 yields the ﬁrst and the second identity of (a) if (1 − |αn |)(1 − |αn−1 |) > 0 and in the case (1 − |αn |)(1 − |αn−1 |) < 0 the ﬁrst and the second identity of (b). The other identities follow then from Remark 4.1. q,τ Lemma 4.3. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Let [(Xk )τk=0 , (Yk )τk=0 ] be a pair of orthonormal systems corresponding to (αj )∞ j=1 and F . Further, let n ∈ N1,τ and let Z ∈ Rq×q . α,n−1

(a) If (1 − |αn |)(1 − |αn−1 |) > 0, then the following statements are equivalent: g α = 0. (i) det gαn−1 Z, Xn−1 n F,l gα = 0. (ii) det Yn−1 , gαn−1 Z n

(iii) det Z

[α,n−1]

F,r

(αn ) = 0.

(b) If (1 − |αn |)(1 − |αn−1 |) < 0, then the following statements are equivalent: f α [α,n−1] = 0. (iv) det gαn−1 Z, Yn−1 n F,l [α,n−1] fα = 0. (v) det Xn−1 , gαn−1 Z n

(vi) det Z

[α,n−1]

(αn ) = 0.

F,r

Szeg˝o Pairs of Orthogonal Rational Matrix-valued Functions

183

gα

Proof. (a) Suppose (1 − |αn |)(1 − |αn−1 |) > 0. Obviously, W := gαn−1 Z belongs n to Rq×q α,n . Using [FKL2, Remark 2.8 and Proposition 2.13] we obtain W [α,n] = −ηn

fαn−1 [α,n−1] Z . gαn

(4.2)

From fαn−1 (αn−1 ) = 0 it follows W [α,n] (αn−1 ) = 0. In view of [FKL2, Remark 2.1], let q × q matrices such that

(Wk )nk=0

W =

n

(4.3)

be the unique sequence of complex (q)

Wk Bα,k

k=0

and for each j ∈ N0,n let (Xjk )nk=0 be the sequence of complex q × q matrices such that n (q) Xj = Xjk Bα,k . (4.4) k=0

We have Xn[α,n] (αn−1 ) = X∗nn + bαn (αn−1 )X∗n,n−1

(4.5)

and ∗ . W [α,n] (αn−1 ) = Wn∗ + bαn (αn−1 )Wn−1

From (4.3) it follows Wn = −bα,n (αn−1 )Wn−1 .

(4.6)

Using (1.3) and (4.2) we get Wn∗ = W [α,n] (αn ) = −ηn = −

fαn−1 (αn ) [α,n−1] Z (αn ) gαn (αn )

1 − αn αn−1 bαn (αn−1 )Z [α,n−1] (αn ). 1 − |αn |2

If αn = αn−1 , then comparing this with (4.6) yields 1 − αn αn−1 [α,n−1] ∗ Z (αn ) = Wn−1 . 1 − |αn |2

(4.7)

If αn = αn−1 we have W = Z and hence, in view of (1.3), equation (4.7) also holds in this case. From (4.4) and [FKL2, Lemma 3.6] we obtain n−1 (q) (q) 0 = (Xn , Xn−1 )F,l = Xnn (Bα,n , Xn−1 )F,l + Xnk Bα,k , Xn−1 k=0

=

(q) Xnn (Bα,n , Xn−1 )F,l

+

Xn,n−1 X−1 n−1,n−1 .

Analogously, we get (q) (W, Xn−1 )F,l = Wn (Bα,n , Xn−1 )F,l + Wn−1 X−1 n−1,n−1 .

F,l

(4.8)

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Hence from (4.8), (4.6), (4.5), and (4.7) we can conclude −1 (W, Xn−1 )F,l = −Wn X−1 nn Xn,n−1 + Wn−1 Xn−1,n−1 −1 = Wn−1 X−1 nn bαn (αn−1 )Xn,n−1 + Xnn Xn−1,n−1 [α,n] ∗ = Wn−1 X−1 (αn−1 ) X−1 nn Xn n−1,n−1 ∗ −1 [α,n] ∗ 1 − αn−1 αn [α,n−1] Z = (αn ) Xnn Xn (αn−1 ) X−1 n−1,n−1 . 2 1 − |αn |

(4.9)

[α,n]

Since we know from Remark 1.2 that det Xn (αn−1 ) = 0 holds we see then from (4.9) that (i) and (iii) are equivalent. In view of [FKL2, Remark 2.11 and Remark 3.5], we also obtain that (ii) and (iii) are equivalent. fα (b) Now let (1−|αn |)(1−|αn−1 |) < 0. Let V := gαn−1 Z, let (Vk )nk=0 be the unique n sequence of complex q × q matrices such that n (q) V = Vk Bα,k k=0

and let

(Ynk )nk=0

be the sequence of complex q × q matrices such that Yn =

n

(q)

Ynk Bα,k .

k=0

Obviously, R :=

n−1 # k=0

(q)

Ynk Bα,k belongs to Rq×q α,n−1 . From [FKL2, Remark 2.8 and

Proposition 2.13] we get V [α,n] = −ηn

gαn−1 [α,n−1] Z gαn

and (1.3) yields then ∗ ∗ 1 − αn αn−1 [α,n−1] Vn = V [α,n] (αn ) = −ηn Z (αn ) . 2 1 − |αn |

(4.10)

[α,n−1]

belongs to Rq×q Since Xn−1 α,n−1 , from [FKL2, Lemma 3.6 and Remark 4.2] and (1.3) we obtain ∗ [α,n−1] [α,n−1] (q) = Xn−1 , Bα,n Ynn + R[α,n−1] , Xn−1 0 = Xn−1 , Yn F,r F,r F,l ∗ ∗ −1 [α,n−1] (q) = Xn−1 , Bα,n Ynn + R(αn−1 ) Xn−1,n−1 . (4.11) F,r

Furthermore, we have (q) R(αn−1 ) = Yn (αn−1 ) − Ynn Bα,n (αn−1 ) = Yn (αn−1 ).

The matrix-valued function H :=

n−1 k=0

(q)

Vk Bα,k

(4.12)

Szeg˝o Pairs of Orthogonal Rational Matrix-valued Functions

185

belongs to Rq×q α,n−1 and satisﬁes (q) H(αn−1 ) = V (αn−1 )−Vn Bα,n (αn−1 ) = V (αn−1 ) =

fαn−1(αn−1 ) Z(αn−1 ) = 0. gαn(αn−1 )

By virtue of [FKL2, Lemma 3.6 and Remark 4.2], (1.3), (4.10), (4.11), and (4.12) it follows ∗ ∗ [α,n−1] [α,n−1] (q) = Xn−1 , Bα,n Vn + H(αn−1 ) X−1 Xn−1 , V n−1,n−1 F,r F,r ∗ ∗ [α,n−1] (q) −1 = Xn−1 , Bα,n Vn = − R(αn−1 ) X−1 Ynn Vn n−1,n−1 F,r ∗ [α,n−1] ∗ −1 ∗ 1−αn αn−1 −1 Y Z = −ηn (α ) X Ynn (αn ) . n n−1 n−1,n−1 1−|αn |2 Since Remark 1.2 yields that det Yn (αn−1 ) = 0 is fulﬁlled we see that (v) and (vi) are equivalent. The equivalence of (iv) and (vi) follows then by application of [FKL2, Remark 2.11 and Remark 3.5]. q,τ Theorem 4.4. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). o pair of orthonormal systems corresponding to Let [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝ τ (αj )∞ and F , and let (E ) be the sequence of its Szeg˝ o parameters. For each n n=1 j=1 n ∈ N1,τ , the following statements hold: (a) If (1 − |αn |)(1 − |αn−1 |) > 0, then −1 [α,n−1] ∗ gαn−1 fαn−1 ∗ En = ηn−1 ZdF Xn−1 ZdF Yn−1 T gαn T gαn

and

En = ηn−1

T

[α,n−1] ∗ Xn−1 dF

f

αn−1

gαn

Z T

∗ Yn−1 dF

g

αn−1

gαn

−1 Z

for all Z ∈ Rq×q α,n−1 which satisfy det Z [α,n−1] (αn ) = 0.

(4.13)

(b) If (1 − |αn |)(1 − |αn−1 |) < 0, then [α,n−1] ∗ −1 fαn−1 gαn−1 ∗ En = ηn−1 ZdF Yn−1 ZdF Xn−1 T gαn T gαn and

En = ηn−1

T

∗ Yn−1 dF

Rq×q α,n−1

g

αn−1

gαn

Z

T

[α,n−1] ∗

Xn−1

dF

f

αn−1

gαn

−1 Z

for all Z ∈ which satisfy (4.13). (c) If (1 − |αn |)(1 − |αn−1 |) > 0, then −1 fαn−1 gαn−1 [α,n−1] ∗ ∗ En = ηn−1 Xn−1 dF Z Yn−1 dF Z T gαn T gαn

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B. Fritzsche, B. Kirstein and A. Lasarow and

En = ηn−1 for all Z ∈

Z ∗ dF T

Rq×q α,n−1

g

αn−1

gαn

[α,n−1]

−1

Xn−1

Z ∗ dF

f

αn−1

gαn

T

Yn−1

which satisfy det Z(αn ) = 0.

(4.14)

(d) If (1 − |αn |)(1 − |αn−1 |) < 0, then −1 gαn−1 [α,n−1] fαn−1 En = ηn−1 Yn−1 dF Z ∗ Xn−1 dF Z ∗ T gαn T gαn and

En = ηn−1

∗

Z dF T

f

αn−1

gαn

−1 Yn−1

T

Z ∗ dF

g

αn−1

gαn

[α,n−1]

Xn−1

for all Z ∈ Rq×q α,n−1 which satisfy (4.14). Proof. Application of Lemma 4.2 and Lemma 4.3 yields the proof of parts (a) and (b). Let Z ∈ Rq×q α,n−1 . In view of Remark 4.1 and [FKL2, Remark 2.4], we have gαn−1 [α,n−1] gαn−1 [α,n−1] Z , Xn−1 = Z, X , gαn gαn n−1 F,l F,r gα gαn−1 [α,n−1] Yn−1 , n−1 Z [α,n−1] = Yn−1 , Z , gαn gαn F,r F,l fαn−1 fαn−1 [α,n−1] [α,n−1] Z , Yn−1 = Z, Yn−1 , gαn gαn F,l F,r and

fαn−1 [α,n−1] fαn−1 [α,n−1] Xn−1 , Z = Xn−1 , Z . gαn gαn F,r F,l

All the matrices stated in the left-hand sides of these equalities are in view of (4.14), [FKL2, Remark 2.4], and Lemma 4.3 nonsingular. Application of Lemma 4.2 completes the proof of parts (c) and (d). The following remark presents particular matrix-valued functions Z which satisfy (4.13) and (4.14), respectively. q,τ Remark 4.5. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Let [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝o pair of orthonormal systems corresponding to τ (αj )∞ o parameters. From j=1 and F , and let (En )n=1 be the sequence of its Szeg˝ Remark 1.2 one can see that for all n ∈ N1,τ , the following statements hold: [α,n−1]

(a) If (1 − |αn |)(1 − |αn−1 |) > 0, then the relations det Xn−1 (αn ) = 0 and [α,n−1] det Yn−1 (αn ) = 0 are satisﬁed. (b) If (1 − |αn |)(1 − |αn−1 |) < 0, then det Xn−1 (αn ) = 0 and det Yn−1 (αn ) = 0.

Szeg˝o Pairs of Orthogonal Rational Matrix-valued Functions

187

We want to draw the attention of the reader to the special case αn−1 = αn where the integral representations of the Szeg˝o parameters stated in Theorem 4.4 can be simpliﬁed. In particular, this situation will be met if one studies orthogonal q × q matrix polynomials. q,τ Corollary 4.6. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Let o pair of orthonormal systems corresponding to (αj )∞ [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝ j=1 τ and F , and let (En )n=1 be the sequence of its Szeg˝ o parameters. Let n ∈ N1,τ and let αn−1 = αn . For all Z ∈ Rq×q α,n−1 which satisfy (4.13), ∗ −1 [α,n−1] [α,n−1] ∗ [α,n−1] (αn−1 ) Xn−1 (αn−1 ) bαn ZdF Yn−1 (4.15) En = − Z T

and En = −

T

[α,n−1] ∗

Xn−1

[α,n−1] −1 ∗ dF bαn Z Yn−1 (αn−1 ) Z [α,n−1] (αn−1 ) . (4.16)

In particular,

En = −

and En = −

T

[α,n−1] ∗ bαn Xn−1 dF Yn−1

T

[α,n−1] ∗

Xn−1

dF bαn Yn−1 .

(4.17)

(4.18)

Proof. In view of (1.1), (2.1), and (4.1), we have ηn−1

fαn−1 fα = ηn−1 n−1 = −bαn−1 = −bαn . gαn gαn−1

Further, using [FKL2, Remark 2.1 and Lemma 3.6] and (1.3), we obtain ∗ [α,n−1] −∗ gαn−1 ∗ ∗ ZdF Xn−1 = ZdF Xn−1 = Z [α,n−1] (αn−1 ) Xn−1 (αn−1 ) . g αn T T Consequently, part (a) of Theorem 4.4 yields (4.15). Equation (4.16) follows analogously. By virtue of part (a) of Remark 4.5, choosing Z = Xn−1 (respectively, Z = Yn−1 ) from (4.15) and (4.16) we get (4.17) and (4.18). Note that, similar as in Corollary 4.6, starting from part (c) of Theorem 4.4 one can also obtain simpler expressions for the Szeg˝ o parameter En in the special case αn−1 = αn . In view of Corollary 3.6, we consider ﬁnally the situation that the Szeg˝ o parameter En is equal to zero for some n ∈ N1,τ . q,τ Corollary 4.7. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). o pair of orthonormal systems corresponding to Let [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝ τ (αj )∞ o parameters. Further, j=1 and F , and let (En )n=1 be the sequence of its Szeg˝ let n ∈ N1,τ and let (1−|αn |)(1−|αn−1 |) > 0. The following statements are equivalent: (i) En = 0. (ii) Xn (αn−1 ) = 0.

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B. Fritzsche, B. Kirstein and A. Lasarow

(iv) For all Z ∈

C

1−|αn |2 fαn−1 1−|αn−1 |2 gαn Rq×q α,n−1 ,

(iii) Xn = −ηn−1

(v) There exists a Z ∈ 0. (vi) Yn (αn−1 ) = C

(vii) Yn = −ηn−1 (viii) For all Z ∈

fαn−1 Xn−1 dF Z ∗ = 0. gαn

T q×q Rα,n−1

(ix) There exists a Z ∈

Yn−1 .

Z ∗ dF

T q×q Rα,n−1

(4.19)

such that (4.14) and (4.19) hold.

1−|αn |2 fαn−1 1−|αn−1 |2 gαn Rq×q α,n−1 ,

Xn−1 .

f

αn−1

gαn

Yn−1 = 0.

(4.20)

such that (4.14) and (4.20) hold.

Proof. (i) ⇔ (ii): This equivalence follows immediately from (2.6) and (2.1). (i) ⇒ (iii): Use part (a) of Corollary 2.12, (1.1), (2.1), (2.8), and (4.1). (iii) ⇒ (iv): This implication is an easy consequence of [FKL2, Remark 2.1 and Lemma 3.6]. (iv) ⇒ (v): From [FKL2, Remark 2.4] and part (a) of Remark 4.5 it follows that (v) is necessary for (iv). (v) ⇒ (i): Apply part (c) of Theorem 4.4. Analogously, in view of (2.7), one can verify that (i), (vi), (vii), (viii), and (ix) are equivalent. Observe that if (1 − |αn |)(1 − |αn−1 |) < 0 is fulﬁlled, the case En = 0 can be similarly characterized as in Corollary 4.7. We omit the details.

References [BGHN1] A. Bultheel, P. Gonz´ alez-Vera, E. Hendriksen, O. Nj˚ astad, A Szeg˝ o Theory for Rational Functions. Report TW 131, Department of Computer Science, K.U. Leuven 1990. [BGHN2] A. Bultheel, P. Gonz´ alez-Vera, E. Hendriksen, O. Nj˚ astad, Orthogonal Rational Functions. Cambridge Monographs on Applied and Comput. Math. 5, Cambridge University Press, Cambridge 1999. [DGK] P. Delsarte, Y. Genin, Y. Kamp, Orthogonal polynomial matrices on the unit circle. IEEE Trans. Circuits and Systems CAS 25 (1978), 145–160. [Dj] M. M. Djrbashian, Orthogonal systems of rational functions on the circle with given set of poles (in Russian). Dokl. Akad. Nauk SSSR 147 (1962), 1278–1281. [DFK] V. K. Dubovoj, B. Fritzsche, B. Kirstein, Matricial Version of the Classical Schur Problem. Teubner-Texte zur Mathematik 129, Teubner, Leipzig 1992. [D] H. Dym, J-contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation. CBMS Regional Conf. Ser. Math. 71, Amer. Math. Soc., Providence, R. I. 1989.

Szeg˝o Pairs of Orthogonal Rational Matrix-valued Functions [FK]

[FKL1]

[FKL2] [FKL3]

[G1]

[G2] [GS] [Kt]

[R] [S]

189

B. Fritzsche, B. Kirstein, Schwache Konvergenz nichtnegativ hermitescher Borelmaße. Wiss. Z. Karl-Marx-Univ. Leipzig, Math.-Naturwiss. R. 37 (1988), 375–398. B. Fritzsche, B. Kirstein, A. Lasarow, On rank invariance of moment matrices of nonnegative Hermitian-valued Borel measures on the unit circle. Math. Nachr. 263–264 (2004), 103–132. B. Fritzsche, B. Kirstein, A. Lasarow, Orthogonal rational matrix-valued functions on the unit circle. Math. Nachr. 278 (2005), 525–553. B. Fritzsche, B. Kirstein, A. Lasarow, Orthogonal rational matrix-valued functions on the unit circle: Recurrence relations and a Favard-type theorem. to appear in: Math. Nachr. Ja. L. Geronimus, On polynomials orthogonal on the unit circle, on the trigonometric moment problem and on associated functions of classes of Carath´ eodory-Schur (in Russian). Mat. USSR-Sb. 15 (1944), 99–130. Ja. L. Geronimus, Orthogonal Polynomials (in Russian). Fizmatgiz, Moskva 1958. U. Grenander, G. Szeg˝ o, Toeplitz Forms and Their Applications. University of California Press, Berkeley-Los Angeles 1958. I. S. Kats, On Hilbert spaces generated by Hermitian monotone matrix functions (in Russian). Zupiski Nauc.-issled. Inst. Mat. i Mech. i Kharkov. Mat. Obsh. 22 (1950), 95–113. M. Rosenberg, The square integrability of matrix-valued functions with respect to a non-negative Hermitian measure. Duke Math. J. 31 (1964), 291–298. G. Szeg˝ o, Orthogonal Polynomials. Amer. Math. Soc. Coll. Publ. 23, Providence, R. I. 1939.

Bernd Fritzsche, Bernd Kirstein Fakult¨ at f¨ ur Mathematik und Informatik Universit¨ at Leipzig Augustusplatz 10 D-04109 Leipzig Germany e-mail: {fritzsche,kirstein}@mathematik.uni-leipzig.de Andreas Lasarow Katholieke Universiteit Leuven Departement Computerwetenschappen Celestijnenlaan 200A B-3001 Heverlee (Leuven) Belgium e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 191–248 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Singularities of Generalized Strings Michael Kaltenb¨ack, Henrik Winkler and Harald Woracek Abstract. We investigate the structure of a maximal chain of matrix functions whose Weyl coeﬃcient belongs to Nκ+ . It is shown that its singularities must be of a very particular type. As an application we obtain detailed results on the structure of the singularities of a generalized string which are explicitly stated in terms of the mass function and the dipole function. The main tool is a transformation of matrices, the construction of which is based on the theory of symmetric and semibounded de Branges spaces of entire functions. As byproducts we obtain inverse spectral results for the classes of symmetric and essentially positive generalized Nevanlinna functions. Mathematics Subject Classiﬁcation (2000). 46C20, 34A55, 30H05. Keywords. generalized string, chain of matrices, de Branges space.

1. Introduction A vibrating string S[L, m] with inhomogeneous mass distribution is given by its length L > 0 and a function m : [0, L) → [0, ∞) which is nondecreasing and continuous from the left. The function m measures the total mass of the part of the string between 0 and x. In the description of the motion of the string the following boundary value problem appears: x y (x) + z y(t)dm(t) = 0 , 0

y (0) = 0, y(L) = 0 if L + m(L) < ∞ . Thereby z is a complex parameter. The concept of the principal Titchmarsh-Weyl coeﬃcient qS of a string S was introduced by I.S.Kac and M.G.Krein, cf. [KaK1]. It turned out to be of fundamental importance. The principal Titchmarsh-Weyl coeﬃcient belongs to the Stieltjes class S, i.e., it is analytic in the open upper half-plane C+ and satisﬁes Im qS (z) ≥ 0, Im zqS (z) ≥ 0, z ∈ C+ .

192

M. Kaltenb¨ack, H. Winkler and H. Woracek

A basic inverse result states that to each function q ∈ S there exists a unique string S[L, m], such that q is the principal Titchmarsh-Weyl coeﬃcient of S[L, m]. A canonical system of diﬀerential equations, or one-dimensional Hamiltonian system, is a 2 × 2-system of diﬀerential equations of the form y (x) = zJH(x)y(x), x ∈ [0, lH ) ,

(1.1)

where H is a locally integrable 2×2-matrix-valued function on [0, lH ) whose values are real and nonnegative matrices. Moreover, J is the matrix 0 −1 . J := 1 0 If x is interpreted as time parameter, it models the motion of a particle under the inﬂuence of a time-dependent potential. The function H is called the Hamiltonian of the system under consideration and describes its total energy. To a canonical system there is associated its Weyl coeﬃcient qH , which is a function belonging to the Nevanlinna class N , i.e., is analytic in C+ and satisﬁes Im qH (z) ≥ 0, z ∈ C+ . A basic inverse result of L. de Branges, cf. [dB1] (a proper formulation can be found, e.g., in [W1]), states that to each function q ∈ N there exists an essentially unique Hamiltonian which has q as its Weyl coeﬃcient. The notions of strings and canonical systems are closely related. In fact, in view of the above inverse results, we know that to each string S[L, m] there exists a unique Hamiltonian Hs such that qS = qHs , and that the behavior of the string is completely determined by the behavior of the canonical system with Hamiltonian Hs . There are also other ways to relate strings and canonical systems. Since qS ∈ S we know that also zqS (z) ∈ N , and therefore that there exists a Hamiltonian H0 such that qH0 = zqS (z). Moreover, it is known that, if q ∈ S, then also zq(z 2 ) ∈ N . Thus we have naturally associated yet another Hamiltonian Hd , namely such that qHd = zqS (z 2 ). Each of the Hamiltonians Hs , H0 and Hd fully describes the string, the most natural choice is Hs . Each of Hs , H0 and Hd can be determined explicitly in terms of the string S[L, m]. A detailed exposition of these topics is given in [KWW3]. During the last decades a theory was developed which deals with generalizations of the notions and theorems mentioned above to an indeﬁnite setting. The class N<∞ of generalized Nevanlinna functions is deﬁned by substituting the positivity condition in the deﬁnition of N by the requirement that a certain kernel function has only a ﬁnite number of negative squares. For the exact deﬁnition see §2.3. This class of functions was intensively studied, for the basic results we refer + to [KL1]. Moreover, a function q is said to belong to the class N<∞ , if q ∈ N<∞ and zq(z) ∈ N . This can be viewed as a generalization of the Stieltjes class. For + example it is known that, if q ∈ N<∞ , then zq(z 2 ) ∈ N<∞ . The theory of canonical systems and the inverse spectral theorem of L. de Branges was generalized to an indeﬁnite setting in [KW1, KW2, KW3]. Thereby the diﬀerential equation y (x) = zJH(x)y is substituted by the family

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of its fundamental matrices ω(x), which form a so-called chain of matrices. The notion of chains of matrices can be axiomatically accessed and generalized to the indeﬁnite situation. It is then proved that to each chain of matrices a function q∞ (ω) ∈ N<∞ is associated, which is called the Weyl coeﬃcient of the chain, and that conversely to each q ∈ N<∞ there exists an essentially unique chain of matrices such that q is its Weyl coeﬃcient. We will recall these notions and results in more detail in §4. The interpretation of an indeﬁnite chain of matrices as the family of fundamental matrices of an indeﬁnite canonical system is work in progress, cf. [KW4]. In contrast to the classical situation, a chain of matrices in the indeﬁnite case has singularities. The peculiarities of indeﬁniteness are reﬂected in the structure of these singularities. A generalization of the notion of a string to the indeﬁnite setting was proposed in [LW]. A generalized string is a triple S[L, m, D] where L > 0, m is a locally square integrable function deﬁned on [0, L) which is nondecreasing and continuous from the left with possible exception of a ﬁnite number of points, and where D is a step function deﬁned on [0, L) which has only a ﬁnite number of points of increase, is nondecreasing and continuous from the left. A point xe ∈ [0, L) is called critical, if either D(xe +) − D(xe ) > 0, m(xe +) − m(xe ) < 0 or lim supx→xe |m(x)| = ∞. The relation f (x) + z f (x) dm(x) + z 2 f (x) dD(x) = 0, f (0−) = 0 . [0,x]

[0,x]

is called the diﬀerential equation of the generalized string. Of course, this equation requires an appropriate interpretation. Also to a generalized string S[L, m, D] a function qS is associated and again called the principal Titchmarsh-Weyl coeﬃcient + of S[L, m, D]. It belongs to the class N<∞ . An inverse theorem is established which states that this notion induces a bijective correspondence between the set of all + generalized strings and N<∞ . By the above inverse results we know that, given a generalized string S[L, m, D], there exist chains of matrices ωs , ω0 and ωd , such that q∞ (ωs )(z) = qS (z), q∞ (ω0 )(z) = zqS (z) and q∞ (ωd )(z) = zqS (z 2 ). Thereby ω0 will be positive deﬁnite, since zqS (z) ∈ N , whereas ωs and ωd will in general be indeﬁnite. The behavior of the generalized string is fully determined by each of these chains. It is the aim of this paper to describe the chain ωs , in particular the structure of its singularities, in terms of the generalized string S[L, m, D]. It turns out that the singularities of ωs correspond exactly to the critical points of S[L, m, D], and that their structure can be explicitly read oﬀ from the behavior of the mass function m and the possible presence of dipoles. We obtain a noteworthy inverse result, which states that the singularities of a chain whose Weyl coeﬃcient belongs to + N<∞ are of a very special kind. In fact, there are just ﬁve diﬀerent types which can occur. We will obtain these results by explicit construction. Assume that q ∈ N<∞ such that also zq(z), zq(z 2) ∈ N<∞ . Let ωs , ω0 , ωd be the chains of matrices with q∞ (ωs )(z) = q(z), q∞ (ω0 )(z) = zq(z), q∞ (ωd )(z) = zq(z 2 ) .

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We will give a method how to construct all three of the chains ωs , ω0 , ωd , once one of them is known. From this we deduce results on the structure of singularities of either of these chains. In the situation that S[L, m, D] is a generalized string and q = qS we can apply this knowledge to obtain what we were aiming for. We are also led to the conclusion that indeed the principal Titchmarsh-Weyl coeﬃcient qS , whose deﬁnition in [LW] might seem to be a bit ‘ad hoc’, is the most natural object to describe the structure of a generalized string S[L, m, D]. Although, of course, S[L, m, D] is also determined by either of zqS (z) or zqS (z 2 ). However, looking at the chains ω0 and ωd , we see that ω0 does not have any singularities and it will be apparent from our results that the singularities of ωd can be of a much more complicated type as those of ωs . Hence the information on S[L, m, D] in ω0 is somewhat hidden and ωd is simply to big to describe S[L, m, D] in a neat way. The present work is divided into three parts. The ﬁrst part consists of Sections 2 and 3. In Section 2 we introduce some classes of functions which are of importance in our investigations, study the relationship between those classes as well as the reproducing kernel spaces generated by such type of functions. These are ﬁrst of all the M-classes of 2 × 2-matrix functions, in particular the subclasses of symmetric and essentially positive matrix functions, cf. Deﬁnition 2.1,Deﬁnition 2.2. Secondly we recall the notions of generalized Nevanlinna functions and of Hermite-Biehler functions, and the appropriate analogues of symmetry and essential positivity on the level of these functions. Moreover, we recall the deﬁnition of a de Branges space of entire functions, cf. Deﬁnition 2.12, and the relation of those spaces to the introduced classes of functions. Some of these results are well known, however, we wish to set up these notations in suﬃcient generality and to collect what is needed in the sequel. In Section 3 we deal with a transformation of matrix functions and its converse. This transformation relates the M-classes of symmetric and essentially positive matrix functions, cf. Theorem 3.2. Although the methods employed in these investigations are mostly elementary, they lead to two striking results on the structure of de Branges spaces, Proposition 3.12 and Proposition 3.14. The second part of this paper consists of Sections 4, 5 and 6. In this part we lift the above-mentioned transformations to the level of chains of matrices and investigate the evolution of singularities. In Section 4 we give the deﬁnition of a chain of matrices, cf. Deﬁnition 4.1, and of symmetric and essentially positive chains of matrices, cf. Deﬁnition 4.3. Moreover, we recall some basic facts concerning these notions. A noteworthy inverse result, which is proved in this section, states that a chain of matrices is symmetric if and only if its Weyl coeﬃcient is symmetric, cf. Proposition 4.4. Section 5 deals with the proper lifting of the transformations of matrix functions to whole chains of matrices. It contains Theorem 5.1 which can be viewed as the core of our present work. It shows explicitly how we can obtain the chain whose Weyl coeﬃcient is q(z) from the chain ω whose Weyl coeﬃcient is zq(z 2 ). As a corollary we obtain another inverse result which states that a chain of matrices is essentially positive if and only if its Weyl coeﬃcient has this property, cf. Proposition 5.6. Also a ﬁrst result on the structure of essentially positive chains

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is deduced, cf. Corollary 5.8. Moreover, we consider the inverse transformation, cf. Theorem 5.10 where we give an explicit construction of the chain whose Weyl coeﬃcient is zq(z 2 ) assuming that the chain with Weyl coeﬃcient q(z) is known. These results also lead to a construction of the chain with Weyl coeﬃcient zq(z) out of the one with Weyl coeﬃcient q as indicated in Remark 5.11. The statement of Remark 5.11 is of fundamental importance, since it tells us how to proceed in order to reach our aim stated above. In Section 6 we recall the classiﬁcation of singularities of a chain of matrices and investigate how singularities appear and transform when switching from the chain with Weyl coeﬃcient q to those whose Weyl coeﬃcients are zq(z 2 ) and −(zq(z))−1 , respectively. In view of our needs in the discussion of generalized strings we restrict ourselves to the case that q ∈ N . However, it is obvious how a more general discussion can be carried out (and how tedious this might be). Finally, in the third part, Section 7, we prove the results on generalized strings we were aiming for, cf. Theorem 7.3. Following the idea which was made explicit in Remark 5.11, they are deduced from the results of Sections 5 and 6.

2. Some classes of functions and their interrelation In this preliminary we set up our notation and collect some results on various classes of functions and their interrelation. Only some of these results are new, some are well known, some are taken from previous work. However, we feel that it is a beneﬁt for the reader to have this collection of preliminaries at hand. 2.1. The class M of matrix functions Main objects of our studies in the present paper are 2 × 2-matrix-valued entire functions of a particular kind. For a function f : C → C we denote by f # the function f # (z) := f (z) . (2.1) If f = f # , we call f real. 2.1. Deﬁnition. Let M be the set of all 2 × 2-matrix-valued functions W = (wij )2i,j=1 : C → C2×2 such that the entries wij are real and entire functions, det W ≡ 1 and W (0) = I. Denote by Msym the subset of M which consists of those functions W such that w11 , w22 are even and w12 , w21 are odd. Let Mep be the set of all functions W ∈ M which have the property that each of their entries has only ﬁnitely many zeros oﬀ the positive real axis. Let J be the matrix

J :=

0 −1 . 1 0

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2.2. Deﬁnition. Let κ ∈ N ∪ {0}. We write W ∈ Mκ , if W belongs to M and if the 2 × 2-matrix-valued kernel HW (w, z) :=

W (z)JW (w)∗ − J z−w

has κ negative squares on C. Throughout this paper we will use the notation $ $ M≤κ := Mν , M<∞ := 0≤ν≤κ

Mν ,

ν∈N∪{0}

and write ind− W = κ to express the fact that W ∈ M belongs to Mκ . Moreover, ep := Msym ∩ Mκ , Mep ∩ Mκ , Msym κ κ := M sym ep ep and Msym ≤κ , M<∞ , M≤κ , M<∞ are deﬁned correspondingly. sym Similarly we can deﬁne classes +Mκ , +Mκ etc., by imposing a restriction on the numbers of positive squares, instead of negative squares, of the kernel HW . For later reference let us explicitly state the following elementary and mostly well-known results. Put 0 1 . V := 1 0 sym

+ + 2.3. Lemma. The classes M, Msym , M<∞ , Msym <∞ , M<∞ , M<∞ are closed with respect to multiplication. Each of the following transformations φj is an involution of M. The subclasses Msym , Mep remain invariant (symbolized by a ✓) or not and the class Mκ remains invariant or is mapped to the class +Mκ according to the following scheme:

Transformation φ1 : W (z) → W (−z)−1 φ2 : W (z) → W (−z) φ3 : W (z) → W (z)−1 φ4 : W (z) → −JW (z)J φ5 : W (z) → V W (z)−1 V φ6 : W (z) → W (z)T

Msym ✓ ✓ ✓ ✓ ✓ ✓

Mep

✓ ✓ ✓ ✓

Mκ ✓ + Mκ + Mκ ✓ ✓ + Mκ

Proof. The fact that M and Msym are closed with respect to multiplication is obvious. The kernel relation (W1 W2 )(z)J(W1 W2 )∗ (w) − J = z−w W2 (z)JW2∗ (w) − J W1 (z)JW1∗ (w) − J W1 (w)∗ + z−w z−w shows that also M<∞ and +M<∞ have this property. = W1 (z)

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Each of the transformations φj is an involution on M. The facts that Msym is mapped into itself by all φj and that Mep is invariant under φ3 , . . . , φ6 is seen by explicitly writing down the matrix φj (W ). The fact that φ1 , φ4 and φ5 map Mκ into itself follows from the kernel relations W (−z)−1 JW (−w)−∗ − J −J + W (z)JW (w)∗ = W (−z)−1 W (w)−∗ , z−w (−z) − (−w) W (z)JW (w)∗ − J ∗ (−JW (z)J)J(−JW (w)J)∗ − J =J J , z−w z−w and W (z)JW ∗ (w) − J (V W −1 V )(z)J(V W −1 V )∗ (w) − J = V W (z)−1 W (w)−∗ V . z−w z−w Moreover, φ2 (Mκ ) = +Mκ since HW (−z) (w, z) = −HW (−w, −z) . From the relations φ3 = φ1 ◦φ2 and φ6 = φ3 ◦φ4 , we ﬁnd that also φ3 (Mκ ) = +Mκ and φ6 (Mκ ) = +Mκ . 2.4. Remark. The simplest examples of matrices in M, besides the constant I, are linear polynomials. An elementary argument shows that a nonconstant linear polynomial W belongs to M if and only it is of the form 1 − lz sin φ cos φ lz cos2 φ W (z) = W(l,φ) (z) := (2.2) −lz sin2 φ 1 + lz sin φ cos φ for some l ∈ R \ {0} and φ ∈ [0, π). The numbers l and φ are uniquely determined by W . Note that for all l ∈ R \ {0} and φ ∈ [0, π) we have W(l,φ) ∈ Mep , that W(l,φ) ∈ Msym if and only if φ = 0 or φ = π2 , and that + M0 ∩ +M1 , l > 0 W(l,φ) ∈ M1 ∩ +M0 , l < 0 Any function W ∈ Mκ generates a Pontryagin space K(W ) by means of the reproducing kernel HW . Recall that this space is obtain as completion of the linear space & ' x x ∈ C2 , : w ∈ C, span HW (w, .) y y equipped with the inner product ∗ x2 x x x := HW (w1 , .) 1 , HW (w2 , .) 2 HW (w1 , w2 ) 1 , y1 y2 y2 y1 see for example [ADRS].

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Let W ∈ M<∞ . It follows from the fact that the entries of W are real, i.e., # = wij , that the mapping satisfy wij # # f f f −→ := g g g# is an anti-isometry of K(W ) onto itself. Moreover, cf. [KW1, Proposition 8.3], the space K(W ) is invariant under the diﬀerence quotient operator (w ∈ C) f (z)−f (w) f z−w −→ g(z)−g(w) . Rw : g z−w

We put

' & 0 : w∈C , , K− (W ) := cls HW (w, z) 1 1 and, similarly, K+ (W ) := cls{HW (w, z) : w ∈ C}. 0

2.2. A characterization of Msym <∞ The fact that W ∈ Msym is reﬂected in a symmetry property of K(W ). κ Denote by O(C) the set of all entire functions and consider the map −F (−z) F (z) → M : G(−z) G(z)

(2.3)

This map is an involution of O(C)2 . if and only if M |K(W ) is an 2.5. Proposition. Let W ∈ Mκ . Then W ∈ Msym κ isometry of K(W ) onto itself. Proof. Assume that W ∈ Msym κ . Then −1 0 −1 0 W (z) W (−z) = 0 1 0 1

and hence HW (−w, −z) =

Since M we obtain

Moreover,

−1 0 −1 0 HW (w, z) . 0 1 0 1

F (−z) −1 0 F (z) , = G(−z) 0 1 G(z)

−1 0 α α M HW (w, z) HW (w, −z) = 0 1 β β −α ∈ K(W ) . = HW (−w, z) β

α1 α2 M HW (w1 , z) , M HW (w2 , z) β1 β2 K(W )

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−α1 −α2 , HW (−w2 , z) = HW (−w1 , z) K(W ) β1 β2 ∗ ∗ α2 −α2 −α1 α1 = = HW (−w1 , −w2 ) HW (w1 , w2 ) β2 β1 β2 β1 α1 α2 = HW (w1 , z) , HW (w2 , z) . β1 β2 K(W )

' & α : w, α, β ∈ C . L := span HW (w, z) β Then L is a dense linear subspace of K(W ) and M |L maps L isometrically onto itself. Hence there exists an isometric continuation of M |L to K(W ) which must be given by (2.3), since point evaluation is continuous. Conversely, assume that (2.3) is an isometry of K(W ) onto itself. As M 2 = id, we obtain F (z) α F (z) α , M HW (w, z) , H = M (w, z) W K(W ) G(z) β G(z) β K(W ) ∗ ∗ −α −F (−w) F (−w) α = = β G(−w) G(−w) β F (z) −α , HW (−w, z) = . K(W ) G(z) β It follows that −1 0 α α HW (w, −z) = M HW (w, z) 0 1 β β α −α −1 0 . = HW (−w, z) = HW (−w, z) β β 0 1 Since α, β were arbitrary, we conclude that −1 0 −1 0 HW (w, −z) = HW (−w, z) . 0 1 0 1 Let

Substituting w = 0 in −1 0 and since

it follows that i.e., that W ∈ Msym .

this relation yields W (z)J − J −1 0 0 W (−z)J − J , = 1 0 1 −z z −1 −1 0 J = −J 0 0 1

0 , 1

−1 0 −1 0 , W (−z) = W (z) 0 1 0 1

This result puts us in position to apply [KWW2, Lemma 2.1], and to obtain a splitting of the space K(W ).

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2.6. Corollary. Deﬁne

) ( F ∈ K(W ) : F odd, G even , G ) ( F K(W )o := ker(I + M ) = ∈ K(W ) : F even, G odd . G Then K(W )e and K(W )o are closed subspaces of K(W ) and ˙ K(W ) = K(W )e [+]K(W )o . K(W )e := ker(I − M ) =

The reproducing kernels of K(W )e and K(W )o are given by and 12 (I − M )HW (w, z), respectively.

1 2 (I

+ M )HW (w, z)

Of particular interest is the situation when K− (W ) = K(W ). T 2.7. Lemma. Let W ∈ Msym <∞ , (F, G) ∈ K(W ), and assume that K− (W ) = K(W ). T Then (F, G) ∈ K(W )e if and only if G is even, and (F, G)T ∈ K(W )o if and only if G is odd. The analogous assertion holds when K+ (W ) = K(W ) and G is replaced by F .

Proof. Assume that K− (W ) = K(W ) and that (F, G)T ∈ K(W ) is such that G is even. Then F (z) + F (−z) F = (I − M ) 0 G and hence (I − M )(F, G)T = 0. The other assertions follow similarly.

2.3. The class N<∞ of generalized Nevanlinna functions Let us recall the notion of matrix-valued generalized Nevanlinna functions: A n × n-matrix-valued function Q is said to belong to Nκn×n , if it is deﬁned and meromorphic on C \ R, satisﬁes Q(z) = Q(z)∗ , and has the property that the n × n-matrix-valued kernel Q(z) − Q(w)∗ LQ (w, z) := z−w has κ negative squares. The following subclasses of generalized Nevanlinna functions were investigated in [KWW2]. A function Q ∈ Nκn×n is said to be (i) symmetric, if Q(−z) = −Q(z), i.e., if Q is odd. (ii) essentially positive, if Q is analytic on C \ [0, ∞) with possible exception of ﬁnitely many poles. The subset of Nκn×n which consists of all symmetric (essentially positive) functions will be denoted by Nκn×n,sym (Nκn×n,ep , respectively). If we deal with scalar-valued functions, i.e., n = 1, then the upper index n × n will be suppressed. Moreover, the scalar function q(z) ≡ ∞ will be regarded as an element of N0 . We will freely ep n×n,sym , N<∞ etc. use self-explanatory notation like N≤κ The M-classes of matrix functions are related to the generalized Nevanlinna classes in several ways, two of them are of importance in the present context.

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The ﬁrst one we deal with is the Potapov-Ginzburg transform. If W is a 2×2matrix function whose entries are real analytic functions such that det W ≡ 1 and w21 ≡ 0, then the Potapov-Ginzburg transform Ψ(W ) is deﬁned as (cf., e.g., [Br]) ⎛ ⎞ Ψ(W )(z) := ⎝

w11 (z) w21 (z) 1 w21 (z)

1 w21 (z) ⎠ w22 (z) w21 (z)

.

2.8. Lemma. We have W ∈ Mκ if and only if Ψ(W ) ∈ Nκ2×2 . Moreover, W ∈ sym Msym (W ∈ Mep (Ψ(W ) ∈ Nκep , respectively). κ κ ) if and only if Ψ(W ) ∈ Nκ Proof. The assertion follows from the kernel relation ∗ −1 w11 (z) Ψ(W )(z) − Ψ(W )(w)∗ −1 w11 (w) HW (w, z) = 0 w21 (z) 0 w21 (w) z−w

Secondly, matrix functions of the class M<∞ operate on N<∞ via fractional linear transformations: If W ∈ M and τ : Ω ⊆ C → C is an analytic function such that w21 (z)τ (z) + w22 (z) ≡ 0, then we deﬁne (W τ )(z) :=

w11 (z)τ (z) + w12 (z) . w21 (z)τ (z) + w22 (z)

Then W τ is a meromorphic function on Ω. If w21 (z)τ (z) + w22 (z) ≡ 0 we set W τ ≡ ∞. Moreover, w11 (z) . (W ∞)(z) := w21 (z) 2.9. Lemma. If W ∈ Mκ and τ ∈ Nν , then W τ ∈ N≤κ+ν . If W ∈ +Mκ and τ ∈ R ∪ {∞}, then −(W τ ) ∈ N≤κ . Proof. This assertion follows from the kernel relation (W τ )(z) − (W τ )(w) (w21 (w)τ (w) + w22 (w)) z−w (V W −1 V )(z)J(V W −1 V )∗ (w) − J −τ (w) τ (z) − τ (w) + . = −τ (z) 1 1 z−w z−w

(w21 (z)τ (z) + w22 (z))

2.10. Corollary. (i) If W ∈ Mκ , then each of the functions w11 (z) w12 (z) , , w21 (z) w22 (z)

−

w11 (z) w21 (z) ,− w12 (z) w22 (z)

(2.4)

belongs to N≤κ . (ii) A function W ∈ M<∞ belongs to Mep <∞ if and only if one of its entries has the property to have only ﬁnitely many zeros in C\[0, ∞). In fact, if W ∈ Mκ and one entry has n zeros in C \ [0, ∞), then each other entry has at most 4n + 6 + 3κ zeros in this region.

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Proof. For the ﬁrst two functions in (2.4) use the ﬁrst part of Lemma 2.9 with the matrix W and the parameters τ = ∞ and τ = 0, respectively. For the second pair of functions use the second part of Lemma 2.9 with the matrix W T and the parameters τ = 0, ∞. The second assertion follows from (2.4) by employing [KWW2, Lemma 4.5]. An inspection of its proof shows the explicit estimate. The estimate in Corollary 2.10, (ii), is very rough, but suﬃcient for our purposes. 2.4. de Branges spaces of entire functions. The class HB<∞ of Hermite-Biehler functions Let us recall the notion of dB-spaces and their connection to the class of HermiteBiehler functions. To make the present work more self-contained, let us recall the notion of an almost Pontryagin space, [KWW1]. 2.11. Deﬁnition. Let L be a linear space, [., .] an inner product on L and O a Hilbert space topology on L. The triplet (L, [., .], O) is called an almost Pontryagin space, if (aPS1) [., .] is O-continuous. (aPS2) There exists a O-closed linear subspace M of L with ﬁnite codimension such that (M, [., .]) is a Hilbert space. 2.12. Deﬁnition. An inner product space (P, [., .]) is called a de Branges space (dB-space, for short), if the following axioms hold true: (dB1) (P, [., .]) is a reproducing kernel almost Pontryagin space on C whose elements are entire functions. (dB2) If F ∈ P, then also F # ∈ P. Moreover, [F # , G# ] = [G, F ], F, G ∈ P . (dB3) If F ∈ P and z0 ∈ C \ R with F (z0 ) = 0, then z − z0 F (z) ∈ P . z − z0 Moreover, if additionally G ∈ P with G(z0 ) = 0, then z − z0 z − z0 F (z), G(z) = [F, G] . z − z0 z − z0 We will assume throughout this paper that also (Z) For every t ∈ R there exists F ∈ P with F (t) = 0. Let P be a dB-space. We deﬁne the set of associated functions as Assoc P := zP + P . A function S belongs to Assoc P if and only if P is closed with respect to the diﬀerence quotient operator (w ∈ C) F (z) →

F (z)S(w) − F (w)S(z) . z−w

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The Hermite-Biehler class HBκ with negative index κ ∈ N ∪ {0} is deﬁned as the set of all entire functions E, such that E and E # have no common non-real zeros, E −1 E # is not constant, and the kernel # (z) E # (w) 1 − EE(z) E(w) S E# (w, z) := i E z−w has κ negative squares on C+ . The Hermite-Biehler class is related to the notion of dB-Pontryagin spaces, i.e., non-degenerated dB-spaces, by the fact that, if P is a dB-Pontryagin space, then its reproducing kernel K is of the form K(w, z) = i

E(z)E(w) − E # (z)E(w) 2π(z − w)

for a (not necessarily unique) Hermite-Biehler function E. Conversely, every Hermite-Biehler E function generates in this way a dB-Pontryagin space which we will denote by P(E), cf. [KW1] The class of Hermite-Biehler functions is also connected with generalized Nevanlinna functions. Let E be an entire function and write E = A − iB with A, B real entire functions. Then A E ∈ HBκ ⇐⇒ ∈ Nκ , B A A ∈ Nκsym , E ∈ HBκsb ⇐⇒ ∈ Nκep . E ∈ HBκsym ⇐⇒ B B The following two special classes of dB-spaces were investigated in [KWW4]. Let M : O(C) → O(C) be deﬁned as O(C) → O(C) M: F (z) → F (−z) The map M is a linear involution of O(C). A dB-space (P, [., .]) is called symmetric, if M induces an isometric involution of P, i.e., if M (P) ⊆ P and [M F, M G] = [F, G], F, G ∈ P . Let (P, [., .]) be a dB-space, and denote by S the operator of multiplication with the independent variable in P. Then P is called semibounded if the inner product [F, G]S := [SF, G], F, G ∈ dom S , has a ﬁnite number of negative squares on dom S. The sets of Hermite-Biehler functions which correspond to these classes of dB-Pontryagin spaces are the following: Deﬁne HBκsym to be the subset of HBκ consisting of all functions E which have the property that E # (z) = E(−z). Then sym P is a symmetric dB-Pontryagin space if and only P = P(E) for some E ∈ HB<∞ . sb Moreover, we denote by HBκ the set of all functions E = A − iB ∈ HBκ such that B has only ﬁnitely many zeros in C \ [0, ∞). Then P is a semibounded dBsb Pontryagin space if and only P = P(E) for some E ∈ HB<∞ .

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We will need the following result which supplements our discussion of symmetric dB-spaces in [KWW4]. A subspace Q of a dB-space P is called a dB-subspace of P, if it is with the topology and inner product inherited from P a dB-space. This is the case if and only if F ∈ Q implies F # ∈ Q and if F ∈ Q, F (w) = 0, implies (z −w)−1 F (z) ∈ Q. A main result in the theory of dB-spaces, cf. [dB1], [KW1], is the ordering theorem for subspaces of P. It states that the set of all dB-subspaces of a given dB-space P is totally ordered with respect to inclusion. 2.13. Lemma. Let P be a symmetric a dB-space and let Q be a dB-subspace of P. Then Q is symmetric. ˜ := Q, we have to show that Q ˜ ⊆ Q. It is straightforward to check Proof. Put Q ˜ that Q is a dB-subspace of P. By the ordering theorem for subspaces of P we have ˜ ⊆ Q or Q ⊆ Q. ˜ In the ﬁrst case we are already done. In the second case either Q ˜ ˜ ˜ = Q. ˜ we have M Q = Q ⊆ Q. Since M is an involution, this implies M Q Matrices of the class M<∞ give rise to Hermite-Biehler functions. If W ∈ Mκ , then the function EW := w22 + iw21 belongs to HB≤κ . It was shown in [KW1] that the projection π− : (F, G)T → G is an isometric isomorphism of K− (W )/K− (W )◦ onto P(EW ). ˜W := w11 − iw12 , then E˜W ∈ HB≤κ and the projection Similarly, if we put E T ˜W ). π+ : (F, G) → F is an isometric isomorphism of K+ (W )/K+ (W )◦ onto P(E It is an important result, cf. [KW1], that for a non-degenerated dB-space P we have 1 ∈ Assoc P if and only if there exists a matrix W ∈ M<∞ , K− (W ) = K(W ) such that P = P(EW ). If Q is a dB-subspace of P, then trivially Assoc Q ⊆ Assoc P. It follows from [dB1] that, if 1 ∈ Assoc P, then also 1 ∈ Assoc Q. The dB-subspaces of a P(EW ) can be obtained from certain subspaces of K(W ). Note that, if we assume that K− (W ) = K(W ), then the projection π− onto the second component is an isometry of K(W ) onto P(EW ). 2.14. Lemma. Let W ∈ M<∞ and assume that K− (W ) = K(W ). Then the projection π− induces an order-preserving bijection of the set of all closed subspaces L of K(W ) which are invariant under the mappings .# (cf. (2.1)) and Rw , w ∈ C, and the set of all dB-subspaces of P(EW ). Thereby ind− π− (L) = ind− L and dim π− (L)◦ = dim L◦ . Under the assumption that K+ (W ) = K(W ) the same assertion holds with π+ and P(E˜W ). Proof. Since K− (W ) = K(W ), the mapping π− is an isometric isomorphism of K(W ) onto P(EW ). Hence it induces an order preserving bijection of the set of all closed subspaces of K(W ) onto the set of all closed subspaces of P(EW ), and hence leaves negative indices and degree of degeneracy invariant. Assume that L is a closed subspace of K(W ) which is closed with respect to .# and Rw . Since π− ◦ .# = .# ◦ π− and π− ◦ Rw = Rw ◦ π− , also π(L) has this

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property. We see that F ∈ π− (L) implies F # ∈ π− (L) and that, if F ∈ π− (L) and F (w) = 0, then (z − w)−1 F (z) ∈ π− (L). Thus π− (L) is a dB-subspace of P(EW ). −1 (Q). Then Conversely, let Q be a dB-subspace of P(EW ) and put L := π− # T # F ∈ Q implies F ∈ Q. Assume that F = π− (G, F ) , F = π− (H, F # )T . It follows that (G# − H, 0)T ∈ K(W ) and thus that G# = H. Thus also L is closed under .# . As 1 ∈ Assoc P(EW ), also 1 ∈ Assoc Q, i.e., Q is invariant under Rw . Again using that K− (W ) = K(W ), we see that also L has this property.

3. The square root transformation In this section we investigate a transformation T√ which assigns to each matrix W ∈ Msym an element of M and its converse transformations T2,γ . These results have consequences on the structure of symmetric and semibounded dB-spaces. Moreover, they are the basic tool for the subsequent sections. 3.1. The transformations T√ and T2,γ 3.1. Deﬁnition. Deﬁne a transformation T√ : Msym → M by w12 (z) − w12 (0)w11 (z) w11 (z) 2 z T√ (W )(z ) := . (0)zw21 (z) zw21 (z) w22 (z) − w12 Let γ ∈ R. Deﬁne a transformation T2,γ : M → Msym by w11 (z 2 ) z(w12 (z 2 ) + γw11 (z 2 )) . T2,γ (W )(z) := w21 (z2 ) w22 (z 2 ) + γw21 (z 2 ) z M

The facts that T√ (W ) is well deﬁned and belongs to M, and that T2,γ (W ) ∈ follow on inspecting the deﬁning formulas.

sym

3.2. Theorem. The transformation T√ maps Msym surjectively onto M. For each W ∈ M we have & ' T√−1 {W } = T2,γ (W ) : γ ∈ R . ep By T√ the class Msym <∞ is mapped onto M<∞ . In fact, ind− T√ (W ) ≤ ind− W . Moreover, the map zf (z 2) f (z) → Φ: g(z) g(z 2 )

is an isometry of K(T√ (W )) onto K(W )e . The proof of this theorem needs some preparation. First we list some elementary properties of T√ and T2,γ , and show that these transformations are in a way inverse to each other.

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3.3. Lemma. (i) Let W ∈ Msym , then T√ (W )(z ) = 2

1

0 1

z

0

z W (z) 0

0 1

1 −w12 (0) . 0 1

(3.1)

ˆ = (w If additionally W ˆij )2i,j=1 ∈ Msym , then ˆ )(z 2 ) = T√ (W )−1 (z 2 ) · T√ (W 1 1 1 w12 (0) 0 z ˆ (z) z 0 W −1 (z)W = 0 1 0 0 1 0 1

−w ˆ12 (0) . 1

(3.2)

Moreover, w (0) w (0) tr T√ (W ) (0)J = −w21 − w12 , (0) + 12 (0) 11 6 2 and we have z T√ (W ) ∞ (z 2 ) = (W ∞)(z) , T√ (W )22 (z 2 ) w22 (z) 1 = · − w12 (0) . 2 √ T (W )21 (z ) w21 (z) z (ii) Let W ∈ M, then

T2,γ (W )(z) =

z 0

0 1 W (z 2 ) 1 0

γ 1

1 z

0

(3.3)

(3.4)

0 . 1

(3.5)

ˆ ∈ M, then If additionally W 1 z 0 = 0 0 1 Moreover, (iii) We have

ˆ )(z) T2,γ (W )−1 (z)T2,ˆγ (W 1 −γ 1 γˆ 2 −1 ˆ 2 z W (z ) W (z ) 1 0 1 0

tr T2,γ (W ) (0)J = γ − w21 (0) .

0 . 1

(3.6)

(3.7)

T√ ◦ T2,γ (W ) = W, W ∈ M , sym (0) ◦ T√ (W ) = W, W ∈ M . T2,w12

Proof. The formulas (3.1), (3.5) and (3.7) are veriﬁed by straightforward computation. The relation (3.3) is proved by comparison of the Taylor coeﬃcients in w12 (z) (z) − w12z2(z) − w12 (0)w11 (z) w11 2 z T√ (W ) (z ) · 2z = (z) w22 (z) − w12 (0)w21 (z) − w12 (0)zw21 (z) w21 (z) + zw21 The relation (3.2) is established by substituting the expression (3.1) for T√ (W )(z 2 ) ˆ )(z 2 ), respectively. The relation (3.6) follows from (3.5). Finally, (3.4) and T√ (W follows from the deﬁnition of T√ (W ). The second relation in (iii) follows from (3.5) and (3.1). Since T2,γ (W )12 (0) = γ, the same source implies the validity of the ﬁrst relation in (iii).

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The kernel of the map T√ can be determined explicitly. ˆ ) if and only if ˆ ∈ Msym . Then T√ (W ) = T√ (W 3.4. Lemma. Let W, W ˆ = W W(l,0) , W for some l ∈ R, where W(l,0) is as in (2.2). ˆ = W W(l,0) . Then Proof. Assume that W ˆ (z) = W (z) 1 lz , w ˆ12 (0) = w12 (0) + l , W 0 1 and hence, by (3.2), ˆ )(z 2 ) T√ (W )−1 (z 2 )T√ (W 1 1 −w12 z 0 1 lz (0) 0 (0) − l w12 z = 0 1 0 1 1 0 1 0 1 1 −w12 1 w12 1 l 1 0 (0) (0) − l . = = 0 1 0 1 0 1 0 1 ˆ ) = I. Then we obtain from (3.2) that Conversely, assume that T√ (W )−1 T√ (W 1 1 w ˆ12 (0) − w12 (0) 0 z ˆ (z) = z 0 W (z)−1 W 0 1 0 1 0 1 1 (w ˆ12 (0) − w12 (0))z . = 0 1 1 = 0

The relationship between the spaces K(W ) and K(T√ (W )) is expressed by the following kernel relation. 3.5. Lemma. For each W ∈ Msym we have w 0 z 0 2 2 HT√ (W ) (w , z ) 2 0 1 0 1 −1 0 HW (w, −z) = (I + M )HW (w, z) . = HW (w, z) + 0 1 (0) ∈ R we have Proof. Since w12 1 1 −w12 (0) J 0 1 0

(3.8)

∗ −w12 (0) =J, 1

and hence we obtain, using (3.1), 2 2 ∗ z 0 T√ (W )(z )JT√ (W )(w ) − J w 0 0 1 0 1 z 2 − w2 1 w 0 w z 0 z 0 ∗ = 2 W (w) J J W (z) − 0 1 0 0 1 0 1 z − w2 1 0 −z 0 −z = 2 W (w)∗ − . W (z) w 0 w 0 z − w2

0 1

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We compute the expression on the right-hand side of (3.8). −1 0 HW (w, −z) HW (w, z) + 0 1 W (z)JW (w)∗ − J −1 0 W (−z)JW (w)∗ − J + = 0 1 z−w −z − w 1 −1 0 J W (w)∗ W (z) (z + w)J − (z − w) = 2 0 1 z − w2 −1 0 J . − (z + w)J − (z − w) 0 1 Since 0 −2z −1 0 (z + w)J − (z − w) J= , 2w 0 1 0

the equality (3.8) follows. We are now in position to prove Theorem 3.2.

Proof of Theorem 3.2. By the ﬁrst relation in Lemma 3.3, (iii), it follows that T√ : Msym → M is surjective. Moreover, for each W ∈ M and all γ ∈ R, we have T2,γ (W ) ∈ T√−1 ({W }). The relation (3.6) shows that T2,ˆγ (W ) = T2,γ (W )W(ˆγ −γ,0) , hence we conclude from Lemma 3.4 that the matrices T2,γ (W ) exhaust T√−1 ({W }). Let W ∈ Msym κ . Then, by the kernel relation (3.8), we have T√ (W ) ∈ M≤κ . Corollary 2.6 together with (3.8) yields that the map Φ is an isometry of K(T√ (W )) onto K(W )e . By (3.4) and Corollary 2.10, (i), the function q(z) = T√ (W ) ∞ has the property that zq(z 2 ) belongs to N≤κ . It follows from [KWW2, Theorem 4.1] that ep q ∈ N≤κ . Since the entries of the ﬁrst column of T√ (W ) can not have common zeros, we see that the entry T√ (W )21 has only ﬁnitely many zeros in C \ [0, ∞). Thus, by Corollary 2.10, (ii), we have T√ (W ) ∈ Mep ≤κ . Conversely, assume that W ∈ Mep . We show that T2,γ (W ) ∈ Msym <∞ <∞ . Since T2,γ (W ) = T2,0 (W )W(γ,0) , it suﬃces to consider the particular case γ = 0. In this case w11 (z 2 ) zw12 (z 2 ) T2,0 (W )(z) = w21 (z2 ) . w22 (z 2 ) z Hence the Potapov-Ginzburg transform computes as w (z2 ) z 11 z w21 2 2 (z ) w21 (z ) Ψ T2,0 (W ) (z) = w22 (z 2 ) z z 2 w21 (z ) w21 (z 2 ) = zΨ(W )(z 2 ) .

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209

2×2,ep Since W ∈ Mep and hence, by [KWW2, Theorem 4.1], <∞ we have Ψ(W ) ∈ N<∞ 2×2,sym . Ψ T2,0 (W ) (z) ∈ N<∞

This shows T2,0 (W ) ∈ Msym <∞ .

The subject of the next proposition is to clarify how linear polynomials are transformed when either of T√ or T2,γ is performed. This result is an important tool for our later investigation of transformations of matrix chains. 3.6. Proposition. ˆ ∈ Msym and assume that W −1 W ˆ = W(l,α) . Then α ∈ {0, π }. If (i) Let W, W 2 ˆ ) = I. If α = π , then α = 0, we have T√ (W )−1 T√ (W 2 ˆ ) = W(l ,φ) , where l = l(1 + w12 T√ (W )−1 T√ (W (0)2 ), φ = Arccot w12 (0) .

ˆ ∈ M, γ, γˆ ∈ R. If W −1 W ˆ = W(l,φ) for some l ∈ R, φ ∈ [0, π), cf. (ii) Let W, W (2.2), then ˆ )(z) T2,γ (W )−1 (z)T2,ˆγ (W ⎞ ⎛ z(ˆ γ −γ)+ 1 − z 2 l sin φ(cos φ − γ sin φ) +z3 l(cos φ−γ sin φ)(cos φ−ˆγ sin φ) ⎠. =⎝ 2 2 −zl sin φ 1 + z l sin φ(cos φ − γˆ sin φ)

(3.9)

ˆ ) = W(L,α) if and only if W −1 W ˆ = W(l,φ) and We have T2,γ (W )−1 T2,ˆγ (W either l = 0, in which case L = γˆ − γ and α = 0 , π or l = 0, φ = 0, γ = γˆ = cot φ, in which case L = l sin2 φ, α = . 2 Proof. The matrix function W(l,α) belongs to Msym if and only if α ∈ {0, π2 }. This proves the ﬁrst assertion in (i). The case that α = 0 was already treated in Lemma 3.4. Assume that α = π2 . Then 1 0 ˆ , w ˆ12 W (z) = W (z) (0) = w12 (0) . −lz 1 It follows that ˆ )(z 2 ) T√ (W )−1 (z 2 )T√ (W 1 1 −w12 z 0 1 0 (0) 0 (0) − l 1 w12 z = 0 1 −lz 1 0 1 0 1 0 1 1 0 1 w12 (0) 1 −w12 (0) = −lz 2 1 0 1 0 1 1 − lw12 (0)z 2 lw12 (0)2 z 2 = = W(l ,φ) (z 2 ) 2 −lz 1 + lw12 (0)z 2 when l = l(1 + w12 (0)) and cot φ = w12 (0).

210

M. Kaltenb¨ack, H. Winkler and H. Woracek ˆ = W(l,φ) . Then We come to the proof of (ii). Assume that W −1 W

⎛ =⎝

z = 0

ˆ )(z) = T2,γ (W )−1 (z)T2,ˆγ (W

−γz 1

1 1 − lz 2 sin φ cos φ lz 2 cos2 φ z 0 −lz 2 sin2 φ 1 + lz 2 sin φ cos φ

1 − z 2 l sin φ(cos φ − γ sin φ) −zl sin2 φ

γˆ 1

= ⎞

z(ˆ γ −γ)+ +z l(cos φ−γ γ ˆ sin φ−ˆ γ sin φ cos φ−γ sin φ cos φ)⎠ 3

2

2

1 + z 2 l sin φ(cos φ − γˆ sin φ)

.

ˆ )(z) = W(L,α) . By the already proved part (i) of Assume that T2,γ (W )−1 (z)T2,ˆγ (W ˆ = W(l,φ) for certain the present proposition, we must have α ∈ {0, π2 } and W −1 W l ∈ R, φ ∈ [0, π). Considering the just proved formula (3.9) we see that either (i) or (ii) holds. The converse follows from (3.9). 3.7. Remark. Note that in case φ = 0 the matrix (3.9) is equal to 1 z(ˆ γ − γ) + lz 3 . 0 1 If φ = 0, l = 0 we can decompose (3.9) as 1 0 1 (cot φ − γ)z 1 −(cot φ − γˆ )z . 0 1 0 1 −lz sin2 φ 1 Then its Potapov-Ginzburg transform is equal to 1 1 cot φ − γ 0 1 1 − . +z 0 −(cot φ − γˆ ) z l sin2 φ 1 1 From the isomorphy of K(W )e and K(T√ (W )) we also obtain a relation between the spaces K− (W ) and K− (T√ (W )). 3.8. Proposition. Let W ∈ Msym κ , then ⊥ 1 dim K− (W )⊥ , dim K− T√ (W ) = 2 and ◦ 1 dim K− (W )◦ . dim K− T√ (W ) = 2 Proof. By [KW1, Corollary 9.7,Proposition 8.3] we have for any matrix W ' & p(z) ∈ K(W ) : p(z) polynomial K− (W )⊥ = 0 & p(z) ' : p(z) polynomial, deg p < dim K− (W )⊥ . = 0 Let Φ be deﬁned as in Theorem 3.2. Then we conclude that Φ maps K− (T√ (W ))⊥ into K− (W )⊥ , and ⊥ dim K− (W )⊥ ≥ 2 dim K− T√ (W ) .

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211

Conversely, if l is the largest odd number ≤ dim K− (W )⊥ − 1, then l l−1 ⊥ z z 2 ⊥ ∈ K− (W ) , and hence ∈ K− T√ (W ) . 0 0 If dim K− (W )⊥ ≡ 0 mod 2, then (dim K− (W )⊥ − 1) − 1 dim K− (W )⊥ l−1 = = −1, 2 2 2 and hence dim K− (T√ (W ))⊥ ≥

1 2

dim K− (W )⊥ . If dim K− (W )⊥ ≡ 1 mod 2, then

l−1 (dim K− (W )⊥ − 2) − 1 dim K− (W )⊥ − 1 = = −1, 2 2 2 and hence dim K− (T√ (W ))⊥ ≥ 12 (dim K− (W )⊥ − 1). This yields the ﬁrst equality. For any matrix W we have '◦ & p(z) ◦ ⊥ ◦ ∈ K(W ) : p(z) polynomial , = K− (W ) = K− (W ) 0 and it follows from [KW1, Proposition 8.3] that the space K− (W )◦ is invariant with respect the diﬀerence quotient operator. Hence & p(z) ' : p(z) polynomial, deg p < dim K− (W )◦ . K− (W )◦ = 0 Using the fact that

˙ K− (W )⊥ ∩ K− (W )o K− (W )⊥ = K− (W )⊥ ∩ K− (W )e [+] ' & zl ˙ : l odd, l < dim K− (W )⊥ [+] = span 0 & zl ' ˙ span : l even, l < dim K− (W )⊥ , [+] 0

the same argument as in the previous paragraph shows that ◦ 1 dim K− (W )◦ . dim K− T√ (W ) = 2

3.2. Two characteristic values of a matrix W ∈ Mep <∞ ep It was shown in [KWW2, Proposition 4.9] that for a function q ∈ N<∞ the limit ep limt→−∞ q(t) exists in R∪{±∞}. If W ∈ M<∞ , we obtain two essentially positive −1 −1 w11 and w21 w22 . We generalized Nevanlinna functions, namely W ∞ = w21 denote the respective limits by m(W ) := lim (W ∞)(t), λ(W ) := lim t→−∞

t→−∞

w22 (t) . w21 (t)

(3.10)

These numbers have interpretations in terms of the transformation T2,γ and the corresponding Pontryagin spaces.

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3.9. Proposition. Let W ∈ Mep <∞ and γ ∈ R. Then m(W ) ∈ R if and only if K− (W ) = K(W ) and K− (T2,γ (W ))◦ = {0}. Thereby K− (T2,γ (W )) = K(T2,γ (W )) if and only if m(W ) = 0. Otherwise, if m(W ) ∈ R \ {0}, we have dimK− (T2,γ (W ))⊥ = 1. sym ˆ Proof. Since W ∈ Mep <∞ we have W := T2,γ (W ) ∈ M<∞ , and, hence, may conˆ ). ˆ ). Keep in mind that W = T√ (W sider the space K(W Assume ﬁrst that m(W ) ∈ R. By (3.4) we have

m(W ) = lim

y→+∞

W ∞ (iy)2 = lim

y→+∞

1ˆ W ∞ (iy) . iy

We conclude from [KW2, Theorem 5.7] that 1 −m(W )z ˆ ˜ W (z) . W (z) := 0 1 ˜ ) = K(W ˜ ). In case m(W ) = 0 the space generated by the linear masatisﬁes K− (W trix in the above relation is one-dimensional and spanned by the constant (1, 0)T , in ˆ ) is orthocomplemented in K(W ˆ) ˆ ) = K(W ˜ ) ⊕ span{(1, 0)T }. Hence K− (W fact K(W ⊥ ˆ and dim K− (W ) = 1. We conclude from Proposition 3.8 that K− (W ) = K(W ). ˆ )⊥ = 0. In the case m(W ) = 0 we have dim K− (W ˆ )◦ = {0}. Again appealing Conversely, assume that K− (W ) = K(W ) and K(W to [KW2, Theorem 5.7] we ﬁnd that there exists a polynomial p(z) such that 1 p(z) ˆ ˜ W (z) W (z) := 0 1 ˜ ) = K(W ˜ ). Thereby dim K− (W ˆ )⊥ = deg p. Since, by Proposition satisﬁes K− (W ⊥ ˆ 3.8, dim K− (W ) ≤ 1 we can choose p(z) = az and thus lim

y→+∞

1ˆ W ∞ (iy) iy 1˜ W ∞ (iy) − a = −a . = lim y→+∞ iy

W ∞ (iy)2 = lim

y→+∞

3.10. Proposition. Let W ∈ Mep <∞ and assume that m(W ) = 0. Then λ(W ) ∈ R if and only if ind− T2,γ (W ) ∞ < ind− T2,γ (W ) for all γ ∈ R. If λ(W ) ∈ R, we have & ' λ(W ) = − sup γ ∈ R : ind− T2,γ (W ) ∞ < ind− T2,γ (W ) . If γ, γˆ ∈ R, then we have

⎧ ⎪ ⎨−1 ind− T2,ˆγ (W ) = ind− T2,γ (W ) + 0 ⎪ ⎩ 1

, γ < −λ(W ) ≤ γˆ , γ, γˆ ≥ −λ(W ) or γ, γˆ < −λ(W ) . , γˆ < −λ(W ) ≤ γ

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213

Proof. According to Proposition 3.9 our assumption m(W ) = 0 implies that K− (W ) = K(W ) and that K− (T2,γ (W )) = K(T2,γ (W )) for all γ ∈ R. Put Wγ := T2,γ (W ) and Eγ := wγ,22 + iwγ,21 . Note that the function q(z) := Wγ ∞ (z) = z W ∞ (z 2 ) does not depend on γ. By (3.4) we have w22 (−y 2 ) 1 wγ,22 (iy) = −γ. w21 (−y 2 ) iy wγ,21 (iy) There are four possibilities: λ(W ) + γ = lim

y→+∞

⎧ =0 ⎪ ⎪ ⎪ ⎨> 0 = ⎪ <0 ⎪ ⎪ ⎩

1 wγ,22 (iy) iy wγ,21 (iy) , , , ,

wγ,21 wγ,21 wγ,21 wγ,21

∈ P(Eγ ) ∈ P(Eγ ), [wγ,21 , wγ,21 ] > 0 ∈ P(Eγ ), [wγ,21 , wγ,21 ] < 0 ∈ P(Eγ ), [wγ,21 , wγ,21 ] = 0

(3.11)

In the ﬁrst two cases (cf. [KW2, Lemma 5.12]) ind− Wγ = ind− Wγ ∞ = ind− q , and in the third ind− Wγ = ind− Wγ ∞ + 1 = ind− q + 1 . Assume now that λ(W ) ∈ R, i.e., we are in one of the ﬁrst three cases of (3.11). Then + , γ ≥ −λ(W ) ind− q ind− Wγ = ind− q + 1 , γ < −λ(W ) Hence, in this case, the assertion of the lemma follows. Consider the case that λ(W ) ∈ R, i.e., that wγ,21 ∈ P(Eγ ) and [wγ,21 ,wγ,21 ] = 0. Then, by the proof of [KW2, Theorem 7.1], for all l < 0 ind− Wγ W(l,0) = ind− Wγ . It follows that ind− Wγ = ind− Wγˆ and that ind− Wγ ∞ < ind− Wγ for all γ, γˆ ∈ R (cf. [KW3]). 3.3. Structure of symmetric and semibounded dB-spaces The above Proposition 3.9 has two consequences on the structure of symmetric and semibounded dB-Pontryagin spaces, which we shall elaborate in the following. The ﬁrst one gives a growth restriction on the elements of a semibounded dB-space. For the proof we use a lemma which supplements [KW5, Theorem 3.17, Corollary 3.18]. 3.11. Lemma. Let q ∈ N<∞ and let B be an entire function of ﬁnite order ρ. If A(z) := q(z)B(z) is entire, then the order of A is also equal to ρ. Moreover, B is of ﬁnite type if and only if A is. If ρ is not an integer, then B being of minimal type is equivalent to A possessing the same property.

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Proof. By [DLLS] we can write q(z) = r(z) · q1 (z) with q1 ∈ N0 and a rational function r. This shows that it suﬃces to prove the assertion for the case κ = 0. By our assumption the function q must be meromorphic in the whole plane and hence we may write according to [L, VII. Lehrsatz 1] z − a0 A z z −1 q(z) = c 1− 1− . (3.12) z − b0 ak bk k=0

Denote the zeros of B by xk . Since q(z) · B(z) is entire, the zero set of B splits up as {xk } = {bk }∪X. Here and for the rest of this proof we understand that a zero is listed as often as its multiplicity states, and also understand set theoretic notations as including multiplicities. It follows that the zeros {yk } of A are given by {yk } = {ak } ∪ X. From [KWW2, Lemma 4.5] we conclude that the convergence exponents of {xk } and {yk } are equal and that the upper densities of these sequences coincide: '% '% 1 %& 1 %& lim sup ρ % xk : |xk | ≤ r % = lim sup ρ % yk : |yk | ≤ r % . r→∞ r r→∞ r Moreover, the values 1 1 lim sup ρ , lim sup xk ykρ r→∞ r→∞ |xk |≤r

|xk |≤r

are together ﬁnite or inﬁnite. If we arrange the sequences (ak ) and (bk ) so that ak < bk < ak+1 , for all l ∈ N the series 1 l 1 l (3.13) − ak bk k

converges. Consider the Hadamard factorization of B: p A z 1 z l B(z) = eP (z) 1− exp xk l xk l=1

= eP (z)

A

p p 1 z l A 1 z l z z 1− exp 1− exp . bk l bk xk l xk l=1

X

l=1

Here P is a polynomial of degree at most ρ and p is the genus of the zeros of B. From A = qB, (3.12) and the convergence of the series (3.13) it follows that the entire function P˜ in the product representation p p A z z 1 z l A 1 z l ˜ 1− exp 1− exp A(z) = eP (z) ak l ak xk l xk X

l=1

l=1

of A must be equal to p a0 z l 1 l 1 l . · P (z) · − P˜ (z) = log c b0 l bk ak l=1

k

Hence P˜ is in fact a polynomial of degree at most ρ. It follows that A is of ﬁnite order at most ρ. Since in this argument the roles of A and B can be exchanged,

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we conclude that the order of A actually equals ρ. The assertion of the lemma now follows from what was said above by Lindel¨ of’s Theorem, see, e.g., [L, I. Lehrsatz 14, 15]. Let us remark that the assumption ρ ∈ Z in the last part of Lemma 3.11 can not be dropped. 3.12. Proposition. Let W ∈ Mep <∞ . Then every entry wij is an entire function of growth at most order 12 , ﬁnite type. In particular, if P is a semibounded dB-space such that 1 ∈ Assoc P := P + zP, then sup

lim sup

F ∈Assoc P

r→∞

log max|z|=r |F (z)| √ <∞ r

(3.14)

ˆ ˆ := T2,0 (W ) ∈ Msym Proof. Consider the matrix W <∞ . Since at most one of K− (W ) ˆ ) is degenerated, we conclude that 1 ∈ Assoc P(E) where E either is and K+ (W w ˆ11 − iw ˆ12 or w ˆ22 + iw ˆ21 . In any case we conclude from [KWW2, Theorem 3.10] ˆ is of exponential type. Lemma 3.11 now implies that E and, hence, one entry of W ˆ that all entries of W are of exponential type. Therefore the entries of W are of growth at most order 12 , ﬁnite type. Let P be a semibounded dB-space and choose an inner product (., .) on P such that (P, (., .)) = P(E) is a semibounded dB-Pontryagin space, this is possible by [KWW4]. Since P and P(E) coincide as sets, we have Assoc P = Assoc P(E). If 1 ∈ Assoc P hence also 1 ∈ Assoc P(E), and we conclude that there exists a matrix W ∈ Mep <∞ such that (0, 1)W = (−B, A). By what was proved in the ﬁrst paragraph the function E is of growth at most order 12 , ﬁnite type. The relation (3.14) now follows from [KW5, Theorem 3.4]. In order to give the second promised structure result on dB-spaces, we need to recall a construction introduced in [KWW4]: If P is a symmetric dB-space, then deﬁne P+ := {F ∈ O(C) : F (z 2 ) ∈ P} . If P+ is endowed with a topology and inner product so that the map F (z) → F (z 2 ) becomes an isometric homeomorphism, this space is a semibounded dB-space. The main result of [KWW4] states that every semibounded dB-space can be obtained in this way and determines the kernel of the assignment Υ : P → P+ . Moreover, if P and hence also P+ is a dB-Pontryagin space, the action of the assignment Υ is explicitly determined in terms of the respective generating Hermite-Biehler functions. This construction on the level of dB-spaces is the exact analogue of the transformations T√ , T2,γ on the level of matrix functions. This follows by comparing the deﬁnition of T√ and T2,γ with [KWW4, Theorem 4.5]. sym 3.13. Lemma. Let W ∈ Msym , ET√ (W ) ∈ HB ep , and <∞ . Then EW ∈ HB

P(EW )+ = P(ET√ (W ) ) .

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ep sym Conversely, let W ∈ Mep , and <∞ and γ ∈ R. Then EW ∈ HB , ET2,γ (W ) ∈ HB

P(ET2,γ (W ) )+ = P(EW ) . Each two spaces P(ET2,γ (W ) ) are not isometrically equal and every dB-Pontryagin space P with P+ = P(EW ) is of this form. It was shown in [KWW4, Proposition 2.6] that every even function F ∈ Assoc P can be obtained as F (z) = G(z 2 ) with G ∈ Assoc P+ . In particular, if 1 ∈ Assoc P, then also 1 ∈ Assoc P+ . The converse does not hold in general. sym and put E+ := A+ − iB+ with 3.14. Proposition. Let E = A − iB ∈ HB<∞

A+ (z 2 ) = A(z), B+ (z 2 ) = zB(z) , so that P(E)+ = P(E+ ). Assume that 1 ∈ Assoc P(E)+ and let W+ ∈ Mep <∞ be such that K− (W+ ) = K(W+ ) and (0, 1)W+ = (−B+ , A+ ). Then 1 ∈ Assoc P(E) if and only if m(W+ ) ∈ R. In this case there exists W ∈ Msym <∞ , K− (W ) = K(W ), such that (0, 1)W = (−B, A). Proof. By its deﬁnition the function B+ has only ﬁnitely many zeros in C \ [0, ∞). sym ˆ Hence W+ ∈ Mep <∞ and thus W := T2,0 (W+ ) ∈ M<∞ . By the deﬁnition of T2,0 ˆ satisﬁes (0, 1)W ˆ = (−B, A). We have 1 ∈ Assoc P(E) if and only if the matrix W ◦ ˆ K− (W ) = {0}, cf. [KW1, Proposition 10.3], [KW2, Lemma 5.11]. This, however, is in view of Proposition 3.9 equivalent to m(W+ ) ∈ R.

4. Chains of matrix functions In this section we investigate chains of matrix functions and introduce the appropriate analogues of the notion of symmetry and semiboundedness on the level of chains of matrices. Let us recall the notion of a maximal chain of matrices as introduced in [KW3]. For a matrix W ∈ M denote by t(W ) the trace function t(W ) := tr(W (0)J). 4.1. Deﬁnition. A mapping ω : I → M<∞ is called a maximal chain of matrices if the following axioms are satisﬁed: (W1) The set I equals (0, M ), 0 < M < ∞, with possible exception of ﬁnitely many points. (W2) The function ω is not constant on any interval contained in I. (W3) For all s, t ∈ I, s ≤ t, we have ω(s)−1 ω(t) ∈ M<∞ and ind− ω(t) = ind− ω(s) + ind− ω(s)−1 ω(t) . (W4) If t ∈ I and for some W ∈ M<∞ , W = I, we have W −1 ω(t) ∈ M<∞ and ind− ω(t) = ind− W + ind− W −1 ω(t), then there exists a number s ∈ I such that W = ω(s). (W5) We have limtM t(ω(t)) = +∞. If I is not connected, there exist numbers s < t, both contained in the last connected component I∞ of I (that is sup I∞ = M ), such that ω(s)−1 ω(t) is not a linear polynomial.

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It is proved in [KW3, Lemma 3.5] that the function ind− ω(t) is constant on each connected component of I and takes diﬀerent values on diﬀerent components. Moreover, by (W3), it is nondecreasing. In particular, it is bounded and attains its maximum on I∞ . Let us deﬁne ind− ω := maxt∈I ind− ω(t). The set of all maximal chains ω with ind− ω = κ will be denoted by Mκ . Moreover, $ $ M≤κ := Mν , M<∞ := Mν . ν≤κ

ν∈N∪{0}

It is already seen from the axiom (W5) that points s, t where the transfer matrix ω(s)−1 ω(t) is a linear polynomial play a special role. This is formalized by the notion of indivisible intervals. Let ω : I → M<∞ be a maximal chain of matrices. An interval (s, t) ⊆ I is called indivisible of type φ ∈ [0, π) if for all s , t ∈ (s, t) ω(s )−1 ω(t ) = W(l(s ,t ),φ) The number L := sup{l(s , t ) : s ≤ t , s , t ∈ (s, t)} is called the length of the indivisible interval (s, t). If (s1 , t1 ) and (s2 , t2 ) are indivisible intervals of types φ1 and φ2 , respectively, which have nonempty intersection, then φ1 = φ2 and (min{s1 , s2 }, max{t1 , t2 }) is again indivisible of the same type. Hence every indivisible interval is contained in a maximal indivisible interval. It can happen that for some s, t ∈ I, s ≤ t, we have ω(s)−1 ω(t) = W(l,φ) for some l < 0. Although in this case (s, t) ⊂ I we shall speak of an indivisible interval of negative length. Chains which can be obtained out of each other by a change of variable will share their important properties. More precisely: Let J1 , J2 be open subsets of R and let ωi : Ji → M<∞ be functions. Then we say that ω2 is a reparameterization of ω1 if there exists an increasing and bijective map α : J2 → J1 such that ω2 = ω1 ◦ α. In this case we write ω2 ∼ ω1 . It is obvious that ∼ is an equivalence relation. A central role in the theory of maximal chains of matrices is played by the Weyl coeﬃcient associated to a maximal chain. It is proved in [KW2] that for all functions τ ∈ N0 the limit q∞ (ω)(z) :=

lim (ω(t) τ )(z)

tsup J

exists locally uniformly on compact subsets of C \ R with respect to the chordal metric and does not depend on the particular choice of τ . It is called the Weyl coeﬃcient of the chain ω. The main result of [KW2] states that the set Mκ / ∼ bijectively corresponds to Nκ via ω/ ∼ −→ q∞ (ω) . The elements of maximal chains can be characterized by means of decompositions of the Weyl coeﬃcient. Recall the following fact from [KW3]: Let ω : I → M<∞ be a maximal chain and let W ∈ M<∞ . Then W ∈ ω(I) if and only if ind− W ≤ ind− ω and there exists a function τ ∈ Nind− ω−ind− W such that q∞ (ω) = W τ .

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Let a function υ : J → M<∞ be given. In the next lemma we give conditions, adapted to our needs, under which υ can be extended to a maximal chain. This result is an immediate consequence of [KW3, Lemma 3.7] and [KW2, Lemma 8.5] with its proof. 4.2. Lemma. Let υ : J → M<∞ be given and assume that υ satisﬁes (W3) and (C1) We have κm := supt∈J ind− υ(t) < ∞. (C2) The following two implications hold true. (a) If lim suptsup J t(ω(t)) < +∞, then limtsup J ω(t) ∈ Mκ . (b) If lim suptsup J t(ω(t)) = +∞ and if there is a number φ ∈ [0, π) such that ω(s)−1 ω(t) = W(l(s,t),φ) , s, t ∈ ω −1 Mκm , then for one (and hence for all) t ∈ ω −1 Mκm ω(t) cot φ ∈ Nκm . Then there exists a maximal chain ω ∈ Mκm and a nondecreasing function λ : J → I such that υ = ω ◦ λ. If lim suptsup J t(ω(t)) = +∞, the chain ω is unique and we have q∞ (ω) = lim υ(t) τ , τ ∈ N0 . tsup J

Otherwise, the set of all extensions in Mκm is parameterized by the set of all functions τ ∈ N0 with the property that ind− lim υ(t) τ = ind− lim υ(t) . tsup J

tsup J

This correspondence is established via the relation lim υ(t) τ . q∞ (ω) = tsup J

For the purposes of the present paper two particular kinds of chains of matrices are of interest. 4.3. Deﬁnition. Let ω ∈ Mκ . We write ω ∈ Msym if ω(t) ∈ Msym <∞ for all t ∈ I. κ ep ep Moreover, we write ω ∈ Mκ if ω(t) ∈ M<∞ for all t ∈ I and if the number of zeros which an entry of ω(t) possesses in C \ [0, ∞) is bounded independently of t ∈ I. The following inverse result gives a connection between the classes Msym <∞ sym and N<∞ . The corresponding result for the class Mep <∞ will be seen later, cf. Proposition 5.6. 4.4. Proposition. Let ω be a maximal chain of matrices. Then ω ∈ Msym <∞ if and sym only if q∞ (ω) ∈ N<∞ . Proof. Assume that ω ∈ Msym <∞ . Then for all t ∈ I the function ω(t) ∞ is odd. Since lim ω(t) ∞ (z) = q∞ (ω) t→sup I

sym we conclude that also q∞ (ω) belongs to N<∞ .

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219

Assume conversely that q∞ (ω) is odd and let t ∈ I be given. Then there exists a function τ ∈ N<∞ , ind− τ = ind− ω − ind− ω(t), such that ω(t)11 (z)τ (z) + ω(t)12 (z) . q∞ (ω)(z) = ω(t) τ (z) = ω(t)21 (z)τ (z) + ω(t)22 (z) It follows that ω(t)11 (−z)[−τ (−z)] − ω(t)12 (−z) q∞ (ω)(z) = −q∞ (ω)(−z) = −ω(t)21 (−z)[−τ (−z)] + ω(t)22 (−z) = − JV ω(t)−1 V J (−z) [−τ (−z)] . Hereby the constant matrices J and V are deﬁned as in Lemma 2.3. With ω(t) and τ we also have (−JV ω(t)−1 V J)(−z) ∈ M<∞ and −τ (−z) ∈ N<∞ , respectively. In fact ind− − JV ω(t)−1 V J (−z) = ind− ω(t)(z), ind− − τ (−z) = ind− τ (z) . Hence ind− q∞ (ω)(z) = ind− ω(t) + ind− τ = ind− − JV ω(t)−1 V J (−z) + ind− − τ (−z) . We conclude that both matrices, ω(t) and (−JV ω(t)−1 V J)(−z), are members of the maximal chain of matrices having q∞ (ω) as its Weyl coeﬃcient. Since ind− (−JV ω(t)−1 V J)(−z) = ind− ω(t) and t((−JV ω(t)−1 V J)(−z)) = t(ω(t)) we conclude that (−JV ω(t)−1 V J)(−z) = ω(t), cf. [KW3, Lemma 3.5]. This means that ω(t) ∈ Msym <∞ . The following technical condition on a maximal chain ω will appear frequently: (K− ) For all t ∈ I we have K− (ω(t)) = K(ω(t)). Recall from [KW2] that ω satisﬁes (K− ) if and only if for one t ∈ I the equality K− (ω(t)) = K(ω(t)) holds. Moreover, the inverse result [KW2, Theorem 5.7] shows that ω satisﬁes (K− ) if and only if limy→+∞ y −1 q∞ (ω)(iy) = 0. We need two general constructions which can be made with chains of matrices. The ﬁrst one formalizes the intuitive idea of linking of chains, compare the discussion after [KW2, Theorem 7.1]. Let υ1 and υ2 be functions into M<∞ deﬁned on open subsets J1 , J2 of R. If both, J1 and J2 are nonempty we deﬁne a function υ1 ! υ2 as follows: Choose increasing bijections αi of Ji onto open sets with the property that inf αi (Ji ) = i − 1, sup αi (Ji ) = i, i = 1, 2 , and deﬁne υ1 ! υ2 : α1 (J1 ) ∪ α2 (J2 ) → M<∞ by + (υ1 ◦ α−1 1 )(t) , t ∈ α1 (J1 ) υ1 ! υ2 (t) := (υ2 ◦ α−1 2 )(t) , t ∈ α2 (J2 )

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Note that this deﬁnition is independent of the choice of α1 , α2 if we identify functions which are equal up to reparameterization. For the function ε with empty domain we set υ !ε = ε!υ = υ. Up to reparameterization the operation ! is associative. The second construction is simply extension by continuity: Assume that the function υ is deﬁned on a set J of the form (a, b) \ {x1 , . . . , xn }. Let L be the set of all those points xi such that the limit limx→xi υ(x) exists. Then we can deﬁne Cυ : J ∪ L → M<∞ by + υ(t) , t∈I Cυ(t) := limx→xi υ(x) , t = xi ∈ L

5. Transformation of matrix chains The transformation T√ can be applied pointwise to a maximal chain of matrices. The outcome will almost be a maximal chain. 5.1. The square root transformation Consider a chain ω : I → M<∞ in Msym κ . Write the index set I as I = (0, σ1 ) ∪ (σ1 , σ2 ) ∪ · · · ∪ (σn , M ) , and put σ0 := 0, σn+1 := M . To each point σi , i = 1, . . . , n + 1, we associate a function ςi according to the table on top of the next page. Deﬁne a function ω√ as ω√ := C T√ ◦ ω|(0,σ1 ) ! ς1 ! T√ ◦ ω|(σ1 ,σ2 ) ! ς2 ! · · · ! ςn ! T√ ◦ ω|(σn ,M) ! ςn+1 5.1. Theorem. Let ω ∈ Msym and assume that ω satisﬁes (K− ). Let be the maxκ imal chain with zq∞ ()(z 2 ) = q∞ (ω)(z). Then there exist functions λ : dom → dom ω√ , µ : dom ω√ → dom such that = ω√ ◦ λ, ω√ = ◦ µ . The proof of this theorem will be carried out in four steps. Before we come to the ﬁrst step, we need to provide two lemmata. The ﬁrst of which is implicitly contained in [dB1], the second one follows from the considerations in [KW2]. We shall however provide complete proofs. 5.2. Lemma. Let ω = (Wt )t∈I ∈ M<∞ and let I1 be a connected component of I. Assume that for all s, t ∈ I1 the transfer matrix Wst = ω(s)−1 ω(t) is a polynomial and that n := sup deg Wst < ∞ . s,t∈I1

Then I1 = (m0 , m1 ] ∪ [m1 , m2 ] ∪ · · · ∪ [mn−1 , mn ) ,

lim T√ (ω(t))

i = 1, . . . , n : α := ind− limtσi T√ (ω(t)) − ind− limtσi T√ (ω(t))

l → limtσi T√ (ω(t)) · W(l,0) ,

α=0 ∃

D

∃ α = 0 ∃

∃

l ∈ (0, L)

L:=t(limtσi T√ (ω(t)))−t(limtσi T√ (ω(t)))

l → lim T√ (ω(t))W(l,0) tσi

E

D

!

l → lim T√ (ω(t))W(l,0)

l∈(0,∞)

221

Deﬁnition of ςi

tσi

tσi

lim T√ (ω(t))

Singularities of Generalized Strings

E

tσi

l∈(−∞,0)

l → limtσi T√ (ω(t)) · W(l,0) , l → limtσi T√ (ω(t)) · W(l,0) ,

l ∈ (−∞, 0) l ∈ (0, ∞)

ε

i =n+1 : l → limtσi T√ (ω(t)) · W(l,0) ,

∃

l ∈ (0, ∞)

ε

where inf I1 = m0 < m1 < · · · < mn−1 < mn = sup I1 , and where the intervals (mi−1 , mi ) are indivisible in ω of certain types φi ∈ [0, π) with φi = φi+1 . Proof. Let us start with the following remark: If E ∈ HB<∞ is a polynomial, then the chain of dB-subspaces of P(E) is given by {0} span{1} span{1, z} · · · span{1, z, . . . , z deg E−1 } = P(E) . If ind− P(E) = 0, every dB-subspace is also positive deﬁnite and in particular non-degenerated. It follows from [dB1] that a polynomial matrix W ∈ M0 can be factorized uniquely as W = W(l1 ,φ1 ) · · · · · W(ln ,φn ) with n = deg W , li > 0, φi ∈ [0, π), φi = φi+1 . Let us come to the proof of the present assertion. Choose s0 , t0 ∈ I1 , s0 < t0 , such that deg Ws0 t0 = n. Since ind− Ws0 t0 = 0, we can factorize Ws0 t0 as Ws0 t0 = W(l1 ,φ1 ) · · · · · W(ln ,φn ) . If t ∈ I1 , t > t0 , we have Ws0 t = Ws0 t0 Wt0 t = W(l1 ,φ1 ) · · · · · W(ln ,φn ) Wt0 t .

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On the other hand we have the factorization Ws0 t = W(l1 ,φ1 ) · · · · · W(lk ,φk ) . By uniqueness of the factorization we obtain k = n, li = li , φi = φi for i = 1, . . . , n − 1 and W(ln ,φn ) = W(ln ,φn ) Wt0 t . This implies φn = φn and Wt0 t = W(l,φn ) for some l > 0. The same argument shows that for all s ∈ I1 , s < s0 , the transfer matrix Wss0 is of the form W(l,φ1 ) . If we choose mi ∈ I1 , i = 1, . . . , n − 1, such that Wmi = Ws0

i A

W(lj ,φj )

j=1

we obtain the desired result.

Note that in the situation of the previous lemma the types φi in the assertion are exactly the types occurring in the factorization of any transfer matrix of maximal degree. 5.3. Lemma. If W ∈ M<∞ and W ∞ = ∞, then 1 p(z) W = 0 1

(5.1)

for some polynomial p. The matrix (5.1) can not be decomposed as W1 W2 with W1 , W2 ∈ M<∞ and ind− W = ind− W1 + ind− W2 diﬀerently than in the form W = [W W(−l,0) ]W(l,0) or W = W(l,0) [W(−l,0) W ]. Proof. The assumption W ∞ = ∞ just means that W21 ≡ 0. As det W = 1, the functions W11 and W22 are zero-free and hence equal to ev1 and ev2 , respectively. Thereby vi# = vi and vi (0) = 0. Every entry of W is of ﬁnite exponential type, and therefore v1 (z) = az (and thus v2 (z) = −az) for some a ∈ R. Since every entry of W is of bounded type in C+ , it follows that a = 0. Since W 0 ∈ N<∞ , the function W12 can have only ﬁnitely many zeros. The same argument as above shows that W12 must be a polynomial. If p is linear, the assertion is clear. Assume that the degree of p is at least 2. Then ind− W > 0. Consider the chain (Wt )t∈J which goes downwards from W as constructed in [KW2, Theorem 7.1]. Its domain is of the form (c− ,ξ1 )∪(ξ1 ,ξ2 )∪···∪ (ξm ,0]. The interval (c− , ξ1 ) must be indivisible of type 0 and inﬁnite length since (1, 0)T belongs to K(W ) and is neutral. Also (ξm , 0] must be indivisible of type 0 and inﬁnite length since ind− W ∞ < ind− W . Since limtξ1 Wt ◦ = limtξn Wt ◦ it follows from the results of [KW3, §5] on intermediate Weyl coeﬃcients that ξ1 = ξn . In the proof of Theorem 5.1 we mainly deal with the function T√ ◦ ω : I → M<∞ .

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223

Step 1: The function T√ ◦ ω satisﬁes (C1). This follows immediately from: 5.4. Lemma. Let ω ∈ Msym <∞ and assume that ω satisﬁes (K− ). Then ind− (T√ ◦ω)(t) is constant on each connected component of I. Proof. Put Et := Eω(t) , t ∈ I. If Et,+ and Et,− are deﬁned by Bt,+ (z 2 ) := zBt (z), At,+ (z 2 ) := At (z) , Bt (z) , At,− (z 2 ) := At (z) , z then, cf. Lemma 3.13, [KWW4, Proposition 4.9], ∼ K((T√ ◦ ω)(t)), P(Et )o ∼ P(Et )e ∼ = P(Et,+ ) = = P(Et,− ) . Bt,− (z 2 ) :=

In particular, we get ind− P(Et ) = ind− P(Et,+ ) + ind− P(Et,− ) . Recall that ind− P(Et ) is constant on I1 , say ind− P(Et ) = κ1 , t ∈ I1 . We shall show that the set & ' Mν := t ∈ I1 : ind− (T√ ◦ ω)(t) = ν is closed in I1 . Let (tn )n∈N , tn ∈ Mν , and assume that tn → t0 ∈ I1 . Then ind− P(Etn ,+ ) = ind− (T√ ◦ ω)(t) = ν, n ∈ N and hence ind− P(Etn ,− ) = κ1 − ν, n ∈ N . Recall that each set HB≤κ is closed with respect to locally uniform convergence. Since limn→∞ ω(tn ) = ω(t0 ) locally uniformly, it follows that also limn→∞ Etn ,± = Et0 ,± . Therefore ind− P(Et0 ,+ ) ≤ ν, ind− P(Et0 ,− ) ≤ κ1 − ν .

(5.2)

However, we must have ind− P(Et0 ,+ ) + ind− P(Et0 ,− ) = κ1 . Hence in both inequalities (5.2) equality must hold, and we conclude that t0 ∈ Mν . B 1 Since I1 = κν=0 Mν and I1 is connected, it follows that all but one of the sets Mν is empty, i.e., that ind− (T√ ◦ ω)(t) is constant on I1 . Step 2: We show that T√ ◦ ω satisﬁes (W3). ˆ t := (T√ ◦ ω)(t). Proof of Step 2. Let t, s ∈ I, t ≤ s, be given. Put Wt := ω(t), W Case 1: Assume ﬁrst that t is not contained in the interior of an indivisible interval of the chain ω. Then P(EWt ) ⊆ P(EWs ) isometrically, and hence also P(EWt )e ⊆ P(EWs )e . It follows that P(EW ˆ t ) = P(EWt )+ ⊆ P(EWs )+ = P(EW ˆs) isometrically. By [KW1, Theorem 12.2] there exists a matrix ˆ ts ∈ M<∞ , ind− W ˆ ts = ind− P(E ˆ ) − ind− P(E ˆ ) W Ws Wt

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such that ˆ (−BW ˆ s , AW ˆ s ) = (−BW ˆ t , AW ˆ t ) Wts . From our assumption K− (Wt ) = K(Wt ), [KW1, Corollary 10.4] and [KWW2, Proposition 2.6] we know that 1 ∈ Assoc P(EW ˆ t ), Assoc P(EW ˆ s ). The matrices ˆ ˆ Wt and Ws are the unique matrices belonging to Mind− P(EWˆ ) and Mind− P(EWˆ ) , s t respectively, with (cf. [KW1, Corollary 10.4]) ˆ t ) = K(W ˆ t ), K− (W ˆ s ) = K(W ˆ s) , K− (W

(5.3)

and ˆ ˆ (−BW ˆ t , AW ˆ t ) = (0, 1)Wt , (−BW ˆ s , AW ˆ s ) = (0, 1)Ws . It follows that (cf. proof of [KW2, Theorem 7.1]) ˆs = W ˆt W ˆ ts . W ˆ ˆ By (5.3) we have ind− P(EW ˆ t ) = ind− Wt and ind− P(EW ˆ s ) = ind− Ws , respectively, i.e., ˆ ts = ind− W ˆ s − ind− Wt , ind− W and we have proved the requirement of (W3) in the considered case. Case 2: Next assume that t belongs to the interior of a maximal indivisible interval (t− , t+ ) of type 0. Then, by [KW3, Proposition 3.16], at least one of t− and t+ ˆt = W ˆ t− and hence the belongs to I, say t− ∈ I. By Proposition 3.6 we have W assertion follows from what we already proved. Case 3: Finally consider the case that t belongs to a maximal indivisible interval (t− , t+ ) of type π2 (regardless whether this indivisible interval is of positive, negative or inﬁnite length). We divide the proof of this case into three subcases. s ≤ t+ : In this case we have Ws = Wt W(l, π2 ) for some l ∈ R \ {0}. It follows from ˆs = W ˆ t W(l ,φ) , i.e., W ˆ ts = W(l ,φ) . Thereby φ ∈ (0, π) and Proposition 3.6 that W sgn l = sgn l. Consider the space P(EWs ). Since s is the right endpoint of an indivisible interval of type π2 we have AWs ∈ P(EWs ). It follows from the fact dom S = span{A}⊥ that P(EWs )o ⊆ dom S. The spaces P(EWs ) and P(EWt ) are equal as sets but not isometrically. However, on the subspace dom S the inner products of P(EWt ) and P(EWs ) coincide. Hence P(EWt )o = P(EWs )o isometrically. We have ind− P(EWs )e + ind− P(EWs )o = ind− P(EWs ) + + 0 , l>0 0 , l>0 = ind− P(EWt )e + ind− P(EWt )o + . = ind− P(EWt ) + 1 , l<0 1 , l<0 ∼ Since sgn l = sgn l, ind− P(EWt )o = ind− P(EWs )o and P(EW ˆ t ) = P(EWt )e , ∼ P(EW )e , it follows that P(E ˆ ) = Ws

s

ˆ ind− P(EW ˆ s ) = ind− P(EW ˆ s ) + ind− Wts .

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ˆ s as W ˆs = W ˆ t+ W ˆ t+ s . Since t+ is not contained s > t+ , t+ ∈ I: We decompose W in the interior of an indivisible interval, it follows from what we already proved ˆ t+ = W ˆ tW ˆ s = ind− W ˆ t+ + ind− W ˆ t+ s . Moreover, we decompose W ˆ tt+ . that ind− W By the above treated case also in this relation negative indices add up. Since ˆ tt+ W ˆ t+ s ) ≤ ind− W ˆ tt+ + ind− W ˆ t+ s , it follows that also in the factorization ind− (W ˆs = W ˆ t (W ˆ tt+ W ˆ t+ s ) negative indices add up. W s > t+ , t+ ∈ I: In this case we must have t− ∈ I. To shorten notation put ˆt , At , etc. be deﬁned correspondingly. Et := EWt and let E The interval [t− , t] is in ω indivisible of type π2 and positive length l > 0. Hence At− = At ∈ P(Et ). Since t+ ∈ I, we have [At , At ]P(Es ) = 0. It follows that P(Et )e P(Et− )e ˆt ) P(E ˆt− ) with codimension 1. The with codimension 1 and hence also P(E ˆ ˆs ) is a degenerated set P(Et ) endowed with the inner product inherited from P(E ˆ dB-subspace P of P(Es ), and we have & & √ ' √ ' ˆt− )[+] ˙ span At− ( z) , P = P(E P◦ = span At− ( z) . ˆ t− s ). Consider the space K(W By [KW2, Lemma 7.6] it contains a constant (cos ψ, sin ψ)T . This constant has the property that the space & ' ˆt− cos ψ + Aˆt− sin ψ ˙ span − B P(Eˆt− )[+] where cos ψ cos ψ ˆt− cos ψ + Aˆt− sin ψ, −B ˆ t− cos ψ + Aˆt− sin ψ = , −B ˆ t s) sin ψ K(W sin ψ − ˆs ). It follows that is a dB-subspace of P(E & ' ˆt− )[+] ˆt− cos ψ + Aˆt− sin ψ = P , ˙ span − B P(E and hence that

& ' & √ ' ˆ t− cos ψ + Aˆt− sin ψ = span At− ( z) , span − B

i.e.,

√ ˆt− (z) cos ψ + Aˆt− (z) sin ψ At− ( z) = λ − B for some λ ∈ C, and that cos ψ cos ψ , = 0. ˆ t s) sin ψ K(W sin ψ − From the deﬁnition of the transformation T√ we obtain ˆt− (z 2 ) = zBt− (z), Aˆt− (z 2 ) = At− (z) + wt ,12 (0)zBt− (z) . B − It follows that

√ ˆt− (z) . At− ( z) = Aˆt− (z) − wt − ,12 (0)B

Comparing this with (5.4) we obtain ψ = Arccot wt − ,12 (0).

(5.4)

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ˆ t− t = W(l ,ψ) with some l > 0. Hence the Proposition 3.6 shows that W T ˆ t− s ) as well as to K(W ˆ t−1t ). In the ﬁrst constant (cos ψ, sin ψ) belongs to K(W − space it is neutral, in the second (one-dimensional) space it is negative. It follows (cf. [ADRS]) that ˆ −1 W ˆ ˆ ind− W t− t t− s = ind− Wt− s . ˆ ts = Wt−1 Ws = Wt−1t Wt− s ) ˆ t = ind− W ˆ t− and hence (W We have ind− W −

ˆ s = ind− W ˆ t− + ind− W ˆ t− s = ind− W ˆ t + ind− W ˆ ts . ind− W We have seen that in any case in the relation Ws = Wt Wts negative indices add. Step 3: We show that T√ ◦ ω satisﬁes (C2). Proof of Step 3. Also this proof is divided into several cases. We use again the ˆ t. notation ω(t) = Wt and (T√ ◦ ω)(t) = W Case 1: ω ends with an indivisible interval (m, sup I) of type 0. ˆt = W ˆ m , t ∈ (m, sup I), and hence Then W ˆ t ) = t(W ˆ m ) < +∞ lim sup t(W tsup I

and ˆm . ˆt = W lim W

tsup I

We are therefore in case (a) of (C2). By the already proved property (W3) we have ˆ t ≤ ind− W ˆ m , t ∈ I, t ≤ m . ind− W ˆ t = ind− W ˆ m . Hence the implication (a) of (C2) For t ∈ (m, sup I) trivially ind− W holds true. Case 2: ω ends neither with an indivisible interval of type zero nor is the last component I∞ of ω of the form (m− , m] ∪ [m, sup I) with an indivisible interval (m− , m) of type 0 and an indivisible interval (m, sup I) of type π2 . ˆ u = maxv∈I ind− W ˆ v} We claim that there exist s, t ∈ Iˆ∞ := {u ∈ I : ind− W −1 ˆs W ˆ t ) is not a linear polynomial. This excludes the occurrence ˆ st (:= W such that W of case (b) of (C2). In case (a) we appeal to [KW3, Lemma 3.4] to obtain the validity of the desired implication. In order to establish our claim assume on the contrary that for some φ ∈ [0, π) ˆ st = W(l(s,t),φ) whenever s, t ∈ Iˆ∞ . By Lemma 5.4 this relation holds we have W especially for s, t ∈ I∞ = {u ∈ I : ind− Wu = maxv∈I ind− Wv }. φ = 0: Choose s, t ∈ I∞ , s < t, such that Wst is not a linear polynomial. By Remark 3.7 we have for some α ∈ R 1 αz + l(s, t)z 3 . Wst = 0 1 Since Wst is not a linear polynomial we must have l(s, t) = 0. Thus ind− Wst > 0, a contradiction.

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φ = 0: We conclude from Remark 3.7 that for all s, t ∈ I∞ , s < t, Wst = W(l1 ,0) W(l2 , π2 ) W(l3 ,0) with some li ≥ 0. By Lemma 5.2 we must be in one ⎧ ⎪ ⎨(m0 , m1 ] ∪ [m1 , m2 ] ∪ [m2 , m3 ) I∞ = (m0 , m1 ] ∪ [m1 , m2 ) ⎪ ⎩ (m0 , m1 ] ∪ [m1 , m2 )

of the following situations: of types 0, π2 , 0 of types π2 , 0 of types 0, π2

Again a contradiction since these are exactly those cases we do not consider in the present step of the proof. Case 3: The last component I∞ of ω is equal to (m0 , m1 ] ∪ [m1 , m2 ) with an indivisible interval (m0 , m1 ] of type 0 and an indivisible interval [m1 , m2 ) of type π2 . ˆ t )t∈I of some type ψ ∈ In this case the interval (m0 , m2 ) is indivisible in (W ˆ t ) = +∞, which rules out the occurrence of (0, π). It follows that limtsup I t(W case (a) in (C2). Fix t0 ∈ (m1 , m2 ). Since [m1 , t0 ] is indivisible of type π2 in (Wt )t∈I and has positive length, we obtain that Am1 ∈ P(Et0 ), [Am1 , Am1 ]P(Et0 ) > 0 . ˆt0 which As we saw in the proof of (W3), the linear combination Sˆψ of Aˆt0 and B ˆt0 ) is linearly dependent with Am1 (√z) and hence belongs to the space P(E [Sˆψ , Sˆψ ] ˆ > 0 . P(Et0 )

It follows from [KW2, Lemma 5.12] that ˆ t0 cot ψ = ind− W ˆ t0 . ind− W We conclude that the implication (b) of (C2) holds true.

We have established that T√ ◦ ω can be extended to a maximal chain by means of Lemma 4.2. If limtM t((T√ ◦ ω)(t)) = +∞, this extension is unique and satisﬁes q∞ () = lim (T√ ◦ ω(t)) ∞ . tM

By (3.4) this yields (5.5) zq∞ ()(z 2 ) = q∞ (ω)(z) . If limtM t((T√ ◦ ω)(t)) < +∞, so that the extension of T√ ◦ ω is not unique, we should choose the parameter τ = ∞ in Lemma 4.2 in order to achieve the relation (5.5). The fact that this choice is permitted needs justiﬁcation. However, ˆ st is not of we saw in the proof of Step 3 that there exist s, t ∈ I∞ such that W the form W(l,0) . As indicated in the proof of [KW2, Lemma 8.5] this implies that ind− [limtM T√ ◦ ω(t)] ∞ = ind− limtM T√ ◦ ω(t). Let be the maximal chain with zq∞ ()(z 2 ) = q∞ (ω)(z). Due to the previous steps there exists a nondecreasing function µ ˆ : I → dom such that T√ ◦ ω = ◦ µ ˆ.

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Step 4: There exists a surjective function µ : dom ω√ → dom such that ω√ = ◦ µ. Proof of Step 4. Write I = (σ0 , σ1 ) ∪ · · · ∪ (σn , σn+1 ). By Lemma 5.4 for each i the set µ ˆ ((σi , σi+1 )) is contained in one connected component of dom . As it is seen from (3.3), the function (t ◦ )(ˆ µ(t)) depends continuously on t ∈ (σi , σi+1 ). Thus µ ˆ is continuous, and it follows that µ ˆ((σi , σi+1 )) is an interval. Put ξi,− := lim µ ˆ(t), ξi,+ := lim µ ˆ(t) . t→σi −

t→σi +

Since T (W ) depends continuously on W and T (I) = I, we have ξ0,+ = 0. We have ξn+1,− = σn+1 if and only if limtσn+1 t((T√ ◦ ω)(t)) = +∞. Otherwise, by (5.5) and the deﬁnition of , we have √

√

q∞ () =

lim (T√ ◦ ω)(t) ∞ = (ξn+1,− ) ∞ .

tσn+1

This implies that the interval (ξn+1,− , sup dom ) is indivisible of type 0 in . Consider a point σi , i ∈ {1, . . . , n}. The essential observation for the following proof is that, by (3.4) and the existence of intermediate Weyl-coeﬃcients (see [KW3, §5]), the function (T√ ◦ ω) ∞ has a continuous extension to [σ0 , σn+1 ]. We divide cases similar as in the deﬁnition of ςi . ξi,− , ξi,+ ∈ dom : Denote by qξi,± the intermediate Weyl coeﬃcient of at the singularity ξi,± . We have qξi,− =

lim

s→ξi,− −

(s) ∞ = lim (T√ ◦ ω)(t) ∞ = lim (T√ ◦ ω)(t) ∞ = t→σi −

=

lim

s→ξi,+ +

t→σi +

(s) ∞ = qξi,+ ,

and hence ξi,− = ξi,+ . ξi,− ∈ dom , ξi,+ ∈ dom : We have qξi,+ = lim (T√ ◦ ω)(t) ∞ = lim (T√ ◦ ω)(t) ∞ = (ξi,− ) ∞ . t→σi −

t→σi +

Since qξi,+ is the Weyl-coeﬃcient of the maximal chain |dom ∩(0,ξi,+ ) , it follows that (ξi,− , ξi,+ ) is indivisible of type 0 and inﬁnite length. ξi,− ∈ dom , ξi,+ ∈ dom : The same argument as above yields qξi,− = (ξi,+ ) ∞ and hence that ind− (ξi,+ ) ∞ < ind− (ξi,+ ). This implies that the interval (ξi,− , ξi,+ ) is indivisible of type 0 and inﬁnite length. ξi,− ∈ dom , ξi,+ ∈ dom : In this case we ﬁnd (ξi,− ) ∞ = (ξi,+ ) ∞ and hence, by Lemma 5.3, 1 p(z) (ξi,− )−1 (ξi,+ ) = 0 1 If ind− (ξi,− ) = ind− (ξi,+ ), the interval (ξi,− , ξi,+ ) is indivisible of type 0 and positive length t( lim T√ (ω(t))) − t( lim T√ (ω(t))) . tσi

tσi

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229

If ind− (ξi,− ) < ind− (ξi,+ ), there exists a singularity ξ of in the interval (ξi,− , ξi,+ ). By Lemma 5.3 the intervals (ξi,− , ξ) and (ξ, ξi,+ ) are indivisible of type 0 and inﬁnite length. The case ind− (ξi,− ) > ind− (ξi,+ ) can not occur. We saw that in any case |(ξi,− ,ξi,+ ) ∼ ςi . The required function µ is now deﬁned in the obvious way. We have constructed the function µ required in Theorem 5.1. To complete the proof of Theorem 5.1 choose for λ a right inverse of µ. ✌ 5.5. Corollary. Let ω ∈ Msym and assume that ω satisﬁes (K− ). Then the maximal κ chain with zq∞ ()(z 2 ) = q∞ (ω)(z) belongs to Mep ≤κ and also satisﬁes (K− ). Proof. We know from Theorem 5.1 that = ω√ ◦ λ. By (3.4) the number of zeros of T√ (ω(t))21 in C \ [0, ∞) equals the number of non-real zeros of ω(t)21 and is therefore, by Corollary 2.10, bounded uniformly by ind− ω. On indivisible intervals of type 0 the entry (s)21 is constant. Hence the number of zeros of (s)21 in C \ [0, ∞) is bounded independently of s. By Corollary 2.10, (ii), the maximum number of zeros that any entry of (s) can have in C\[0, ∞) is bounded independently of s. The fact that satisﬁes (K− ) readily follows from the relation of Weyl coeﬃcients and [KW2, Theorem 5.7]. We deduce an inverse result for the class Mep <∞ , the analogue to Proposition 4.4. 5.6. Proposition. Let be a maximal chain. Then ∈ Mep <∞ if and only if ep q∞ () ∈ N<∞ . Proof. Assume ﬁrst that ∈ Mep <∞ . For every t ∈ dom the function (t) ∞ ep belongs to N≤ind . Moreover, the number of poles γ((t) ∞) of (t) ∞ − which are located in C \ [0, ∞) is equal to the number of zeros of (t)21 . Hence it is bounded independently of t ∈ dom . We have limt→sup dom (t)∞ = q∞ () ep and hence [KWW2, Proposition 4.10] implies that q∞ () ∈ N<∞ . ep Conversely, let be given such that q∞ () ∈ N<∞ and assume ﬁrst that additionally limx→−∞ q∞ ()(x) = 0. By [KWW2, Theorem 4.1] the function sym zq∞ ()(z 2 ) belongs to N<∞ . Let ω ∈ Msym <∞ be the maximal chain with q∞ (ω) = 2 zq∞ (ω)(z ). Since lim

y→∞

1 q∞ (ω)(iy) = lim q∞ ()(−y 2 ) = 0 , y→∞ iy

we know that ω satisﬁes (K− ). Corollary 5.5 implies that ∈ Mep <∞ . The general case is reduced to the already proved particular instance by a possible application of one of the transforms TJ or Tα of [KW2, Section 10]. 5.7. Remark. Note that Proposition 5.6 could most likely also be deduced without help of the transformation T√ by employing the the theory of isometric embeddings

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of dB-spaces into ‘L2 -spaces’ induced by distributions associated to integral representations of generalized Nevanlinna functions. For this not yet fully developed theory see [KW2, §4,§6], [KWW2, §5]. 5.8. Corollary. Let ∈ Mep <∞ and assume that limx→−∞ q∞ ()(x) = 0. Then there exist only ﬁnitely many indivisible intervals of type 0 in . In fact, the number of such intervals is bounded by 2 ind− + 1. Proof. We know that = ω√ ◦ λ where ω : I → M<∞ , I = (σ0 , σ1 ) ∪ · · · ∪ (σn , σn+1 ), is the maximal chain with q∞ (ω)(z) = zq∞ ()(z 2 ). By Remark 3.7 the parts (T√ ◦ ω)|(σi ,σi+1 ) of ω√ can not contain indivisible intervals of type 0. Thus contains at most 2n + 1 indivisible intervals of type 0. 5.2. The inverse transformation We employ the rather detailed discussion of Theorem 5.1 to deﬁne and investigate the inverse of the square root transformation. Let ∈ Mep <∞ . Then by [KWW2, Proposition 4.9] a function Λ : dom → R ∪ {±∞} is well deﬁned by Λ(t) := − lim

x→−∞

(t)22 (x) . (t)21 (x)

The following lemma will play a central role. 5.9. Lemma. Let ∈ Mep <∞ and assume that limx→∞ q∞ ()(x) = 0. Then there exist pairwise disjoint open intervals (ak , bk ), k = 0, . . . , n, with ak < bk ≤ ak+1 , such that $& ' dom \ [t− , t+ ] : (t− , t+ ) maximal indivisible of type 0 in =

n $

' & (ak , bk ) ∪ ak : ak ∈ dom , ak = bk−1

k=0

and that the function Λ has the following properties: (i) Λ|(ak ,bk ) is ﬁnite, nondecreasing and continuous from the left. (ii) If α ∈ {ak : ak ∈ dom , ak = bk−1 }, then either limt→α± Λ(t) ∈ R and limt→α+ Λ(t) − limt→α− Λ(t) < 0, or lim supt→α |Λ(t)| = ∞. The intervals (ak , bk ) are uniquely determined by these properties. For each k the function τk (t) := Λ(t) − (t)21 (0), t ∈ (ak , bk ), is strictly increasing and continuous from the left. The proof of this lemma will also show how the inverse of the square root transformation acts. Let us describe this action precisely. Let ∈ Mep <∞ , assume that limx→∞ q∞ ()(x) = 0, and let (ak , bk ) be the intervals from the above lemma. For each k ∈ {1, . . . , n} deﬁne k : R → M<∞ as follows: If t ∈ ran τk , put k (t) := T2,Λ(t) ◦ (τk−1 (t)) .

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231

Since τk is strictly increasing and continuous from the left, we have $ inf ran τk , sup ran τk = ran τk ∪˙ (αl , βl ] , l

with at most countably many pairwise disjoint intervals (αl , βl ]. If t ∈ (αl , βl ] deﬁne ⎧ lim tαl k (t) · W(t−αl ,0) , t ∈ (αl , βl ] ⎪ ⎪ ⎪ t∈ran τk ⎨ limtsup ran τk k (t) · W(t−sup ran τk ,0) , t ∈ (sup ran τk , ∞) k (t) := t∈ran τk ⎪ ⎪ ⎪ ⎩ limtinf ran τk k (t) · W(t−inf ran τk ,0) , t ∈ (−∞, t, inf ran τk ) t∈ran τk

For k = 0 we deﬁne a function 0 : (0, ∞) → M<∞ in exactly the same manner. 5.10. Theorem. Let ∈ Mep <∞ and assume that limx→∞ q∞ ()(x) = 0. Then ω := 0 ! 1 ! · · · ! n ∈ Msym <∞ , and q∞ (ω)(z) = zq∞ ()(z 2 ) . Proof of Lemma 5.9 and Theorem 5.10. We use the same notation as in the ﬁrst paragraph of the proof of Step 4. Consider the continuous and nondecreasing function µ ˆ : dom ω → dom . Put $& ' L := dom ω \ (t− , t+ ] : (t− , t+ ) maximal indivisible of type 0 . Then, by Lemma 3.4, µ ˆ|L is injective. The set L has the property that it is closed in dom ω with respect to monotonically increasing limits. We show that the function (ˆ µ|L )−1 is continuous from the left: Assume that sn " s, sn , s ∈ µ ˆ(L), and put tn := (ˆ µ|L )−1 , t := (ˆ µ|L )−1 . Then (tn )n∈N is increasing and bounded above by t. Hence limn→∞ tn =: t0 exists and belongs to dom ω. Therefore it belongs to L. We have µ ˆ (t0 ) = lim µ ˆ (tn ) = s = µ ˆ(t) , n→∞

and therefore t0 = t. For k = 1, . . . , n we deﬁne numbers ak := ξk,+ and bk := ξk+1,− . Then (ak , bk ) ⊆ µ ˆ (σk , σk+1 ) ⊆ [ak , bk ] . By Remark 3.7 (ak , bk ) does not contain any indivisible interval of type 0. Moreover, by Step 4, every interval in dom ω which has empty intersection with B n k=0 (ak , bk ) is indivisible of type 0. It follows that dom = $&

in particular

∪˙ Bn

n $

' & (ak , bk )∪˙ ak : ak ∈ dom , ak = bk−1 ∪˙

k=0

' [t− , t+ ] : (t− , t+ ) maximal indivisible of type 0 ,

k=0 (ak , bk )

⊆ L.

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Let t ∈ L. Then t is not the right endpoint of an indivisible interval of type 0 and hence ω(t)21 ∈ P(Eω(t) ). Hence lim

y→+∞

1 ω(t)22 (iy) = 0, iy ω(t)21 (iy)

and it follows from (3.4) that ω(t)12 (0) = (Λ ◦ µ ˆ)(t), t ∈ L . It readily follows that Λ is real and nondecreasing on (ak , bk ). Moreover, since ω(t)12 (0) depends continuously on t, it follows from Λ(t) = ω (ˆ µ|L )−1 12 (0), t ∈ (ak , bk ) , that Λ is continuous from the left. By Lemma 3.3 we also obtain T2,Λ(t) ◦ (t) = ω ◦ (ˆ µ|−1 L ) (t), t ∈ (ak , bk ) . Since for all t ∈ dom ω the relation ω(t)21 (0) = (t)21 (0) holds, we see that τk (t) = t ◦ ω ◦ (ˆ µ|−1 L ) (t), t ∈ (ak , bk ) . Thus τk is nondecreasing, injective and continuous from the left. We conclude from the above discussion that, what is missing from k |ran τk to all of ω are just indivisible intervals of type 0. However, the deﬁnition of k on R \ ran τk just ﬁlls in indivisible intervals of type 0. Note that t T2,Λ(t) ◦ (τk−1 (t)) = t, t ∈ ran τk . Thus the ﬁlled in indivisible intervals have the proper length.

5.11. Remark. We know from [KWW2, Corollary 3.4, Proposition 4.7] that, if q ∈ ep N<∞ , then zq(z) ∈ N<∞ if and only if q ∈ N<∞ . Under an additional regularity condition on q, the previous results lead to an explicit method to construct the maximal chain with Weyl-coeﬃcient zq(z) out of the one with Weyl-coeﬃcient q. To see this recall from [KW2, Lemma 10.1] that, if ω ∈ Mκ and if we deﬁne υ(t) := −Jω(t)J, then υ ∈ Mκ and the respective Weyl-coeﬃcients are related by q∞ (υ) = −q∞ (ω)−1 . Denote this transformation by TJ , let T√ be the square root ep we transformation and T2 its inverse as studied above. Then for each q ∈ N<∞ have −1 T√ −1 TJ T TJ q(z) 2 zq(z 2 ) zq(z) zq(z 2 ) zq(z) In order to justify the application of Theorem 5.1 and Theorem 5.10, we have to assume that 1 = 0. lim q(x) = lim x→−∞ x→−∞ xq(x) This just means that the chain ω whose Weyl-coeﬃcient is zq(z 2 ) does not start with an indivisible interval. The same procedure can be applied in order to construct the chain with ep Weyl-coeﬃcient z −1 q(z) out of the chain corresponding to q(z) ∈ N<∞ . This is

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233

exactly what will be needed in the discussion of generalized strings. In fact, under the assumption that 1 q(x) lim = lim = 0, x→−∞ q(x) x→−∞ x we have 2 √ TJ −1 T2 −z TJ q(z ) T q(z) q(z) 2 q(z) q(z ) z z In both cases the essential part of the sequence is T√ ◦ TJ ◦ T2 .

6. Evolution of singularities In the following discussion we recall some facts and notions on singularities of maximal chains from [KW3]. If ω ∈ M<∞ , dom ω = (0, σ1 )∪(σ1 , σ2 )∪· · ·∪(σn , M ), then the numbers σi are called the singularities of the chain ω. One reason is that limt→σi |t(ω(t))| = ∞, another one is that the σi are exactly the points of increase of ind− ω(t). A ﬁrst characteristic value attached to a singularity is κ(σi ) := lim ind− ω(t) − lim ind− ω(t) ∈ N . tσi

tσi

Another characteristic of a singularity is whether or not there is an indivisible interval to the left or to the right of it. Put σi+ := sup {t ∈ I : (σi , t) indivisible} ∪ {σi } , σi− := inf {t ∈ I : (t, σi ) indivisible} ∪ {σi } . We call σi of polynomial type, if σi− < σi < σi+ . Moreover, σi is called left dense, right dense or dense, if σi− = σi < σi+ , σi− < σi = σi+ or σi− = σi = σi+ , respectively. A deeper insight in the structure of a singularity is obtained by considering the chain of dB-spaces associated with the chain ω. If ω satisﬁes (K− ), then K(ω(t)) ∼ = K− (ω(t)) ∼ = P(Eω(t) ). Moreover, if s, t ∈ dom ω, s ≤ t, then P(Eω(s) ) ⊆ P(Eω(t) ) and this inclusion is isometric unless for some > 0 the interval (s − , t) is indivisible. Conversely, if t ∈ I, then every non-degenerated dB-subspace of P(Eω(t) ) is of the form P(Eω(s) ). It is an important observation that the singularities of ω correspond to the degenerated dB-spaces in this chain. In fact, if we put :& ' ' & Pσ− := cls P(Eω(t) ) : t < σi− , Pσ+ := P(Eω(t) ) : t > σi+ , i

i

then every dB-space Pσ− P Pσ+ is degenerated. Conversely, unless (σi− , σi+ ) i i is indivisible with negative length, there exists a degenerated dB-space Pσ− ⊆ P ⊆ i Pσ+ . More exactly: Put δ := dim Pσ+ /Pσ− ∈ N ∪ {0}, and δ− := dim P◦σ− , δ+ := i i i i dim P◦σ+ . If δ > 1, then there exist degenerated dB-spaces P1 , . . . , Pδ−1 with i

Pσ− P1 P2 · · · Pδ−1 Pσ+ . i

i

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It was proved in [KW3] that the isotropic parts of the members of a chain of subsequent degenerated dB-spaces show a very particular behavior. If Q1 , . . . , Qn are degenerated dB-spaces, Qi Qi+1 with codimension 1, then there exists an index imax ∈ {1, . . . , n} such that Q◦1 Q◦2 · · · Q◦imax ⊇ Q◦imax +1 · · · Q◦n , where in each inclusion the codimension is at most 1 and only in the middle inclusion equality can hold. The singularity σi is of polynomial type, left dense, right dense or dense, if and only if δ− = 0 = δ+ , δ− > 0 = δ+ , δ− = 0 < δ+ or δ− > 0 < δ+ , respectively. We can have δ = 0 only if σi is dense. The case δ = 1, δ− = δ+ = 0, just means that (σi− , σi+ ) is indivisible with negative length. We will visualize this inner structure of a singularity in the following way: For example Pσ − P1 P2 Pσ + should describe a singularity which is right dense with δ = 3. It is our aim in this section to describe the evolution of singularities when performing the transformation T√ ◦TJ ◦T2 . In view of our needs in the investigation of generalized strings we will content ourselves with a sound discussion of the case that this transformation is applied to a chain with ind− = 0. In the ﬁrst step we deal with the transformation T2 . To this end let ∈ Mep 0 be given, assume that limx→−∞ q∞ ()(x) = 0, and let ω ∈ Msym <∞ be such that q∞ (ω)(z) = zq∞ ()(z 2 ). We have to investigate the structure of the chain of dB-spaces arising from P(Eω(t) ). By Lemma 2.13 all these spaces P are symmetric. Moreover, since ind− = 0, we know that always P+ is a Hilbert space, see Lemma 3.13. In particular, by [KWW4, Lemma 2.4] for every dB-space P in this chain we have dim P◦ ≤ 1. By the structure theory of degenerated dB-spaces developed in [KW3] this knowledge already has a big inﬂuence on the kind of singularities that may appear. 6.1. Proposition. Let ∈ Mep 0 and assume that limx→−∞ q∞ ()(x) = 0. More2 over, let ω ∈ Msym <∞ be such that q∞ (ω)(z) = zq∞ ()(z ) and write dom ω = (σ0 , σ1 ) ∪ · · · ∪ (σn , σn+1 ). We have for every k ∈ {1, . . . , n} κ(σk ) = 1, δ(σk ) ≤ 3, δ± (σk ) ≤ 1 . Let ak , bk , k = 0, . . . , n, and Λ be as in Lemma 5.9. According to the following table the structure of σk can be read oﬀ the behavior of Λ at ak and bk−1 and the fact whether or not there is an indivisible interval between bk−1 and ak . Thereby we set l := t((ak )) − t((bk−1 )) , Λ(bk−1 −) :=

lim

t→bk−1 −

Λ(t), Λ(ak +) := lim Λ(t) , t→ak +

and, in case both of these limits are ﬁnite, m := Λ(ak +) − Λ(bk−1 −).

m<0

∈R

∈R

+∞

∈R

bk−1 < ak

bk−1 = ak

∈R

∈R

Λ bk−1 −

Λ ak +

Singularities of Generalized Strings

Structure of σk

✓

=

>?

=

>?

✓

∈R

Pσ −

✓

Pσ −

−∞

✓

Pσ − k

∈R

−∞

✓

Pσ − k

+∞ −∞ +∞ −∞

P1

W = W(m,0) 1 mz + lz 3 W = 0 1 Pσ− ,e = Pσ+ ,e

Pσ +

k

k

P1

k

∈R

@

W

k

+∞

@

W

✓

235

Pσ + k

k

Pσ− ,e = P1,e = Pσ+ ,e k

k

Pσ− ,e = Pσ+ ,e

Pσ +

k

k

Pσ + k

k

Pσ− ,e = P1,e = Pσ+ ,e k

k

✓ ✓

Pσ − k

Pσ +

Pσ− ,e = Pσ+ ,e k

k

k

Proof. As we already noted for every dB-space P in the chain arising from P(Eω(t) ), we must have dim P◦ ≤ 1. From this it is immediate that δ± (σk ) ≤ 1. The fact that δ(σk ) ≤ 3 is clear from the structure of the isotropic parts of a chain of subsequent degenerated dB-spaces. The proof of the remaining assertions of the present proposition is based on the following observations: (i) The limit Λ(ak +) is ﬁnite if and only if Pσ+ is non-degenerated. Similarly, k the limit Λ(bk−1 −) is ﬁnite if and only if Pσ− is non-degenerated. k (ii) We have Pσ+ ,+ = P(E(ak ) ) and Pσ− ,+ = P(E(bk−1 ) ). k k (iii) If P1 P2 are two degenerated symmetric dB-spaces such that P1,+ and P2,+ are non-degenerated, then P1,+ P2,+ . (iv) If bk−1 = ak , then δ(σk ) ≤ 1. (v) There exist at most two degenerated dB-spaces P with Pσ− ⊆ P ⊆ Pσ+ . k

k

ad (i): We have Λ(ak +) > −∞ if and only if limtak τk (t) > −∞. This implies that ζ+ := limtak (ˆ µ|L )−1 > σk . Since in this case (σk , ζ+ ) is indivisible, we have Pσ+ = P(Eω(ζ+ ) ). The same argument applies to λ(bk−1 −). k

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ad (ii): We have Pσ+ ,e = k

However, and, since

µ ˆ(σk+ )

:

: P(Eω(t) ) e = P(Eω(t) )e .

+ t>σk

+ t>σk

P(Eω(t) )e ∼ = P(Eω(t) )+ = P(E(ˆµ(t)) ) , = ak , : P(E(ˆµ(t)) ) = P(E(ak ) ) . + t>σk

The same argument applies to P(E(bk−1 ) ). ad (iii): Assume that P1,+ = P2,+ =: P. By [KWW4, Proposition 4.7] there exists exactly one degenerated dB-space Q with Q+ = P, a contradiction. ad (iv): If bk−1 = ak , then, by (ii), Pσ+ ,+ = Pσ− ,+ . By [KWW4, Theorem 3.11] k k this implies that dim Pσ+ /Pσ− ≤ 1. k

k

ad (v): Since dim P◦ ≤ 1, this follows from the structure of a chain of subsequent degenerated dB-spaces. We will now go through the cases listed in the above table. Λ(bk−1 −), Λ(ak +) ∈ R, bk−1 = ak : We know that δ− = δ+ = 0 and that Pσ− ,+ = Pσ+ ,+ It follows from Lemma 3.4 that ω(σk− )−1 ω(σk+ ) = W(α,0) for k k some α. However, in the present case we have α = lim τk (t) − lim τk−1 (t) = Λ(ak +) − Λ(bk−1 −) = m . tak

tbk−1

Λ(bk−1 −), Λ(ak +) ∈ R, bk−1 < ak : Again we know that δ− = δ+ = 0. We have T√ (ω(σk− ))−1 T√ (ω(σk+ )) = W(l,0) . Since ω(σk− ) = T2,Λ(bk−1 −) ((bk−1 )) and ω(σk+ ) = T2,Λ(ak +) ((ak )), the assertion follows from Remark 3.7. Λ(bk−1 −) = +∞, Λ(ak +) ∈ R, bk−1 = ak : It follows from (iv) that δ ≤ 1. Moreover, we know from (i) that δ− = 1 and δ+ = 0, which implies that δ = 0. Λ(bk−1 −) = +∞, Λ(ak +) ∈ R, bk−1 < ak : We have δ− = 1, δ+ = 0, and know that Pσ− ,+ = Pσ+ ,e . Since (σk , σk+ ) is indivisible of type 0, it follows that ω(σk+ )21 ∈ k

k

P(Eω(σ+ ) ), and hence that Pσ+ ,e ⊆ dom SPσ+ . This space is a dB-subspace with k

k

k

codimension 1 in Pσ+ and is degenerated since δ− > 0. However, it contains the k same even functions than Pσ+ , and thus can not be equal to Pσ− . By (v) it must k k be the only space which lies strictly between Pσ+ and Pσ− . k

k

Λ(bk−1 −) ∈ R, Λ(ak +) = −∞, bk−1 = ak : It follows from (iv) that δ ≤ 1. We know, moreover, from (i) that δ− = 0 and δ+ = 1, which also implies that δ = 0. Λ(bk−1 −) ∈ R, Λ(ak +) = −∞, bk−1 < ak : We have δ− = 0, δ+ = 1, and Pσ+ ,+ = k Pσ− ,+ . Let P1 be the smallest degenerated dB-space which contains Pσ− . Since k

k

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237

(σk− , σk ) is indivisible of type 0, we have P1 = span(Pσk ,− ∪ {ω(σk− )21 }), and therefore P1,e = Pσ− ,e . This rules out the possibility that δ = 1. k

Λ(bk−1 −) = +∞, Λ(ak +) = −∞, bk−1 = ak : We have δ− = δ+ = 1 and Pσ− ,+ = k Pσ+ ,+ . Thus (iii) implies that Pσ− = Pσ+ . k

k

k

Λ(bk−1 −) = +∞, Λ(ak +) = −∞, bk−1 < ak : We have δ− = δ+ = 1 and Pσ− ,+ = k Pσ+ ,+ . Thus δ > 0, and by (v) it follows that δ = 1. k

It remains to show that κ(σk ) = 1. To this end recall that ind− Pσ− = k maxt<σk ind− ω(t) and ind− Pσ+ + δ+ = mint>σk ind− ω(t), cf. [KW3]. k Consider the case that δ− = δ+ = 0. If δ = 1 the assertion is clear. If δ = 3 it follows since l > 0. Assume that not both of δ± are equal to 0. If δ = 1, we have ind− Pσ+ = ind− Pσ− , and again κ(σk ) = 1. Consider the fourth of the cases in k k the table. Then ind− Pσ− = ind− P1 since Pσ− ,e and P1,e have the same negative k k index. The increase of negative squares from P1 to Pσ+ is then 1. In the last k remaining case, the same argument shows that ind− P1 = ind− Pσ+ , and thus k that κ(σk ) = 1. Next we deal with TJ . It is a consequence of Lemma 2.14 that an application of this transformation does not change the structure of singularities. 6.2. Corollary. Let ω ∈ M<∞ and assume that K− (ω(t)) = K(ω(t)) and K+ (ω(t)) = K(ω(t)). Moreover, let υ := TJ (ω). Let π+ , π− be the projections of K(ω(t)) onto −1 . Then φ induces the ﬁrst and second component, respectively, and put φ := π+ ◦π− an order preserving bijection of the chain of all dB-subspaces of P(Eω(t) ) onto the chain of all dB-subspaces of P(Eυ(t) ). Thereby for all dB-subspaces Q of P(Eω(t) ) ind− φ(Q) = ind− Q, dim φ(Q)◦ = dim Q◦ . If ω ∈ Msym <∞ , then also υ is symmetric, and we have φ(Qe ) = φ(Q)o , φ(Qo ) = φ(Q)e .

(6.1)

˜ω(t) . Hence the ﬁrst Proof. With the notation of Lemma 2.14 we have Eυ(t) = E assertion is an immediate corollary of Lemma 2.14. It remains to investigate the −1 symmetric situation. However, by Lemma 2.7, φ = π+ ◦ π− maps even to odd −1 functions and odd to even functions. Since φ has the same property, the relation (6.1) follows. We can now deduce which singularities are created by an application of T√ ◦TJ ◦T2 . 6.3. Proposition. Let ∈ Mep 0 and assume that lim q∞ ()(x) = lim

x→−∞

x→−∞

1 = 0. q∞ ()(x)

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M. Kaltenb¨ack, H. Winkler and H. Woracek

−1 Moreover, let υ ∈ Mep . Let ak , bk , <∞ be such that q∞ (υ)(z) = −(zq∞ ()(z)) k = 0, . . . , n, and Λ be as in Lemma 5.9. Then υ has exactly n singularities, say γ1 < · · · < γn . We have for every k ∈ {1, . . . , n}

κ(γk ) = 1, δ(γk ) ≤ 2, δ± (γk ) ≤ 1 . The structure of γk can be read oﬀ the behavior of Λ at ak and bk−1 and the fact whether or not there is an indivisible interval between bk−1 and ak . Let l, Λ(bk−1 −), Λ(ak +) and m be as in Proposition 6.1. Moreover, put

Structure of σk

Λ ak +

Λ bk−1 −

α := (bk−1 )21 (0), p(z) := −(mz + lz 2 ), m := m(1 + α2 ), φ := − Arccot α .

∈R

∈R

=

bk−1 = ak bk−1 < ak

>?

@

W

=

>?

@

W

+∞

∈R

∈R

−∞

+∞

−∞

W = W(m ,φ) 1−αp(z) −α2 p(z) W= p(z) 1+αp(z)

Proof. Let ω := T2 () and ω ˜ := TJ (ω), so that υ = T√ (˜ ω ). By Theorem 5.1 and Corollary 6.2 singularities of υ can only occur at singularities of ω. We shall go through the diﬀerent possibilities of singularities of ω and show that each of them gives rise to a singularity of υ with the asserted structure. We will use the same notation as in the previous proofs. The ﬁrst two cases require explicit computation. Λ(bk−1 −) ∈ R, Λ(ak +) ∈ R, bk−1 = ak : Then we have ω(σk− )−1 ω(σk+ ) = W(m,0) , ˜ (σk+ ) = W(m, π2 ) . Moreover, ω ˜ (σk− )21 (0) = −ω(bk−1 )21 (0) = and thus ω ˜ (σk− )−1 ω −(bk−1 )21 (0) = −α. We obtain from Proposition 3.6 that υ(ˆ µ(σk− ))−1 υ(ˆ µ(σk+ )) = W(m ,φ) . Λ(bk−1 −) ∈ R, Λ(ak +) ∈ R, bk−1 < ak : In this case 1 0 ω ˜ (σk− )−1 ω ˜ (σk+ ) = −p(z) 1 and ω ˜ (σk− )12 (0) = ω ˜ (σk+ )12 (0) = −α. By (3.2) it follows that 1 − αp(z) −α2 p(z) − −1 + µ(σk )) = υ(ˆ µ(σk )) υ(ˆ p(z) 1 + αp(z)

Singularities of Generalized Strings

239

For the investigation of the remaining cases note that, by Corollary 6.2, we just have to inspect the behavior of the odd parts of the dB-spaces arising from the chain ω. Λ(bk−1 −) = +∞, Λ(ak +) ∈ R, bk−1 = ak : By Proposition 6.1 Pσ− ,o is degenk erated, Pσ+ ,o is non-degenerated and dim Pσ+ ,o /Pσ− ,o = 1. Hence we have a k k k singularity with δ− = 1, δ+ = 0 and δ = 1. Λ(bk−1 −) = +∞, Λ(ak +) ∈ R, bk−1 < ak : We have Pσ− ,o = P1,o and this space is k degenerated. The space Pσ+ ,o is non-degenerated. Thus we also have a singularity k with δ− = 1, δ+ = 0 and δ = 1. The remaining cases are treated exactly in the same manner. This knowledge on the structure of the singularities implies that in any case δ ≤ 2, δ± ≤ 1 and, cf. [KW3, Corollary 2.12] and its proof, that the increase of the negative index is equal to 1.

7. On generalized strings Recall [KaK1] that a string S[L, m] is given by its length L, 0 ≤ L ≤ ∞, and a nonnegative and nondecreasing function m deﬁned on [0, L) which may be chosen to be left-continuous. This concept can be generalized as follows: Let m be a function deﬁned on [0, L) which is nondecreasing and left-continuous except a ﬁnite number of points from [0, L), and let D be a nondecreasing and left-continuous step function deﬁned on [0, L) which is constant except a ﬁnite number of growth points. Note that D corresponds to the so-called dipoles in [KL2]. Some point xe ∈ [0, L) is called a dipole if D(xe +) − D(xe ) > 0, a negative jump, if m(xe +) − m(xe ) < 0, and a singularity, if m(x) → +∞ for x " xe , or m(x) → −∞ for x $ xe , and (xe − ,xe + ) m(t)2 dt < ∞. The point xe ∈ [0, L) is called critical if it is a dipole, a singularity or a negative jump. The point 0 is critical if it is a dipole or if −∞ ≤ m(0+) < 0, the point L is never a critical point. Observe that at a critical point can be both a dipole and a singularity or a negative jump. The triple S[L, m, D] is called generalized string. The relation 2 f (x) + z f (x)dm(x) + z f (x)dD(x) = 0, f (0−) = 0. (7.1) [0,x]

[0,x]

is the diﬀerential equation of a generalized string. Of course, this equation requires some explanation if S[L, m, D] has singularities among its critical points. The appropriate interpretation is given by means of canonical systems, and for the sake of completeness we continue to recall some basic facts about these systems. Let H be a real, symmetric and non-negative 2 × 2–matrix function on the interval [0, lH ): h1 (x) h3 (x) , x ∈ [0, lH ), H(x) = h3 (x) h2 (x)

240

M. Kaltenb¨ack, H. Winkler and H. Woracek

with locally integrable functions h1 , h2 and h3 . A canonical system is a boundary value problem of the form Jf (x) = −zH(x)f (x), x ∈ [0, lH ), f1 (0) = 0, (7.2) 0 −1 , and a complex parameter z. Here the with f (x) = (f1 (x) f2 (x))T , J = 1 0 diﬀerential equation in (7.2) is considered to hold almost everywhere on [0, lH ). Weyl’s limit point case prevails at the point lH for the canonical system (7.2) if and only if lH tr H(x)dx = ∞, (7.3) 0

and from now on we assume that for each Hamiltonian H the relation (7.3) holds. The fundamental matrix function w11 (x, z) w12 (x, z) W (x, z) = w21 (x, z) w22 (x, z) of a canonical system (7.2) with Hamiltonian H is the unique solution of the integral equation x W (x, z)J − J = z W (s, z)H(s)ds. (7.4) 0

Note that W (0, z) = I, and that x → W (x, z), 0 ≤ x < lH is a maximal chain of matrix functions belonging to M0 .

0 and z ∈ C+ the limit For each ω ∈ N Q(z) := lim

x→lH

w11 (x, z)ω(z) + w12 (x, z) w21 (x, z)ω(z) + w22 (x, z)

(7.5)

exists, is independent of ω, and, as a function of z, belongs to the set of Nevanlinna functions N0 , cf. [dB2]. The function Q is called the Titchmarsh-Weyl coeﬃcient of the canonical system (7.2) or of the Hamiltonian H. Note that W (·, z) is a maximal matrix chain, and that Q coincides with its Weyl coeﬃcient. Let ξφ := (cos φ, sin φ)T for some φ ∈ [0, π). The open interval Iφ ⊂ [0, lH ) is called H-indivisible of type φ if the relation ξφT JH = 0, a.e. on Iφ ,

(7.6)

holds, see [Ka], [dB1]. This notion is the same as introduced for maximal chains in Section 4. In particular, det H = 0 a.e. on Iφ . If (x1 , x2 ) is a H-indivisible of type φ and length l, the fundamental matrix W satisﬁes the relation W (x2 , z) = W (x1 , z)W(l,φ) (z), where the factor W(l,φ) (z) is deﬁned in relation (2.2), that is, H-indivisible intervals are also indivisible. A Hamiltonian H is called trace normed if h1 + h2 = 1 a.e. on [0, ∞). For the class of trace normed Hamiltonians a basic inverse result in [dB1] can be formulated as follows (see [W1]): Each function Q ∈ N0 is the Titchmarsh-Weyl coeﬃcient

Singularities of Generalized Strings

241

of a canonical system with a trace normed Hamiltonian H on [0, ∞). This correspondence is bijective if two Hamiltonians which coincide almost everywhere are identiﬁed. Let QH denote the Titchmarsh-Weyl coeﬃcient corresponding to some Hamiltonian H, and let = JHJ T . H (7.7) Then

* (x, z) = JW (x, z)J T (7.8) W is the fundamental matrix corresponding to H, and the relation (7.5) implies that QH (z) = −(QH (z))−1

(7.9)

If H is of diagonal form, that is H = diag(h1 , h2 ), then QH ∈ N0sym . The following proposition will be of use in what follows. For the following we need the fact that any Q ∈ N0 \ {∞} admits a unique integral representation: t 1 a + bz + − dσ(t) , (7.10) 1 + t2 R t−z 1 where a, b ∈ R, b ≥ 0, σ is a nonnegative Borel measure on R with R 1+t 2 dσ(t) < ∞. σ is called the spectral measure of Q(z). 7.1. Proposition. Let Q = ∞ be some Nevanlinna function with a semibounded spectral measure, supp σ ⊂ [c, ∞), c ∈ R. Let H be some Hamiltonian corresponding to Q with left-continuous components hi , 1 ≤ i ≤ 3 and det H = 0. If W denotes the corresponding fundamental matrix, the relations h3 (x) w12 (x, z) w22 (x, z) = lim − = lim − , h2 (x) z→−∞ w11 (x, z) z→−∞ w21 (x, z) h3 (x) w11 (x, z) w21 (x, z) = lim − = lim − , h1 (x) z→−∞ w12 (x, z) z→−∞ w22 (x, z)

x > 0, h2 (x) > 0,

(7.11)

x > 0, h1 (x) > 0,

(7.12)

hold. Proof. The existence of a Hamiltonian H corresponding to Q with the required properties was shown in [W3], Theorem 3.1. We continue in two steps Step 1: First we assume supp σ ⊆ (0, ∞) and b = limy→∞ Q(iy)/iy = 0. According to Corollary 3.2 of [W3], the function v(x) =

h3 (x) , h2 (x)

x ∈ (0, lH )

is nondecreasing with and left-continuous. A rescaling allows to assume without loss of generality that the Hamiltonian H is of the form v(x)2 v(x) . (7.13) H(x) = v(x) 1

242

M. Kaltenb¨ack, H. Winkler and H. Woracek

We are going to show that v(x) = lim − z→−∞

w22 (x, z) , w21 (x, z)

x > 0.

(7.14)

If v(x) = v(x0 ) for some x0 < x, the interval (x0 , x) is H-indivisible with constant v. It follows with l = x − x0 that 1 − zlv −zl w21 (x0 , z) w21 (x, z) = , zlv 2 w22 (x, z) w22 (x0 , z) 1 + zlv which leads to the relation −

w22 (x, z) 1 =v+ , w21 (x, z) zl + N (z)

N (z) = −

w22 (x0 , z) +v w21 (x0 , z)

−1 .

(7.15)

As w22 (x0 , z)/w21 (x0 , z) is a Nevanlinna function, also N (z) is a Nevanlinna function, and the relation (7.14) follows from the relation (7.15). Now we assume that v(x) > v(t) for all 0 < t < x. The relation (7.4) implies that w21 (x, z) 1 . (7.16) = −z(v(x)w (x, z) + w (x, z)) 21 22 w22 −v(x) (x, z) It follows that −w22 (x, z) = v(x)w21 (x, z), which implies that x w21 (t, z)dv(t). w22 (x, z) = 1 − v(x)w21 (x, z) +

(7.17)

0

We will show that lim w21 (x, z) = ∞

z→−∞

(7.18)

and lim

z→−∞

w21 (t, z) = 0, w21 (x, z)

0 ≤ t < x,

(7.19)

then the relation (7.17) implies the relation (7.14). Because of w21 (x, z) = −z(v(x)w21 (x, z) + w22 (x, z)) x = −zv(x)w21 (x, z) − z + z v(t)w21 (t, z)dt, 0

one ﬁnds with integration by parts that (x, z) = −z 1 + w21

w21 (t, z)dv(t) ,

(7.20)

(x − u)w21 (u, z)dv(u).

(7.21)

x

0

and it follows that

w21 (x, z) = −zx − z

x

0 (·, z) is positive in a Let z < 0. The relation (7.20) implies that the function w21 neighborhood of 0. Then the relation (7.21) implies that w21 (·, z) is positive and

Singularities of Generalized Strings nondecreasing, and the relations (7.18) and

w21 (x, z) ≥ −zw21 (t, z) follow. As

x t

243

x

(x − u)dv(u)

(7.22)

t

(x − u)dv(u) > 0, the relation (7.22) implies the relation (7.19).

Step 2: Now we assume that supp σ ⊂ (c, ∞) for some c < 0. Recall from [W2] ˜ deﬁned by that then the Hamiltonian H ˜ H(x) = W (x, −c)H(x)W (x, −c)T

(7.23)

˜ corresponds to the Titchmarsh-Weyl coeﬃcient Q(z) given by ˜ Q(z) = Q(z − c), and the fundamental matrix ˜ (x, z) = W (x, z − c)W (x, −c)−1 . W In particular, supp σ ˜ ⊂ (0, ∞), and it follows that there exists a nondecreasing ˜ is of the form (7.13). Moreover, the relation function v˜ such that the Hamiltonian H (7.14) is satisﬁed. It follows that w22 (x, −c)˜ h3 (x) v (x) − w12 (x, −c) = , h2 (x) −w21 (x, −c)˜ v (x) + w11 (x, −c) and that w21 (x, z − c) w ˜21 (x, z)w11 (x, −c) + w ˜22 (x, z)w21 (x, −c) = . w22 (x, z − c) w ˜21 (x, z)w12 (x, −c) + w ˜22 (x, z)w22 (x, −c) Together with (7.14) the last relation implies lim −

z→−∞

h3 (x) w22 (x, z) = . w21 (x, z) h2 (x)

The second relation in (7.11) follows from −

w12 (x, z) w22 (x, z) 1 + = →0 w11 (x, z) w21 (x, z) w11 (x, z)w21 (x, z)

(z → −∞),

where the relation det W (x, z) = 1 has been used. The relations (7.12) can be shown in a similar way. If limy→∞ Q(iy)/iy = b > 0, the Hamiltonian is of the form diag(1, 0) on the interval (0, b), and it is easy to see that the relation (7.12) for x ∈ (0, b) holds. If a generalized string S[L, m, D] is given, a Hamiltonian H0 of a canonical system can be constructed as follows: Deﬁne a new scale by x(t) = t + dD(u), L0 = L + dD(u). (7.24) [0,t)

[0,L)

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M. Kaltenb¨ack, H. Winkler and H. Woracek

Let x(t+) = x(t) + D(t) − D(t−). If te is a dipole of S[L, m, D], the interval (x(te ), x(te +)) is assumed to be maximal H0 -indivisible of type π/2, that is, H0 = diag(0, 1) on (x(te ), x(te +)). Deﬁne 1 −m0 (x(t)) , (7.25) m0 (x(t)) = m(t), H0 (x(t)) = −m0 (x(t)) m0 (x(t))2 and put H0 (x(te )) = diag(0, 1) if the point te is a singularity of S[L, m, D]. If L + [0,L) m(t)2 dt < ∞, it is assumed that H0 = diag(0, 1) on (L0 , ∞). Summing up, there exists a (possibly empty) ﬁnite sequence of n maximal H B0 -indivisible intervals Dk of type π/2 such that Dk < Dk+1 , and with D = nk=1 Dk and I := [0, L0 ) \ D the Hamiltonian H0 is given as follows. ⎧ ⎪ 1 −m0 (x) ⎪ ⎪ if x ∈ I, ⎪ ⎨ −m (x) m (x)2 0 0 (7.26) H0 (x) = ⎪ 0 0 ⎪ ⎪ if x ∈ D. ⎪ ⎩ 0 1 Moreover, the construction of H0 implies that the limit point case prevails. Conversely, if a Hamiltonian H0 of the form (7.26) is given, the corresponding generalized string S[L, m, D] can be recovered as follows: For x ∈ I, let x t(x) := χ{h1 =0} (u)du, L := t(L0 ), (7.27) 0

and

t m(t) := m0 (x), D(t) :=

χ{h1 =0} (u)du.

(7.28)

0

If Q ∈ N0 with spectral measure σ has the property that supp σ ∩ (−∞, 0] consists of ﬁnitely many isolated points, there exists a Hamiltonian H of the form (7.26) such that Q is its Titchmarsh-Weyl coeﬃcient, see [W3]. Let f be the solution of the relation (7.2) with Hamiltonian H0 . For t(x) given by the relation (7.27), the function f (t) := f2 (x)

(7.29)

is the solution of the relation (7.1). To justify the relation (7.29), note that if 0 is a dipole or a singularity the relations Jf = −zHf , f1 (0) = 0, and H0 (0) = diag(0, 1) imply that f2 (0) = 0. Otherwise, the relation f (0) = f2 (0) = zm(0)f (0) holds and matches with the equation (7.1). On intervals where m0 is deﬁned and bounded, the relation Jf = −zHf implies that f2 satisﬁes the diﬀerential equation df2 = −zf2 dm0 . (7.30) Now consider a maximal H-indivisible interval Dk = (a, b) of type π/2. Then f2 (x) = 0, f1 (x) = −zf2 (x) if x ∈ (a, b),

Singularities of Generalized Strings

245

which yields f2 (a) = f2 (b), f1 (b) − f1 (a) = −z(b − a)f2 (a). The relation (7.31) yield

f2 (x)

(7.31)

= zf1 (x) − zm0 (x)f2 (x) for x = a and x = b and the relations

f2 (b) − f2 (a) + z(m0 (b) − m0 (a))f2 (a) + z 2 (b − a)f2 (a) = 0.

(7.32)

In particular, as f2 is constant on (a, b), the function f is well deﬁned by the relation (7.29). Recall [LW] that the solution f of (7.1) can also be characterized in terms of m. Namely, if xe is singularity of m, for a solution f of (7.1) on [0, xe ) with xe − f (0−) = 0 the limits f (xe −) and m(t)f (t)dt exist. Moreover, if x > xe , 0

there is exactly one solution f on (xe , x) of (7.1) such that f (xe −) = f (xe +) x and m(t)f (t)dt is ﬁnite. This solution f coincides with the function deﬁned xe −

in (7.29). Let Q0 be the Titchmarsh-Weyl coeﬃcient of the canonical system with the Hamiltonian (7.26). The function QS (z) := z −1 Q0 (z)

(7.33)

is called the principal Titchmarsh-Weyl coeﬃcient of the generalized string S[L, m, D], see [LW]. Let Nκ+ be the set of all functions q ∈ Nκ such that zq(z) ∈ N0 is a Nevanlinna function. The basic inverse result from [LW] can be formulated as follows: If S[L, m, D] is a generalized string with κ critical points, then its principal Titchmarsh-Weyl coeﬃcient QS belongs to the class Nκ+ . Conversely, each function Q ∈ Nκ+ is the principal Titchmarsh-Weyl coeﬃcient of a generalized string with κ critical points, which is uniquely determined by Q. Let a generalized string S[L, m, D] with principal Titchmarsh-Weyl coeﬃcient QS ∈ Nκ+ be given. Then QS is also the Weyl coeﬃcient of some matrix chain υ, and now the problem arises how the singularities of which are characterized in Proposition 6.3 can be described in terms of the generalized string S[L, m, D]. As Q0 ∈ N0ep implies in particular that the corresponding spectral measure σ is semibounded, Proposition 7.1 can be applied to the Hamiltonian H0 of (7.26): 7.2. Corollary. Let W0 be the fundamental matrix of the canonical system with Hamiltonian H0 from the relation (7.26). Then 0 w12 (x, z) 0 (x, z) , z→−∞ w11

m0 (x) = lim

x ∈ I.

Moreover, 0 w12 (x, z) = ∞, z→−∞ w0 (x, z) 11

lim

x ∈ D.

(7.34)

246

M. Kaltenb¨ack, H. Winkler and H. Woracek

7.3. Theorem. Let QS ∈ Nκ+ be the principal Titchmarsh-Weyl coeﬃcient of some generalized string S[L, m, D], and assume that υ is the maximal chain with Weyl coeﬃcient QS . Then the function m0 from the relation (7.25) is equal to the function Λ from Lemma 5.9, and the maximal chain υ has κ singularities, which correspond to the critical points of S[L, m, D]. The ﬁve possible cases concerning the structure of a singularity of the chain υ which are described in Proposition 6.3 correspond to a negative jump in case 1, a dipole in case 2, and to a singularity in the last 3 cases. Proof. Let be the maximal chain with Weyl coeﬃcient 1 q∞ ()(z) = − . zQS (z) Then q∞ () ∈ N0 , and the representation formulas for Nκ+ functions from [KL1] imply that limz→∞ q∞ ()(z) = 0. One ﬁnds from the relations (7.9) and (7.33) *0 is equal to the chain from Proposition 6.3, and the that the the matrix chain W relations (7.34) and (7.8) imply that m0 is equal to the function Λ from Lemma 5.9. Note that the intervals (ak , bk ) from Lemma 5.9 are the maximal intervals of I which contain no critical point. If W0D denotes the factor in the chain W0 which corresponds to a maximal H0 -indivisible interval (x1 , x2 ) of type π/2 and length d, that is, W0 (x2 , z) = W0 (x1 , z)W0D (z), then 1 0 D . (7.35) W0 (z) = −zd 1 Roughly speaking it can be said that W0D corresponds to a dipole interval of length d. If ∆m0 = m0 (x2 +) − m0 (x1 ), the corresponding factor in the maximal chain with Titchmarsh-Weyl coeﬃcient Qd (z) = zQS (z 2 ) is equal to 1 0 D,∆ , (7.36) Wd (z) = −z∆m0 − z 3 d 1 and factor in the chain with Titchmarsh-Weyl coeﬃcient QS (z) is equal to WsD,∆ (z) = I − (z 2 d + z∆m0 )(t(x1 ), 1)T (t(x1 ), 1)J.

(7.37)

Note that both matrix functions generate 1 negative square if d > 0 or if d = 0 and ∆m0 < 0.

References [ADRS] D. Alpay, A. Dijksma, J. Rovnyak, H. de Snoo: Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Oper. Theory Adv. Appl. 96, Birkh¨ auser Verlag, Basel 1997. [dB1] L. de Branges: Hilbert spaces of entire functions, Prentice-Hall, London 1968. [dB2] L. de Branges: Some Hilbert spaces of entire functions II, Trans. Amer. Math. Soc. 99 (1961), 118–152.

Singularities of Generalized Strings

247

[Br]

P. Bruinsma: Interpolation problems for Schur and Nevanlinna pairs, Doctoral Dissertation, University of Groningen 1991.

[DLLS]

A. Dijksma, H. Langer, A. Luger, Yu. Shondin: A factorization result for generalized Nevanlinna functions of the class Nκ , Integral Equations Operator Theory 36 (2000), 121–125.

[Ka]

I.S. Kac: Linear relations, generated by a canonical diﬀerential equation on an interval with a regular endpoint, and expansibility in eigenfunctions (Russian), Deposited in Ukr NIINTI, No. 1453, 1984. (VINITI Deponirovannye Nauchnye Raboty, No. 1 (195), b.o. 720, 1985).

[KaK1]

I.S. Kac and M.G. Krein: On the spectral functions of the string, Supplement II to the Russian edition of F.V. Atkinson, Discrete and continuous boundary problems. Mir, Moscow, 1968 (Russian) (English translation: Amer. Math. Soc. Transl. (2) 103 (1974), 19–102).

[KaK2]

I.S. Kac and M.G. Krein: R-functions–analytic functions mapping the upper halfplane into itself, Supplement II to the Russian edition of F.V. Atkinson, Discrete and continuous boundary problems. Mir, Moscow, 1968 (Russian) (English translation: Amer. Math. Soc. Transl. (2) 103 (1974), 1–18.

[KWW1] M. Kaltenb¨ ack, H. Winkler, H. Woracek: Almost Pontryagin spaces, Operator Theory Adv.Appl, to appear. [KWW2] M. Kaltenb¨ ack, H. Winkler, H. Woracek: Generalized Nevanlinna functions with essentially positive spectrum, J.Oper.Theory, to appear. [KWW3] M. Kaltenb¨ ack, H. Winkler, H. Woracek: Strings, dual strings, and related canonical systems, submitted. [KWW4] M. Kaltenb¨ ack, H. Winkler, H. Woracek: De Branges spaces of entire functions symmetric about the origin, submitted. [KW1]

M. Kaltenb¨ ack, H. Woracek: Pontryagin spaces of entire functions I, Integral Equations Operator Theory 33 (1999), 34–97.

[KW2]

M. Kaltenb¨ ack, H. Woracek: Pontryagin spaces of entire functions II, Integral Equations Operator Theory 33 (1999), 305–380.

[KW3]

M. Kaltenb¨ ack, H. Woracek: Pontryagin spaces of entire functions III, Acta Sci.Math. (Szeged) 69 (2003), 241–310.

[KW4]

M. Kaltenb¨ ack, H. Woracek: Pontryagin spaces of entire functions IV, in preparation.

[KW5]

M. Kaltenb¨ ack, H. Woracek: De Branges spaces of exponential type: General theory of growth, Acta Sci.Math. (Szeged), to appear. ¨ M.G. Krein, H. Langer: Uber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Πκ zusammenh¨ angen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187–236.

[KL1]

[KL2]

M.G. Krein, H. Langer: On some extension problems which are closely connected with the theory of hermitian operators in a space Πκ , III. Indeﬁnite analogues of the Hamburger and Stieltjes moment problems, Part (1): Beitr¨ age zur Analysis 14 (1979), 25–40. Part (2): Beitr¨ age zur Analysis 15 (1981), 27–45.

[LW]

H. Langer, H. Winkler: Direct and inverse spectral problems for generalized strings, Integral Equations Operator Theory 30 (1998), 409–431.

248 [L] [W1] [W2] [W3]

M. Kaltenb¨ack, H. Winkler and H. Woracek B. Levin: Nullstellenverteilung ganzer Funktionen, Akademie Verlag, Berlin 1962. H. Winkler: The inverse spectral problem for canonical systems, Integral Equations Operator Theory, 22 (1995), 360–374. H. Winkler: On transformations of canonical systems, Operator Theory Adv. Appl. 80 (1995), 276–288. H. Winkler: Canonical systems with a semibounded spectrum, Operator Theory Adv.Appl. 106 (1998), 397–417.

Michael Kaltenb¨ ack Institut f¨ ur Analysis und Scientiﬁc Computing Technische Universit¨ at Wien Wiedner Hauptstr. 8–10/101 A–1040 Wien Austria e-mail: [email protected] Henrik Winkler Institute der Wiskunde en Informatica Rijksuniversiteit Groningen Postbus 800 9700 AV Groningen The Netherlands e-mail: [email protected] Harald Woracek Institut f¨ ur Analysis und Scientiﬁc Computing Technische Universit¨ at Wien Wiedner Hauptstr. 8–10/101 A–1040 Wien Austria e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 249–256 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Orthogonal Polynomials on the Unit Circle with Respect to a Rational Weight Function Levon V. Mikaelyan Abstract. We use the analogue of the Christoﬀel’s formula for orthogonal polynomials on the unit circle introduced in [5] to construct a system of orthogonal polynomials on the unit circle with respect to weights of the type % % % p(z) %2 % g(z) % , where p(z) and g(z) are arbitrary polynomials. Exact formulas are established for Toeplitz determinants of these weights. Mathematics Subject Classiﬁcation (2000). Primary 42C05; Secondary 47B35. Keywords. Orthogonal polynomials, Christoﬀel’s formula, Toeplitz determinants .

For a given non-negative polynomial (x) on (a, b) the well-known formula of Christoﬀel ([6]) represents the polynomials orthogonal on the interval (a, b) with respect to the measure (x)dα(x) through the orthogonal polynomials with respect to the measure dα(x). In [5] (see also [2], [3]) the analogue of the above-mentioned formula for polynomials orthogonal on the unit circle was obtained. The objective of this note is to calculate the normalizing constants contained in this formula (Theorem 1 below) and give some applications of this result and Theorem 1. We need some deﬁnitions and notation in order to formulate the analogue of the Christoﬀel’s formula. For a non-decreasing function µ(t) on [−π, π] we denote by dµ(t) the measure generated by µ(t) on [−π, π]. This measure will be called of inﬁnite type if the range of µ(t) consists of an inﬁnite number of diﬀerent values. For a polynomial Pk (z) = a0 + a1 z + · · · + ak z k of degree k we introduce the following notation P˙k (z) = a0 z k + a1 z k−1 + · · · + ak , and we take for every number α = 0,

α˙ = 1/α.

The author supported in part by the grant NFSAT MA 070-02 / CRDF 12011.

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L.V. Mikaelyan

It is well known (see [4]) that every non-negative trigonometrical polynomial Q(eit ), t ∈ [−π, π], can be represented in the form Fn n % A %2 % % j=1 (z − αj )(1 − zαj ) 2 Q(z) = %a (z − αj )% = |a| , (1) zn j=1 where |αj | = 0, and, from now on, let us assume that the variables t and z are connected via the formula z = eit . Under these agreements the following theorem holds (see [5]): Theorem 1. Let dµ(t) be a measure of inﬁnite type on [−π, π]; Q(z) be a trigonometrical polynomial of the form (1), where α1 , α2 , . . . , αn are pairwise distinct points and neither of them lies on the unit circle. If d˜ µ(t) = Q(z)dµ(t), (2) ∞ ∞ and {ϕk (z)}k=0 and {ψk (z)}k=0 are systems of orthogonal polynomials on the unit circle with respect to the measures dµ(t) and d˜ µ(t) respectively, then for all k = 0, 1, . . . ck z n ψk (z) = 2 Fn × |a| j=1 (z − αj )(1 − zαj ) % % % ϕk (z) ϕk+1 (z) . . . ϕk+n (z) z −1 ϕ˙ k+1 (z) . . . z −n ϕ˙ k+n (z) % % % % ϕk (α1 ) ϕk+1 (α1 ) . . . ϕk+n (α1 ) α−1 ˙ k+1 (α1 ) . . . α−n ˙ k+n (α1 ) %% 1 ϕ 1 ϕ % % ϕk (α˙ 1 ) ϕk+1 (α˙ 1 ) . . . ϕk+n (α˙ 1 ) α˙ −1 ˙ k+1 (α˙ 1 ) . . . α˙ −n ˙ k+n (α˙ 1 ) %% 1 ϕ 1 ϕ % (3) % %, . . . . .. . . .. .. .. .. .. .. % % . % % % ϕk (αn ) ϕk+1 (αn ) . . . ϕk+n (αn ) α−1 ϕ˙ k+1 (αn ) . . . α−n ϕ˙ k+n (αn ) % n n % % % ϕk (α˙ n ) ϕk+1 (α˙ n ) . . . ϕk+n (α˙ n ) α˙ −1 ϕ˙ k+1 (α˙ n ) . . . α˙ −n ϕ˙ k+n (α˙ n ) % n 1 where ck = 0 are normalizing constants. Taking into account that the paper [5] is in Russian and is not easily accessible, for completeness reasons, we bring the proof of this theorem. Proof. The determinant of the right-hand expression in (3) multiplied by z n is a polynomial of degree not greater than 2n + kFand vanishes for z = α1 , α2 , . . . , αn n and α˙ 1 , α˙ 2 , . . . , α˙ n ; hence it is divisible by j=1 (z − αj )(1 − zαj ), so ψk (z) is a polynomial of the degree not greater than k. Therefore, for the proof of the theorem it suﬃces to show that ψk (z) is a polynomial of the degree k and satisﬁes the equations π

Jk,j :=

−π

ψk (z)z −j d˜ µ(t) = 0

(4)

for j = 0, 1, . . . , k − 1. To prove (4), we ﬁrst notice that ψk (z) can be represented in the following manner: n n ck ψk (z) = µp,k ϕk+p (z) + νq,k z −q ϕ˙ k+q (z) , (5) Q(z) p=0 q=1

Orthogonal Polynomials on the Unit Circle

251

where µp,k and νq,k are some constants. From (5) we have π π n n µp,k ϕk+p (z)z −j dµ(t) + νq,k z −q−j ϕ˙ k+q (z)dµ(t) . Jk,j = ck p=0

−π

−π

q=1

Now, for 0 ≤ j ≤ k − 1, π ϕk+p (z)z −j dµ(t) = 0 −π

π

and −π

z −q−j ϕ˙ k+q (z)dµ(t) = 0

(6)

for all p = 0, 1, . . . , n and q = 1, 2, . . . , n, since the system of polynomials {ϕk (z)}∞ k=0 is orthogonal with respect to dµ(t). Note that the equations of the second part of (6) remain true also for j = k. Let us prove that ψk (z) is a polynomial of the degree k. For the leading (k) coeﬃcient ψk of ψk (z) we have from (5) (k+n)

(k)

ψk =

ck µn,k ϕk+n Fn , |a|2 j=1 (−αj )

(7)

(k+n)

where ϕk+n = 0 is the leading coeﬃcient of ϕk+n (z); so it suﬃces to show that µn,k is diﬀerent from zero. From (3) we obtain µn,k = (−1)n mkn , where % % ϕk (α1 ) % % ϕk (α˙ 1 ) % % .. k mn = % . % % ϕk (αn ) % % ϕk (α˙ n )

% . . . α−n ˙ k+n (α1 ) %% 1 ϕ . . . α˙ −n ˙ k+n (α˙ 1 )%% 1 ϕ % .. .. %. . . % −1 −n ϕk+1 (αn ) . . . ϕk+n−1 (αn ) αn ϕ˙ k+1 (αn ) . . . αn ϕ˙ k+n (αn ) %% ϕk+1 (α˙ n ) . . . ϕk+n−1 (α˙ n ) α˙ −1 ˙ k+1 (α˙ n ) . . . α˙ −n ˙ k+n (α˙ n ) % n ϕ 1 ϕ ϕk+1 (α1 ) ϕk+1 (α˙ 1 ) .. .

. . . ϕk+n−1 (α1 ) α−1 ˙ k+1 (α1 ) 1 ϕ −1 . . . ϕk+n−1 (α˙ 1 ) α˙ 1 ϕ˙ k+1 (α˙ 1 ) .. .. .. . . .

(8)

(9)

Assuming the contrary, we can ﬁnd constants a0 , a1 , . . . , an−1 and b1 , b2 , . . . , bn , not all of them zero, such that B(z) =

n−1 n 1 n a z ϕ (z) + bq z n−q ϕ˙ k+q (z) p k+p z n p=0 q=1

vanishes for z = α1 , α2 , . . . , αn and α˙1 , α˙2 , . . . , α˙n . Hence B(z) is of the form Q(z)q(z), where q(z) is a polynomial of the degree not greater than k − 1. It is easy to see from (6) that π B(z)u(z)dµ(t) = 0 −π

for all polynomials u(z) of degree not greater than k − 1; so π Q(z)|q(z)|2 dµ(t) = 0, −π

which contradicts the condition of dµ being a measure of inﬁnite type.

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To calculate the normalizing constants ck let us assume that the system of 1 polynomials {ϕk (z)}∞ k=0 is orthonormal with respect to the measure 2π dµ with the leading coeﬃcient of ϕk (z) being positive for all k = 0, 1, . . .. Assume also 1 that the same is true for {ψk (z)}∞ µ. The k=0 with respect to the measure 2π d˜ ∞ ∞ systems {ϕk (z)}k=0 and {ψk (z)}k=0 are uniquely determined under this normalization (see [4]). For this normalization we get π 1 Ik := ψk (z)ψk (z)d˜ µ(t) = 1. 2π −π From (5) we conclude that π n n µp,k ϕk+p (z)ψk (z)dµ(t) + νq,k 2πIk = ck −π

p=0

−π

q=1

and in view of (6); 1 Ik = ck µ0,k 2π

π

z −q ϕ˙ k+q (z)ψk (z)dµ(t) ,

π

−π

ϕk (z)ψk (z)dµ(t).

For µ0,k , from (3) we have µ0,k = mk0 , where % % ϕk+1 (α1 ) ϕk+2 (α1 ) . . . ϕk+n (α1 ) α−1 ˙ k+1 (α1 ) 1 ϕ % % ϕk+1 (α˙ 1 ) ϕk+2 (α˙ 1 ) . . . ϕk+n (α˙ 1 ) α˙ −1 ϕ˙ k+1 (α˙ 1 ) 1 % % .. .. .. .. .. mk0 = % . . . . . % %ϕk+1 (αn ) ϕk+2 (αn ) . . . ϕk+n (αn ) α−1 ϕ˙ k+1 (αn ) n % %ϕk+1 (α˙ n ) ϕk+2 (α˙ n ) . . . ϕk+n (α˙ n ) α˙ −1 ϕ˙ k+1 (α˙ n ) n

(10)

% . . . α−n ˙ k+n (α1 ) %% 1 ϕ . . . α˙ −n ˙ k+n (α˙ 1 )%% 1 ϕ % .. .. %. . . % −n . . . αn ϕ˙ k+n (αn ) %% . . . α˙ −n ϕ˙ k+n (α˙ n ) %

(11)

1

Now, if we rearrange the polynomial ψk (z) as a linear combination of the polynomials ϕj (z), j = 0, 1, . . . , k, i.e., ψk (z) =

k

dj ϕj (z) ;

(12)

j=0

from (10) and (12) we obtain ck mk0 dk = 1. (k) ϕk

(13)

(k) ψk

and be the leading coeﬃcients of ϕk (z) and ψk (z) respectively. Let From (7), (8) and (12) it is easy to conclude that (k+n)

(k)

ψk

(k)

= dk ϕk =

ck mkn ϕk+n Fn . |a|2 j=1 αj (k)

Finally, in view of (13) and the last relation, taking into account that ϕk , (k+n) (k) ϕk+n and ψk are positive; we obtain G G H (k) (k+n) H (k) n k H ϕk Πj=1 αj 1 H (k) I ϕk ϕk+n mn , −iθk I , ψ = (14) ck = |a|e k (k+n) |a| mk0 Πnj=1 αj ϕk+n mk0 mkn

Orthogonal Polynomials on the Unit Circle

253

where mkn and mk0 are deﬁned by (9) and (11) respectively, and θk = arg

mkn n Πj=1 αj

.

Let us now give some applications of these results. First we consider weights h(z) on the unit circle, which are ratios of nonnegative trigonometrical polynomials. It is clear that such weights admit the following representation: % Fn (z − αj ) %2 % j=1 % h(z) = % (z = eit ), (15) % g(z) where g(z) is an ordinary polynomial having zeros only inside the unit circle and with a positive leading coeﬃcient. Using the fact that, if g(z) is a polynomial of degree m, then the orthonormal polynomials ϕk (z) on the unit circle with respect to the measure 1 dt 2π|g(z)|2 (with respect to the weight 1/2π|g(z)|2 ) have the form (see [4]) ϕk (z) = z k−m g(z),

k = m, m + 1, . . . ;

(16)

from Theorem 1 we immediately obtain Theorem 2. Let the weight h(z) be representable in the form (15), where g(z) is a polynomial of the degree m with positive leading coeﬃcient having zeros inside the unit circle only, then the polynomials ψk (z), k = m, m + 1, . . . , orthogonal on the unit circle with respect to the weight h(z), admit the following representation: ck z n × j=1 (z − αj )(1 − zαj )

ψk (z) = Fn % % % % % % % % % % % % %

. . . z k−m+n g(z) z −1 g(z) ˙ ... −1 . . . αk−m+n g(α ) α g(α ˙ ) 1 1 ... 1 1 −1 . . . α˙ k−m+n g( α ˙ ) α ˙ g( ˙ α ˙ 1 1) . . . 1 1 .. .. .. .. . . . . −1 k−m+1 k−m+n g(α ) α g(α ) . . . α g(α ) α g(α ˙ ) . . . αk−m n n n n n n n 1 −1 k−m+1 k−m+n g( α ˙ ) α ˙ g( α ˙ ) . . . α ˙ g( α ˙ ) α ˙ g( ˙ α ˙ ) . . . α˙ k−m n n n n n n n 1 z k−m g(z) z k−m+1 g(z) g(α1 ) αk−m+1 g(α1 ) αk−m 1 1 k−m g(α˙ 1 ) α˙ 1 g(α˙ 1 ) α˙ k−m+1 1 .. .. . .

where ck = 0 are the normalizing constants.

% % % % % % % % , (17) % % −n α1 g(α ˙ n ) %% α˙ −n ˙ α˙ n ) % 1 g( z −n g(z) ˙ α−n g(α ˙ 1) 1 α˙ −n g( ˙ α ˙ 1) 1 .. .

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The special case of this theorem, when g(z) ≡ 1, yields the following statement: Corollary 3. The system of the polynomials {ψk (z)}∞ k=0 , orthogonal on the unit circle with respect to the weight n % A %2 % % Q(z) = %a (z − αj )% , j=1

has the form

% % % % % ck %% ψk (z) = % Q(z) % % % % %

zk αk1 α˙ k1 .. .

z k+1 αk+1 1 α˙ k+1 1 .. .

... ... ... .. .

z k+n αk+n 1 α˙ k+n 1 .. .

z −1 α−1 1 α˙ −1 1 .. .

... ... ... .. .

z −n α−n 1 α˙ −n 1 .. .

αkn α˙ kn

αk+1 n α˙ k+1 n

... ...

αk+n n α˙ k+n n

α−1 n α˙ −1 n

... ...

α−n n α˙ −n n

% % % % % % % %, % % % % %

(18)

where ck = 0 are the normalizing constants. Now, let us show some applications of the second formula in (14). Assume that dµ(t) is a measure of inﬁnite type on [−π, π], the constants π 1 µ ˆk = e−ikt dµ(t) 2π −π are its Fourier-Stieltjes coeﬃcients and {ϕk (z)}∞ k=0 is the system of polynomials 1 dµ(t). If ϕk (z) are orthonormal on the unit circle with respect the measure 2π (k) normalized in such a way that their leading coeﬃcients ϕk are positive, then, as it is known (see for instance [4]): (k)

[ϕk ]2 =

Dk−1 [µ] , Dk [µ]

k = 0, 1, . . . ,

where Dk [µ] = detˆ µi−j ki,j=0 are the Toeplitz determinants associated with the measure dµ(t) , D−1 [µ] = 1. According to the last relation, from (14) we deduce the following: Proposition 4. If the measures dµ(t) and d˜ µ(t) satisfy the conditions of Theorem 1, then for the Toeplitz determinants associated with these measures holds: J Dk−1 [˜ µ] Dk−1 [µ]Dk+n−1 [µ] mkn = 2 k n , k = 0, 1, . . . , Dk [˜ µ] Dk [µ]Dk+n [µ] |a| m0 Πj=1 αj where mkn and mk0 are as in (9) and (11). In particular, if we assume that dµ(t) =

1 dt, |g(z)|2

where g(z) is a polynomial of the degree m with the leading coeﬃcient gm > 0, (k) then from (16) we obtain ϕk = gm for k = m, m + 1, . . . .

Orthogonal Polynomials on the Unit Circle

255

Hence for the Toeplitz determinants associated with the measure % Fn (z − αj ) %2 % % j=1 d˜ µ(t) = % % dt, g(z) we get from (14) the following representation: Dk−1 [˜ µ] g 2 mk = k mn n Dk [˜ µ] m0 Πj=1 αj for k = m, m + 1, . . . . Applying (9) and (11) to this special case, we obtain % % k−m+1 k−m+2 k−m+n % α1 g(α1 ) α−1 ˙ 1 ) . . . α−n ˙ 1 ) %% 1 1 g(α 1 g(α % k−m+1 g(α1 ) α1k−m+2 g(α1 ) . . . αk−m+n % α˙ g(α˙ 1 ) α˙ 1 g(α˙ 1 ) . . . α˙ 1 g(α˙ 1 ) α˙ −1 ˙ α˙ 1 ) . . . α˙ −n ˙ α˙ 1 ) %% 1 g( 1 g( % 1 % % .. . . . . k . . .. .. .. .. .. .. m0 = % % . % % −n % αk−m+1 g(αn ) αk−m+2 g(αn ) . . . αk−m+n g(αn ) α−1 g(α ˙ n ) %% n ) . . . α1 g(α n n % n 1 ˙ % α˙ k−m+1 g(α˙ ) α˙ k−m+2 g(α˙ ) . . . α˙ k−m+n g(α˙ ) α˙ −1 g( ˙ α˙ ) . . . α˙ −n g( ˙ α˙ ) % n

n

n

n

n

n

1

n

1

n

and % k−m % α1 g(α1 ) % k−m % α˙ % 1 g(α˙ 1 ) % .. k mn = % . % % αk−m g(αn ) % n % α˙ k−m g(α˙ ) n

n

% . . . α−n ˙ 1 ) %% 1 g(α . . . α˙ −n ˙ α˙ 1 ) %% 1 g( % .. .. %. . . % −1 −n k−m+1 k−m+n−1 αn g(αn ) . . . αn g(αn ) α1 g(α ˙ n ) . . . α1 g(α ˙ n ) %% α˙ k−m+1 g(α˙ n ) . . . α˙ k−m+n−1 g(α˙ n ) α˙ −1 ˙ α˙ n ) . . . α˙ −n ˙ α˙ n ) % n n 1 g( 1 g( αk−m+1 g(α1 ) 1 α˙ k−m+1 g(α˙ 1 ) 1 .. .

. . . αk−m+n−1 g(α1 ) α−1 ˙ 1) 1 1 g(α k−m+n−1 . . . α˙ 1 g(α˙ 1 ) α˙ −1 g( ˙ 1) 1 ˙ α .. .. .. . . .

Remark 5. If the determinants in formulas (17) and (18) are expanded according to the Laplace’s formula with respect to the ﬁrst (n + 1) columns, then the multipliers in the corresponding sum will be the Vandermonde determinants, or will diﬀer from Vandermonde determinants by a multiplier, which is easy to calculate. Consequently, another representation of orthogonal polynomials can be obtained from (17) and (18). In this way, using the last two equations one can obtain Day’s formula for Toeplitz determinants of rational functions, given the condition that they are non-negative on the unit circle (see [1]). Remark 6. In the case, if in the theorems above and the corollary some α-s have multiplicity greater than one, one should proceed as follows: i) If αj , (|αj | = 1) has multiplicity m (m > 1), i.e., if there are m multipliers (z − αj )(1 − zαj ) in the representation of Q(z), then the rows 2kj + 2, 2kj + 4, . . . , 2kj + 2m − 2 of the right-hand determinant of (3), (17), (18) must be replaced by derivatives of the orders 1, 2, . . . , m − 1 of the ﬁrst row at the point αj , and the rows 2kj + 3, 2kj + 5, . . . , 2kj + 2m − 1 must be replaced by the same derivatives at the point α˙j . Here 2kj is the number of the ﬁrst row, which contains αj .

256

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ii) If |αj | = 1, i.e., αj = α˙j and there are m multipliers (z − αj )(1 − zαj ) = −

(z − αj )2 αj

in the representation of Q(z), then the rows 2kj , 2kj + 1, . . . , 2kj + 2m − 1 must be replaced by derivatives of orders 0, 1, . . . , 2m − 1 of the ﬁrst row at the point αj . Remark 7. We would like to mention that one can use Theorem 1 for the construction of an algorithm for ﬁnding orthogonal polynomials on the unit circle with respect to the given rational weight. Acknowledgements I would like to express my gratitude to the organizers of the Colloquium on Operator Theory in Vienna and especially to Professor Heinz Langer, whose eﬀorts made possible my participation. I also convey my thanks to the Referees for their valuable suggestions.

References [1] A. B¨ ottcher, B. Silbermann, Analysis of Toeplitz Operators. Springer-Verlag, Berlin, 1990. [2] B.L. Golinski˘ı, An analogue of the Christoﬀel formula for orthogonal polynomials on the unit circle and some applications. Izv. Vyssh. Uchebn. Zaved. Mat. 1:(2) (1958), 33–42. [3] E. Godoy, F. Marcellan, An analog of the Christoﬀel formula for polynomial modiﬁcation of a measure on the unit circle. Boll. Un. Mat. Ital. A (7) 5 (1991), 1–12. [4] U. Grenander, G. Szeg˝ o, Toeplitz forms and their applications. Univ. Calif. Press, Berkeley, Los Angeles, 1958; Russ. transl.: Izdat. Inostr. Lit., Moscow, 1961. [5] L. Mikaelyan, The analogue of Christoﬀel formula for orthogonal polynomials on unit circle. Dokl. Akad. Nauk Arm. SSR, 67:5 (1978), 257–263. [6] G. Szeg˝ o, Orthogonal Polynomials. Volume 23 of AMS Colloquium Publications, Providence, R.I., 1959; Russ. transl.: Izdat. Fiz.-Mat. Lit., Moscow 1962. Levon V. Mikaelyan Dept. of Applied Mathematics Yerevan State University A. Manoukian 1 375025 Yerevan Armenia e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 257–274 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Bi-dimensional Moment Problems and Regular Dilations Dan Popovici To Professor Heinz Langer

Abstract. Given a map f on Z2+ × X (X is just a non-empty set) into a Hilbert space H we provide necessary and suﬃcient conditions in order to ensure the existence of a commuting pair (S, T ) of contractions on H having regular dilation such that S m T n f (0, 0, x) = f (m, n, x),

(m, n) ∈ Z2+ , x ∈ X.

Such moment problems are strongly related to the theory of harmonizable and stationary processes. Isometric or unitary solutions are also characterized in terms of the initial data. Mathematics Subject Classiﬁcation (2000). Primary 47A57, 47A20, 60G10; Secondary 43A35, 47A45, 47B15. Keywords. Moment problem, regular dilation, unitary extension, spectral measure, positive deﬁnite function, harmonizable process, stationary process.

1. Introduction One of the most important directions in the theory of non-selfadjoint operators was opened by the theorem of Sz.-Nagy [20] on the existence of a unitary dilation for every Hilbert space contraction. Among numerous applications of this last theorem we want to mention invariant subspace theory, interpolation theory or prediction theory for stochastic processes ([22], [4], [7]). More recently, several authors have contemplated the idea to extend known results for single operators to systems of (at least two) commuting operators on the same Hilbert space. It is expected that such generalizations would provide a larger set of applications. The problem of ﬁnding a unitary (or isometric) dilation of a This work was supported by the EU Research Training Network “Analysis and Operators” with contract no. HPRN-CT-2000-00116.

258

D. Popovici

system of several commuting contractions was proposed and solved by Sz.-Nagy under the additional assumption that the given system is doubly commuting ([21]). A special kind of a joint dilation for n commuting contractions, called regular and based on the fact that a certain operator function is positive deﬁnite on Zn , was introduced and developed by Brehmer in [1]. We should mention here the work of Sz.-Nagy [19] and Halperin [5] which brought more light on the subject. Regular dilations have been recently studied, for example, in [2], [3], [23], [8], [9], [11]. It is our aim in the present paper to give detailed solutions to the following problem: Problem 1.1. Provide necessary and suﬃcient conditions on a function f : Z2+ × X → H (X is a non-empty set and H a Hilbert space) in order to ensure the existence of a commuting pair (S, T ) of contractions on H having regular dilation and such that f (m, n, x) = S m T n f (0, 0, x),

(m, n) ∈ Z2+ , x ∈ X.

Similar one-dimensional problems have been proposed and completely solved in [14], [15], [16], [17], [18] or [12]. For the multi-dimensional case we refer to [13] and [6]. From the applications point of view we should mention the connection of such problems with the theory of harmonizable stochastic processes1: An H-valued discrete process {f (n)}n on Z is weakly harmonizable if and only if it has a stationary dilation. In other words, there exist a Hilbert space K containing H (as a closed subspace) and a K-valued discrete stationary process K {g(n)}n such that f (n) = PH g(n), n ∈ Z. In particular, if an operator function Φ is positive deﬁnite on Z with Φ(0) = IH then the process {Φ(n)h}n is weakly harmonizable, for every h ∈ H. In addition, Φ has the form + T n, n≥0 Φ(n) = , T ∗|n|, n < 0 for a certain contraction T on H ([21]). Passing to the bi-dimensional case, we consider H-valued two-time parameter discrete weakly harmonizable processes on Z2 of the form {Φ(m, n)h}(m,n) (h ∈ H is ﬁxed), where Φ is a given positive deﬁnite function on Z2 . In particular, we assume that Φ has the form ⎧ m n S T , m, n ≥ 0 ⎪ ⎪ ⎪ ⎨S ∗|m| T n , m < 0, n ≥ 0 ΦS,T (m, n) = , (1.1) ⎪T ∗|n| S m , m ≥ 0, n < 0 ⎪ ⎪ ⎩ ∗|m| ∗|n| S T , m, n < 0 deﬁned in terms of a certain pair (S, T ) of commuting contractions on H. ΦS,T is positive deﬁnite on Z2 if and only if (S, T ) has a regular unitary dilation (U, Z). In this case {U m Z n h}(m,n)∈Z2 is a stationary dilation of {ΦS,T (m, n)h}(m,n)∈Z2 . 1 for

the theory of harmonizable processes we refer to [7]

Bi-dimensional Moment Problems

259

We choose to characterize two-time parameter weakly harmonizable processes of the form {ΦS,T (m, n)h}(m,n) deﬁned on Z2+ , rather than on Z2 . In fact, we consider a family {{f (m, n, x)}(m,n) } (indexed by X) of two-time parameter processes and provide necessary and suﬃcient conditions in order to ensure an expression as above, but deﬁned in terms of the same (for all processes in the family) pair (S, T ) of commuting contractions having regular dilation. We also characterize the cases when (S, T ) is a pair of commuting isometries (i.e., the process is stationary2 ) or unitary operators on H. Various kind of applications conclude the paper.

2. Notation and preliminaries Let H and K be complex Hilbert spaces. Denote by L(H, K) (L(H) if H = K) the Banach space of all linear and bounded operators between H and K. If {Hα }α∈I is K a family of (closed) subspaces in H then its closed linear span is expressed by α∈I Hα . A function Φ : Z2 → L(H) is said to be positive deﬁnite if Φ(m − p, n − q)hm,n ; hp,q ≥ 0, (m,n),(p,q)∈Z2

for every sequence {hm,n } ⊂ H with ﬁnite support. A unitary dilation of a commuting pair (S, T ) of contractions on H is a commuting pair (U, Z) of unitary operators on a Hilbert space K containing H (as a closed subspace) such that S m T n h = PH U m Z n h,

(m, n) ∈ Z2+ , h ∈ H

(PH is the orthogonal projection of K onto H). Such a dilation is said to be minimal if L K= U m Z n H. (m,n)∈Z2

A unitary dilation (U, Z) on K for a commuting pair (S, T ) of contractions on H is said to be regular if, in addition, S ∗m T n h = PH U ∗m Z n h,

(m, n) ∈ Z2+ , h ∈ H.

It was stated by Brehmer ([1], [19], [5]) that a pair (S, T ) of commuting contractions has regular unitary dilation if and only if IH − S ∗ S − T ∗ T + S ∗ T ∗ ST ≥ 0. 2A

process {f (m, n)}(m,n)∈Z2 ⊂ H is supposed to be stationary if f (m + 1, n), f (p + 1, q) = +

f (m, n + 1), f (p, q + 1) = f (m, n), f (p, q), (m, n), (p, q) ∈ Z2+ .

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D. Popovici

If (S, T ) has a regular unitary dilation (U, Z) on K then, by Stone’s theorem, there exists an L(K)-valued spectral measure E on the bitorus T2 such that m n ¯m µ U Z = ¯n dE(λ, µ), (m, n) ∈ Z2 . (2.1) λ T2

Setting, for any Borel subset σ of T2 , ω(σ) = PH E(σ)|H ,

(2.2)

we obtain an L(H)-valued semispectral measure on T2 , uniquely determined by (S, T ) such that ¯mµ ΦS,T (m, n) = λ ¯n dω(λ, µ), (m, n) ∈ Z2 . (2.3) T2

Conversely, suppose that (2.3) holds for an L(H)-valued semispectral measure on T2 . Then, by Naimark’s theorem, there exists a larger Hilbert space K and an L(K)-valued spectral measure E on T2 such that (2.2) holds. The pair (U, Z) deﬁned on K by ¯ U= µ ¯dE(λ, µ) λdE(λ, µ) and Z = T2

T2

is clearly a regular unitary dilation of (S, T ) and satisﬁes (2.1). Consequently, the following conditions are equivalent: (i) (S, T ) has a regular unitary dilation; (ii) There exists an L(H)-valued semispectral measure ω on T2 such that (2.3) holds; (iii) ΦS,T is positive deﬁnite. In other words our problem for a function f : Z2+ × X → H can be reformulated as follows: Find necessary and suﬃcient conditions in order to ensure the existence of an L(H)-valued semispectral measure ω on T2 such that ¯mµ f (m, n, x) = ¯n dω(λ, µ)f (0, 0, x), (m, n) ∈ Z2+ , x ∈ X λ T2

and

Φ

T2

¯ λdω(λ,µ),

T

(m, n) = ¯ 2 µdω(λ,µ)

¯mµ ¯n dω(λ, µ), λ T2

(m, n) ∈ Z2 .

A (regular) unitary dilation for a pair (V, W ) of commuting isometries on H is, in fact, a unitary extension. If (U1 , Z1 ) on K1 and (U2 , Z2 ) on K2 are minimal unitary extensions for (V, W ) then there exists a unique (up to an isomorphism) unitary operator ϕ : K1 → K2 which leaves H invariant (ϕ|H = IH ) and intertwines U1 and U2 (ϕU1 = U2 ϕ), respectively Z1 and Z2 (ϕZ1 = Z2 ϕ). According to

Bi-dimensional Moment Problems

261

[10], a model for the minimal unitary extension of (V, W ) is given on K = H ⊕ 2 Hker(V W )∗ (T) by the formulas f (z) − f (0) (2.4) U (h, f (z)) := V h + W ∗ f (0), V (I − W W ∗ )f (z) + W ∗ z and f (z) − f (0) Z(h, f (z)) := W h + V ∗ f (0), W (I − V V ∗ )f (z) + V ∗ , (2.5) z 2 h ∈ H, f ∈ Hker(V W )∗ (T).

3. General solutions Let X be a non-empty set, H a complex Hilbert space and f : Z2+ × X → H. The ﬁrst result is a generalization of the one in [13]: Theorem 3.1. The following conditions are equivalent: (i) Problem 1.1 has a solution; (ii) cx(m,n),(p,q) f (m + p, n + q, x)2 (m,n),(p,q),x

≤

cx(m,n),(m ,n ) cy(p,q),(p ,q ) f ((m − p)+ + m , (n − q)+ + n , x),

(m,n),(m ,n ),x (p,q),(p ,q ),y

f ((m − p)− + p , (n − q)− + q , y), for every ﬁnite family {cx(m,n),(m ,n ) } of complex numbers; cx(m,n),(m ,n ) cy(p,q),(p ,q ) f ((m − p)+ + m , (n − q)+ + n , x), (iii) (m,n),(m ,n ),x (p,q),(p ,q ),y

f ((m − p)− + p , (n − q)− + q , y) ≥ 0, for every ﬁnite family {cx(m,n),(m ,n ) } of complex numbers; cxm,n f (m + 1, n, x) ≤ cxm,n f (m, n, x), (iv) (m,n),x

(m,n),x

cxm,n f (m, n

(m,n),x

and

≤

cxm,n f (m, n, x)

(m,n),x

cxm,n f (m + 1, n, x)2 +

(m,n),x

+ 1, x) ≤

cxm,n f (m, n + 1, x)2

(m,n),x

cxm,n f (m, n, x)2

(m,n),x

for every ﬁnite family

+

cxm,n f (m + 1, n + 1, x)2 ,

(m,n),x

{cxm,n }

of complex numbers.

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D. Popovici

Proof. (i) ⇒ (ii) Suppose that Problem 1.1 has a solution (S, T ) (on H) having a regular unitary dilation (U, Z) (on K). Then, for every ﬁnite family {cx(m,n),(p,q) } of complex numbers, cx(m,n),(p,q) S m+p T n+q f (0, 0, x)2 (m,n),(p,q),x

=

cx(m,n),(p,q) PH U m Z n S p T q f (0, 0, x)2

(m,n),(p,q),x

≤

(m,n),(m ,n ),x

=

cx(m,n),(m ,n ) U m Z n S m T n f (0, 0, x),

cy(p,q),(p ,q ) U p Z q S p T q f (0, 0, y)

(p,q),(p ,q ),y

cx(m,n),(m ,n ) cy(p,q),(p ,q )

(m,n),(m ,n ),x (p,q),(p ,q ),y

=

U m−p Z n−q S m T n f (0, 0, x), S p T q f (0, 0, y) +

+

cx(m,n),(m ,n ) cy(p,q),(p ,q ) S (m−p) T (n−q) S m T n f (0, 0, x),

(m,n),(m ,n ),x (p,q),(p ,q ),y −

−

S (m−p) T (n−q) S p T q f (0, 0, y). (ii) ⇒ (iii) is obvious. (iii) ⇒ (iv) Fix a ﬁnite family {cxm,n } of complex numbers. The inequalities of (iv) can be obtained if the family {cx(m,n),(m ,n ) } in (iii) take, respectively, one of the following forms: ⎧ (a) x ⎪ ⎨−cm −1,n , if m = n = 0; m ≥ 1 cx(m,n),(m ,n ) = cxm ,n , if m = 1, n = 0 ; ⎪ ⎩ 0, for the other cases ⎧ (b) x ⎪ ⎨−cm ,n −1 , if m = n = 0; n ≥ 1 cx(m,n),(m ,n ) = cxm ,n , if m = 0, n = 1 ; ⎪ ⎩ 0, for the other cases ⎧ (c) ⎪ cxm −1,n −1 , if m = n = 0; m , n ≥ 1 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎨−cm −1,n , if m = 0, n = 1; m ≥ 1 cx(m,n),(m ,n ) = −cxm ,n −1 , if m = 1, n = 0; n ≥ 1 , ⎪ ⎪ ⎪ cxm ,n , if m = n = 1 ⎪ ⎪ ⎪ ⎩0, for the other cases ((m, n), (m , n ) ∈ Z2+ , x ∈ X).

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Observe that the sum in (iii) reduces to the terms for which m, n, p, q ∈ {0, 1}. In view of the formulas + 0, a=0 + (a − b) = 1 − b, a = 1 and

+ −

(a − b) =

b, a = 0 0, a = 1

(b ∈ {0, 1})

we only have to change some summation variables to get the conclusion. (iv) ⇒ (i) Let Hf :=

L

f (m, n, x).

(m,n),x

By (iv), operators S cxm,n f (m, n, x) →0 cxm,n f (m + 1, n, x) ∈ Hf Hf (m,n),x

and Hf

(m,n),x

T

cxm,n f (m, n, x) →0

(m,n),x

cxm,n f (m, n + 1, x) ∈ Hf

(m,n),x

are well deﬁned contractions on H . They commute since f

S0 T0 f (m, n, x) = f (m + 1, n + 1, x) = T0 S0 f (m, n, x),

(m, n) ∈ Z2+ , x ∈ X.

It is easily shown (inductively) that formulas f (m, n, x) = S0m T0n f (0, 0, x),

(m, n) ∈ Z2+ , x ∈ X

hold true. The last inequality of (iv) becomes S0 h2 + T0 h2 ≤ h2 + S0 T0 h2 , for every h in the linear space generated by {f (m, n, x) | (m, n) ∈ Z2+ , x ∈ X}, so for every h ∈ Hf . Deduce that IHf − S0∗ S0 − T0∗ T0 + S0∗ T0∗ S0 T0 ≥ 0, and the pair (S0 , T0 ) has regular unitary dilation. It can be extended (by 0 on H Hf ) to a commuting pair (S, T ) of contractions on H having regular unitary dilation. Obviously, f (m, n, x) = S m T n f (0, 0, x), as required.

(m, n) ∈ Z2+ , x ∈ X,

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Remark 3.2. • There is another method, similar to the one used in [13] (cf. [14]) to build a solution to Problem 1.1 starting from the inequality (ii) of Theorem 3.1, but without using (iv). On the linear space F0 of all ﬁnite families {cx(m,n),(p,q) } of complex numbers we deﬁne (in view of (ii)) the semi-inner product {cx(m,n),(m ,n ) }, {dy(p,q),(p ,q ) } (3.1) := cx(m,n),(m ,n ) dy(p,q),(p ,q ) f ((m − p)+ + m , (n − q)+ + n , x), (m,n),(m ,n ),x (p,q),(p ,q ),y

f ((m − p)− + p , (n − q)− + q , y), {cx(m,n),(m ,n ) }, {dy(p,q),(p ,q ) } ∈ F0 . By factorization and completion F0 becomes a Hilbert space F . The map ϕ cx(m,n),(p,q) f (m + p, n + q, x) ∈ H F {cx(m,n),(p,q) } → (m,n),(p,q),x

is a well-deﬁned contraction (consequence of (ii)). Operators A

(3.2)

B

(3.3)

F {cx(m,n),(p,q) } → {cx(m+1,n),(p,q) } ∈ F and F {cx(m,n),(p,q) } → {cx(m,n+1),(p,q) } ∈ F

have isometric adjoints and the pair (ϕA∗ ϕ∗ , ϕB ∗ ϕ∗ ) is a solution to Problem 1.1. • A solution to Problem 1.1 is unique if and only if H = Hf . • If f : Z2+ × X → H satisﬁes one of the equivalent conditions of Theorem 3.1 then it can be extended to a function F : Z2 × X → H such that

cxm,n F (m, n, x)2

(m,n),x

≤

cxm,n cyp,q F (m − p, n − q, x), f (0, 0, y),

(m,n),(p,q),x,y

for every ﬁnite family {cxm,n }(m,n)∈Z2 ,x∈X of complex numbers. More precisely, let F (m, n, x) := ΦS,T (m, n)f (0, 0, x),

(m, n) ∈ Z2 , x ∈ X

be such a function deﬁned in terms of a solution (S, T ) of Problem 1.1. Then, for every ﬁnite family {cxm,n } of complex numbers and any regular unitary

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265

dilation (U, Z) of (S, T ), cxm,n ΦS,T (m, n)f (0, 0, x)2 (m,n),x

≤

cxm,n U m Z n f (0, 0, x)2

(m,n),x

=

cxm,n U m Z n f (0, 0, x),

(m,n),x

cyp,q U p Z q f (0, 0, y)

(p,q),y

=

cxm,n cyp,q U m−p Z n−q f (0, 0, x), f (0, 0, y)

(m,n),(p,q),x,y

=

cxm,n cyp,q ΦS,T (m − p, n − q)f (0, 0, x), f (0, 0, y).

(m,n),(p,q),x,y

The ﬁrst application generalizes the one in [14]: Corollary 3.3. Let f : Z+ × X → H. The following conditions are equivalent: (i)

There exists a contraction S on H such that f (m, x) = S m f (0, x),

(ii)

m ≥ 0, x ∈ X;

cxm,n f (m + n, x)2

(m,n),x

≤

cxm,n cyp,q f ((m − p)+ + n, x), f ((m − p)− + q, y),

(m,n),(p,q),x,y

for every ﬁnite family {cxm,n } of complex numbers; cxm,n cyp,q f ((m − p)+ + n, x), f ((m − p)− + q, y) ≥ 0, (iii) (m,n),(p,q),x,y

for every ﬁnite family {cxm,n } of complex numbers; cxm f (m + 1, x) ≤ cxm f (m, x), (iv) m,x

for every ﬁnite family

m,x

{cxm,n }

of complex numbers.

Proof. It is easy to observe that a solution to Problem 1.1 (if it exists) for the map Z2+ × X (m, n, x) → f (m, x) ∈ H can be taken of the form (S, IH ) such that S is a contraction on H. Finally, the link between two ﬁnite families {cxm,n } and {cx(m,n),(p,q) } of complex numbers is given by cx(m,n),(p,q) . cxm,n = p,q

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D. Popovici

For a given double sequence of operators {Am,n }(m,n)∈Z2+ ⊂ L(K, H), if we re-write the equivalent conditions of Theorem 3.1 for the function f

Z2+ × K (m, n, k) → Am,n (k) ∈ H and replace any ﬁnite family {cx(m,n),(p,q) } of complex numbers with the ﬁnite double sequence {k(m,n),(p,q) } ⊂ K given by k(m,n),(p,q) := ck(m,n),(p,q) k, (m, n), (p, q) ∈ Z2+ , k∈K

then we obtain an improvement and an extension of [15]: Corollary 3.4. The following conditions are equivalent: (i)

There exists a commuting pair (S, T ) of contractions on H having regular unitary dilation and such that (m, n) ∈ Z2+ ;

Am,n = S m T n A0,0 ,

(ii)

Am+p,n+q (k(m,n),(p,q) )2

(m,n),(p,q)

≤

A(m−p)+ +m ,(n−q)+ +n (k(m,n),(m ,n ) ),

(m,n),(m ,n ) (p,q),(p ,q )

A(m−p)− +p ,(n−q)− +q (k(p,q),(p ,q ) ), for every ﬁnite sequence {k(m,n),(m ,n ) } of vectors in K; (iii) A(m−p)+ +m ,(n−q)+ +n (k(m,n),(m ,n ) ), (m,n),(m ,n ) (p,q),(p ,q )

A(m−p)− +p ,(n−q)− +q (k(p,q),(p ,q ) ) ≥ 0, for every ﬁnite sequence {k(m,n),(m ,n ) } of vectors in K; (iv) Am+1,n (km,n ) ≤ Am,n (km,n ), m,n

m,n

Am,n+1 (km,n ) ≤

m,n

and

≤

m,n

Am,n (km,n )

m,n

Am+1,n (km,n )2 +

m,n

Am,n+1 (km,n )2

m,n

Am,n (km,n )2 +

Am+1,n+1 (km,n )2 ,

m,n

for every ﬁnite sequence {km,n } of vectors in K.

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267

A similar use of Corollary 3.3 provides: Corollary 3.5. ([12]) Let {Am }m≥0 ⊂ L(K, H). The following conditions are equivalent: (i) There exists a contraction S on H such that

(ii)

Am = S m A0 , m ≥ 0; Am+n (km,n )2 ≤ A(m−p)+ +n (km,n ), A(m−p)− +q (kp,q ), m,n

m,n,p,q

for every ﬁnite sequence {km,n } of vectors in K; (iii) A(m−p)+ +n (km,n ), A(m−p)− +q (kp,q ) ≥ 0, m,n,p,q

for every ﬁnite sequence {km,n } of vectors in K; (iv) Am+1 (km ) ≤ Am (km ), m

m

for every ﬁnite sequence {km } of vectors in K. The following corollary may also have some independent interest: Corollary 3.6. Let P, Q be linear and bounded operators on H. Then (P, Q) is a commuting pair of orthogonal projections if and only if P (a + b) + P Q(c + d) ≤ a + P b + Qc + P Qd and Q(a + b) + P Q(c + d) ≤ a + Qb + P c + P Qd, for every a, b, c, d ∈ H. Proof. We can use Corollary 3.4 for the ⎧ ⎪ ⎪IH , ⎪ ⎨P, Am,n = ⎪Q, ⎪ ⎪ ⎩ P Q,

sequence {Am,n } ⊂ L(H) given by if if if if

m=n=0 m = 0, n = 0 . m = 0, n = 0 m, n = 0

The assumption in Corollary 3.4 (i) take the following form: there exists a commuting pair (S, T ) of contractions on H having regular unitary dilation and such that S m = P (m > 0),

T n = Q (n > 0),

S m T n = P Q (m, n > 0).

We obtain that S = P and T = Q are both idempotent contractions on H, hence orthogonal projections. We also observe that a pair of commuting orthogonal projections has regular unitary dilation (being double commuting). The # conclusion follows #if we take, for any#ﬁnite sequence {hm,n } ⊂ H, a = h0,0 , b = m≥1 hm,0 , c = n≥1 h0,n and d = m,n≥1 hm,n .

268

D. Popovici

4. Isometric solutions In order to obtain a commuting pair of isometries (brieﬂy bi-isometry) as solution to Problem 1.1 it is necessary and suﬃcient that equality holds in Theorem 3.1 (ii). More precisely, we have: Theorem 4.1. Let f be as above. The following conditions are equivalent: There exists a bi-isometry (V, W ) on H as a solution to Problem 1.1; (ii) cx(m,n),(p,q) f (m + p, n + q, x)2 (i)

(m,n),(p,q),x

=

cx(m,n),(m ,n ) cy(p,q),(p ,q ) f ((m − p)+ + m , (n − q)+ + n , x),

(m,n),(m ,n ),x (p,q),(p ,q ),y

f ((m − p)− + p , (n − q)− + q , y), for every ﬁnite family {cx(m,n),(m ,n ) } of complex numbers; (iii)

f (m + 1, n, x), f (p + 1, q, y) = f (m, n + 1, x), f (p, q + 1, y) = f (m, n, x), f (p, q, y), (m, n), (p, q) ∈ Z2+ , x, y ∈ X.

Proof. (i) ⇒ (ii) We proceed as in the corresponding part in the proof of Theorem 3.1. We just have to observe that, for every commuting pair (V, W ) of isometries on H, +

−

V m W n h, V p W q h = V (m−p) W n h, V (m−p) W q h +

+

−

−

= V (m−p) W (n−q) h, V (m−p) W (n−q) h, (m, n), (p, q) ∈ Z2+ , h ∈ H. (ii) ⇒ (iii) Fix a ﬁnite family {cxm,n } of complex numbers and apply (ii) to the (ﬁnite) family {cx(m,n),(m ,n ) } given by ⎧ x ⎪ ⎨−cm −1,n , if m = n = 0; m ≥ 1 , cx(m,n),(m ,n ) = cxm ,n , if m = 1, n = 0 ⎪ ⎩ 0, for the other cases for (m, n), (m , n ) ∈ Z2+ and x ∈ X. The left-hand side of (ii) becomes cx(0,0),(p,q) f (p, q, x) + cx(1,0),(p,q) f (p + 1, q, x)2 (p,q),x

=−

(p,q),x

p≥1,q≥0,x

cxp−1,q f (p, q, x) +

(p,q),x

cxp,q f (p + 1, q, x)2 = 0.

Bi-dimensional Moment Problems

269

Similarly, the corresponding right-hand side can be re-written as cx(m,n),(m ,n ) cy(p,q),(p ,q ) (m,n),(p,q)∈{(0,0),(1,0)} (m ,n ),(p ,q ),x,y

f ((m − p)+ + m , (n − q)+ + n , x), f ((m − p)− + p , (n − q)− + q , y) cxm −1,n cyp −1,q f (m , n , x), f (p , q , y)

=

m ,p ≥1 n ,q ≥0

−

cxm −1,n cyp ,q f (m , n , x), f (p + 1, q , y)

m ≥1 n ,p ,q ≥0

−

p ≥1 m ,n ,q ≥0

cxm ,n cyp −1,q f (m + 1, n , x), f (p , q , y)

+

cxm ,n cyp ,q f (m , n , x), f (p , q , y)

m ,n ,p ,q ≥0

=

cxm,n f (m, n, x)2 −

(m,n),x

We deduce that

cxm,n f (m + 1, n, x)2 .

(m,n),x

cxm,n f (m, n, x) =

(m,n),x

for every ﬁnite family

cxm,n f (m + 1, n, x),

(m,n),x

{cxm,n }

of complex numbers. Equivalently,

f (m + 1, n, x), f (p + 1, q, y) = f (m, n, x), f (p, q, y) and, analogously, f (m, n + 1, x), f (p, q + 1, y) = f (m, n, x), f (p, q, y) for every (m, n), (p, q) ∈ Z2+ and x, y ∈ X. (iii) ⇒ (i) By Hf

V

cxm,n f (m, n, x) →0

(m,n),x

and

H f

(m,n),x

cxm,n f (m + 1, n, x) ∈ Hf

(4.1)

cxm,n f (m, n + 1, x) ∈ Hf

(4.2)

(m,n),x W cxm,n f (m, n, x) →0

(m,n),x

we deﬁne (in view of (iii)) a commuting isometric pair (V0 , W0 ) on Hf which satisﬁes f (m, n, x) = V0m W0n f (0, 0, x), (m, n) ∈ Z2+ , x ∈ X. It can be extended to a commuting isometric pair (V0 ⊕ IHHf , W0 ⊕ IHHf ) on H with the same property.

270

D. Popovici

Remark 4.2. • The method used in Remark 3.2 for an alternate solution to Theorem 3.1 can be modiﬁed correspondingly to obtain another proof for the implication (ii) ⇒ (i) of Theorem 4.1. Following the same notations we obtain a Hilbert space F (with the inner product deﬁned by (3.1)) and operators A and B (introduced in (3.2) and (3.3)) having isometric adjoints. The map ϕ F {cx(m,n),(p,q) } → cx(m,n),(p,q) f (m + p, n + q, x) ∈ Hf (m,n),(p,q),x

is unitary (being isometric-by (ii)-and surjective). The commuting isometric pair (ϕA∗ ϕ∗ , ϕB ∗ ϕ∗ ) is a solution to our problem on Hf and can be obviously extended to a solution on H. • A pair (V, W ) satisfying Theorem 4.1 (i) is uniquely determined if and only if H = Hf . Particular choices for the map f will lead to diﬀerent applications, corresponding to the ones in the previous section: Corollary 4.3. Let f : Z+ × X → H. The following conditions are equivalent: (i) There exists an isometric operator V on H such that

(ii)

f (m, x) = V m f (0, x),

m ≥ 0, x ∈ X;

cxm,n f (m + n, x)2

(m,n),x

=

cxm,n cyp,q f ((m − p)+ + n, x), f ((m − p)− + q, y),

(m,n),(p,q),x,y

for every ﬁnite family {cxm,n } of complex numbers; (iii)

f (m + 1, x), f (n + 1, y) = f (m, x), f (n, y),

m, n ≥ 0, x, y ∈ X.

Corollary 4.4. Let {Am,n }(m,n)∈Z2+ be a sequence of operators between K and H. The following conditions are equivalent: (i) There exists a commuting pair (V, W ) of isometries on H such that

(ii)

Am,n = V m W n A0,0 ,

Am+p,n+q (k(m,n),(p,q) )2

(m,n),(p,q)

=

(m, n) ∈ Z2+ ;

A(m−p)+ +m ,(n−q)+ +n (k(m,n),(m ,n ) ),

(m,n),(m ,n ) (p,q),(p ,q )

A(m−p)− +p ,(n−q)− +q (k(p,q),(p ,q ) ), for every ﬁnite sequence {k(m,n),(m ,n ) } of vectors in K; (iii)

A∗m+1,n Ap+1,q = A∗m,n+1 Ap,q+1 = A∗m,n Ap,q ,

(m, n), (p, q) ∈ Z2+ .

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271

Corollary 4.5. ([12]) Let {Am }m≥0 be a sequence of operators between K and H. The following conditions are equivalent: There exists an isometry V on H such that

(i)

Am = V m A0 , m ≥ 0; Am+n (km,n )2 = A(m−p)+ +n (km,n ), A(m−p)− +q (kp,q ),

(ii)

m,n,p,q

(m,n),(p,q)

for every ﬁnite sequence {km,n } of vectors in K; A∗m+1 An+1 = A∗m An ,

(iii)

5. Unitary solutions f Let f : Z2+ × X → H, Hp,q :=

m, n ≥ 0.

K m≥p,n≥q,x∈X

f (m, n, x),

(p, q) ∈ Z2+

f H0,0 .

and H := In order to obtain a commuting pair of unitary operators as a solution to Problem 1.1 attached to f we proceed as follows: f

• ﬁnd a bi-isometric solution (according to Theorem 4.1); • extend its restriction to Hf to a commuting pair of unitary operators on H. Unfortunately, even if f satisﬁes the equivalent conditions of Theorem 4.1, it is not always possible to perform this last step: Remark 5.1. Suppose that there exists a commuting pair (U, Z) of unitary operators on H such that f (m, n, x) = U m Z n f (0, 0, x),

(m, n) ∈ Z2+ , x ∈ X.

(5.1)

˜ , Z) ˜ Then (U |Hf , Z|Hf ) is a commuting pair of isometric operators on Hf . If (U is its minimal unitary extension (given by (2.4) and (2.5)) on the Hilbert space ˜ = Hf ⊕ H 2 ˜ H ker(UZ|Hf )∗ (T) then the (Hilbert) dimension of H (which clearly depends only on f and not on the particular choice of (U, Z)) is not greater than the dimension of H. Consequently, if, for example, Hf = H and one of the operators V0 and W0 deﬁned by (4.1) and (4.2) is not unitary then our problem cannot have a bi-unitary solution. The following theorem is an improvement of a recent result in [6]: Theorem 5.2. The following conditions are equivalent: (i) There exists a commuting pair (U, Z) of unitary operators on H such that (5.1) holds true; (ii) (a) f veriﬁes one of the equivalent conditions of Theorem 4.1; f ) ≤ dim(H Hf ). (b) ℵ0 dim(Hf H1,1

272

D. Popovici

Proof. Since the restriction of a commuting pair of unitary operators on H to one of its invariant subspaces (namely Hf ) is a commuting pair of isometries, we can always suppose that the equivalent conditions of Theorem 4.1 hold. In other words, the problem reduces to the following: ﬁnd necessary and suﬃcient conditions on f such that the pair (V0 , W0 ) given on Hf by (4.1) and (4.2) extends to a commuting pair of unitary operators on H. ˜ , Z) ˜ be the minimal unitary extension of (V0 , W0 ) given on H ˜ = Hf ⊕ Let (U 2 ∗ f HHf Hf (T) by (2.4) and (2.5) (remark that ker(V0 W0 ) = H V0 W0 Hf = 1,1

f ). Hf H1,1 f ) ≤ dim(H Hf ) then there exists an isometric operator If ℵ0 dim(Hf H1,1 ˜ Hf → H Hf . Deduce that IHf ⊕ Y is a unitary operator between H ˜ and Y :H f f ∗ ˜ ˜ ˜ H := H ⊕ Y (H H ). Consequently, ((IHf ⊕ Y )U (IHf ⊕ Y ) , (IHf ⊕ Y )Z(IHf ⊕ Y )∗ ) is a commuting pair of unitary operators acting on H (⊂ H) that extends (V0 , W0 ). Take

˜ (IHf ⊕ Y )∗ and Z = IHH ⊕ (IHf ⊕ Y )Z(I ˜ Hf ⊕ Y )∗ U = IHH ⊕ (IHf ⊕ Y )U and observe that (U, Z) is a commuting pair of unitary operators on H which veriﬁes (5.1). Conversely, suppose that (V0 , W0 ) can be extended to a commuting pair (U, Z) of unitary operators on H. Then (U, Z) contains a K minimal unitary extension (U0 := U |H0 , Z0 := Z|H0 ) of (V0 , W0 ) on H0 := (m,n)∈Z2 U m Z n Hf . Moreover, f ) = dim(H0 Hf ) ≤ dim(H Hf ). ℵ0 dim(Hf H1,1

We conclude the paper with some remarks: Remark 5.3. • Suppose that f satisﬁes one of the equivalent conditions of Theorem 4.1 and f = Hf then dim Hf ≤ ℵ0 . If H1,1 f ) dim(H Hf ) ≥ 0 = ℵ0 dim(Hf H1,1 f = Hf we obtain the same and our problem has a unitary solution. If H1,1 conclusion since, in this case, f ℵ0 dim(Hf H1,1 ) = ℵ0 ≤ dim H = dim(H Hf ).

• Jab lo´ nski and Stochel showed in [6] that it is possible to produce unitary solutions to our problem if |X| = 1, f veriﬁes the conditions of Theorem 4.1 (iii) and dim(H Hf ) ≥ dim Hf . This is, however, a consequence of Theorem 5.2: as mentioned above we can consider the case dim Hf > ℵ0 . Then f ). dim(H Hf ) ≥ dim Hf ≥ ℵ0 dim(Hf H1,1

Bi-dimensional Moment Problems

273

• If it exists, a pair (U, Z) of commuting unitary operators on H such that (5.1) holds is uniquely determined by f if and only if f H = Hf = H1,1 . f Suppose that Hf = H1,1 . Then operators V0 and W0 deﬁned on Hf by (4.1) and (4.2) are unitary and (V0 , W0 ) has a unique unitary extension to H if and only if H = Hf . If the pair (U, Z) is unique then we can freely suppose that this unitary extension of (V0 , W0 ) is minimal. Let ϕ be any unitary operator on H Hf . Then ((IHf ⊕ ϕ)U (IHf ⊕ ϕ)∗ , (IHf ⊕ ϕ)Z(IHf ⊕ ϕ)∗ ) is a minimal unitary extension of (V0 , W0 ). By uniqueness, we conclude that (U, Z) commutes with any operator of the form IHf ⊕ A, A being linear and bounded on H Hf . In particular,

U (IHf ⊕ (x ⊗ x))U ∗ (x) = (IHf ⊕ (x ⊗ x))(x) or, equivalently, x, U xU x = x, for any unit vector x ∈ H Hf (if H = Hf then V0 = U and W0 = Z f = V0 W0 Hf = Hf ; x ⊗ x is deﬁned on H Hf by are unitary and H1,1 x ⊗ x(y) := y, xx). Hence, U (H Hf ) = H Hf and, analogously, Z(H Hf ) = H Hf . We obtain that Hf is reducing for (U, Z) and, consequently f = Hf . (V0 = U |Hf , W0 = Z|Hf ) should be a unitary pair. But then H1,1 • The conclusion of Theorem 5.2 holds for any particular choice of f : the same kind of solutions can be obtained for the moment problems proposed in Corollaries 4.3-4.5. Acknowledgement The work on this paper was partly done during the author’s postdoctoral visit at the Department of Analysis and Scientiﬁc Computing, Technical University of Vienna. The author would like to express his appreciation to Professor Heinz Langer for his kind support. Very helpful remarks of the referees are also acknowledged with gratitude.

References ¨ [1] S. Brehmer, Uber vertauschbare Kontraktionen des Hilbertschen Raumes. Acta Sci. Math. 22 (1961), 106–111. [2] R.E. Curto, F.H. Vasilescu, Standard operator models in the polydisc. Indiana Univ. Math. J. 42 (1993), 791–810. [3] R.E. Curto, F.H. Vasilescu, Standard operator models in the polydisc II. Indiana Univ. Math. J. 44 (1995), 727–746. [4] C. Foia¸s, A.E. Frazho, I. Gohberg, M.A. Kaashoek, Metric Constrained Interpolation, Commutant Lifting and Systems. Operator Theory: Advances and Applications 100, Birkh¨ auser-Verlag, Basel, 1998.

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[5] I. Halperin, Sz.-Nagy–Brehmer dilations. Acta Sci. Math. 23 (1962), 279–289. [6] Z.J. Jablo´ nski, J. Stochel, Subnormality and operator multidimensional moment problems. J. London Math. Soc. 71 (2005), 438–466. [7] Y. Kakihara, Multidimensional second order stochastic processes. Series on Multivariate Analysis 2, World Scientiﬁc Publ. Co., Inc., River Edge, NJ, 1997. [8] A. Olofsson, Operator valued n-harmonic measure in the polydisc. Studia Math. 163 (2004), pp. 203–216. [9] D. Popovici, Matrix representations for regular dilations. Preprint (2004). [10] D. Popovici, On the unitary extension for a system of commuting isometries. Preprint (2004). [11] D. Popovici, Regular dilations on Kre˘ın spaces. Preprint (2004). [12] D. Popovici, Z. Sebesty´en, Positive deﬁnite functions and Sebesty´en’s operator moment problem. Glasgow Math. J. 47 (2005), 471–488. [13] D. Popovici, Z. Sebesty´en, Sebesty´en moment problem: The multidimensional case. Proc. Amer. Math. Soc. 132 (2004), 1029–1035. [14] Z. Sebesty´en, Moment theorems for operators on Hilbert space. Acta Sci. Math. 44 (1982), 165–171. [15] Z. Sebesty´en, Moment theorems for operators on Hilbert space II. Acta Sci. Math. 47 (1984), 101–106. [16] Z. Sebesty´en, A moment theorem for contractions on Hilbert spaces. Acta Math. Hung. 47 (1986), 391–393. [17] Z. Sebesty´en, Moment problem for dilatable semigroups of operators. Acta Sci. Math. 45 (1983), 365–376. [18] Z. Sebesty´en, Moment type theorems for ∗-semigroups of operators on Hilbert space I. Ann. Univ. Sci. Budapest, E¨ otv¨ os Sect. Math. 26 (1983), 213–218. [19] B. Sz.-Nagy, Bemerkungen zur vorstehenden Arbeit des Herrn S. Brehmer. Acta Sci. Math. 22 (1961), 112–114. [20] B. Sz.-Nagy, Sur les contractions de l’espace de Hilbert. Acta Sci. Math. 15 (1953), 87–92. [21] B. Sz.-Nagy, Transformations de l’espace de Hilbert, fonctions de type positif sur un groupe. Acta Sci. Math. 15 (1954), 104–114. [22] B. Sz.-Nagy, C. Foia¸s, Harmonic analysis of operators on Hilbert space. NorthHolland, Amsterdam, 1970. [23] D. Timotin, Regular dilations and models for multi-contractions. Indiana Univ. Math. J. 47 (1998), 671–684. Dan Popovici Department of Mathematics and Computer Science University of the West Timi¸soara B-dul Vasile Pˆ arvan 4 RO-300223 Timi¸soara Romania e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 275–298 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Isospectral Vibrating Systems, Part 2: Structure Preserving Transformations Uwe Prells and Peter Lancaster Abstract. The study of inverse problems for n×n-systems of the form L(λ) := M λ2 + Dλ + K is continued. In this paper it is assumed that one vibrating system is speciﬁed and the objective is to generate isospectral families of systems, i.e., systems which reproduce precisely the eigenvalues of the given system together with their multiplicities. Two central ideas are developed and used, namely, standard triples of matrices, and structure preserving transformations. Mathematics Subject Classiﬁcation (2000). Primary 74A15; Secondary 15A29. Keywords. Vibrating systems, inverse problems, damping.

1. Introduction A broad spectrum of physical problems concerning vibrations is covered by classical models of second order constant-coeﬃcient diﬀerential equations. In many cases it is natural to refer to the three coeﬃcient matrices as the mass (M ), damping (D), and stiﬀness matrices (K) and we will use this terminology. However, by admitting the possibility of complex coeﬃcients (not just real matrices), the number of physical problems covered is enlarged (admitting gyroscopic eﬀects, for example) and, not only this, it turns out that a broader mathematical analysis is obtained which casts some light on those problems in which M, D, K are real. This leads to our ﬁrst deﬁnition: Deﬁnition 1. A (vibrating) system is a triple of n × n complex matrices {M, D, K} for which M is nonsingular. When convenient, the quadratic matrix function L(λ) := λ2 M + λD + K may also be described as a vibrating system. It is well known that the solutions of the corresponding diﬀerential equations can be described entirely in terms of the solutions of the algebraic eigenvalue

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problem: ﬁnd those λ ∈ C and non-zero x ∈ Cn for which L(λ)x = (λ2 M + λD + K)x = 0. In Part 1 of this work (reference [6]) it was shown how, in some cases, the three coeﬃcient matrices can be recovered if (well-deﬁned) complete information is given on the eigenvalue and eigenvector structures (known as the spectral data). Here, the objective is to start with one system, say (M0 , D0 , K0 ), and, implicitly, a complete set of corresponding spectral data, and to show how to generate isospectral vibrating systems, i.e., systems (M, D, K) which share the same set of eigenvalues (including their multiplicity structures). And this is to be done without explicit reference to eigenvalues and eigenvectors. There is an interesting contrast with the approach taken by Chu et al. [2], where partial eigenstructure is prescribed in a scheme to generate monic real symmetric systems. Our objective will be achieved using two parallel (but closely linked) notions. The ﬁrst is the tool of “standard triples” of matrices for L(λ) introduced and developed in great detail in [4] and earlier works cited there (see Section 3). The second is the idea of “structure preserving transformations” developed in this context in [3] (and introduced here in Section 2). Computational procedures for solution of the problem for general systems (with no symmetries) are contained in Section 5. Sections 6, 7, and 8 are concerned with the important (and more complex) problems in which M, D, K are required to be Hermitian (or real symmetric). It is of interest to ask when there is an isospectral system of simplest possible form – in which all three coeﬃcients are diagonal. This is the topic of Section 9. We conclude this introduction with a discussion of the basic idea of linearization. It is very well known that linear second order systems can be transformed to ﬁrst order in an elementary way; generally known as “linearization”. Those linearizations associated with vibrating systems are our present concern. In the ﬁrst two sections their properties under transformations of three types are reviewed, namely, “equivalence”, “strict equivalence”, and “similarity” in decreasing order of generality. There have been several recent publications concerning equivalence and similarity transformations of vibrating systems. The authors’ formulation of these ideas together with new results appear in Sections 2 and 4 (especially Theorem 7 and its Corollary). Deﬁnition 2. Let L(λ) be a vibrating system. A 2n × 2n matrix pencil λX − Y is a linearization of L(λ) if 6 7 L(λ) 0 = E(λ)(λX − Y )F (λ) (1) 0 In for some matrix polynomials E(λ), F (λ) with constant non-zero determinants. Notice that it follows from this deﬁnition that the matrix X of a linearization is necessarily nonsingular. (This follows from the assumption that M is nonsingular.) The transformation of λX − Y on the right-hand side of (1) is known as an equivalence transformation. The “equivalence class” of all pencils equivalent to

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277

λX − Y in this sense determines the set of all linearizations of L(λ). A vital property of equivalence transformations is that they preserve the set of eigenvalues and their multiplicity structures (i.e., they have a common Jordan structure). Thus a linearization of a vibrating system shares its “spectrum” (its eigenvalues – with their multiplicities); it is isospectral. Deﬁne the right and left companion matrices, CR and CL , of a vibrating system by: 7 7 6 6 0 I 0 −KM −1 , CL = . (2) CR = −M −1 K −M −1 D I −DM −1 The ﬁrst lemma is very well known: Lemma 1. The following pencils are linearizations of the vibrating system L(λ): 1. λA − B with 7 7 6 6 −K 0 D M , (3) , B := A := 0 M M 0 2.

λI2n − A−1 B = λI2n − CR ,

3.

λI2n − BA−1 = λI2n − CL .

2. Structure preserving transformations If E and F are nonsingular 2n-by-2n matrices in C2n×2n , then the pencils λA − B and E(λA − B)F are said to strictly equivalent. Clearly, a strict equivalence is also an equivalence in the sense used in Deﬁnition 2 et seq., and so strict equivalence also preserves the spectrum. The set of all pencils strictly equivalent to λA − B in this sense form an equivalence class. (The relation of “strict equivalence” is reﬂexive, symmetric and transitive.) Now consider the linearization λA − B of (3) and let λA − B be a strictly equivalent pencil, i.e., there exist E and F such that A = EAF, B = EBF . The question is, when does λA −B deﬁne a vibrating system? The answer is, when A , B have the deﬁning properties of A and B, namely, when there exist M , D , K with M nonsingular such that 7 7 6 6 D M −K 0 , B = . (4) A = 0 M M 0 Deﬁnition 3. (cf. Garvey et al. [3].) Let λA − B be deﬁned from a vibrating system as in (3). Let λA − B be obtained from λA − B by a strict equivalence. Then this strict equivalence is said to be structure preserving if and only if A , B have the form (4) and M is nonsingular. For brevity, a structure preserving strict equivalence of this kind is called an SPE. When the linearizations of two vibrating systems are connected in this way they will be said to be related by an SPE. Observe that strict equivalence

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transformations of L(λ) itself, say L(λ) → SL(λ)T where S and T are nonsingular, are included in this deﬁnition. Such a transformation corresponds to an SPE with 7 7 6 6 T 0 S 0 . , F = E= 0 T 0 S Deﬁnition 4. (Lancaster and Prells [8].) Let CR be the right companion matrix (2) of L(λ). The nonsingular 2n-by-2n matrix SR is called a (right ) structure −1 CR SR = CR is a (right) companion matrix. preserving similarity (SPS) if SR (Clearly, a similar deﬁnition can be made in terms of the left companion matrix.) When the companion matrices of two vibrating systems are connected in this way they will be said to be related by an SPS. The close relationship between SPE and SPS transformations is the subject of the easily proved result: Theorem 2. Two vibrating systems are related by an SPE if and only if they are related by an SPS. Proof. Consider a vibrating system L0 (λ) = λ2 M0 + λD0 + K0 . Let 7 7 6 6 D0 M 0 −K0 0 , B0 := , A0 := 0 M0 M0 0

(5)

(0)

and let CR be the corresponding right companion matrix. Consider the SPE λA − B = E(λA0 − B0 )F . Since the structure is preserved A and B have the form of equations (3).This implies that A is nonsingular and −1 A−1 B = (EA0 F )−1 (EB0 F ) = F −1 A−1 EB0 F = F −1 (A−1 0 E 0 B0 )F,

i.e., CR = F −1 CR F , so that the companion matrices CR and CR are similar. Thus, the systems are related by an SPS. (0) Conversely, let CR = S −1 CR S be an SPS (with the deﬁnitions above for (0) CR , CR , A, B, A0 , B0 ). Then (0)

(0)

λA − B = A(λI − CR ) = A(λI − S −1 CR S) = (AS −1 )(λI − CR )S (0)

(0)

= (AS −1 A−1 0 )(λA0 − B0 )S, and this is an SPE.

Corollary 3. Let A, B be deﬁned by a vibrating system as in (3) and let E(λA − (0) B)F = λA0 − B0 be an SPE, then E deﬁnes an SPS for CL , (CL = ECL E −1 ) (0) and F deﬁnes an SPS for CR , (CR = F −1 CR F ). Proof. It has been shown in the above argument that, given the SPE deﬁned by matrices E and F , the matrix F deﬁnes an SPS for CR . In a similar way it follows that E deﬁnes an SPS for CL . These results suggest that one might equally well work with SPS transformations as with SPE’s (which require more parameters). Indeed, if one is interested only in monic systems (with M = I), SPS transformations tell the whole story.

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Otherwise, SPS transformations tell us about the products M −1 D and M −1 K and it remains to specify M itself, and hence D and K. In contrast, SPE’s are deﬁned in terms of the explicit coeﬃcients M , D, and K. This point becomes more signiﬁcant when working with systems for which the three coeﬃcients are Hermitian (or real symmetric) in which case the corresponding symmetry is reﬂected explicitly in A and B of (3).

3. Standard pairs and triples The following notions of “standard pairs and triples” play an important unifying role in the discussion of linearizations (see [4] and earlier references given there). Our deﬁnitions diﬀer in some respects from earlier sources. They are consistent with [6], and are formulated so that they are better suited to inverse problems – they do not refer explicitly to the matrix coeﬃcients of the vibrating system. Deﬁnition 5. A pair of matrices U ∈ Cn×2n and T ∈ C2n×2n form a standard pair for a vibrating system if (a) the dimension of each of T does not exceed n, and 7 6 eigenspace U is nonsingular. (b) the 2n × 2n matrix UT Note that the technical condition (a) is vital if T is to relate to a vibrating system (see Theorem 1.7 of [4]). Deﬁnition 6. Three matrices U ∈ Cn×2n and T ∈ C2n×2n and V ∈ C2n×n form a standard triple if (U, T ) is a standard pair, U V = 0 and the n × n matrix U T V is nonsingular. Notice that, if (U, T, V ) is a standard triple, then 7 7 6 6 0 U . V = UT V UT Since U T V is nonsingular (by deﬁnition) we may deﬁne U T V = M −1 so that the equation 7 6 6 7 0 U V = (6) M −1 UT is satisﬁed. In other words, given only a standard pair, the third member of a triple can be obtained by ﬁrst assigning a nonsingular M and then solving equation (6) for V . Important examples of standard triples associated with vibrating systems involve the companion matrices of (2) and are: 7 7 6 6 0 In 0 U1 = In 0 , T1 = CR = , V1 = , (7) M −1 −M −1 K −M −1 D and U2 =

0 M

−1

6 , T2 = CL =

0 In

−KM −1 −DM −1

7

6 , V2 =

In 0

7 ,

(8)

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(see [4] for details). It is easy to see that, if (U, T, V ) is a standard triple then the three matrices obtained by a (generalized) similarity, U1 = U P −1 , T1 = P T P −1 , V1 = P V,

(9)

also form a standard triple. For example, the similarity from (U1 , T1 , V1 ) to (U2 , T2 , V2 ) in the examples of (7) and by the matrix A of 7 6 (8) is determined D M . equation (3), i.e., in this case P = A = M 0 Another important case is obtained by choosing P in (9) to be a matrix for which P T P −1 is in Jordan canonical form. The resulting (very special) standard triple is labelled a Jordan triple (see [4] and [6]). Now let (U, T, V ) be a standard triple for a vibrating system and consider the n × n matrices Γ0 , Γ1 , Γ2 , . . . deﬁned by Γj = U T j V,

j = 0, 1, 2, . . . .

(10)

They are known as the moments of the triple. Observe that, by Deﬁnition 6, Γ0 = 0 and Γ1 is nonsingular. (Also, if it is known that T is nonsingular then moments can be deﬁned for negative integers j.) It is an important fact that any transformation to a similar triple does not aﬀect the moments: Γj = U T j V = (U P −1 )(P T j P −1 )(P V ) = (U P −1 )(P T P −1 )j (P V ). Thus, for a vibrating system the moments are the same whether they are found from the triples of (7), (8), or from a Jordan triple. This invariance property suggests that the moments reﬂect intrinsic properties of the vibrating system. Indeed, it has been shown in Theorem 2 of [6] that the moments generate a unique corresponding vibrating system through the recursive equations M = Γ−1 1 , D = −M Γ2 M, K = −M (Γ3 )M + DΓ1 D.

(11)

Furthermore, for any standard pair, (U, T ), M U T 2 + DU T + KU = 0.

(12)

Theorem 4. Let (U, T, V ) be any standard triple and deﬁne moments by (10). Then equations (11) generate a unique corresponding vibrating system. Furthermore, any two standard triples for the same system are similar (in the sense of equations (9)). Proof. The ﬁrst statement is obtained by adapting the proof of Theorem 2 of [6] to standard triples, rather than Jordan triples. The second follows from Theorem 1.25 of [4], for example. Naturally, the moments generated by a standard triple may now be described without ambiguity as the moments of a vibrating system. Corollary 5. If (U, T, V ) is a standard triple for a vibrating system L(λ), then λI − T is a linearization for L(λ).

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281

Proof. Note that the standard triple (7) has CR as the main matrix and (Lemma 1) λI −CR is a linearization. Also, if (U, T, V ) is any standard triple, then the second statement of the theorem implies that CR and T are similar. Hence, λI − CR = P (λI − T )P −1 and it follows from Deﬁnition 2 that λI − T is a linearization. It is useful to note that the transforming matrix 7 in the last step of 6 P used U . To see this, ﬁrst use this proof can be written explicitly in the form P = UT −1 (12) to verify that CR P = P T , and hence CR = P T P . The following technical lemma concerning the moments will assist in the proof of a theorem showing how SPE (and SPS) transformations can be obtained from standard triples. 6 7 Γ 0 Γ1 Lemma 6. (a) = A−1 . Γ1 Γ2 7 7 6 6 −K 0 Γ1 Γ2 A−1 . = A−1 (b) 0 M Γ2 Γ3 Proof. (a) Using equations (11) we have 7−1 6 7 6 6 Γ0 D M 0 M −1 −1 = = A = M 0 M −1 −M −1 DM −1 Γ1 (b) Using equations (11) again, 6 7 6 Γ1 Γ 2 M −1 = Γ2 Γ3 −M −1 DM −1

Γ1 Γ2

−M −1 DM −1 M −1 (DM −1 D − K)M −1

Multiplying on left and right with A gives 6 7 6 76 Γ 1 Γ2 0 −KM −1 D A A= M Γ2 Γ3 I −DM −1

M 0

7

from which (b) follows.

6 =

−K 0

7 .

7

0 M

. 7 ,

4. Constructing SPE and SPS transformations The next theorem is the main result of this paper. First recall the standard triple (7) for a vibrating system. To study isospectral systems it is convenient to write associated standard triples in the form (X, CR , Y ) where X and Y are still to be determined. The second matrix, CR , is then common to all of these systems, so they must be isospectral. Furthermore, CR automatically satisﬁes the condition (a) of Deﬁnition 5. Now it will also be shown how SPE and SPS transformations can be constructed from such a set of standard triples. In the process, a constructive method for ﬁnding isospectral families of systems will appear.

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Theorem 7. Let a vibrating system L0 (λ) be given and matrices A0 and B0 be formed as in (5), and write C0 = A−1 0 B0 . Consider any standard triple of the form (X, C0 , Y ). 7 6 X are nonsingular and Then the 2n × 2n matrices A0 Y B0 Y XC0 and 7−1 6 −1 −1 X E := Y C0 Y A0 , F := (13) XC0 determine an SPE of the given system. Furthermore, if the moments of the transformed system L(λ) are written Γj = XC0j Y , then the transformed coeﬃcients are M = (Γ1 )−1 , D = −M Γ2 M, K = −M Γ3 M + DΓ1 D.

(14)

Conversely, every system L(λ) isospectral with L0 (λ) can be obtained from some choice of X and Y in a standard triple (X, C0 , Y ) Before going into the proof of this theorem, the importance of two properties of the standard triple (X, C0 , Y ) are emphasized. Namely, that XY = 0,

and

det(XC0 Y ) = 0.

(15)

Keep in mind that A0 and B0 (and hence C0 ) are prescribed by the coeﬃcients M0 , D0 , K0 of a given vibrating system. The design of isospectral systems depends only on these two conditions and, generally, they can be satisﬁed in inﬁnitely many ways. The ﬁrst can be interpreted as a biorthogonality condition imposed on the rows of X and columns of Y , and the second as a non-degeneracy condition which, generically, will be satisﬁed once the ﬁrst condition is resolved. This interpretation is particularly useful in the discussion of systems with Hermitian (or real symmetric) coeﬃcients. Conditions (15) are also natural generalizations of corresponding conditions used in Part I of this work, [6]. There, the analysis is conﬁned to the case of Jordan triples, a special case of the present formulation. In Part I the data consists of suitable sets of eigenvalue and eigenvector data. Here, the data consists of the coeﬃcients M0 , D0 , K0 of a given system and the spectral information remains implicit. Proof of Theorem 7. Consider the following product and use (15) to obtain 7 7 6 6 X 0 XC0 Y Y C0 Y = . XC0 Y XC02 Y XC0 But we also have det(XC0 Y ) = 0 and it follows that the product is nonsingular. So each factor on the left is nonsingular and E, F can be deﬁned as in (13). So we may deﬁne the strict equivalence λA − B = E(λA0 − B0 )F. It is to be shown that this is an SPE.

(16)

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Since A = EA0 F it follows that A is nonsingular and 6 7−1 −1 X , A = EA0 F = Y C0 Y XC0 7−1 6 7−1 6 XY XC0 Y Γ0 Γ1 = = . XC0 Y XC02 Y Γ1 Γ2 Now use Part (a) of Lemma 6 to obtain 6 D A= M

M 0

7 ,

and the structure of A0 is preserved. For the transformation of B0 , observe that B0 = E −1 BF −1 =

A0 Y

B0 Y

Multiply on the left and on the right by 6 7 X Y A−1 and 0 XC0 respectively, to obtain 6 Γ1 Γ2

Γ2 Γ3

7

6 =

Γ0 Γ1

Γ1 Γ2

7

6 B

C0 Y 6

B

Γ0 Γ1

7

X XC0

.

,

Γ1 Γ2

7 .

Using Part (a) of Lemma 6 it follows that 6 7 Γ1 Γ 2 B=A A. Γ2 Γ3 Finally, Part (b) of Lemma 6 yields B=

6

−K 0

0 M

7 .

Thus, the structure of B0 is also preserved and we have an SPE. The formulae (13) are just a repetition of (11). For the converse, if L and L0 are isospectral systems they share a common ˆ J, Yˆ ), say. But J is also a Jordan Jordan form J, and so L has a Jordan triple (X, ˆ −1 , T JT −1, T Yˆ ) obtained by form for C0 , say C0 = T JT −1, and the triple (XT similarity is also a standard triple for L. Since T JT −1 = C0 , the result is obtained ˆ −1 and Y = T Yˆ . if we deﬁne X = XT Notice also that an application of Corollary 3 gives: Corollary 8. With E and F deﬁned as in Theorem 7, E deﬁnes an SPS for CL , (0) (0) (CL = ECL E −1 ) and F deﬁnes an SPS for CR , (CR = F −1 CR F ).

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Example 1. Consider the vibrating system deﬁned by 7 7 6 6 6 1 0 1 1 1 , D0 = , K0 = M0 = 0 1 1 1 0

0 2

7 ,

with (truncated) eigenvalues −0.9567 ± 0.6412i,

and

− 0.0433 ± 1.2272i.

(Note that, although D is singular, the damping is pervasive.) Making the (almost arbitrary) choice 7 6 1 2 0 −3 , X= 0 1 1 2 6 7 X it is found that is nonsingular. Since C0 is a companion matrix, condition XC0 (a) of Deﬁnition 5 is satisﬁed and it6follows 7that (X, C0 ) form a standard pair. 1 0 , and compute the third member of Assign the mass matrix M = 0 1 the standard triple (cf. equation (6)): ⎤ ⎡ −5 0 1⎢ 1 3 ⎥ ⎥. Y = ⎢ ⎣ 1 −7 ⎦ 5 −1 2 Now moments Γ1 , Γ2 and Γ3 can be computed (as in equation (10)), and the new (exact) coeﬃcients come from (14): 7 7 6 7 6 6 −1.4 −4.0 −0.8 −5.4 1 0 . , K= , D= M= −0.2 −2 1.6 2.8 0 1 Calculations conﬁrm that the spectrum is unchanged. Alternatively, after ﬁnding Y as above, E and F could be computed from equations (13) and then the coeﬃcients of the system are obtained from A = EA0 F , B = EB0 F .

5. Algorithms The computational procedures for ﬁnding isospectral systems resulting from Theorem 7 are summarized in this section. Some simpliﬁcations arise if one is interested only in monic systems. The reader will easily make these adjustments when required. Algorithm 1. The SPE method. Data: Coeﬃcients M0 , D0 , K0 of a vibrating system and corresponding matrices 7 7 6 6 D0 M 0 −K0 0 , B0 := . A0 := 0 M0 M0 0 Step 1. Compute the matrix CR of equation (2).

Isospectral Vibrating Systems 6 Step 2. Choose an n × 2n matrix X for which

285

7

X XCR

is nonsingular.

Step 3. Choose a new mass matrix M and solve for Y : 6 7 6 7 X 0 Y = . XCR M −1 Step 4. Compute E :=

Y

CR Y

−1

A−1 0 ,

6 F :=

X XCR

7−1

Step 5. Compute A = EA0 F , and EB0 F and read oﬀ the new coeﬃcients D and K. End Algorithm 2. The moment method. Data: Coeﬃcients M0 , D0 , K0 of a vibrating system. Steps 1, 2, and 3: As steps 1, 2, and 3 of Algorithm 1. 2 3 Step 4. Compute the moments Γ1 = XCR Y , Γ2 = XCR Y, Γ3 = XCR Y.

Step 5. Compute the new coeﬃcients M = Γ−1 1 ,

D = −M Γ2 M,

K = −M Γ3 M + DΓ1 D.

End Of course, these summaries cannot be described as eﬃcient general purpose algorithms. At a superﬁcial level, Algorithm 1 may seem more direct. On the other hand, Algorithm 2 may gain in eﬃciency because it is formulated in terms of matrices of size n rather than 2n, as in the ﬁrst algorithm.

6. Systems with symmetries In this section the study of systems isospectral with a given system (M0 , D0 , K0 ) is continued, but now all of these matrices are Hermitian (in particular, they may be real and symmetric). Isospectral systems are to be generated which preserve the symmetry of (M0 , D0 , K0 ). Here, an important role is played by the nonsingular Hermitian matrix 7 6 D0 M 0 . (17) A0 = M0 0 Continuing the practice of Section 4, C0 will denote the right companion matrix associated with the given system. Matrix A0 is always indeﬁnite with inertia {n, n, 0}, and it has the important property that it symmetrises C0 in the sense that A0 C0 = C0∗ A0 = (A0 C0 )∗ . (The * denotes the complex-conjugate transposed matrix.) The matrix C0 is said to be self-adjoint in the indeﬁnite inner product deﬁned by A0 , see [5]. Indeed, it is easy to prove the more general statement:

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Lemma 9. The matrix A0 of (17) has the property that, for j = 0, 1, 2, . . ., A0 C0j = (A0 C0j )∗ = (C0j )∗ A0 . Let (X, C0 , Y ) be a standard triple generated as in the algorithms of Section 5, for example. It will now be shown that if X and Y satisfy a geometric constraint, then the system generated will be self-adjoint (in the sense that the coeﬃcients are Hermitian). But ﬁrst note the following lemma: Lemma 10. Let (X, T, Y ) be a standard triple for a vibrating system (M, D, K). Then M, D and K are Hermitian if and only if the moments Γ1 , Γ2 , and Γ3 are Hermitian. Proof. The proof is an easy veriﬁcation using the formulae of (11) (or see [6]). Proposition 11. Let (X, C0 , Y ) be a standard triple. Then the corresponding vi∗ brating system has Hermitian coeﬃcients if Y = A−1 0 X . ∗ ∗ Proof. If Y = A−1 0 X then X = Y A0 and

Γj = XC0j Y = Y ∗ (A0 C0j )Y. But, by Lemma 9, A0 C0j = (A0 C0j )∗ and it follows that Γj is Hermitian for each j. Then it follows from Lemma 10 that the corresponding system has Hermitian coeﬃcients. Now the condition X = Y ∗ A0 is reformulated in terms of X alone, and the geometric interpretation will be clariﬁed. Since X ∗ = A0 Y , the deﬁnition of Y in terms of X and C0 (from Section 3) gives 6 7−1 6 7 X 0 , X ∗ = A0 XC0 M −1 whence

6

X XC0

7

∗ A−1 0 X =

6

7

0 M −1

.

Thus, Hermitian coeﬃcients are generated provided the two following conditions are satisﬁed: ∗ XA−1 0 X ∗ X(C0 A−1 0 )X

= 0,

(18)

= M

−1

∗

.

(19)

These equations say that, ﬁrst, the subspace ImX must be isotropic with respect to the known indeﬁnite matrix A−1 0 and, at the same time, (if M is to be positive deﬁnite) this subspace must be positive with respect to the Hermitian matrix C0 A−1 0 . (Here, it is simply convenient to suppose that the ubiquitous condition M > 0 applies.) Naturally, equations (18) and (19) are the symmetrised form of the fundamental equations (15). They are also the analogues of very similar conditions formulated in [6] in terms of canonical forms for A0 and C0 .

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A procedure for generating Hermitian (or real and symmetric) isospectral systems can now be formulated as follows: Algorithm 3. Hermitian, or real symmetric systems. Step 1. From the given Hermitian (or real symmetric) system {M0 , D0 , K0 } with M0 > 0 formulate 7 6 0 I C0 = −M0−1 K0 −M0−1 D0 and 7 6 0 M0−1 A−1 . = 0 M0−1 −M0−1 D0 M0−1 Step 2. Compute an n-dimensional subspace S which is A−1 0 -isotropic and is also C0 A−1 0 -positive. Step 3. Form the columns of matrix X ∗ from basis vectors for S and compute ∗ Y = A−1 0 X . Step 4. Compute the moments Γ1 = XC0 Y,

Γ2 = XC02 Y,

Γ3 = XC03 Y.

Step 5. Compute the new coeﬃcients M = Γ−1 1 ,

D = −M Γ2 M,

K = −M Γ3 M + DΓ1 D.

End From the computational point of view, the most serious challenge here is to complete Step 2. To the authors’ knowledge algorithms for this step are not generally available. The ﬁrst essential is a general-purpose algorithm for ﬁnding a family of n-dimensional subspaces which are isotropic with respect to a given 2n × 2n matrix with inertia {n, n, 0}. Then it is necessary to scan these subspaces to ﬁnd one (or more) satisfying the positivity condition of Step 2. If the context is that of real and symmetric systems then C0 and A0 are real, of course, and the algorithms may be in real arithmetic. Analysis of the relevant algorithms may be assisted by using the spectral decompositions: 7−1 7∗ 7 6 7 6 6 6 X X X X J P C0 = , A−1 = . 0 XJ XJ XJ XJ Deﬁnitions of J and P can be found in [6] and the complete theory in [5], for example. Example 2. It is shown here that, by deﬁning S = span{e1 , e2 , . . . , en } at Step 2 of Algorithm 3, the original problem can be reproduced. First, it is easily seen −1 that this subspace is A−1 0 -isotropic and C0 A0 -positive. Indeed, it is found that −1 −1 ∗ X = I 0 and X(C0 A0 )X = M0 so that (from (19)), M = M0 .

288

U. Prells and P. Lancaster 7 0 . Step 3 gives Y = M0−1 At Step 4 it is found that 6

(1) Γ1 = M0−1 , Γ2 = −M0−1 D0 M0−1 , Γ3 = −M0−1 K0 + (M0−1 D0 )2 M0−1 and then that M = M0 , D = D0 , K = K0 . Example 3. Data: M0 =

6

1 0 0 1

7

6 ,

D0 =

1 0

0 2

7

6 ,

K0 =

4 3

3 6

7 .

The truncated eigenvalues are found to be (in truncated form), −0.6959 ± 1.1943i, and −0.8041 ± 2.6841i. Step 1 yields ⎤ ⎤ ⎡ ⎡ 0 0 1 0 0 0 1 0 ⎢ 0 ⎢ 0 0 0 0 0 1 ⎥ 1 ⎥ −1 ⎥ ⎥ ⎢ C0 = ⎢ ⎣ −4 −3 −1 0 ⎦ , A0 = ⎣ 1 0 −1 0 ⎦ . −3 −6 0 −2 0 1 0 −2 For Step 2 it is found that, for example, the image (or range) of the (truncated) matrix ⎤ ⎡ 0.2923 −1.3743 ⎢ −1.6097 −0.1742 ⎥ ⎥ X∗ = ⎢ ⎣ −0.4729 −0.1959 ⎦ −0.1742 0.4204 −1 is A−1 0 -isotropic and C0 A0 -positive. Then ⎡ −0.4729 ⎢ −0.1742 Y =⎢ ⎣ 0.7651 −1.2613

step 3 yields ⎤ −0.1959 0.4204 ⎥ ⎥. −1.1784 ⎦ −1.0151

The calculations of Steps 3 and 4 produce 7 7 6 6 46.9830 14.7497 11.9099 −6.6010 , , D= M= 14.7497 −31.3384 −6.6010 4.2471 7 6 47.5991 56.5223 . K= 56.5223 69.3272 Then it can be veriﬁed that, indeed, this system and the system M0 , D0 , K0 are isospectral.

7. Monic Hermitian systems with D ≥ 0 Hermitian systems arise in many practical problems and, frequently, the conditions M > 0 and D ≥ 0 hold. In this case, the system is easily reduced to monic form, i.e., with M = I. In this section it is shown that, under these conditions, a direct

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attack can be made on the problem (of generating isospectral systems) using the methods of the preceding section. Note ﬁrst of all that, in the monic case, 7 7 6 6 0 I 0 I −1 A0 = , C0 = . (20) I −D −K −D First, let us examine the possibility of generating an A−1 0 -isotropic subspace of the form ImX ∗ with X = I Z for some n × n matrix Z. It is easily seen that such a subspace is A−1 0 -isotropic if and only if 0 = Z + Z ∗ − ZDZ ∗ .

(21) ∗

Suppose that D has rank h and consider a factorization D = W W where W is an n × h matrix of full rank. Solution sets for (21) are easily formulated in the two extreme cases h = 0 and h = n: • When h = 0 then D = 0 and every skew-symmetric matrix Z is a solution of (21). • When h = n it is easily veriﬁed that every matrix of the form Z = (W −1 )∗ (I − U )W −1 ,

(22)

where U is a unitary matrix, is a solution of equation (21). The next proposition gives a generalization of these families of solutions to the general case 0 ≤ h ≤ n. First, for any n × h matrix W of rank h deﬁne a unitary completion of W to * for which W ∗ W * = 0 and W *∗ W * = In−h . Then it can be an n × (n − h) matrix W be shown that the matrix * E := W W is nonsingular. * be a unitary Proposition 12. Given the full-rank factorization D = W W ∗ , let W completion of W , and construct the nonsingular matrix E as above. Then for any unitary matrix U of size h, any skew-Hermitian matrix S of size n − h, and an arbitrary matrix N of size (n − h) × h, the matrix 7 6 UN∗ Ih − U Z = (E −1 )∗ E −1 (23) 1 ∗ −N N N − S 2 is a solution of (21). Proof. The proof is by veriﬁcation. Substitute for Z from (23) in (21) and simplify. It can be shown that the number of free (complex) parameters in the representation (23) is 12 n(n − 1), and that this is independent of h. The parameters

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will be further constrained on applying the second condition, namely, that the subspace S =Im(X ∗ ) is C0 A−1 0 positive. Note that, in the monic case, 7 6 I −D −1 , C0 A0 = −D D2 − K so, with X =

I

Z , the positivity condition becomes I − ZD − DZ ∗ + Z(D2 − K)Z ∗ > 0.

7 2 i and −i 2 K0 = diag(1, −1). The eigenvalues (rounded) are λ1 = −3.0523, λ2 = 0.4476 and λ3 = λ4 = −0.6977 + 0.4952i. Since D0 is positive deﬁnite there is a solution of (21) of the form Z = (W −1 )∗ (I2 − U )W −1 where U is unitary. From the singular value decomposition, D = W0 Σ2 W0∗ , it is found that D = W W ∗ with 7 6 (2) i 3/2 i/ √ W = W0 Σ = 3/2 −1/ 2 6

Example 4. Consider the monic Hermitian system given by D0 =

and the choice

6 U=

cos(2π/3) − sin(2π/3) sin(2π/3) cos(2π/3)

yields Z = (W −1 )∗ (I2 − U )W −1 =

6

1 i

7

0 1

7 .

It is now a matter of computation to verify that, for X = [I2 Z], we have XP X ∗ = I2 − ZD0 − D0 X ∗ + Z(D02 − K0 )Z ∗ = I2 and hence ImX ∗ is C0 A−1 0 -positive. After ﬁrst computing Γ2 and Γ3 in Step 4 of Algorithm 3, new monic system matrices are obtained: 7 6 3 i . D = diag(4, 0), K = −i 0 This monic system has, indeed, the same eigenvalues as the initial system. In this section we have focused on the case of standard triples (X, C0 , Y ) for which X = [In Z], i.e., for which the ﬁrst n-by-n block of X is nonsingular. In the next section the case where the second n-by-n block is nonsingular will lead to another family of monic isospectral systems.

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8. Isospectral families of Hermitian, or real symmetric systems The notations of Section 7 are maintained here. Once again, as the context is Hermitian systems with positive deﬁnite leading coeﬃcient, there is no signiﬁcant loss in assuming that the system is already reduced to monic form. In generating a parametrized set of isospectral systems, keep in mind that, from (18) and (19), such a system corresponds to an n-dimensional subspace of C2n , namely Im X ∗ . Decompose X = X1 X2 , where X1 , X2 ∈ Cn×n . Using C0 and A−1 for the monic system, the isotropy condition (18) takes the 0 form X1 X2∗ + X2 X1∗ = X2∗ D0 X2 . (24) Now a general complex matrix C can be written in the form C = A + iB where 1 A = 12 (C + C ∗ ) and B = 2i (C − C ∗ ) are Hermitian. Using this, it follows from the last equation that 1 X1 X2∗ = (X2∗ D0 X2 ) + S (25) 2 where S is an arbitrary skew-Hermitian matrix. In applying this technique, it is important to recognize that it is the ndimensional subspace Im(X ∗ ) which entirely determines the new system (isospectral with the (A0 , C0 ) system). Furthermore, Im (X ∗ ) =Im (X ∗ A) for any nonsingular n × n matrix A. Using this idea, if we conﬁne attention to subspaces for which X2 is nonsingular then we can, without further loss of generality, set X2 = I. However, it is convenient to introduce another real parameter = 0 and set X2 = I. In this case, using (25) (and absorbing a real parameter into S), we may take X1 = 12 D0 + S and 7 6 1 D0 + S ∗ 2 . (26) X = I Now reconsider Algorithm 3, and notice ﬁrst that in Step 1 we can set M0 = I. For Step 2 a parametrized set of subspaces is to be deﬁned with the necessary isotropic and positivity properties. Let S be an arbitrary skew-Hermitian matrix (n ≥ 2), be a real parameter, and deﬁne X by (26). Then 7 76 1 6 0 1 I ∗ 2 D0 + S D − S I =0 XA−1 X = 0 0 2 I I −D0 and the isotropic property is veriﬁed independently of and S. Then, with a little computation, 76 6 1 I −D0 −1 ∗ X(C0 A0 )X = 2 D0 − S I −D0 −K0 + D02 1 1 = ( D0 + S)( D0 + S)∗ − 2 K0 , 2 2

1 2 D0

I

+S

7

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and it is apparent from this last expression that, provided only that and S are cho∗ sen so that 12 D0 + S is nonsingular, X(C0 A−1 0 )X > 0 for all suﬃciently small . Then it is found in Step 3 (with M = I) that 7 6 7 76 1 6 I D0 + S 0 I 2 = . Y = − 21 D0 + S I I −D0 Now all the necessary information is generated to complete Steps 4 and 5, and hence a family of isospectral systems determined by ∈ R and skew-Hermitian matrices S. It is clear that, if the given system is real and symmetric, then real and symmetric isospectral systems are determined by choosing a real skew-symmetric matrix S. The following examples are of this kind. Example 5. (Data are ⎡ 1 ⎢ 0 D0 = ⎢ ⎣ 0 0

taken from Example 7 of [8].) Let M0 = I4 , ⎤ ⎡ 0 0 0 5 −2 0 0 ⎢ −2 4 −2 0 0.5 0 0 ⎥ ⎥ , K0 = ⎢ ⎣ 0 −2 5 −3 0 1 0 ⎦ 0 0 0.5 0 0 −3 3

⎤ ⎥ ⎥. ⎦

The spectrum of this system is −0.3951 ± 2.8146i, −0.4247 ± 2.3936i, −0.3302 ± 1.5813i, −0.3500 ± 0.4081i. Assign the arbitrary skew-symmetric matrix ⎡ 0 1 1 ⎢ −1 0 1 S=⎢ ⎣ −1 −1 0 −1 −1 −1

⎤ 1 1 ⎥ ⎥, 1 ⎦ 0

and it is found experimentally that the positivity condition is satisﬁed for 0 ≤ ≤ 18 . For the purpose of comparison, this technique has been applied to generate isospectral systems at = 1/16 and at = 1/8. The results are then scaled to produce monic systems. First compare the damping matrices D(): ⎤ ⎡ .8210 −.0985 .2144 .0484 ⎢ 1.0410 −.1285 −.0760 ⎥ ⎥, D(1/16) = ⎢ ⎣ .5334 .2769 ⎦ .6046 ⎤ ⎡ 1.440 −.6155 .8599 .0396 ⎢ 1.5039 −.4797 −.4254 ⎥ ⎥, D(1/8) = ⎢ ⎣ .3818 1.1442 ⎦ −.3297 Note that the positive deﬁnite property of D0 is maintained at = 1/16 and lost at = 1/8.

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The stiﬀness matrices are: ⎤ ⎡ 3.7321 −1.7603 −.5248 3.6499 ⎢ 5.0111 −1.2465 −.2472 ⎥ ⎥, K(1/16) = ⎢ ⎣ 3.3212 −.5021 ⎦ 5.0464 ⎤ ⎡ 5.8149 −2.9548 −.8040 6.0288 ⎢ 5.3074 −.1764 −2.2789 ⎥ ⎥. K(1/8) = ⎢ ⎣ 2.3600 −.2299 ⎦ 7.4087 The technique developed and illustrated here can include Example 3 as a limiting case (i.e., in the limit as → 0) and, consequently, determines smooth isospectral continuations of the given system. However, some care must be taken if n is odd, for then the perturbing skew-symmetric matrix S is necessarily singular and (see (26)) X does not have full rank at = 0. Now Example 1 is revisited to construct real and symmetric isospectral systems. Example 6. As in Example 1 take 7 6 6 1 0 1 M0 = , D0 = 0 1 1

1 1

7

6 ,

K0 =

1 0

0 2

7 ,

with (truncated) eigenvalues −0.9567 ± 0.6412i, and − 0.0433 ± 1.2272i. 7 6 0 1 and, in eﬀect, there is only one free parameter, Since n = 2, we take S = −1 0 namely . It is easily seen that the positivity condition is satisﬁed if −0.6 ≤ ≤ 0.5. The damping and stiﬀness matrices are tabulated below for = −0.1 and = 0.1. Observe that the singular damping matrix D0 is perturbed to a positive deﬁnite matrix by a negative shift of the parameter, but not by the positive shift. 7 7 6 6 1.6477 .0504 .8988 −.9235 K(−0.1) = D(−0.1) = .0504 1.2153 −.9235 1.1012 7 7 6 6 2.4715 −.0503 1.1013 −1.1276 K(0.1) = D(0.1) = −.0503 .8103 −1.1276 .8987

9. Systems in reduced form A wide class of systems of practical interest are intimately related to a diagonal system, or to a system which is “close to diagonal”. They will be investigated in this section. Observe ﬁrst that, for each companion matrix appearing in a standard triple there is a class of isospectral systems. If a relation ↔ is deﬁned on n×n systems by

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saying L1 (λ) ↔ L2 (λ) when they are isospectral, then ↔ determines an equivalence relation, and it is natural to ask for canonical systems in each equivalence class. The fact that this is an equivalence relation is most easily veriﬁed by noting that all companion matrices associated with one of these classes have the same Jordan form (and this is the true canonical representation). Deﬁnition 7 A system L(λ) is said to be reduced or in reduced form if L(λ) = (Iλ − J1 )(Iλ − J2 ) = λ2 − λ(J1 + J2 ) + J1 J2

(27)

where J1 and J2 are in Jordan canonical form. (Deﬁnition 7 includes a similar notion of “canonical systems” introduced in [8] under more restrictive hypotheses. There is a thorough treatment of the conditions under which L(λ) in (27) is self-adjoint in [7].) Not all equivalence classes contain such a reduced form. For example, a system with n = 3 and three distinct double eigenvalues each with an associated block of size two has no reduced form in its equivalence class (such an example is easily constructed). For this reason, the word “reduced” is used here, rather than “canonical”. If J1 and J2 are consistent with a real, or a Hermitian system then, of course, there is interest in isospectral families and reduced forms which preserve these properties. In many cases, when there is a reduced form (27) the class has an associated Jordan matrix 6 7 J1 0 J= , (28) 0 J2 but this is not always the case. For example, when n = 1 the system L(λ) = λ2 is already in reduced form (with J1 = J2 = 0), but 7 6 7 6 J1 0 0 1 = . J= 0 J2 0 0 To put this another way, it is not the case that a structure like (28) for J implies that there is a reduced form like (27) in the associated equivalence class. For example, if n = 1 and J1 = J2 = λ0 , then (λI − J1 )(λI − J2 ) = (λ − λ0 )2 , 7 6 7 6 λ0 0 λ0 1 and not . but the Jordan form of this system is 0 λ0 0 λ0 Suﬃcient conditions for a Jordan form with the structure of (28) to have a reduced form in its equivalence class of systems are contained in: Proposition 13. An isospectral equivalence class contains a reduced system if the class has a corresponding Jordan matrix J of the form (28) where J1 and J2 are n × n Jordan forms and either (a) J1 and J2 have no common eigenvalues, or (b) J1 and J2 commute and det(J1 − J2 ) = 0.

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Proof. Case (a) follows because the ﬁrst factor in (27) is nonsingular at each eigenvalue of the second, and vice versa. This ensures that chains of right (left) eigenvectors for J2 (resp. J1 ) are inherited by L(λ). For case (b) observe that the determinantal condition implies that the matrix 7 6 I I X= J1 J2 is nonsingular. Then (with L as in (27)) 7 6 J2 J1 CR X = J12 + (J2 J1 − J1 J2 ) J22

6 and

XJ =

J1 J12

J2 J22

7 .

Since J1 and J2 commute it follows that CR X = XJ (so that CR = XJX −1). Thus, J is characteristic of this equivalence class. Notice that in case (b) J1 and J2 may have common eigenvalues. Deﬁnition 8. A system is said to be regular if the following properties hold: 1. The non-real eigenvalues (if any) arise in conjugate pairs with the same multiplicities. (Denote the corresponding Jordan matrices by Jc and Jc .) 2. The real eigenvalues, if any, (say 2r in number) can be divided into two subsets of size r with corresponding r × r Jordan matrices Js and Jt . 3. det(Js − Jt ) = 0. 4. Js and Jt commute. Property 1 is enjoyed by two important classes of systems, namely, those with real coeﬃcient matrices and those with Hermitian coeﬃcients (and this is the main reason for introducing this deﬁnition). Of course, Properties 2, 3, and 4 are void if there are no real eigenvalues. Property 2 excludes some systems; for example, those with exactly one real eigenvalue which is not semisimple and has algebraic multiplicity two. Notice that there are no constraints on the multiplicities of the non-real eigenvalues (other than the fact that their algebraic multiplicity cannot exceed n − r). Given Properties 1-4, it is possible to construct an associated 2n × 2n Jordan matrix as in (28) with 7 7 6 6 Jc 0 Jc 0 J1 = , J2 = . 0 Js 0 Jt Now form the reduced system L(λ) of (27) and it is easily seen that either or both of the conditions (a) and (b) of Theorem 10 hold. Then it is easily veriﬁed that: Proposition 14. The isospectral equivalence class of a regular system contains a real reduced system: 7 7 6 6 0 0 Jc + Jc Jc Jc D = −(J1 + J2 ) = − , K = J1 J2 = . 0 Js + Jt 0 Js Jt If in addition, all eigenvalues are semisimple, then there is a real diagonal reduced system.

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One might ask whether a reduced system will be included in the isospectral families of Sections 7 and 8. As the parametrized systems are all Hermitian it is, of course, necessary that the reduced form be symmetric so, as in Proposition 12, the answer can be “yes” if all real eigenvalues are semisimple or if there are no real eigenvalues, as in: Example 7. The systems generated in Example 6 include a reduced system – up to a congruence transformation. If we choose (truncated) = −0.1728 then we ﬁnd ⎤ ⎡ −0.0864 0.9136 7 6 ⎢ −1.0864 −0.0864 ⎥ D +S ⎥ X∗ = 2 0 =⎢ ⎦ ⎣ −0.1728 0 I2 0 −0.1728 and 6 V1 =

I2 S − 2 D0

7

from which we may compute 7 6 0.8123 0.0149 Γ1 = U1 T V1 = , 0.0149 1.1280

⎤ −0.1728 0 ⎢ 0 −0.1728 ⎥ ⎥ =⎢ ⎣ 0.0864 1.0864 ⎦ −0.9136 0.0864 ⎡

6

−0.6545 0.8422 Γ2 = U1 T V1 = 0.8422 −1.3157 7 6 0.0075 −1.7931 , Γ3 = U1 T 3 V1 = −1.7931 0.8645 2

7 ,

and hence we have 7 7 6 6 1.0265 −0.9519 1.2314 −0.0163 , , D= M= −0.9519 1.0591 −0.0163 0.8867 7 6 1.7653 0.0699 . K= 0.0699 1.2395 It can now be veriﬁed that KM −1 D = DM −1 K and hence the three matrices M, D, K can be diagonalized by a congruence transformation [1]. Deﬁne a matrix of eigenvectors of the undamped system: 7 6 −0.5655 −0.7018 . X0 := 0.8165 −0.6792 Because of the commutativity assumption above, this matrix will also diagonalize D. Indeed, X0T M X0 = I2 , X0T DX0 = diag(1.9134, 0.0866) = −Λ − Λ∗ and X0T KX0 = diag(1.3264, 1.5079) = ΛΛ∗ where Λ := diag(−0.9567 + 0.6412i, −0.0433 + 1.2272i). Instead of this indirect derivation of the reduced system it is also possible to obtain the same result by using Ud∗ := U1∗ (X0T )−1 which is A−1 0 -isotropic and U = I . leaves M0 = I2 invariant, i.e., Ud∗ C0 A−1 d 2 0

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10. Conclusions The spectral theory of vibrating systems has been reviewed and and re-examined from the point of view of inverse spectral problems: i.e., the construction of systems with spectral characteristics deﬁned implicitly via a given system. In Part 1 of this work (see [8]) the spectral characteristics were prescribed explicitly in the form of complete sets of eigenvalue and eigenvector data. If no symmetry properties are required of the systems generated, the problem has a relatively easy solution summarized in the ﬁve-step procedures summarized in Section 5. If symmetries are imposed on the coeﬃcients of all systems considered, then the situation is more involved. However, an explicit construction is given in Section 6 for the determination of isospectral families of symmetric systems (whether complex Hermitian or real symmetric). If eﬃcient general-purpose algorithms are to be created, then an eﬃcient procedure is required for the determination of ndimensional subspaces of a 2n-dimensional space which are neutral with respect to a known real symmetric indeﬁnite matrix (see Step 2 of Algorithm 3 in Section 6). The construction of symmetric systems with positivity conditions imposed on the coeﬃcient matrices remains an essentially open problem, although methods developed in [8] should cast some light on this problem. Special cases of problems with positivity constraints have been discussed in Sections 7 and 8. Finally, in Section 9 it has been clariﬁed under what circumstances an isospectral family may contain a diagonal (or “close to diagonal”) system. Acknowledgements The ﬁrst named author gratefully acknowledges Prof. Seamus D. Garvey for many stimulating discussions and the support of the EPSRC in this work through the research grant GR/S31679/01 on “Model Compaction and Model Reduction Methods for Large-Scale Dynamic Systems in Engineering”. The second named author gratefully acknowledges support from Prof. N. J. Higham of the University of Manchester and from the Natural Sciences and Engineering Research Council of Canada.

References [1] T.K. Caughey and M.E. O’Kelly, Classical normal modes in damped linear system, Journal of Applied Mechanics, Transaction of the ASME, 32, 1965, 583–588. [2] M.T. Chu, Y.-C. Kuo and W.-W. Lin, On inverse quadratic eigenvalue problems with partially prescribed eigenstructure SIAM J. Matrix Anal. Appl., 25, 2004, 995–1020. [3] S.G. Garvey, M.I. Friswell and U. Prells, Co-ordinate Transformations for Second Order Systems, Part I: General Transformations, J. Sound & Vibration, 258(5), 2002, 885–909. [4] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

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[5] I. Gohberg, P. Lancaster and L. Rodman, Matrices and Indeﬁnite Scalar Products, Birkh¨ auser, Basel, 1983. [6] P. Lancaster, Isospectral vibrating systems. Part 1: The spectral method, Linear Algebra Appl., 409, 2005, 51–69. [7] P. Lancaster and J. Maroulas, Inverse eigenvalue problems for damped vibrating systems, J. Math. Anal. Appl. 123, 1987, 238–261. [8] P. Lancaster and U. Prells, Inverse problems for vibrating systems, J. Sound & Vibration, 283, 2005, 891–914. [9] P. Lancaster and M. Tismenetsky, The Theory of Matrices Second Edition, Academic Press, Orlando, 1985. Uwe Prells School of Mechanical, Materials, Manufacturing Engineering & Management University of Nottingham Nottingham NG7 2RD United Kingdom e-mail: [email protected] Peter Lancaster Department of Mathematics and Statistics University of Calgary Calgary AB T2N 1N4 Canada e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 299–335 c 2005 Birkh¨ auser Verlag Basel/Switzerland

A Functional Description for the Commutative W J ∗-algebras of the Dκ+-class Vladimir Strauss Dedicated to Professor Heinz Langer

Abstract. We consider the action in Krein spaces of weakly closed J-symmetric operator algebras with identity possessing an invariant maximal nonnegative subspace, presented as a direct sum of a ﬁnite-dimensional neutral subspace and a uniformly positive subspace. A relation between these algebras (that in addition are assumed to be commutative) and function spaces of the 2 type L∞ σ ∩ Lν is established. Mathematics Subject Classiﬁcation (2000). Primary 46C20, 47B50; Secondary 47B40, 47A60. Keywords. Indeﬁnite metric, operator algebras, model representation, functional calculus.

Introduction This work has a direct connection with the paper [40]. It is assumed the reader is familiar with the elements of Krein space geometry and operator theory (see [9], [2], [16], [26]). In this paper the terminology introduced in [3] will be used. One of the purposes of this paper is the study of relations between commutative W J ∗ -algebras (i.e., weakly closed J-symmetric operator algebras with identity) in separable Krein spaces and certain function spaces. It is assumed that a W J ∗ -algebra has a maximal non-negative invariant subspace, presented as a direct sum of a ﬁnite-dimensional neutral subspace and a uniformly positive subspace. As is known ([6], [39]) this algebra generates a spectral function Eλ with a peculiar spectral set Λ that provides a resolution of spectral type for the operators in the algebra. In particular to every operator A one can associate a scalar function This work was completed with the support of project CONICIT (Venezuela) No 97000668.

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fA (λ) (the portrait of A) such that

AE(∆) =

fA (λ)dEλ , ∆

where ∆ runs over the set of all closed intervals of the real line disjoint from Λ. The main results (Theorems 5.1, 5.4 and Corollary 5.5) show that the set of all portraits corresponding to operators in the algebra contains, as a principal part, 2 ∞ a space L∞ σ ∩ Lν . Here the space Lσ is generated by a bounded measure while 2 the space Lν is generated by an unbounded one. Points at which the unbounded measure has an inﬁnite rate of growth belong to the set of peculiarities Λ mentioned above but, generally speaking, do not exhaust the set. Section 1 contains mainly deﬁnitions and well-known results used in the course of the paper. In particular, in Subsection 1.2 some function spaces are described and Subsection 1.4 deals with a model representation of a resolution of the identity that is simultaneously J-orthogonal and similar to an orthogonal resolution of the identity. In Section 2 a notion of unbounded elements conformed with a resolution of the identity is introduced and studied. Roughly speaking a resolution deﬁnes a correspondence between a Banach space and a (vector-valued) function space. From this point of view unbound elements correspond to measurable functions that do not belong to the function space. Section 3 represents a short description of results from [40] that are used in Sections 4 and 5. In Section 4 a functional description is given for an algebra with a spectral resolution having only one peculiarity, while Section 5 deals with the general case. In Subsection 4.3 a functional calculus for J-self-adjoint operators is constructed. Historical and bibliographical remarks are situated in Section 6.

1. Preliminaries 1.1. Krein spaces ˙ − Let H be a Krein space with an indeﬁnite sesquilinear form [·, ·], let H = H+ [+]H be its canonical decomposition, let P+ and P− be canonical projections: H+ = P+ H, H− = P− H, let J = P+ − P− be a canonical symmetry, and let (·, ·) = [J·, ·] be a canonical scalar product. Note that any one of these canonical objects uniquely determines the others. Everywhere below we ﬁx on H a unique form [x, y] = (Jx, y). At the same time let us note that in the question we consider, a concrete choice of Hilbert scalar product is not really essential. One needs only to ﬁx the topology (deﬁned by the above-mentioned scalar product) and the structure of J. Below non-negative (especially maximal non-negative ) subspaces will play an important role (see [22] for references). The set of all maximal non-negative subspaces of the Krein space H is denoted M+ (H). A subspace L is called pseudo-regular ([15]) if it can be presented in the form ˙ 1, L = Lˆ+L

(1.1)

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where Lˆ is a regular subspace and L1 is a neutral subspace (i.e., L1 is an isotropic part of L). Proposition 1.1. ([4], [5]) Let: • L+ be a pseudo-regular subspace belonging to ∈ M+ (H); • L0 be the isotropic subspace of L+ ; • (·, ·) be a scalar product on L0 , such that the norm (x, x) is equivalent to the original one; [⊥] • L− =L+ ; and let ˆ + +L ˆ − +L ˙ 0 , L− = L ˙ 0, L+ = L (1.2) ˆ ˆ where L+ and L− are uniformly deﬁnite subspaces. Then one can deﬁne on H a canonical scalar product (·, ·) such that: ⎫ a) on L0 : (·, ·) ≡ (·, ·) ⎪ ⎪ ⎬ ˆ + , L0 ⊥ L ˆ− b) L0 ⊥ L (1.3) ˆ+ c) on L : (·, ·) = [·, ·] ⎪ ⎪ ⎭ ˆ− : (·, ·) = −[·, ·] d) on L Deﬁnition 1.2. If a canonical scalar product of a Krein space H has the properties (1.3), it is said to be compatible with Decomposition (1.2) and the choice of the scalar product (·, ·) on L1 . Deﬁne a special case of pseudo-regular subspaces: a non-negative (non-positive) subspace L is called a subspace of the class h+ ( h− ) if it is pseudo-regular and dim L1 < ∞ for L1 as in (1.1). In Pontryagin spaces every subspace is pseudoregular and every semi-deﬁnite subspace belongs to class h+ or h− . Here the term “operator” means “bounded linear operator”. By the symbol B # we denote the operator J-adjoint (J-a.) to an operator B. For an operator A the symbol σ(A) denotes its spectrum treated in the same way as in [13] or [3]. If an operator family Y is such that the condition A ∈ Y implies A# ∈ Y, then this family is said to be J-symmetric. An operator algebra A is said to be W J ∗ -algebra if it is closed in the weak operator topology, J-symmetric and contains the identity I. The symbol Alg Y means the minimal W J ∗ -algebra which contains Y. One of the most important directions in the development of the operator theory is connected to the existence of invariant maximal semi-deﬁnite subspaces for certain operator sets and the study of the properties of the operators in such sets. A subspace L is said to be A-invariant (Y-invariant) if it is invariant with respect to the operator A (operator family Y). Deﬁnition 1.3. A J-symmetric operator family Y belongs to the class Dκ+ if there is a subspace L+ in H, such that • L+ is Y-invariant, • L+ ∈ M+ (H) ∩ h+ , [⊥] • dim(L+ ∩ L+ ) = κ.

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Remark 1.4. If a J-symmetric family Y ∈ Dκ+ and L+ is a Y-invariant sub[⊥] space corresponding to Deﬁnition 1.3, then the pseudo-regular subspace L+ is Y-invariant too. 1.2. Some function spaces Assume that σ(t) is a non-decreasing function deﬁned on the segment [−1; 1], continuous in the points −1; 0; 1, continuous (at least) from the left in all other points of the segment and having an inﬁnite number of growth points, where zero is one of these points. The mentioned function generates on [−1; 1] the LebesgueStieltjes measure µσ and spaces (L2σ , L∞ σ , etc.) of complex-valued functions. At the same time we shall consider also some spaces of vector-valued functions so from time to time we shall note after a symbol of a space a symbol of a range for the functions forming this space, for instance, L2σ (C). Next, let us consider a slightly diﬀerent construction. Let G(t) be a µσ -measurable function deﬁned a.e. on [−1; 1] and such that • a.e. G(t) ≥ 1, −τ 1 • −1 G(t)dσ(t) < ∞, τ G(t)dσ(t) < ∞ for every τ ∈ (0; 1], 1 • −1 G(t)dσ(t) = ∞. Set

⎫ if τ ∈ [−1; 0); ⎪ ⎪ ⎬ 1 − τ G(t)dσ(t), if τ ∈ (0; 1]. ⎪ τ ⎪ ⎭ −1 (1/G(t))dσ(t) for τ ∈ [−1; 1].

+ τ ν(τ )

=

η(τ )

=

−1

G(t)dσ(t),

(1.4)

The function ν(t) is non-decreasing in both segments [−1; 0) and (0; 1] but it is unbounded in neighborhoods of zero. Deﬁne for it a corresponding function space. Let f (t) and g(t) be arbitrary functions continuous in [−1; 1] and vanishing in some neighborhoods (diﬀerent in the general case for f (t) and g(t)) of zero. 1 Then the integral −1 f (t)g(t)dν(t) is well deﬁned and generates a structure of pre-Hilbert space on the set of all such functions. The completion of the space will be denote L2ν (or L2ν (C)). In a similar way one can deﬁne the space L1ν . At the same time the function η(t) is non-decreasing on the whole interval [−1; 1], hence η(t) deﬁnes on this interval the ordinary Lebesgue-Stieltjes measure µη that is absolutely continuous with respect to µσ . Thus, the space L2η and others are deﬁned as usual. 2 1 Note that due to (1.4) the spaces L∞ σ and Lν , as well as the spaces Lσ and 2 Lη , form compatible pairs or Banach pairs (for details see [8] or [24]). Thus, the 2 spaces L1σ + L2η and L∞ σ ∩ Lν are well deﬁned. In particular, the standard norm on 1 2 Lσ + Lη is given by the formula f =

inf

f1 +f2 =f

{f1 L1σ + f2 L2η }.

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2 1 2 The space L∞ σ ∩ Lν can be considered as adjoint to the space Lσ + Lη if the duality 1 between these space is given by the formula f (t), g(t) = −1 f (t)g(t)dσ(t), where 2 f (t) ∈ L1σ + L2η and g(t) ∈ L∞ σ ∩ Lν . Let us pass to some notations relating to direct integrals of Hilbert spaces and corresponding model descriptions of self-adjoint operators (see [33], §41; [10], Chapter 7; [11], Chapter 4.4; [34], Chapter VII). Let E be some separable Hilbert space (E can be ﬁnite-dimensional), let σ(t) be as above. Consider a mapping t → Et , t ∈ [−1; 1], where Et ⊂ E, dim(Et ) is a µσ -measurable (but not necessarily ﬁnite a.e.) function, and if dim(Et1 ) = dim(Et2 ), then Et1 = Et2 . Denote by Mσ (E) the space of the vector-valued functions f (t) : t → Et µσ -measurable in the weak sense, deﬁned a.e. and ﬁnite a.e. on the segment [−1; 1]. Next, the symbol L2σ (E) means 1 here a Hilbert space of functions f (t) ∈ Mσ (E), such that −1 f (t)2E dσ(t) < ∞ Introduce some notation related with multiplication operators by scalar function. Everywhere below we assume a scalar function ϕ(t) to be deﬁned a.e. on [−1; 1], µσ -measurable and a.e. bounded. For f (t) ∈ Mσ (E) set

(Φf )(t) = ϕ(t)f (t).

(1.5)

It is clear that (Φf )(t) ∈ Mσ (E), so equality (1.5) deﬁnes on Mσ (E) the continuous operator Φ (= the multiplication operator by the function ϕ(t)). If ϕ(t) satisﬁes some additional conditions one can consider the operator Φ as acting simultaneously on diﬀerent spaces. If, for instance, ϕ(t) is continuous then the operator Φ is well deﬁned on every space Mσ (E) independently of σ (t) and E. If ϕ(t) ∈ L∞ σ (C) then L2σ (E) can also be taken as a domain of Φ. So, if it is necessary, we’ll mention simultaneously the operator Φ and its domain using the notation {Φ, D(Φ)}, say, {Φ, L2σ (E)}. 1.3. Spectral functions with peculiarities Spectral resolution for diﬀerent operator classes is one of the important problems in the operator theory. Let Λ={λk }n1 be a ﬁnite set of real numbers and let RΛ be the family {X} of all Borel subsets of R such that ∂X ∩ Λ = ∅, where ∂X is the boundary of X in R. Let E : X → E(X) be a countably additive (with respect to weak topology) function, that maps RΛ to a commutative algebra of projections in a Hilbert space H, E(R) = I. E(X) is called a spectral function (on R) with the peculiar spectral set Λ, the mention of Λ can be omitted. The symbol Supp(E) means the minimal closed subset S ⊂ R, such that E(X) = 0 for every X: X ⊂ R\S and X ∈ RΛ . Besides the symbol E we shall use also the symbol Eλ , λ ∈ R as notation for a spectral function, where Eλ = E((−∞, λ)). Note that the notion of peculiar set has no any direct connection with the behavior of the spectral function and it means only that some points on R are distinguished. See below Deﬁnition 1.8 for some explanations. A spectral function E that acts in a Krein space, is said to be J-orthogonal (J-orth.sp.f.) if E(X) is a J-ortho-projection for every X ∈ RΛ .

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Let us recall the deﬁnition of a scalar spectral operator with real spectrum ([14]). An operator A acting in a Hilbert space is said to be a scalar spectral operator if there exists a spectral function E with empty peculiar spectral set Λ, ¯ and AE(X) = such that for every X ∈ RΛ : E(X)A = AE(X), σ(A|E(X)H ) ⊂ X ξE(dξ) in the weak sense. X Now let E be a spectral function with peculiar spectral set Λ. A scalar function f (ξ) is said to be deﬁned almost everywhere (with respect to E), to have ﬁnite value almost everywhere, etc., if the corresponding property holds almost everywhere in the weak sense on an arbitrary set X ∈ RΛ , X ∩ Λ = ∅. We shall assume that the function f (ξ) is not deﬁned at Λ. The following theorem was announced in [39] and proved in [6]. Theorem 1.5. Let Y ∈ Dκ+ be a commutative family of J-s.a. operators with real spectra. Then there exists a J-orth.sp.f. E with a ﬁnite peculiar spectral set Λ (Λ may be the empty set), such that the following conditions hold ⎫ a) Eλ ∈ Alg Y for all λ ∈ R\Λ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b) there is a non-negative subspace L+ , corresponding to ⎪ ⎪ ⎪ ⎪ Deﬁnition 1.3, for which the descomposition E(∆)H = ⎪ ⎪ ⎪ ⎪ ˙ ⎪ E(∆)L+ [+]E(∆)L holds, ∆ being any closed segment − ⎪ ⎪ ⎪ ⎪ ∆ ⊂ R satisfying ∆ ∈ RΛ and ∆ ∩ Λ=∅; ⎪ ⎪ ⎪ ⎪ ⎪ c) for every operator A ∈ Y, there exists a deﬁned almost ⎪ ⎪ ⎪ ⎪ everywhere and (uniformly) bounded function φ(λ) such ⎪ ⎪ ⎪ ⎪ that for every interval ∆ ⊂ R, ∆ ∈ RΛ , ∆ ∩ Λ=∅, the ⎪ ⎪ ⎪ ⎪ ⎪ descomposition AE(∆) = ∆ φ(λ)E(dλ) is valid; ⎪ ⎪ ⎬

= (1.6) d) the subspace H CLin {E(∆)H} is pseudo-regular ⎪ ∆∈RΛ , ∆∩Λ=∅ ⎪ ⎪ ⎪ ⎪ and its isotropic part has ﬁnite dimension; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∈ Λ the corresponding subspace H = e) for every point λ ⎪ 0 λ0 ⎪ M ⎪ E(∆)H is a subspace of eigenvectors and root vectors ⎪ ⎪ ⎪ ⎪ λ0 ∈∆ ⎪ ⎪ ⎪ associated with an eigenvalue, or is a part of such a sub- ⎪ ⎪ ⎪ ⎪ ⎪ space for every operator A ∈ Y ; ⎪ ⎪ ⎪ ⎪ ⎪ f) if λ0 ∈ Λ, then either lim sup Eλ = ∞ or at least for ⎪ ⎪ ⎪ ⎪ λ→λ0 ⎪ ⎪ one A ∈ Y the operator A|Hλ0 is not a spectral operator of ⎪ ⎪ ⎭ scalar type. A spectral function E with a peculiar spectral set Λ satisfying Conditions (1.6) are called an eigen spectral function (e.s.f.) of the operator family Y. Note that the restriction that all operators from Y have real spectra is not very strong due to the following proposition.

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Proposition 1.6. If a commutative family Y of J-s.a. operators belongs to Dκ+ and ˙ , where H σ(A0 )\R = ∅ at least for one A0 ∈ Y, then the subspaces H = H [+]H and H are Y-invariant, σ(A|H ) ⊂ R for all A ∈ Y, the subspace H has ﬁnite dimension and the family Y|H belongs to the class Dκ+ with κ < κ (it is possible that κ = 0). If a commutative family Y ∈ Dκ+ of J-s.a. operators is such that σ(A0 )\R = ∅ at least for one A0 ∈ Y, then, generally speaking, the subspace H from Proposition 1.6 is not uniquely determined, but there exists a subspace Hmax that is maximal in the following sense: Hmax satisﬁes the conditions of Proposition 1.6 and for every subspace H satisfying the same conditions H ⊂ Hmax . For the operator family in question we, ﬁrst, ﬁnd an e.s.f. Eλ for the operator family Y|Hmax , and, second, set E(C\R) : = I − E(R). The constructed J-orthogonal spectral function is said (as before) to be an e.s.f. of Y. Deﬁnition 1.7. Let Eλ be an e.s.f. of an operator family Y and let an operator A ∈ Y and a function φ(λ) be connected by the system of equalities from (1.6c). Then the function φ(λ) is said to be the portrait of the operator A and the operator A is said to be the original of φ(λ) in Y (with respect to Eλ ). Deﬁnition 1.8. Let a spectral function E with a peculiar spectral set Λ be an e.s.f. of Y. If λ ∈ Λ then λ will be called a peculiarity (of Y). Let λ be a peculiarity. Fix a set X ∈ RΛ : X ∩ Λ = {λ}. The peculiarity λ is called regular if the operator family {E(X ∩ Y )}Y ∈RΛ is bounded, otherwise it is called singular. Note that the notion of regular and singular peculiarities is correctly deﬁned since the boundedness of the family {E(X ∩ Y )}Y ∈RΛ does not depend on X. 1.4. On a model representation for J-orthogonal spectral functions without peculiarities Let us introduce an analog of L2σ (E) that can be used for a model representation of Krein spaces. Assume that the scalar functions σ+ (t) and σ− (t) are such that t t ρ+ (λ)dσ(λ), σ− (t) = ρ− (λ)dσ(λ), ρ2+ (λ) = ρ+ (λ), σ+ (t) = −1

ρ2− (λ) = ρ− (λ),

t

σ(t) = −1

−1

ρ+ (λ) + ρ− (λ) − ρ+ (λ)ρ− (λ) dσ(λ),

where σ(λ) is the same as in the previous subsection, and set J -L2σ (E) : = L2σ+ (E+ ) ⊕ L2σ− (E− ),

[f (t), g(t)] : = (f+ (t), g+ (t)) − (f− (t), g− (t)),

where f (t) = f+ (t), ) + (f− (t), g(t) = g+ (t) + g− (t), f+ (t), g+ (t)) ∈ L2σ+ (E+ ), f− (t), g− (t) ∈ L2σ− (E− ). The space J -L2σ (E) is said to be a standard Krein space. As a slight abuse of the previous notation put also Mσ (E) : = Mσ+ (E+ )⊕Mσ− (E− ). A standard Krein space will be used for a model representation of J-orth.sp.f. Eλ without peculiarities. For simplicity, everywhere below we will assume that E−1 = 0,

E+1 = I,

E−1 = E−1+0 .

(1.7)

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Deﬁnition 1.9. Let Eλ be a J-orth.sp.f. and let its set of peculiarities be empty. A space J -L2σ (E) is said to be a model space for Eλ if for some canonical scalar product on H there is an isometric J-isometric operator W : J -L2σ (E) → H, such that for every λ ∈ [−1; 1] Eλ = W Xλ W −1 , where Xλ = {Xλ , J -L2σ (E)}. The operator W is said to be an operator of similarity. Proposition 1.10. Every J-orth.sp.f. Eλ with empty set of peculiarities has a model space J -L2σ (E).

2. Unbounded elements in Banach spaces 2.1. General case For future reference note the following simple algebraic fact. Proposition 2.1. Let L be a vector space, let f1 , f2 , . . . , fm , f be linear functionals m : on L. Then f ∈ Lin{fj }m Ker fj ⊂ Ker f . 1 if and only if j=1

Now we turn to the notion of unbounded elements. Assume that B is a Banach space, Pt is a resolution of the identity (= a spectral function with the empty set of peculiar points) deﬁned on the segment [−1; 1], continuous in zero (with respect to the weak topology) and ⎫ a) P−1 = 0, P1 = I; ⎪ ⎪ ⎪ ⎬ b) for every t ∈ [−1; 1] there exist the unilateral limits (2.1) w − lim Pµ and w − lim Pµ , where for deﬁniteness Pt−0 = ⎪ ⎪ ⎪ µ→t−0 µ→t+0 ⎭ Pt ; Set Pλ,µ = I + Pλ − Pµ+0 , where λ ∈ [−1; 0), µ ∈ (0; 1]. Next, let xλ,µ be a mapping of the numerical set [−1; 0) × (0; 1] into B (λ ∈ [−1; 0), µ ∈ (0; 1]). The function xλ,µ is said to be conformed with Pt if the following condition is fulﬁlled: for every λ, α ∈ [−1; 0), µ, β ∈ (0; 1] the equality Pλ,µ xα,β = xγ,δ holds, where γ = min{λ, α}, δ = max{µ, β}. Assume now that the space B and the resolution of the identity Pt have additionally the following property ⎫ if sup {xλ,µ } < ∞ then there is an element x ∈ B⎪ ⎪ ⎬ λ∈[−1;0) µ∈(0;1] (2.2) such that for every λ ∈ [−1; 0), µ ∈ (0; 1] the equality⎪ ⎪ ⎭ xλ,µ = Pλ,µ x holds. It is clear that the element x from (2.2) is uniquely deﬁned by xλ,µ and can be found by the formula x = w − lim xλ,µ . λ→−0 µ→+0

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Deﬁnition 2.2. A function xλ,µ conformed with Pλ is said to be an unbounded element conformed with Pλ (or, if it cannot produce a misunderstanding, an unbounded element) if sup {xλ,µ } = ∞. λ∈[−1; 0) µ∈(0;1]

Note that unbounded elements conformed with Pt exist if and only if zero is a point of growth for Pt , i.e., P+ − P− = 0 for every >0. Everywhere below in this Section the mentioned condition for Pt is assumed to be fulﬁlled. For brevity everywhere below we denote by symbols x

, y , etc. unbounded elements. For λ ∈ [−1; 0), µ ∈ (0; 1] we set xλ,µ : = Pλ,µ x

.

2 , . . . , x

k , conformed with a (common) Deﬁnition 2.3. Unbounded elements x

1 , x resolution of the identity Pλ are said to be linearly independent modulo B (or Mod B) if every non-trivial linear combination of them is an unbounded element from B. In a similar way two unbounded elements x

and y is said to be equal modulo B ( x = y | Mod B) if x

− y ∈ B. Although values of functionals from B ∗ are not deﬁned for unbounded elements from B, at least for some functionals there is a natural way for including the unbounded elements in their domains. Let Bλ,µ : = Pλ,µ B, where λ ∈ [−1; 0) and ∗ can be identiﬁed with the space {(Pµ+0 − Pλ )B}⊥ , i.e., µ ∈ (0; 1]. The space Bλ,µ ∗ are extended to the subspace (Pµ+0 − Pλ )B we assume that functionals from Bλ,µ ∗ by zero. Although generally speaking the norms of functionals from Bλ,µ are not preserved during this procedure, it is easy to re-deﬁne the norm on B and, consequently, also the norm on B ∗ without changing the norm topology in such a ∗ manner that the norm of every functional from Bλ,µ coincides with the norm of ∗ its above-mentioned extension to all of B. The linear span of the subspaces Bλ,µ ∗ denote by the symbol M. If f ∈ Bλ,µ , then deﬁne f x

: = f Pλ,µ x

= f xλ,µ . Thus, for every functional f ∈ M and an unbounded element x

the value f x

is well deﬁned. In what follows { x}⊥ means the set of all functionals from f ∈ M such that f x

= 0. Lemma 2.4. Let unbounded elements x

, x

1 , x

2 , . . . , x

m be linearly independent modulo B. Then sup {|f x

|} = ∞. M f∈ m xj }⊥ , f =1 j=1 { xj }⊥ . By hypothesis one has Proof. Suppose the contrary and set M = ∩m j=1 { {|f x

|} < ∞, so by the theorem of Hahn-Banach there is an element sup f ∈M ,f =1 ∗∗

for every f ∈ M . Then by virtue of Proposition 2.1 y ∈ B , such that f y = f x ∗∗ ∗ the element Pλ,µ (y − x

) (considered as a functional on Bλ,µ ) is a linear combination

1 , Pλ,µ x

2 , . . . , Pλ,µ x

m for all λ ∈ [−1; 0), µ ∈ (0; 1]. Thus, of the functionals Pλ,µ x ∗∗ ∗∗ for all such λ and µ, Pλ,µ y ∈ Bλ,µ . It is clear that Pλ,µ y is a bounded vector-valued function conformed with Pλ , so thanks to Condition (2.2) there is an element z ∈ B, ∗∗ such that Pλ,µ y = Pλ,µ z for all λ ∈ [−1; 0), µ ∈ (0; 1]. Finally note that for every

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∗∗ element f ∈ M there exist λ and µ, such that yf = Pλ,µ yf , hence yf = f z. Thus, there is z ∈ B, such that f z = f x

for every f ∈ M . Then Proposition 2.1 implies that the element z − x

is a linear combination of the elements x

1 , x

2 , . . . , x

m considered as functionals on M . If z = x

+ Σm β x

on M , then (recall that j j j=1 ∗ ⊂ M for all λ ∈ [−1; 0), µ ∈ (0; 1]) Pλ,µ z = Pλ,µ x

+ Σm

j . Since Bλ,µ j=1 Pλ,µ βj x z ∈ B, then for λ → −0 and µ → +0 the last equality implies the representation z=x

+ Σm

j considered now in the space B. This is a contradiction! j=1 βj x

Lemma 2.5. Assume that B is a separable Banach space, x

1 , x

2 , . . . , x

k are a collection of unbounded elements conformed with Pλ and linearly independent modulo B, and that {Cλ,µ }λ∈[−1;0), µ∈(0;1] is a family of vector subspaces of B ∗ possessing the following properties ⎫ ∗∗ a) Pλ,µ Cλ,µ ⊂ Cλ,µ ; ⎪ ⎪ ⎪ ⎪ ⎪ b) if 0 ∈ (λ2 , µ2 ) ⊂ (λ1 , µ1 ), then Cλ1 ,µ1 ⊂ Cλ2 ,µ2 ; ⎬ ∗ ∞ (2.3) c) for every functional f ∈ Bλ,µ there is a sequence {fm }1 ⊂ ⎪ ⎪ Cλ,µ with properties w∗ − lim fm = f , lim fm = f , ⎪ ⎪ ⎪ m→∞ m→∞ ⎭ where w∗ − lim is a limit considered in w∗ -topology. Then for every functional f ∈ B ∗ and an arbitrary collection of numbers {αj }kj=1 there is a sequence {gm }∞ 1 ⊂ ∪λ,µ Cλ,µ , such that , a) w∗ − lim gm = f ; m→∞

b)

j = αj . limm→∞ gm x

Proof. First, we show that for every f ∈ B ∗ there is a sequence {um }∞ 1 ⊂ ∪λ,µ Cλ,µ converging to f in the w∗ -topology. Assume (with no loss of generality) that f = 1. It is well known (see [21] Part V , §7, Th. 6) that a bounded closed ball is metrizable in the w∗ -topology. Due to Conditions (2.3) and the equality f = ∗ w∗ − lim Pλ,µ f the functional f belongs to the closure (in the w∗ -topology) λ→−0, µ→+0

of the intersection of M and the unitary ball from B ∗ . Since in the metrizable topology the sequential closure of a subset coincides with its standard closure, the ∞ existence of the desired sequence {um }∞ 1 is now evident. Second, let {um }1 be (j) ﬁxed. Denote um x

j : = αm , j = 1, 2, . . . , k. Lemma 2.4 implies that for every (j) natural integer m and j = 1, 2, . . . , k there is a functional wm ∈ M, such that (j) (j) (j) (j) |wm x

j | ≥ 2m (|αj − αm | + 1), wm x

l = 0 for l = j, wm = 1.

Set (j)

(j) vm = (j)

(αj − αm ) (j)

wm x

j

(j) · wm .

Then vm ≤ 1/2m . Third, Condition (2.3c) implies the existence of a functional (j) (j) (j) (j) zm ∈ ∪λ,µ Cλ,µ such that zm < 1/2m−1, |(zm − vm ) xl | < 1/2m, l = 1, 2, . . . . (j) k As a last step one can set gm = um + Σj=1 zm .

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Remark 2.6. The proof of Lemma 2.5 shows that under Conditions (2.2), for ∗ any sequence {um }∞ 1 ⊂ ∪λ,µ Cλ,µ converging (in the w -topology) to an arbitrary ∗ functional f ∈ B and for every collection of numbers {αj }k1 , there is a “correcting” sequence {vm }∞ xj = 1 ⊂ ∪λ,µ Cλ,µ , such that limm→∞ vm = 0, limm→∞ (um +vm ) αj , j = 1, 2, . . . , k. Now show that the concept of unbounded elements is applicable to the space L1σ + L2η , where the relation between σ(t) and η(t) is given by (1.4). Proposition 2.7. If B = L1σ + L2η and Pt = {Xt , L1σ + L2η }, where Xt is the multiplication operator by the indicator of the set (−∞; t) (denoted χ(−∞;t) (τ )), then for B and Pt Conditions (2.1) and (2.2) are fulﬁlled. Proof. For the given Pt the fulﬁllment of Condition (2.1) is evident, so we shall concentrate on Condition (2.2) only. Consider on the segment [−1; 1] a µσ -measurable, a.e. ﬁnite function x(t) satisfying the condition sup

λ∈[−1; 0), µ∈(0;1]

where

{xλ,µ (t)L1σ +L2ν } < ∞,

+ x(t), xλ,µ (t) = 0,

(2.4)

if t ∈ [−1; λ) ∪ (µ; 1]; if t ∈ [λ, µ].

By the theorem of Banach-Steinhaus Condition (2.4) can be re-written in the 2 following form: for every function f (t) ∈ L∞ σ ∩ Lν λ 1 sup {| x(t)f (t)dσ(t) + x(t)f (t)dσ(t)|} < ∞. (2.5) λ∈[−1; 0), µ∈(0;1] −1 µ We need to show that x(t) ∈ L1σ + L2η . Note that this is equivalent to |x(t)| ∈ L1σ + L2η , therefore we suppose (with no loss of generality) that x(t) ≥ 0. Under this hypothesis set + x(t), if x(t) < G(t); x1 (t) = x2 (t) = x(t) − x1 (t). G(t), if x(t) ≥ G(t); Now we show that x1 (t) ∈ L2η .

(2.6)

Suppose the contrary, i.e., x1 (t) ∈ L2η . Then there is a decreasing sequence of positive numbers {αn }∞ 1 , limn→∞ αn = 0, such that −α1 1 −α2 α1 2 x1 (t)dη(t) = β1 ≥ 1, x21 (t)dη(t) = β2 ≥ 1, + + −1

α1

...

−α1

,

αn−1

αn

...

α2

x21 (t)dη(t) = βn ≥ 1,

+ −αn−1

−α1

.

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For t ∈ [−αn−1 ; −αn ) ∪ (αn ; αn−1 ] set f (t) = Then

x1 (t) √ , n = 1, 2, . . . , α0 = 1. G(t) · n · βn

∞ 1 |f (t)| G(t)dσ(t) = < ∞, 0 ≤ f (t) ≤ 1, 2 n −1 n=1 1 #∞ √ βn 2 so f (t) ∈ L∞ σ ∩Lν but −1 x1 (t)f (t)dσ(t) = n=1 n = ∞. The latter contradicts (2.5). Thus, (2.6) has been proved. To complete the proof we need to demonstrate that x2 (t) ∈ L1σ . Suppose the contrary, i.e., x2 (t) ∈ L1σ . Set + 0, if x(t) ≤ G(t); f (t) = 1, if x(t) > G(t). 1

2

By Condition (2.6) we have 1 2 |f (t)| G(t)dσ(t) ≤ −1

1

−1

|x1 (t)|2 dη(t) < ∞ ,

2 therefore f (t) ∈ L∞ σ ∩ Lν . Also 1 1 xλ,µ (t)f (t)dσ(t) ≥ x2 (t)dσ(t) = ∞, lim λ→−0, µ→+0 −1 −1

but the latter contradicts (2.5).

Remark 2.8. The proof of Proposition 2.7 gives the following result for a µσ -measurable function x(t) (Cf. [8], Theorem 5.2.1): Let + x(t), if |x(t)| < G(t); x1 (t) = x2 (t) = x(t) − x1 (t). i arg x(t) , if |x(t)| ≥ G(t); G(t) · e Then x(t) ∈ L1σ + L2η if and only if simultaneously x1 (t) ∈ L2η and x2 (t) ∈ L1σ . 2.2. Some remarks for the case of Hilbert spaces Here and up to the end of the section H means a ﬁxed separable Hilbert space and Pt means a ﬁxed orthogonal spectral function. For every bounded Pt -measurable function φ(t) deﬁne on H the following operator Φ : 1 Φx : = φ(t)dPt x . −1

Using the previous notation, re-write the last formula as λ 1 φ(t)dPt x

+ φ(t)dPt x

. (Φ x)λ,µ = −1

(2.7)

µ

Representation (2.7) allows the possibility of treating the operator Φ in a more general sense: unbounded elements conformed with Pt can be naturally included to the domain of Φ. Note that the image of an unbounded element can be both a bounded element and an unbounded element.

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311

Until the end of the present section a non-decreasing function σ(t), such that the µσ -measurability on [−1; 1] coincides with Pt -measurability, will be ﬁxed. Note that the existence of σ(t) follows from the separability of H (see [1] Section 76, Theorem 1). We give an additional notation. Let { xj }k1 be a ﬁxed family of unbounded elements conformed with Pt and linearly independent modulo H. Both the unbounded elements and the ordinary vectors from H can be considered as functions deﬁned on [−1; 0)×(0; 1] and taking values in H (see (2.2)). The linear span of vectors from H and unbounded elements from { xj }k1 consistently taken as functions

Additionally H

will be considered as a Hilbert on [−1; 0) × (0; 1] is denoted H. space, where H is a subspace with the the same scalar product that was given on H initially and unbounded elements from { xj }k1 are mutually orthogonal and

is said to be a expansion of H (generated by { orthogonal to H. The space H xj }k1 ). k Next, using the system { xj }1 introduce the function ν(t): + #k

j 2 , if t ∈ [−1; 0); σ(t) + j=1 Pt x (2.8) ν(t) = #k 2 −σ(1) + σ(t) − j=1 (Pt − I) xj , if t ∈ (0; 1]. The connection between Pt and σ(t) implies that the function ν(t) introduced in (2.8) has Representation (1.4). In this case the function G(t) from (1.4) can be calculated directly through ν(t) and σ(t). Proposition 2.9. ([40]) Let { xj }k1 be a system of unbounded elements, forming

together with H the space H, let φ(t) be a µσ -measurable function, let Φ be the 2

⊂ H if and only if φ(t) ∈ L∞ operator deﬁned by Formula (2.7). Then ΦH σ ∩ Lν .

⊂ H, and For future applications both the cases ΦH

⊂H

ΦH

(2.9)

are important. Proposition 2.10. ([40]) Let { xj }k1 be a system of unbounded elements, generating

Then there are no more than k 2 µσ -measurable together with H the space H. 2 functions φ(t) linearly independent modulo L∞ σ ∩ Lν , such that Condition (2.9) is fulﬁlled. Remark 2.11. Let { xj }k1 be a system of unbounded elements, generating together

and let F be a vector space of scalar µσ -measurable functions, with H the space H ¯ ∈ F. If m is the such that (2.9) holds for every φ(t) ∈ F, and if φ(t) ∈ F then φ(t) 2 maximal number of functions from F linearly independent modulo L∞ σ ∩ Lν , then ∞ there are m real-valued functions from F linearly independent modulo Lσ ∩ L2ν . Indeed, if the dimension of some real vector space is m then the dimension (as a complex space) of its complexiﬁcation is also m.

312

V. Strauss

3. A function model for a J-symmetric family of the class Dκ+ Now pass to a function model of a J-symmetric family Y ∈ Dκ+ with real spectrum and non-empty set of peculiarities. This model is deﬁned with the help of an e.s.f. Eλ of Y. By virtue of Theorem 1.5, it is clear that the general situation can be reduced to the case of J-orth.sp.f. Eλ with a unique spectral peculiarity in zero. Furthermore, the case of a regular peculiarity is trivial because, under these conditions, all operators from Y are spectral in the sense of Dunford and have a ﬁnite-dimensional nilpotent part. Thus, we can assume Eλ satisﬁes: ⎫ a) E−1 = E−1+0 =0, E+1 = I; ⎪ ⎬ b) Λ = {0}; (3.1) ⎪ c) sup {Eλ } = ∞. ⎭ λ∈[−1;1]\{0}

Introduce some notation. Let ⎫

∩H

[⊥] , H2 = H⊥ ∩ H,

H0 = JH1 , Pj be an⎬ H1 = H 1 orthoprojection (in the sense of Hilbert spaces) onto Hj , ⎭

λ : = Eλ | . j = 0, 1, 2, E H

(3.2)

Note that by Condition (3.1c) the inequality H1 = {0} holds. In addition to (3.2) set

↑ = H0 ⊕ H2 , E

λ = Eλ | , E

↑ = (P0 + P2 )Eλ | ↑ . H λ H H

(3.3)

It is necessary to take in account that, generally speaking, the subspace H2 is indeﬁnite. Since J-orth.sp.f. Eλ belongs to the class Dκ+ , there is an Eλ -invariant pair of J-orthogonal maximal semi-deﬁnite pseudo-regular subspaces L+ and L− with ﬁnite-dimensional isotropic part, moreover by Condition (1.6b) we can assume that for every closed interval ∆ ⊂ [−1; 1]\{0} the subspace (E(∆)H) ∩ L+ is positive and the subspace (E(∆)H)∩L− is negative. Thanks to the last hypothesis the following subspaces are well deﬁned

− =

+ = CLin {E(∆)L+ }, H CLin {E(∆)L− }. (3.4) H ∆⊂[−1;1]\{0}

∆⊂[−1;1]\{0}

Set

+ , H− = H2 ∩ H

− , (3.5) H2+ = H2 ∩ H 2 and assume that a fundamental scalar product on H is also canonical for the

⊕ H0 and on the last space is compatible (see Deﬁnition 1.2) with the subspace H given decompositions of the corresponding subspaces

− = H1 +H

+ = H1 +H ˙ + and H ˙ −. (3.6) H 2

Thus,

2

⎛

⎞ 0 V −1 0 ˙ H1 + ˙ H2 = ⎝V 0 0 ⎠, J|H0 + (3.7) 0 0 J2 where the operator V : H0 → H1 is isometric, and J2 is a canonical symmetry of the form [·, ·] on H2 .

A Functional Description

313

Let J -L2σ (E) be a standard Krein space (see Section 1 ) and let { gj (t)}kj=1 ⊂ Mσ (E) be a system of unbounded elements conformed with the operator-valued

2 (E) the function Xτ and linearly independent modulo J -L2σ (E). Denote by J -L σ 2 k gj (t)}j=1 . Deﬁne on linear span generated by the space J -Lσ (E) and the system {

2 (E) structures of Hilbert and Krein spaces in the following way: on J -L2 (E) J -L σ σ both structures coincide with the original structures, functions of the system { gj (t)}kj=1 are by deﬁnition positive (as elements of the Krein space), mutually orthogonal and J-orthogonal, normalized and J-normalized, orthogonal and J 2 (E) is said to be the expansion of J -L2 (E) orthogonal to J -L2σ (E). The space J -L σ σ k (generated by the collection { gj (t)}j=1 ). Theorem 3.1. If a J-orth.sp.f. Eλ satisﬁes Condition (3.1) and a scalar product on H is compatible with (3.6), then there are, ﬁrst, a subspace J -L2σ (E) and a system

2 (E) of unbounded elements { gj (t)}kj=1 of this space forming together the space J -L σ

2 (E) → H,

W L2 (E) = H2 , and, second, an isometric J-isometric operator W : J -L σ σ such that for every λ ∈ [−1; 1]

λ = W · X # · (W )−1 , E λ

↑ = W ↑ · Xλ · (W ↑ )−1 , E λ

W ↑ = (I2 ⊕ V )W,

(3.8)

2 (E)}, k = dim H0 = dim H1 . where Xλ = {Xλ , J -L σ Deﬁnition 3.2. If for Decomposition (3.2), (3.5) a relation between a J-orth.sp.f.

2 (E) is given by Formulae (3.8), then Eλ satisfying Condition (3.1) and a space J -L σ 2

(E) is said to be a basic model space for Eλ (compatible with (3.2), (3.3), J -L σ (3.6)) and the operator W is said to be an operator of similarity corresponding to this space. ∗

∗

Remark 3.3. Let H be the factor-space de H generated by H1 . Then H is a ∗

λ induces on this space the J-orth.sp.f. Eλ . Note that the unique Krein space and E ∗

peculiarity of Eλ is regular and can be removed, so, as a slight abuse of terminology, ∗

one can say that Eλ has no peculiarities. It is clear that W |J -L 2σ (E) in (3.8) exists if ∗

∗

∗

and only if there exists a J-isometric operator W : J -L2σ (E) →H , such that Eλ = ∗

∗

W Xλ W −1 . Taking this reasoning into account one can say that an arbitrariness in the choice of a basic model space for J-orth.sp.f. Eλ satisfying (3.1) does not depend on the choice of P2 but essentially depends on an arbitrariness in the choice ∗

of a basic model space for J-orth.sp.f. Eλ with the empty set of peculiarities. This situation was considered in Subsection 1.4. Up to this point we discussed model representations not for a family Y ∈ Dκ+ but for a J-orth.sp.f. Eλ ∈ Dκ+ with unique spectral peculiarity in zero. Now let Y ∈ Dκ+ be a J-symmetric commutative operator family. Then its linear span contains a family (not uniquely deﬁned) Yr of J-s.a. operators, such that the linear span of Yr coincides with the linear span of Y. Then (see Theorem 1.5) the

314

V. Strauss

family Yr has an e.s.f. Eλ . This spectral function is also said to be an e.s.f. of Y. Note (see Proposition 1.6 and accompanying comments) that the case E(R) = I is admissible. Theorem 3.4. Assume that Y ∈ Dκ+ is a commutative J-symmetric family, whose e.s.f. Eλ satisﬁes Condition (3.1), that a canonical scalar product on H is com 2 (E) is a basic model space for Eλ , and W is a correpatible with (3.6), that J -L σ sponding operator of similarity. Then for every operator A ∈ Y there is a function ϕ(t) such that

↑ = W ↑ · Φ · (W ↑ )−1 , A

= W ·Φ ¯ # · W −1 A

(3.9)

↑ and the operator W ↑ are deﬁned via

↑ : = (P0 ⊕ P1 )A| ↑ , the space H where A H

2 (E)}, and Φ ¯ is the multiplication operator by ϕ(t) (3.3), (3.8), Φ = {Φ, J -L ¯ acting σ 2

in the space J -Lσ (E). Deﬁnition 3.5. If Y ∈ Dκ+ is a commutative J-symmetric family and Eλ (satisfying Condition (3.1)) is its e.s.f., then a basic model space for Eλ is also said to be a basic model space for Y. Remark 3.6. Assume that Y ∈ Dκ+ is a commutative J-symmetric family, its

2 (E) and W are, respectively, a basic model e.s.f. Eλ satisﬁes Condition (3.1), J -L σ space and a corresponding operator of similarity for Y. If under these conditions an operator A ∈ Y and a function ϕ(t) are related by Formulae (3.9), then according to Deﬁnition 1.7 the function ϕ(t) is a portrait of the operator A (with respect to Eλ ), and the operator A is an original of ϕ(t) in Y (not unique in the general case). It is clear that the portrait of an operator does not depend on the choice of

2 (E) but does depend of the choice of Eλ and can be the basic model space J -L σ found via Representation (1.6c). Remark 3.7. Theorem 3.4 raises a natural problem concerning a characterization of functions that can be portraits for operators from a given commutative Jsymmetric operator family Y ∈ Dκ+ with a ﬁxed choice of J-orth.sp.f. Eλ . A partial answer to the problem is contained in Propositions 2.9 and 2.10. Indeed, under the present conditions the function ν(t) from (2.8) has the form ⎧ t ⎪ ⎪ G(t)dσ(t), t ∈ [−1; 0), ⎪ ⎪ ⎨ −1 ν(t) = (3.10) 1 ⎪ ⎪ ⎪ ⎪ ⎩ − G(t)dσ(t), t ∈ (0; 1], t

where G(t) = 1 +

k j=1

gj (t)2E .

A Functional Description

315

4. Functional representation of commutative W J ∗ -algebras of the class Dκ+ with a unique singular peculiarity 4.1. Preliminary remarks Below the symbol A means a W J ∗ -algebra belonging to the class Dκ+ . Let Eλ be its e.s.f. In this section we assume that all J-s.a. operators from A have real spectra and that Eλ satisﬁes Conditions (3.1). Let ϕ(t) be a continuous function vanishing in a neighborhood of zero. Set 1 ϕ(t)dEλ = ϕ(t)dEλ . (4.1) Bϕ : = −1

Supp(ϕ)

By Property (1.6a) Bϕ ∈ A. Denote by A0 the weak closure of the set of operators having the form (4.1), where ϕ(t) runs through the set of all functions of the indicated type. Our immediate aim is to characterize the subalgebra A0 in terms of the portraits of the operators from A0 . For this we shall use the notation from (3.2) and (3.5). First, for every A ∈ A0 A|H1 = 0,

AH ⊂ H.

(4.2)

Indeed, for J-s.a. operator from A0 the fulﬁllment of Condition (4.2) is evident, but the family A0 is J-symmetric and linear, so the rest is straightforward.

2 (E) be a basic model space for A generated by J-L2 (E) and a system Let J-L σ σ of unbounded elements { gj (t)}kj=1 and let W be a corresponding operator of similarity. If a function ψ(t) is the portrait of an operator A ∈ A0 , then by Proposition 2.9 and Remark 3.7 2 ψ(t) ∈ L∞ (4.3) σ ∩ Lν , where ν(t) is deﬁned by Formulae (3.10). 2 1 2 Simultaneously with the space L∞ σ ∩ Lν consider the space Lσ + Lη , where 1 2 η(t) is given by Formulae (1.4) and (3.10). Note that the space Lσ + Lη has not 2

2 (E). only the direct relation with the space L∞ σ ∩ Lν but also with the space J-L σ

2 (E) and L1 +L2 be related by Conditions (1.4) Proposition 4.1. Let the spaces J-L σ η σ and (3.10). Then for every function ψ(t) ∈ L1σ + L2η there are vector-functions g−1 (t), g0 (t), f−1 (t), f0 (t), . . . , fk (t) ∈ L2σ (E), such that ψ(t) = (g−1 (t), f−1 (t))E + (g0 (t), f0 (t))E +

k

( gj (t), fj (t))E .

j=1

Proof. Fix a function g0 (t) ∈ L2σ (E), such that g0 (t)E = 1, and a decomposition ψ(t) = ψ1 (t) + ψ2 (t), where ψ1 (t) ∈ L1σ , ψ2 (t) ∈ L2η . Let + 0, if ψ1 (t) = 0; θ(t) = ¯ ψ1 (t)/|ψ1 (t)|, if ψ1 (t) = 0.

316

V. Strauss

The desired system can be constructed as follows. Set f−1 (t) = θ(t)g0 (t) |ψ1 (t)|, g−1 (t) = g0 (t) |ψ1 (t)|, f0 (t) = (ψ¯2 (t)/G(t))g0 (t), fj (t) = (ψ¯2 (t)/G(t)) gj (t), j = 1, 2, . . .. Introduce an additional notation. Let (see (3.8)) ej = W gj (t),

hj = W ↑ g j (t) = V −1 ej ,

ιlj (t) = [ gl (t), g j (t)],

l, j = 1, 2, . . . , k. Then for the operator Bϕ from (4.1) 1 ϕ(t)ιlj (t)dσ(t). (Bϕ hl , ej ) =

(4.4)

(4.5)

−1

It is clear that ιlj (t) ∈ Mσ (C). Note that functions from the collection {ιlj (t)}k1 can be both bounded and unbounded elements of the space L1σ + L2η . Denote by S(Eλ ) the set of operators {S} acting on H and satisfying the following conditions ⎫ a) S = P1 SP0 ; ⎪ ⎬ k k (4.6) 1 2 b) if αlj ιlj (t) ∈ Lσ + Lη , then αlj (Shl , ej ) = 0. ⎪ ⎭ l,j=1

l,j=1

Conditions (4.6) imply that S(Eλ ) is a weakly closed vector set with nilpotent elements and for every S1 , S2 ∈ S(Eλ ), S1 S2 = 0. One of the essential questions concerns the linear dimension of S(Eλ ). Conditions (4.6) give the following simple result. Proposition 4.2. dim S(Eλ ) is equal to the maximal number of functions in the collection {ιlj (t)}k1 linearly independent modulo L1σ + L2η . The following example is from [40]. Example 4.3. Take the space H coinciding with l2 , denote by {um }∞ m=1 the canonical basis of this space, i.e., u1 = (1, 0, 0, . . . ), u2 = (0, 1, 0, . . . ), . . ., deﬁne the canonical symmetry J by the equalities Ju0 = u1 , Ju1 = u0 ; Ju2m+1 = u2m+1 , Ju2m = −u2m , m = 1, 2, . . .; and the spectral function Eλ by the relations a) Eλ = 0 for every λ < 0; b) if λ > 0 then (I − Eλ )u0 =

(u2(m−1) − u2m−1 ), (I − Eλ )u1 = 0,

1/m∈[λ;1)

(I −Eλ )u2(m−1) = u2(m−1) +u1 , (I −Eλ )u2m−1 = u2m−1 +u1 for 1/m ∈ [λ; 1) and (I − Eλ )u2(m−1) = (I − Eλ )u2m−1 = 0 for 1/m ∈ [λ; 1). Here k = 1 and ι11 (t) = 0, so for this case dim S(Eλ ) = 0, but for some subclasses of Dκ+ the inequality dim S(Eλ ) > 0 holds. Recall (see [3]) that a W J ∗ algebra A belongs to the class H if there is at least one A-invariant subspace L+ ∈ M+ (H) and all A-invariant maximal non-negative subspaces belong to the class h+ . In [7] (Theorem 5.6) the following result was proved.

A Functional Description

317

Theorem 4.4. Assume that H is a separable space, that A ∈ H is a J-s.a. operator, and that Eλ is its e.s.f. with a set of peculiarities Λ. Then there exist a ﬁnite set ⊂ R, Λ ⊂ Λ and Eλ -measurable sets X+ , X− ⊂ R, X+ ∩ X− = ∅, X+ ∪ X− = Λ R\Λ, such that for every interval ∆ ∈ RΛ ∩ R each of the subspaces E(∆ ∩ X+ )H and E(∆ ∩ X− )H

(4.7)

either degenerates to {0}, or is deﬁnite (positive and, respectively, negative). Note that this theorem can be used also for a W J ∗ -algebra from H because in the statement of the theorem the condition A ∈ H can be replaced by the condition Eλ ∈ H. Proposition 4.5. If A ∈ H, then dim S(Eλ ) ≥ 1. Proof. The set S(Eλ ) is determined by the behavior of Eλ in a neighborhood (no matter how small) of zero, so one can assume that (see Theorem 4.4) for some decomposition [−1; 1]/{0} = X+ ∪ X− and every closed interval ∆ ⊂ [−1; 1]/{0} each of the subspaces E(X+ ∩ ∆)H and E(X− ∩ ∆)H is either trivial or deﬁnite. By this hypothesis gj (t)2 = ±[ gj (t), g j (t)] for a.e. t ∈ [−1; 1]/{0}, j = 1, 2, . . . , k with the choice of sign depending of the fulﬁllment of one of the conditions t ∈ X+ and t ∈ X− . Thus, a.e. k % % ιjj (t)% (4.8) G(t) = 1 + % j=1

and thanks to Remark 2.8 one can conclude that the collection {ιjj (t)}k1 contains functions that do not belong to L1σ +L2η . The rest follows from Proposition 4.2. It is clear that, generally speaking, dim S(Eλ ) ≤ k 2 . Let us show that this estimate is sharp. Example 4.6. We shall not wholly describe H and Eλ indicating only that H is a Pontryagin space with κ = 2, dim H1 = 2 and adducing a basic model space for Jorth.sp.f. Eλ . So, let L2σ (E) coincide with the space L2t (C) of scalar functions with g2 (t) = |t|−3/2 + i|t|−5/4 . Let the standard Lebesgue measure and let g1 (t) = t−2 , −4 us show that dim S(Eλ ) = 4. Here G(t) = (1 + t + |t|−3 + |t|−5/2 ), ι11 (t) = t−4 , ι12 (t) = ¯ι21 (t) = |t|−7/2 − i|t|−13/4 , ι22 (t) = |t|−3 + |t|−5/2 . The collection {ιlj (t)}k1 #k is linearly independent. In fact, let ψ(t) = l,j=1 αlj ιlj (t) ∈ L1σ + L2η . With no loss #k of generality one can suppose that l,j=1 |αlj | < 1. Then there is a neighborhood of zero where the inequality |ψ(t)| < G(t) holds, so (see Proposition 2.7 and Remark 2.8) ψ(t) ∈ L2η , but a direct veriﬁcation shows that this is impossible if #k #k l,j=1 |αlj | > 0. Thus, l,j=1 |αlj | = 0. 4.2. Main results 2 We introduce an additional notation. Let ϕ(t) ∈ L∞ σ ∩ Lν . Then Gϕ (Eλ ) means the totality of operators from A0 that are an original of the function ϕ(t).

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Theorem 4.7. A ∈ A0 if and only A satisﬁes Condition (4.2) and there is a function 2 ϕ(t) ∈ L∞ σ ∩ Lν such that ⎫ a) ϕ(t) and A are related by Conditions (3.9); ⎪ ⎪ ⎪ k ⎪ ⎪ ⎪ 1 2 ⎬ αlj ιlj (t) ∈ Lσ + Lη , then b) if ψ(t) = (4.9) l,j=1 1 ⎪ k ⎪ ⎪ ⎪ ⎪ αlj (Ahl , ej ) = ϕ(t)ψ(t)dσ(t). ⎪ ⎭ −1 l,j=1

Proof. The necessity of Conditions (4.2) and (3.9) for A ∈ A0 was proved before, #k so we turn to Condition (4.9b). Let ψ(t) = l,j=1 αlj ιlj (t) ∈ L1σ + L2η . By virtue of Proposition 4.1 the function ψ(t) can be represented using vector-functions from

2 (E). With the notation from Proposition 4.1 and Theorem 3.1 set J -L σ y−1 = W g−1 (t), y0 = W g0 (t), xj = W fj (t), j = −1, 0, 1, . . . , k.

(4.10)

By the deﬁnition of the subalgebra A0 there is a sequence of continuous functions {ϕm (t)}∞ 1 vanishing near zero, i.e., ϕm (t) ≡ 0 in some neighborhood of zero (generally speaking, this neighborhood depends on m), m = 1, 2, . . ., such that 1 (Ahl , ej ) = lim ϕm (t)d(Et hl , ej ), l, j = 1, 2, . . . , k (4.11) m→∞

and

−1

1 (Ay−1 ,x−1 ) = lim ϕm (t)d(Et y−1 , x−1 ), m→∞ −1 1 ϕm (t)d(Et y0 , x0 ), (Ay0 ,x0 ) = lim m→∞ −1 1 ϕm (t)d(Et hj , xj ), j =1,2,. . . , k. (Ahj , xj )= lim m→∞

−1

On the other hand (see (4.10)) (Ay−1 , x−1 ) =

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.12)

1

−1

ϕ(t)d(g−1 (t), f−1 (t))E dσ(t)

(4.13)

and the rest of the expressions from (4.12) can be re-written in terms of a functional representation in a similar way. Moreover Formulae (4.5) and (4.11) imply 1 k αlj (Ahl , ej ) = lim ϕm (t)ψ(t)dσ(t). (4.14) l,j=1

m→∞

−1

Now the necessity of (4.9b) follows from Formulae (4.12), (4.13), (4.14) and the special choice of the vector collection y−1 , y0 , x−1 , x0 , . . . , xk . Pass to the suﬃciency of (4.9). Let A satisﬁes (4.9). Choose a maximal linear independent modulo L1σ + L2η subset {ιlj (t)}(l,j)∈M of the set {ιlj (t)}k1 . Suppose that M = ∅. Let B be L1σ + L2η and let an operator-valued function Pt be the multiplication operator by χ[−1;t) (t) acting in L1σ + L2η . Take as vector sets Cλ,µ

A Functional Description

319

satisfying Conditions (2.3) the sets of continuous functions vanishing in the corresponding neighborhoods of zero. Then by Lemma 2.5 and Proposition 2.7 there is a sequence of continuous functions {ϕm (t)}∞ 1 vanishing near zero and such that for every function ψ(t) ∈ L1σ + L2η the equality 1 1 lim ϕm (t)ψ(t)dσ(t) = ϕ(t)ψ(t)dσ(t), m→∞

−1

holds and for (l, j) ∈ M

−1

1

lim

m→∞

−1

ϕm (t)ιlj (t)dσ(t) = (Ahl , ej ).

(4.15)

2 (E) and j = 1, 2, . . . , k, then evidently [f (t), g(t)]E ∈ L1 If f (t), g(t) ∈ J-L σ σ 1

t = A

and and [f (t), g (t)]E ∈ L2η . The latter implies that w − lim ϕm (t)dE m→∞ −1 1

↑ = A

↑ . Taking into account (4.15) one can conclude that ϕm (t)dE w − lim t m→∞ −1 1 ϕm (t)dEt = A. Finally note that if M = ∅, it only simpliﬁes the also w − lim m→∞

−1

construction of the sequence {ϕm (t)}∞ 1 because in this case there are no Conditions (4.15). Corollary 4.8. Gϕ (Eλ ) = ∅ for every function ϕ(t) ∈ L1σ + L2η . If B ∈ Gϕ (Eλ ) is a ﬁxed operator, then Gϕ (Eλ ) = {B + S}S∈S(Eλ ) , in particular, G0 (Eλ ) = S(Eλ ). Corollary 4.9. The set S(Eλ ) does not depend on a choice of canonical scalar product in H. Corollary 4.10. The set A0 coincides with the strong sequential closure of the set of all operators represented by (4.1). Proof. Let A ∈ A0 and let ϕ(t) be the portrait of A. For ϕ(t) construct a sequence {ϕm (t)}∞ m=1 of continuous functions vanishing near zero such that a) ϕm (t)L∞ ≤ ϕ(t)L∞ ; σ σ µσ b) ϕm (t) → ϕ(t) for m → ∞; c) ϕm (t) − ϕ(t)L2ν → 0 for m → ∞. 1 1

↑ = A

↑ , where A

and A

↑ are ϕm (t)dEt = A, s − lim ϕm (t)dE Then s − lim t m→∞

−1

m→∞

−1

the same as in (3.9). The rest follows from Lemma 2.5 and Remark 2.6.

Theorem 4.11. For the given algebra A there is a ﬁnite collection of J-s.a. operators A1 , A2 , . . . , Al ∈ A, such that every operator B ∈ A has a representation B = S + Q(A1 , A2 , . . . , Al ) + C,

(4.16)

where S ∈ A is a nilpotent operator, S|H

= S |H

= 0, Q(t1 , t2 , . . . , tl ) is a polynomial of l variables, C ∈ A0 . #

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Proof. Consider the totality of portraits corresponding to operators from A. By Proposition 2.10 this totality contains only a ﬁnite number of functions linearly 2 independent modulo L∞ σ ∩ Lν and by virtue of Remark 2.11 one can chose these functions real-valued. Let {ϕ1 (t), ϕ2 (t), . . . , ϕl (t)} be a maximal system of real2 valued portraits linearly independent modulo L∞ σ ∩ Lν . Fix some J-s.a. operators A1 , A2 , . . . and Al from A that are originals for functions ϕ1 (t), ϕ2 (t), . . . and ϕl (t) respectively. Let B ∈ A. Then for B there is a linear combination α1 A1 + α2 A2 + . . . + αl Al of these operators, such that the portrait ϕ(t) of the operator B − 2 α1 A1 − α2 A2 − · · · − αl Al belongs L∞ σ ∩ Lν . By Theorem 4.7 there is an operator C ∈ Gϕ (Eλ ). Thus one need only to prove that the operator S : = B − α1 A1 − α2 A2 − . . . − αl Al − C satisﬁes Conditions (4.16). By the construction S|H

= 0

[⊥] . On the other hand by the deﬁnition of e.s.f. of the algebra A the and SH ⊂ H

[⊥] is the root space or its part for S (this is true for every operator subspace H

[⊥] and H1 ⊂ Ker S, the rest is trivial. belonging to A). Since H1 ⊂ H Remark 4.12. Recall (see [1] Part VI, Section 92), that a commutative W ∗ -algebra is monogenic, i.e., it contains an operator A, such that Alg A coincides with the algebra. For commutative W J ∗ -algebras this is not true. In particular, if the sub [⊥] has inﬁnite dimension, then in general the nilpotent part of the algebra space H A can have an inﬁnite number of algebraically independent elements. An idea for the construction of an algebra with the desired property can be found in [7]. Next, Representation (4.16) contains the polynomial Q(t1 , t2 , . . . , tl ). In the proof of Theorem 4.11 given above Q(t1 , t2 , . . . , tl ) is a linear function. If one wants to minimize l in Representation (4.16) one needs to consider polynomials of general type. In fact, it can occur that one of the operators from {Aj }l1 has a representation of the type (4.16) (say, Al = S + Q (A1 , A2 , . . . , Al−1 ) + C ) and can be excluded from the right part of (4.16). Let us give an example where l cannot be reduced to one. Example 4.13. We describe a main idea omitting some details. Here H is a Pontryagin space with κ = 3, dim H1 = 3, L2σ (E) coincides with L2t (C), unbounded elements take the form: g1 (t) = t−1 , g 2 (t) = t−4/5 , g 3 (t) = t−2 . Then a maxi2 mal collection of functions-portraits for A linearly independent modulo L∞ σ ∩ Lν , 6/5 is formed by three functions, say: ϕ1 (t) ≡ 1, ϕ2 (t) = t , ϕ3 (t) = t. Using the 2 relations ϕ1 (t) = −(ϕ1 (t) + ϕ2 (t))2 + 2(ϕ1 (t) + ϕ2 (t)) | Mod L∞ σ ∩ Lν , ϕ2 (t) = 2 ∞ 2 (ϕ1 (t) + ϕ2 (t)) − (ϕ1 (t) + ϕ2 (t)) | Mod Lσ ∩ Lν ) one can reduce the number l (in Representation (4.16)) of generating operators for the algebra in question to l = 2, but a further reduction of l is impossible because (ϕ2 (t))2 = (ϕ3 (t))2 = 2 ϕ2 (t)ϕ3 (t) = 0 | Mod L∞ σ ∩ Lν . 1 Up to this point the integral −1 ϕ(t)dEλ was deﬁned exclusively for a continuous function ϕ(t) vanishing near zero. However it is clear that there is a more extended variety of functions such that the integral in question has some natural sense.

A Functional Description

321

Proposition 4.14. Let ϕ(t) be a µσ -measurable function. Then the integral

1 −1

ϕ(t)dEλ

(4.17)

converges unconditionally in the weak topology as an improper integral with the singularity in zero if and only if the function ϕ(t) satisﬁes the following conditions: ⎫ 2 a) ϕ(t) ∈ L∞ ⎪ σ ∩ Lν ; ⎬ 1 k (4.18) b) |ϕ(t)| |ιlj (t)|dσ(t) < ∞. ⎪ ⎭ −1 l,j=1

1 Proof. Since −1 ϕ(t)dEλ |H1 = 0, the necessity of (4.18a) follows from Proposition 2.9, and the necessity of (4.18b) is a consequence of Formula (4.5) and the equivalence in the case of scalar functions between unconditional and absolute convergence of integrals. The suﬃciency of Condition (4.18) for the weak unconditional convergence of the integral (4.17) can be justiﬁed as follows. For the component 1 P2 −1 ϕ(t)dEt |H2 the problem of convergence is solved in a way similar to that for an orthogonal spectral function in a Hilbert space. Next, |ϕ(t)| gj (t) ∈ L2σ (E) for ev 1 2 ery function ϕ(t) ∈ Lν , so (recall Notation (4.4)) the integrals ϕ(t)d(Et hj , x) −1 1 ϕ(t)d(Et x, ej ), j = 1, 2, . . . , k, converge absolutely. Finally, integrals and 1 −1 ϕ(t)d(Et hl , xj ), l, j = 1, 2, . . . , k converge absolutely by (4.18b). −1

Remark 4.15. Since weak convergence and strong convergence of the improper in1 tegral P2 −1 ϕ(t)dEt |H2 can only take place simultaneously, the weak convergence of the improper integral (4.17) implies its strong convergence. Remark 4.16. Conditions (4.18) can be simpliﬁed by diﬀerent ways, depending on properties of Eλ . In particular, if k = 1 and ι11 (t) ≡ 0 (see Example 4.3), then Conditions (4.18) are reduced to Condition (4.18a). Note also that due to the #k inequality l,j=1 |ιlj (t)| < k 2 G(t) the condition 1 ϕ(t) ∈ L∞ σ ∩ Lν

(4.19)

is suﬃcient (but, generally speaking, not necessary) for the unconditional strong (≡weak) convergence of the improper integral (4.17). Moreover, if Eλ ∈ H, then # in some neighborhood of zero (see (4.8)) the inequality G(t) < 1 + kl,j=1 |ιlj (t)| holds, therefore in this case Condition (4.19) is necessary and suﬃcient for the convergence in the indicated sense of the integral (4.17).

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4.3. Monogenic algebras Here a problem of a functional representation for A generated by two operators A and I (A = A# ) is considered. The operator A is assumed to be ﬁxed. It is clear that A = Alg A so the e.s.f. of A is simultaneously an e.s.f. of A. Everywhere in this subsection Eλ means the e.s.f. of A. Condition (3.1) is assumed to be fulﬁlled.

[⊥] = ∞, then there is a regular subspace H4 ⊂ Ker A, Proposition 4.17. If dim H

[⊥] is ﬁnite. such that its co-dimension with respect to H

[⊥] = This proposition is easy to prove. In practice it means that the case dim H ∞ is negligible. Proposition 4.17 is mentioned here for case of reference. Proposition 4.18. Let B ∈ A and B|H1 = 0. Then B = S + B0

(4.20)

where B0 ∈ A0 , S is a nilpotent operator, and S|H

= 0. Proposition 4.18 can be proved using a scheme similar to the one used in the proof of Theorem 4.11. Comparing (4.16) with (4.20), note that the summand Q(A) in (4.20) is absent due to the condition B|H1 = 0. Start to construct a functional calculus for A. Remark 4.19. If the integral (4.17) weakly converges in the unconditional sense, set 1 ϕ (A) : = ϕ(t)dEλ . (4.21) −1

If ϕ(ξ) is a function regular in some neighborhood of σ(A), then, generally speaking, the operator ϕ(A) standardly deﬁned by the Riesz integral does not coincide with ϕ (A). If k0 is the maximal length of Jordan chains of the operator A|H1 then 1 2k 2 0 tk0 ∈ L∞ dEλ unconditionally converges in the strong σ ∩Lν and the integral −1 λ 1 topology but it can occur (see Proposition 4.18) that A2k0 = Z2k0 + −1 λ2k0 dEλ , where the operator Z2k0 = 0 has the same properties as the operator S from (4.20). But there exists a natural number m0 , such that 1 λm dEλ , m ≥ m0 , (4.22) Am = −1

where the integral has the same sense as above. This implies the representation 1 ϕ(λ)dEλ , (4.23) ϕ(A) = −1

valid for every function ϕ(ξ) regular in some neighborhood of σ(A) and having in zero a root with multiplicity no less that m0 . Below we use Formula (4.23) not only for the regular function ϕ(t) but also accept it as deﬁnition in the case when ϕ(t) is an arbitrary µσ -measurable function satisfying in some neighborhood of zero the condition ϕ(t) = O(tm0 ). (4.24)

A Functional Description

323

Proposition 4.20. Let ϕ(t) be a ﬁxed function satisfying Conditions (4.18). Then there is a positive constant cϕ , such that for every function θ(t) ∈ L∞ σ and the

function ψ(t) = θ(t)ϕ(t) the inequality ψ (A) ≤ cϕ θ(t)L∞ holds. σ

Proof. The deﬁnition of ψ (A) implies

ψ (A)hj = 2

ψ(t) gj (t)2L2 (E) σ

k + |

≤

θ(t)2L∞ σ

×

−1

l=1

1

ψ(t)ιlj (t)dσ(t)|2

ϕ(t) gj (t)2L2 (E) σ

ψ (A)x2 = ψ(t)f (t)2L2 (E) + σ

k | j=1

k + (

|ϕ(t)||ιlj (t)|dσ(t))

2

−1

l=1

1

;

1

−1

[ψ(t)f (t), g j (t)]E dσ(t)|2

⎛

≤ θ(t)2L∞ · f (t)2L2 (E) · ⎝ϕ(t)2L∞ + σ σ σ

⎛ · x2 · ⎝ϕ(t)2L∞ + = θ(t)2L∞ σ σ

k j=1

k j=1

1

−1

⎞ 1 −1

ϕ(t) ¯ gj (t)2E dσ(t)⎠ ⎞

ϕ(t) ¯ gj (t)2E dσ(t)⎠ ,

where x ∈ H2 , f (t) = W −1 x. Since H0 is a ﬁnite-dimensional subspace and

ψ (A)|H1 = 0, the rest is trivial.

Proposition 4.21. Let ψ(t) be a continuous function vanishing near zero. Then there is a sequence of polynomials {Pj (t)}∞ j=1 , such that lim {Pj (A)−ψ(A) = 0. j→∞

Proof. Let m0 be as deﬁned in (4.22). Take a sequence {Qj (t)}∞ j=1 of polynomials uniformly converging on [−1; 1] to the continuous function ψ(t) · t−m0 vanishing near zero. Set Pj (t) : = Q(t) · tm0 , ψj : = tm0 · (Qj (t) − ψ(t) · t−m0 ), j = 1, 2, . . . , ϕ(t) = tm0 . Then by Proposition 4.20 ψ(A)−Pj (A) = ψj (A) ≤ cϕ max {Qj (t) − ψ(t) · t−m0 }. t∈[−1;1]

Corollary 4.22. The subalgebra A0 coincides with the closure in the strong topology of the set of all operators {Q(A)}, where Q(t) runs through the set of all polynomials satisfying (4.24). Theorem 4.23. Let B ∈ A (= Alg A). Then B = Q(A) + C,

(4.25)

where Q(t) is a polynomial, C ∈ A0 . Conversely, if B is represented by (4.25), then B ∈ A.

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V. Strauss

Proof. Using the same reasoning as for the proof of Theorem 4.11 and Representation (4.25) one can assume that B is a nilpotent operator and B|H

= 0.

(4.26)

Since H1 ∩ Ker A = {0}, then B|Ker A = 0 and thanks to Proposition (4.13) one

it implies that n ≥ k. For

[⊥] = n < ∞. Since H1 ⊂ H, can assume that dim H simplicity we re-denote the union of the systems {ej }kj=1 and {hj }kj=1 , setting y1 = e1 , y2 = e2 , . . . , yk = ek , yk+1 = h1 , yk+2 = h2 , . . . , y2k = hk , and, if n > k, complete this system in an arbitrary way to a orthonormalized basis {yj }n+k j=1 in [⊥]

the space H ⊕ H0 . Next, ﬁx the maximal number of linearly independent relations of the form ψυ (t) =

k

(υ)

αlj ιlj (t) ∈ L1σ + L2η ,

υ = 1, 2, . . . , Υ.

(4.27)

l,j=1

It can occur that such relations are absent, i.e., the system {ιlj (t)}k1 is linearly independent modulo L1σ + L2η . Consider this case later, i.e., for now assume that 1 ≤ Υ ≤ k 2 . Note also that the set of conditions 2k k

(υ)

αlj (Syl , yj ) = 0,

υ = 1, 2, . . . , Υ,

(4.28)

l=k+1 j=1

is equivalent to Condition (4.6b). By virtue of the condition B ∈ Alg A and Condition (4.26) there is a sequence of polynomials {Rp (t)}∞ 1 , such that Rp (A)|H1 = 0,

p = 1, 2, . . . ,

and a) lim (Rp (A)yl , yj ) = (Byl , yj ), p→∞ 1 b) lim Rp (t)ψυ (t)dσ(t) = 0, p→∞

(4.29)

l, j = 1, 2, . . . , n + k;

⎫ ⎪ ⎬

υ = 1, 2, . . . , Υ.

⎪ ⎭

−1

(4.30)

Condition (4.29) means that the number t0 = 0 is a root of all polynomials Rp (t) and its multiplicity for Rp (t) is no less than k0 , where k0 , as above, is the maximum length of Jordan chains for the operator A|H1 . Thus, the operator Rp (A) from (4.29) and (4.30) represents a (ﬁnite) linear combination of operators from the set {Ak0 +j }∞ j=0 . The latter operators have the representation (4.20) Am = Sm + (Am )0 ,

(Am )0 ∈ A0 ,

m = ko , k0 + 1, . . . .

(4.31)

The elements of Decomposition (4.31) are not, generally speaking, uniquely deﬁned. This indeﬁniteness is not convenient in what follows, so we ﬁx a decision making for picking Sm and (Am )0 in (4.31). First, choose a decomposition Ak0 = Sk0 + (Ak0 )0 , ﬁx it and, second, set Sk0 +l = Al Sk0 ,

(Ak0 +l )0 = Al (Ak0 )0 .

(4.32)

A Functional Description

325

By Corollary 4.10 Al (Ak0 )0 = s − lim Al ϕm (A) = s − lim ψm (A), where {ϕm (t)}∞ 1 m→∞

m→∞

is some sequence of continuous functions vanishing near zero, ψm (t) = tl ϕm (t), so Al (Ak0 )0 ∈ A0 . Hence it is shown that (4.32) is a uniquely deﬁned representation of the type (4.31). Thus, every operator Rp (A), satisfying Condition (4.29) has the representation Rp (A) = Zp + Dp , (4.33) where (see (4.32)) Zp ∈ Z : = Lin{Sk0 +l }∞ l=0 , Dp ∈ GRp (Eλ ). Then Z is a ﬁnite-dimensional linear space consisting of nilpotent operators {Z}. For every [⊥]

[⊥] ⊕ H0 is Z-invariant and Z|H2 = 0. Let dim Z = K. Z the subspace H2 = H Since the operators from Z commute, it implies that K < (k + n)2 . Choose and ﬁx (k + n)2 − K independent conditions k+n

(m)

βlj zlj = 0,

m = 1, 2, . . . , (k + n)2 − K,

(4.34)

l,j=1

where zlj : = (Zyl , yj ), that hold if and only if Z ∈ Z. For the elements of De(p) (p) (p) composition (4.33) set rlj : = (Rp (A)yl , yj ), zlj : = (Zp yl , yj ), dlj : = (Dp yl , yj ), l, j = 1, 2, . . . , n + k. These values satisfy the following relations (see (4.27), (4.28)): ⎫ (p) (p) (p) ⎪ a) zlj + dlj = rlj , l, j = 1, 2, . . . , k + n; ⎪ ⎪ ⎪ k+n ⎪ (m) (p) ⎪ ⎪ 2 ⎪ ⎪ βlj zlj = 0, m = 1, 2, . . . , (k + n) − K; b) ⎪ ⎬ l,j=1 (4.35) 2k k 1 ⎪ (υ) (p) ⎪ ⎪ ⎪ αlj dlj = Rp (t)ψυ (t)dσ(t), υ = 1, 2, . . . , Υ, c) ⎪ ⎪ ⎪ −1 ⎪ l=k+1 j=1 ⎪ ⎪ ⎭ (p) d) dlj = 0 if l ≤ k or l >2k or j > k. Set blj : = (Byl , yj ), l, j = 1, 2, . . . , k + n, and consider the system a) zlj + dlj = blj , l, j = 1, 2, . . . , k + n; k+n (m) b) βlj zlj = 0, m = 1, 2, . . . , (k + n)2 − K; c)

l,j=1 2k

k

l=k+1 j=1

(υ) αlj dlj

= 0, υ = 1, 2, . . . , Υ;

d) dlj = 0 for l ≤ k or l >2k or j > k.

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.36)

Relations (4.36) is a system of linear equations with respect to the unknowns {zlj , dlj }k+n l,j=1 . It has the same principal matrix as System (4.35) and the righthand side of System (4.36) is obtained from the right-hand side of System (4.35) via passage to the limit as p → ∞. Since System (4.35) is solvable, System (4.36)

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is solvable by the well-known theorem of Kronecker-Capelli. Take some solution of (4.36). Then the operator Z given by the conditions Z|H2 = 0,

[⊥]

ZH2

[⊥]

⊂ H2 ,

(Zyl , yj ) = zlj ,

l, j = 1, 2, . . . , k + n,

belongs to Z. Hence there is a polynomial R(t) satisfying Condition (4.29) and such that R(A) = Z + (R(A))0 , where R(A))0 ∈ GR (Eλ . Moreover, the solvability of System (4.36) implies the existence of the operator D ∈ S(Eλ ), such that (Dyl , yj ) = dlj , l, j = 1, 2, . . . , k + n and, ﬁnally, the representation B = R(A) − (R(A))0 + D. Thus, the theorem has been proved under Hypothesis (4.27). If the system {ιlj (t)}k1 is linearly independent modulo L1σ + L2η , then the set S(Eλ ) is subjected only Condition (4.6a), i.e., the operator D satisﬁes Conditions D|H2 = 0, [⊥] [⊥] DH2 ⊂ H2 and Condition (4.36d), then D ∈ S(Eλ ). The latter means that if there are no Conditions (4.27), the proof follows the similar way, as above, with the exclusion of Equality (4.35c) and Equation (4.36c). Corollary 4.24. Alg A is equal to the sequential closure of the set {Q(A)) in the strong topology, where {Q(t)} runs through the totality of (all) polynomials. Consider now a problem of representation of a monogenic uniformly closed algebra. Let Cσ0 be the space of functions {φ(t)} continuous on Supp(µσ ) and satisfying the condition φ(0) = 0. The norm on Cσ0 is deﬁned in the standard way, i.e., φCσ0 = max {|φ(t)|}. Then the space Cσ0 ∩L2ν is well deﬁned as a Banach t∈Supp(µσ )

space with the norm φCσ0 ∩L2ν = max{φCσ0 , φL2ν }. Corollary 4.25. An operator B belongs to the closure of the set {Q(A)} (as above Q(t) runs through the totality of polynomials) in the uniform topology if and only if there exists a function ϕ(t) ∈ Cσ0 ∩ L2ν and a polynomial Q(t), such that B = Q(A) + C,

C ∈ Gϕ (Eλ ).

(4.37)

Proof. Indeed, if B belongs to the uniform algebra, generated by A and I, then B ∈ Alg A, therefore B has Representation (4.25). On the other hand it is clear that the portrait of B under the given conditions must belong to the set of functions continuous on Supp(µσ ). So, the necessity of Condition (4.37) has been established. In turn, the suﬃciency of (4.37) follows from the corresponding results for selfadjoint operators, from the ﬁnite dimensionality of the spaces H0 and H1 and from Remark 2.6.

5. W J ∗ -algebras with an arbitrary number of peculiarities Now we start to consider the problem of the functional description of the algebra A assuming that some J-s.a. operators from A have non-real spectra and that its e.s.f. Eλ does not satisﬁes Conditions (3.1b),c)). We assume (with no loss of generality) that Supp(Eλ ) ⊂ (−1; 1). Then for the set of spectral peculiarities Λ

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the relation Λ ⊂ (−1; 1) holds, E−1 = E−1+0 = 0, Eλ = const for λ ≥1, but, generally speaking, E1 = I. If the latter holds, the operator I − E1 is a J-orthoprojection onto a ﬁnite-dimensional A-invariant subspace, such that there exists at least one J-s.a. operator A ∈ A with σ(A|(I−E1 )H ) ∩ R = ∅. For the construction of a functional representation for A one needs to modify 1 1 2 the description of some key function spaces, say L∞ σ ∩ Lν , Lσ + Lη , etc. Let the

be the same as in (1.6d). The notation from (3.2) also remains valid. subspace H ∗

∗

Consider (cf. Remark 3.3) the factor-space H : = H/H1 . Then H is a Krein ∗

λ induces on this space the J-orth.sp.f. Eλ without peculiarities. Asspace and E ∗

sume (with no loss of generality) that Eλ is both orthogonal and J-orthogonal. ∗

Then Eλ is similar to the operator-valued function Xλ (the multiplication operator by the indicator of the set (−∞; t) denoted χ(−∞;t) (τ )) acting in some standard Krein space J -L2σ (E). The corresponding operator of similarity is simultaneously symmetric and J-symmetric. Fix J -L2σ (E) and the corresponding function σ(t) giving the measure on the standard Krein space. Note that σ(t) is continuous in all points from Λ. Let {αj }n−1 j=1 ⊂ (−1; 1) be a set of numbers taken in ascending order. The set (−1; 1)\Λ is decomposed as the union of disjoint open intervals. Choose {αj }n−1 j=1 ⊂ (−1; 1) such that every point from Λ belongs to one and only one of these intervals. (j) Set α0 = −1 and αn = 1. Then the spectral function Eλ : = Eλ |(Eαj+1 −Eαj )H 2,(j)

has one and only one point of peculiarity, j = 1, 2, . . . n−1. Denote J -Lσ (E) the subspace of J -L2σ (E) that contains all functions from J -L2σ (E) vanishing outside (j) the interval [αj ; αj+1 ). There are two options for Eλ . The ﬁrst is that its unique

2,(j) (E) : = J -L2,(j) (E) and ﬁx a corresponding peculiarity is regular. Then set J -L σ σ 2,(j)

operator of similarity W (j) : J -L (E) → (Eαj+1 − Eαj )H. The second is that the σ

2,(j) (E) for E (j) peculiarity is singular. Then construct a basic model space J -L λ σ 2,(j) expanding J -Lσ (E) with the evident modiﬁcations related with the position of the peculiarity, that now does not coincide with zero. In this option ﬁx some

2,(j) (E) and corresponding operator of similarity W (j) that here acts between J -L σ

Now set (Eαj+1 − Eαj )H.

2,(0) (E) ⊕ · · · ⊕ J -L

2,(n) (E), W : = W (0) ⊕ · · · ⊕ W (n) . (5.1)

2 (E) : = J -L J -L σ σ σ There are again two options. The ﬁrst is that Λ has only regular peculiarities and the second one appears if there is a singular peculiarity in Λ. In the ﬁrst case the

2 (E) coincides with J -L2 (E) and a functional representation for A| is space J -L σ σ H in fact the same as for a commutative W ∗ -algebra. This case is quite simple and will be considered later. So assume that Λ contains at least one singular peculiarity and hence the sub 2 (E) there is a system of functions { space H1 is nontrivial. Then in J -L gj (t)}kj=1 σ

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2 (E) that are linearly independent modulo J -L2σ (E) and by the deﬁnition of J -L σ form an orthonormalized system. With no loss of generality assume that g j (t) ≡ 0 on [αl ; αl+1 ) if this interval contains a regular singularity. Let us use formulas from (4.4) taking into account however that now W is deﬁned by (5.1) and W ↑ : = JW . Continuing to use the old notation deﬁne G(t) by (4.8). Then thanks to the hypothesis assumed above G(t) ≡ 1 on [αl ; αl+1 ) if this interval contains a regular singut larity. Deﬁne ν(t) in the following way: set ν(t) : = −1 G(t)dσ(t) for every t that t is less than the ﬁrst singular peculiarity, say λj0 ; next, set ν(t) : = αj +1 G(t)dσ(t) 0 for every t greater than λj0 and less than the next singular peculiarity, say λj1 , etc. Finally, set t η(t) : = (1/G(τ ))dσ(τ ), t ∈ [−1; 1]. −1

Thus, the space L1σ + L2η is well deﬁned. 2 1 2 ∗ The space L2ν can be deﬁned via the equality L∞ σ ∩ Lν = (Lσ + Lη ) or via the following description. Let f (t) and g(t) be arbitrary continuous scalar functions deﬁned on [−1; 1] and vanishing near all singular peculiarities from Λ. 1 Let (f (t), g(t) : = −1 f (t)¯ g(t)dσ(t). The set of all these functions equipped with the given scalar product forms a pre-Hilbert space. Its completion is by deﬁnition the space L2ν . Pass to descriptions of some operator subalgebras of A. Let ϕ(t) be a continuous scalar function vanishing near Λ. Set 1 Bϕ = ϕ(t)dEλ , (5.2) −1

where the improper integral has the obvious meaning. Let S(Eλ ) be a collection of operators {S} satisfying the conditions a) S = P1 SP0 ; k k b) if αlj ιlj (t) ∈ L1σ + L2η , then αlj (Shl , ej ) = 0. l,j=1

l,j=1

These conditions coincide formally with ones given in (4.6) but really are a generalization of the last due to the arbitrariness in the location of Λ. Up to this point it was assumed that Λ contains at least one singular peculiarity. Now we describe how to introduce function and operator sets needed below if there are no singular peculiarities in Λ. In particular, in this case ν(t) ≡ 2 ∞ 1 2 1 η(t) ≡ σ(t), so L∞ σ ∩ Lν = Lσ and Lσ + Lη = Lσ , moreover S(Eλ ) = {0}. At the same time Formula (5.2) can be applied independently to the characterization in question of Λ. Denote AΛ the weak closure of the operator set {Bϕ } generated by (5.2). Let Gψ (Eλ ) be the subset of the operators from AΛ which are originals for ψ(t) with respect to Eλ .

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Theorem 5.1. An operator A belongs to AΛ if and only if the following conditions hold: ⎫ a) AEλ = Eλ A for every λ ∈ [−1; 1]\Λ; ⎪ ⎪ ⎪

⎪ b) A|H ⎪

[⊥] = 0, AH ⊂ H; ⎪ ⎪ ∞ 2 ⎪ c) there is a function ϕ(t) ∈ Lσ ∩ Lν , such that ϕ(t) is the⎪ ⎪ ⎬ portrait of A with respect to Eλ ; (5.3) #k 1 2 ⎪ d) if H1 = {0} and ψ(t) = l,j=1 αlj ιlj (t) ∈ Lσ + Lη , then ⎪ ⎪ ⎪ 1 ⎪ k ⎪ ⎪ ⎪ ⎪ αlj (Ahl , ej ) = ϕ(t)ψ(t)dσ(t). ⎪ ⎭ −1 l,j=1

This theorem is, evidently, a generalization of Theorem 4.7 and is its corollary if one applies Theorem 4.7 separately for every singular peculiarity from Λ. If J-orth.sp.f. Eλ has no singular peculiarities, then H1 = {0} and Theorem 5.1 coincides in fact with the corresponding theorem for self-adjoint operators. 2 Corollary 5.2. For every function ϕ(t) ∈ L∞ σ ∩ Lν the inequality Gϕ (Eλ ) = ∅ holds and if B ∈ Gϕ (Eλ ) is a ﬁxed operator, then Gϕ (Eλ ) = {B + S}S∈S(Eλ ) ; in particular, G0 (Eλ ) = S(Eλ ).

Corollary 5.3. S(Eλ ) coincides with the strong sequential closure of the set of operators given by (5.2). Theorem 5.4. For a commutative W J ∗ -algebra A ∈ Dκ+ there is a ﬁnite collection of J-s.a. operators A1 , A2 , . . . , Al ∈ A, such that every operator B ∈ A has the representation B = S + Q(A1 , A2 , . . . , Al ) + C, (5.4) where S ∈ A is a nilpotent operator, S|H = S # |H = 0, Q(t1 , t2 , . . . , tl ) is a polynomial in l variables, and C ∈ AΛ . Proof. If E(R) = I, one can take into account Theorem 5.1 and repeat the scheme used for the proof of Theorem 4.11. If E(R) = I, then the ﬁrst step step is the fol1 , A 2 , . . . , A l ∈ A = A|E(R)H , such that every operlowing. Find J-s.a. operators A = S + Q( A 1 , A 2 , . . . , A l )+ ator B ∈ A has a representation of the type (5.4), say B ˇ : = A|H with H3 : = (I − E(R))H. Since Next, consider the algebra A C. 3 H3 is ﬁnite-dimensional, the latter algebra has a ﬁnite number of generators, so ˇ such that there exists a collection of J-s.a. operators Aˇl+1 , Aˇl+2 , . . . , Aˇn ∈ A, ˇ has a representation B ˇ ∈ A ˇ = Q( ˇ Aˇl+1 , Aˇl+2 , . . . , Aˇn ), where every operator B j E(R)x if j = ˇ l+1 , tl+2 , . . . , tn ) is a polynomial. Let x ∈ H. Put Aj x : = A Q(t ˇ 1, 2, . . . , l, and Aj x : = Aj (I − E(R))x whenever j = l + 1, l + 2, . . . , n. The rest is straightforward. Corollary 5.5. Let A ∈ Dκ+ be a commutative W J ∗ -algebra and let F (A) be the set of all portraits corresponding to operators in A. Then there is a ﬁnite

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collection of J-s.a. operators A1 , A2 , . . . , Al ∈ A with corresponding portraits fA1 (λ), fA2 (λ), . . . , fAl (λ), such that f (λ) ∈ F(A) if and only if f (λ) = Q(fA1 (λ), fA2 (λ), . . . , fAl (λ)) + φ(λ), 2 where Q(t1 , t2 , . . . , tl ) is a polynomial in l variables, and φ(λ) ∈ L∞ σ ∩ Lν .

Theorem 5.6. Let W J ∗ -algebra A ∈ Dκ+ be such that A = Alg A, A = A# . Then every operator B ∈ A has a representation B = Q(A) + C,

(5.5)

where Q(t) is a polynomial, C ∈ AΛ . Conversely, if an operator B is as in (5.5), then B ∈ A. Proof. With no loss of generality one can take the e.s.f. Eλ of A as an e.s.f of the algebra A. Next, let Λ = {λ1 , λ2 , . . . , λm }, and let open intervals ∆1 , ∆2 , . . . , ∆m ⊂ (−1; 1) be disjoint and such that ∆j ∩Λ = {λj }, j = 1, 2, . . . , m. Then the algebra A|E(∆j )H coincides with Alg((A− λj I)|E(∆j )H ). If λj is a singular peculiarity, then Theorem 4.23 is applicable, so B|E(∆j )H = Qj (A|E(∆j )H ) + Cj ,

Cj ∈ AΛ |E(∆j )H .

(5.6)

If λj is a regular spectral peculiarity, then there exists the limit E({λj }) = s − lim E((λj − ; λj + )) and the algebra A|E({λj })H is ﬁnite-dimensional, so →+0

B|E({λj })H = Qj (A)|E({λj })H .

(5.7)

Finally, if E(R) = I, then the algebra A|(I−E(R))H is also non-trivial and ﬁnitedimensional, so B|(I−E(R))H = Qm+1 (A)|(I−E(R))H . (5.8) Now let Q(t) be a polynomial “interpolating” the set of polynomials {Qj (t)}m+1 1 in the following sense. If λj is a singular peculiarity, then, ﬁrst, the integral ∆j (Q(t) − Qj (t))dEt unconditionally converges in the strong topology and, second, Q(A|E(∆j )H ) − Qj (A|E(∆j )H ) = ∆j (Q(t) − Qj (t))dEt |E(∆j )H , where Qj (t) corresponds (5.6). If λj is a regular spectral peculiarity, then Q(A|E({λj })H ) = Qj (A|E({λj })H ), where Qj (t) corresponds to (5.7). Finally, if E(R) = I, then Q(A)|(I−E(R))H = Qm+1 (A)|(I−E(R))H . Note that if E(R) = I, then the latter condition for Q(t) as well as Representation (5.8) must be omitted. The rest is straightforward. Corollary 5.7. Let A = A# ∈ Dκ+ and let F (A) be the set of all portraits corresponding to operators in Alg A. Then f (λ) ∈ F(A) if and only if f (λ) = Q(λ) + φ(λ), 2 where Q(λ) is a polynomial and φ(λ) ∈ L∞ σ ∩ Lν .

Up to this point we have assumed that A is a W J ∗ -algebra. Now we omit this preestablished condition and pass to conditions guaranteeing a weakly closed algebra to be a W J ∗ -algebra.

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Theorem 5.8. Let an algebra A ∈ Dκ+ be weakly closed, commutative and J-symmetric. Then A is a W J ∗ -algebra if and only if it is non-degenerate, i.e., for every vector x ∈ H there is an operator A ∈ A, such that Ax =0. Proof. One need only show that the identity belongs to a weakly closed commutative non-degenerate J-symmetric algebra A ∈ Dκ+ because the remaining assertions of the theorem are trivial. Let, as above, a subspace L+ ∈ M+ (H) ∩ h+ [⊥] be Y-invariant and L1 = L+ ∩ L+ . Denote Ωλ (A) the subspace of all eigenvectors and root vectors of an operator A that correspond to λ ∈ σp (A). If A = A# , λ ∈ σ(A|L1 ), λ = 0, then (recall that L1 is ﬁnite-dimensional) there is a polynomial Q(t) with real coeﬃcients such that Q(0) = 0 (0 here is a number!), Q(A)x = x for every x ∈ Ωλ (A) or (if λ ∈ R) x ∈ Ωλ (A) and Q(A)y = 0 for y ∈ Ωµ (A) if µ ∈ σ(A|L1 ), µ = λ, λ. Thus (A is non-degenerate and commutative!), there exists a J-s.a. operator A such that A|L1 = I|L1 . Fix this A. Let Eλ be the e.s.f. of A. Note that all spectral peculiarities of Eλ belongs to σ(A|L1 ), and if λ ∈ R and λ ∈ σp (A), then λ ∈ σ(A|L1 ) or λ ∈ σ(A|L1 ). But by the hypothesis σ(A|L1 ) = {1}, therefore the spectrum of A is real and the unique spectral peculiarity (if any) of Eλ is λ0 = 1. Moreover, the algebra A|E1/2 H has a pair of J-orthogonal uniformly deﬁnite invariant subspaces and therefore is a W ∗ -algebra with respect to a suitable canonical scalar product. So, the proof can be reduced to the testing of the condition I − E1/2 ∈ A. Let ⎧ −1/2, t ≤ −1/4; ⎪ ⎪ ⎨ 2n N (t) = 1 − (t − 1) ; ψ(t) = 2t, t ∈ (−1/4; 1/2); ⎪ ⎪ ⎩ 1, t ≥ 1/2; + ϕ(t) = ψ(t) − N (t); θ(t) =

ϕ(t)/t,

t = 0;

2(1 − n), t = 0.

For a suitable choice of n (see Remark 4.19, Formula (4.23), Condition (4.24) ∞ and Proposition 4.20) there is a sequence of polynomials {Ql (t)}1 , such that lim Ql (A) − θ(A) = 0, where θ(A) = R\{1} θ(t)dEt , and the integral is treated l→∞

in the same way as in (4.23). This implies that lim AQl (A) − Aθ(A) = 0. l→∞ At the same time Aθ(A) = ϕ(A) = R\{1} ϕ(t)dEt , so ϕ(A) ∈ A. Let B : = ϕ(A) + N (A). Since N (0) = 0, then B ∈ A. A direct veriﬁcation shows that 1/2 BE1/2 = (−1/2)E−1/2 + 0 2tdEt , B(I − E1/2 ) = (I − E1/2 ). This implies that s − lim B l = (I − E1/2 ). l→∞

Note that the condition of commutativity of the algebra A in the assertion of Theorem 5.8 is essential even if A is deﬁned on a ﬁnite-dimensional space.

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Example 5.9. Assume that x1 , x2 , x3 , x4 is a basis in a four-dimensional space. Make up an algebra A = {A} spanning four operators: A1 : A1 x1 = x2 , A1 x2 = x4 , A1 x3 = A1 x4 = 0; A2 : A2 x1 = A2 x2 = A2 x4 = 0, A2 x3 = x4 ; A3 : A3 x1 = x3 , A3 x2 = A3 x3 = A3 x4 = 0; A4 : A4 x1 = x1 , A4 x2 = x2 , A4 x3 = 0, A4 x4 = x4 . Give a canonical symmetry J by the following equalities: Jx1 = x4 , Jx4 = x1 , Jx2 = x2 , Jx3 = x3 . # # A direct veriﬁcation shows that A# 1 = A1 , A2 = A3 , A4 = A4 , so the given algebra is J-symmetric. It is also clear that A is non-degenerate. A calculation gives A1 A2 = A1 A3 = A2 A1 = A22 = A2 A4 = A3 A1 = A3 A2 = A23 = A4 A3 = 0, A1 A4 = A4 A1 = A1 , A3 A4 = A3 , A4 A2 = A2 , A24 = A4 . Finally, only two products lead to a new operator: A21 = A2 A3 = A5 , where A5 : A5 x1 = x4 , A5 x2 = A5 x3 = A5 x4 = 0. Since A1 A5 = A5 A1 = A2 A5 = A5 A2 = A3 A5 = A5 A3 = A25 = 0, A5 A4 = A4 A5 = A5 , every operator A ∈ A is a linear combination of the operators A1 , A2 , A3 , A4 and A5 . Thus I ∈ A.

6. Closing remarks Spectral functions are a traditional source for creating a functional calculus for diﬀerent classes of normal operators in Hilbert spaces, so it is natural to use the same technique in Krein spaces. The ﬁrst theorem concerning the existence of spectral functions for π-s.a. operators was published by M.Krein and H. Langer in [23] (see also [26] for detailed proofs). The spectral functions introduced in Subsection 1.3 are a particular case of generalized spectral functions [12]. Our deﬁnition is motivated by the spectral measures arising in Operator Theory in Indeﬁnite Metric Spaces but formally is independent (see [37]) of this Theory. A functional calculus based on the spectral resolution of so-called deﬁnitizable Js.a. and J-unitary operators was given for the ﬁrst time by H. Langer [25], [26]. Jonas [18] has results of a similar nature. The class Dκ+ that is considered in the present paper as well as the class K(H) (introduced by Azizov [3]) diﬀer from the class of deﬁnitizable operators (see [7] for discussion) and represent another natural generalization of the class of self-adjoint operators in Pontryagin spaces, since (as follows from one of the theorems of Naimark [30]) every commutative family of π-s.a. (or π-unitary) operators belongs to the class Dκ+ . The conception of unbounded elements was introduced in [35] (see also [36]). A connection between 2 the spaces of the type L∞ σ ∩ Lν and operator algebras was pointed out for the ﬁrst time in [31] (for Pontryagin spaces of range 1) and [25]. Some part of Sections 4 and 5 represents a detailed account of results announced in [39]. Subsection 4.3 is a generalization of results announced in [38], where the case of a monogenic algebra in an arbitrary Pontryagin space was analyzed. The subject of this paper is very close to the theory of model representations for π-s.a. operators and algebras (a majority of works consider the case of range 1) [32], [29], [41], [28] (see also [17] for

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more references), [27]. The papers [19] and [20] deal with the case of a self-adjoint cyclic operator acting in arbitrary Pontryagin space. Acknowledgment The author would like to thank the Organizing Committee of the Vienna Colloquium for the opportunity to participate in this notable scientiﬁc event. He also is very grateful to Prof. Tom Berry for his assistance to making the paper more readable.

References [1] N.I. Akhiezer, I.M. Glazman, Theory of linear operators in Hilbert space. Pitman: London, 1981. [2] T.Ya. Azizov, I.S. Iokhvidov, Linear operators in spaces with an indeﬁnite metric. “Matematicheskii analiz” (Itogi nauki i tehniki, VINITI) 17 (1979) Moscow, 113–205 (Russian). [3] T.Ya. Azizov, I.S. Iokhvidov, Foundation of the Theory of Linear Operators in Spaces with Indeﬁnite Metric. Nauka: Moscow, 1986 (Russian); Linear Operators in Spaces with Indeﬁnite Metric. Wiley: New York, 1989 (English). [4] T.Ya. Azizov, V.A. Strauss, Spectral decomposition of self-adjoint and normal operators in Krein space. Research Paper No 827 (2002) Department of Mathematics and Statistics, University of Calgary, Calgary, Canada. [5] T.Ya. Azizov, V.A. Strauss, Spectral decompositions for special classes of self-adjoint and normal operators on Krein spaces. Spectral Theory and its Applications, Proceedings dedicated to the 70th birthday of Prof. I. Colojoar˘ a, Theta 2003, 45–67. [6] T.Ya. Azizov, V.A. Strauss, On a spectral decomposition of a commutative operator family in spaces with indeﬁnite metric. MFAT 11 (2005) No 1, 10–20. [7] T.Ya. Azizov, L.I. Sukhocheva, V.A. Shtraus (=Strauss), Operators in Krein space. Matematicheskie Zametki 76 (2004) No 3, 324–334 (Russian); Math. Notes 76 (2004) No 3, 306–314 (English). [8] J. Bergh, J. L¨ ofstr¨ om, Interpolation spaces. An Introduction. Springer Verlag, NY, 1976. [9] J. Bognar, Indeﬁnite inner product spaces. Springer-Verlag, NY, 1974. [10] M.S. Birman, M.Z. Solomyak, Spectral theory of self-ajoint operators in a Hilbert space. Leningrad University, 1980 (Russian). [11] O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics. Volume 1. Springer-Verlag, NY, 1979. [12] I. Colojoar˘ a, C. Foia¸s, Theory of generalized spectral operators. Gordon and Breach, 1968. [13] N. Dunford, J.T. Schwartz, Linear Operators. General Theory. John Wiley & Sons, 1958. [14] N. Dunford, J.T. Schwartz, Linear Operators. Part III. Spectral operators. John Wiley & Sons, 1971.

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[15] A. Gheondea, Pseudo-regular spectral functions in Krein spaces. J. Operator Theory 12 (1984), 349–358. [16] I.S. Iokhvidov, M.G. Krein, H. Langer, Introduction to the spectral theory of operators in spaces with an indeﬁnite metric. Akademie-Verlag, Berlin, 1982. [17] R.S. Ismagilov, M.A. Naimark, Representations of groups and algebras in spaces with indeﬁnite metric. “Matematicheskii analiz” (Itogi nauki i tehniki 1968, VINITI) (1969), Moscow, 73–105 (Russian). [18] P. Jonas, On the functional calculus and the spectral function for deﬁnitizable operators in Krein space. Beitr. Anal. 16 (1981), 121–135. [19] P. Jonas, H. Langer, A model for π-selfadjont operators in π1 -spaces and a special linear pencil. Integral Equations Opererator Theory 8 (1985) No 1, 13–35. [20] P. Jonas, H. Langer, B. Textorius, Models and unitary equivalence of ciclic selfadjoint operators in Pontrjagin spaces. Operator Theory and complex analysis (Sapporo, 1991), 252–284, Oper. Theory Adv. Appl. 59 Birkh¨ auser, Basel, 1992. [21] L.V. Kantorovich, G.P. Akilov, Functional analysis, 2nd edition. Nauka (Moscow) 1977. [22] E. Kissin, V. Shulman, Representations of Krein spaces and derivations of C ∗ algebras. Pitman Monographs and Surveys in Pure and Applied Mathematics 89 Addison-Vesely Longman, 1997. [23] M.G. Krein, H. Langer, On the spectral function of a self-adjoint operator in a space with indeﬁnite metric. Dokl. Akad. Nauk SSSR, 152 (1963), 39–42 (Russian). [24] S.G. Krein, Yu.I. Petunin, Ye.M. Semyonov, Interpolation of linear operators. Nauka, Moscow, 1978 (Russian). [25] H. Langer, Spectraltheorie linearer Operatoren in J-r¨ aumen und enige Anwendungen auf die Shar L(λ) = λ2 I +λB +C. Habilitationsschrift, Tech. Univer., Dresden, 1965. [26] H. Langer, Spectral functions of deﬁnitizable operators in Krein space. Lect. Not. Math. 948 (1982), 1–46. [27] H. Langer, B. Textorius, L-resolvent matrices of symmetric linear relations with equal defect numbers; applications to canonical diﬀerential relations. Integral Equations Operator Theory 5 (1982) No 2, 208–243. [28] S.N. Litvinov, Description of commutative symmetric algebras in the Pontryagin space Π1 . DAN UzSSR (1987) No 1, 9–12 (Russian). [29] A.I. Loginov, Complete commutative symmetric algebras in Pontryagin spaces Π1 . Mat. Sbornik 84 (1971) No 4, 575–582 (Russian). [30] M.A. Naimark, On commutative unitary operators in a space Πκ . Dokl. Akad. Nauk SSSR 149 (1963), 1261–1263 (Russian). [31] M.A. Naimark, On commutative operator algebras in a space Π1 . Dokl. Akad. Nauk SSSR 156 (1964), 734–737 (Russian). [32] M.A. Naimark, Commutative algebras of operators in a space Π1 . Rev. roum. math. pures et appl. 9 (1964) No 6, 499–529.(Russian) [33] M.A. Naimark, Normed Algebras. Wolters-Nordhoﬀ Publishing, Groningen, The Nedherlands, 1972. [34] M. Reed, B. Simon Methods of Modern Mathematical Physics. I: Functional Analysis. 2nd Edition, Acad. Press, Inc., 1980.

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[35] V.A. Strauss, Some singularities of the spectral function of a π-selfadjoint operator. Collection: Funktsionalnii analiz. Operator Theory, State Pedagogical Institute of Uliyanovsk, Ulyanovsk, USSR (1983), 135–146 (Russian) [36] V.A. Strauss, A model representation for a simplest π-selfadjoint operator. Collection: Funktsionalnii analiz. Spectral theory, State Pedagogical Institute of Uliyanovsk, Ulyanovsk, USSR (1984), 123–133 (Russian). [37] V.A. Strauss, Integro-polynomial representation of regular functions of an operator whose spectral function has critical points. Dokl. Akad. Nauk Ukrain. SSR. Ser. A 8 (1986), 26–29 (Russian). [38] V.A. Strauss, Functional representation of an algebra genereted by a selfadjoint operator in Pontryagin space. Funktsionalnyi analiz i ego prilozheniya 20 (1986) No 1, 91–92 (Russian), English translation: Funct. Anal. Appl. [39] V.A. Strauss, The structure of a family of commuting J-self-adjoint operators Ukrain. Mat.Zh., 41 (1989), No. 10, 1431–1433, 1441 (Russian). [40] V.A. Strauss, On models of functional type for a special class of commutative symmetric operator families in Krein spaces. Submitted to Proceedings 4th Workshop Operator Theory in Krein Spaces and Applications, Berlin, 2004. [41] V.S. Shulman, Symmetric Banach algebras of operators in a space of type Π1 . Mat. Sb. (N.S.) 89 (1972) No 2, 264–279 (Russian). Vladimir Strauss Department of Pure & Applied Mathematics USB, Sartenejas-Baruta Apartado 89.000 Caracas 1080-A, Venezuela e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 337–350 c 2005 Birkh¨ auser Verlag Basel/Switzerland

On Normal Extensions of Unbounded Operators: IV. A Matrix Construction Franciszek Hugon Szafraniec to Heinz again, with pleasure

Abstract. A condition for an unbounded operator to have a normal extension, which is a matrix operator, is given. The circumstances under which this condition may become necessary are discussed as well and ﬁnally a question is posed. By the way some substantial facts concerning inﬁnite operator matrices with unbounded entries are gathered. Mathematics Subject Classiﬁcation (2000). Primary 47B15, 47B12 . Keywords. Normal operator, subnormal operator, matrix operator.

Bounded subnormals constitute a vital class in the theory of operators in Hilbert space. Furthermore, unbounded subnormal operators are present in quantum mechanic, see [5, Ch. III, Paragraph 10] and [12], which lays stress on their importance. Though the unbounded case bears the central ideas of subnormality, it causes a lot of hassle in implementing them. Some basic facts concerning the theory can be found in the trilogy [7], [8] and [9] as the most recent and far going results are in [10]. Unfortunately, all of them concern the case when an operator in question has invariant domain. The only results without this troublesome restriction are in [3] and [4]. In [1] Andˆ o has answered to the need of providing a universal construction of a normal extension, independent of an operator one deals with. While Andˆ o’s approach has been intended as a way of shedding more light on normal extensions of bounded operators, ours, which bases upon that in [1], creates a chance to add one more criterion for subnormality of unbounded operators to the very few existing so far.

This work was supported by the KBN grant 2 P03A 037 024.

338

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So as to achieve the goal a weighty piece of theory of operators coming from inﬁnite matrices with unbounded entries has to be built up from scratch 1 . Description of the adjoint of those operators is especially underlined and it turns out column operators come into play in a natural way. This part may be of independent interest. Let us make Proclamation. All the Hilbert spaces are complex and the operators considered in them are densely deﬁned.

Operator matrices versus matrix operators

N∞ 1. Let K = n=0 Hn be the orthogonal sum of Hilbert spaces Hn and let Pn stand for the orthogonal projection of K onto Hn , n ≥ 0. Denote by Kﬁn the space of all vectors f ∈ K such that each Pn f ∈ Hn and Pn f = 0 for all but a ﬁnite number of indices n. We identify g ∈ Hn with 0 ⊕ · · · ⊕ 0 ⊕ g ⊕ 0 ⊕ · · · ∈ K if g is properly ) ) placed. This, in return, legitimizes us to write Pn f ∈ Hn for f ∈ K. Remark 1. Let A be a closable operator 2 in K such that Pn D(A) ⊂ D(A),

n = 0, 1, . . .

(1)

Then for f ∈ D(A) and g ∈ K such that Pn g ∈ D(A∗ ) for all n Af, gK = = =

∞ O

Pn Af,

n=0 ∞

∞ O

Pn g K =

n=0

∗

Pk f, Pk A Pn gHk =

=

∞

Af, Pn gK

n=0

∞ O ∞ ∞ O Pk f, Pk A∗ Pn gK n=0 k=0

n=0 k=0 ∞ ∞ O

Pn Af, Pn gHn =

n=0

f, A∗ Pn gK =

n=0 ∞ ∞

∞

k=0 ∞ ∞

(2) Pn APk f, Pn gHn

n=0 k=0

Pn APk f, g K .

n=0 k=0 1

A common association with inﬁnite matrices is that of Jacobi ones; for some treatment in this matter we suggest to look into [11], [13] (scalar entries) and [2], [6] (bounded operator entries), the latter considers a bit more general matrices. Matrices with unbounded operator entries are dealt with rather occasionally, usually of ﬁnite dimension, and a thorough exposition of the topic would be diﬃcult to point out in the literature. 2 Notation: D(A), R(A), N (A), A ¯ and A∗ stands for the domain, range, kernel, closure and adjoint of an operator A in a Hilbert space H, resp. If D ⊂ D(A) is dense in H then A|D denotes the restriction of A to D, that is an operator in H with domain D.

On Normal Extensions of Unbounded Operators

339

The last equality is a heuristical step, anyway if this happens then Af =

∞ ∞ O

Pn APk f,

f ∈ D(A),

n=0 k=0

as long as condition (1) holds. This shows the way the operator A determines a matrix (Pi APj )∞ i,j=0 of operators or, in other words, an operator matrix. Formula (2) suggests the way back: from matrix to operator; however it requires to agree much in advance on the domain of the adjoint of the prospective operator; also the condition (1) would be diﬃcult to meet when the operator is already closed. The case presented below tries to minimize this annoyance. Suppose a matrix A = (Ai,j )∞ i,j=0 of closable operators Ai,j from (a dense subspace of) Hj into Hi is given. Suppose, moreover, a sequence D = (Dn )∞ n=0 of linear spaces such that each Dn is dense in Hn and each Dk ⊂ D(An,k ) for all n, is chosen. Apparently, the linear space Dﬁn = Kﬁn ∩ def

∞ O

Dn

n=0

is dense in K. Under these formalities we deﬁne in two steps a relevant operator in K: 1 the linear space DA,D is composed of all these f ∈ Dﬁn for which 3 * ∞ An,k Pk f 2Hn < +∞, n=0

k

if DA,D is dense in K, we say the couple (A, D) determines an operator; 2 if (A, D) determines an operator, set D(MA,D ) = DA,D and deﬁne * def

def

MA,D f =

∞ O

An,k Pk f,

f ∈ D(MA,D ).

n=0 k

Then the operator MA,D is said to be determined by the couple (A, D) and one may think of it as a matrix operator. Remark 2. Notice, under our circumstances, always DA,D ⊂ Dﬁn . So, Dﬁn ⊂ DA,D

⇐⇒

D(MA,D ) = Dﬁn ;

if this happens the couple (A, D) certainly determines an operator. In particular, this is the case the matrix A has only ﬁnite number of diagonals diﬀerent from 0. × ∞ ∗ When one deﬁnes the × operation on matrices by A× = (A× i,j )i,j= , Ai,j = Aj,i def

−

− def

for all i, j, then apparently A×× = A where A = ( Ai,j )∞ i,j=0 . 3

Dropping limits under the summation sign brings to mind it to be ﬁnite.

def

340

F.H. Szafraniec

2. Now a need may appear to know more about the adjoint of an operator determined by a matrix A. In the sequel we are going to shed some additional light on this. For this suppose another sequence D× = (Dn× )∞ n=0 of linear spaces such that each Dn× is dense in Hn and each Dk× ⊂ D(A× n,k ) for all n is chosen. Proposition 3. If the couples (A, D) and (A× , D × ) determine operators then the operators MA,D and MA× ,D× satisfy ∗ MA,D ⊂ MA × ,D × , ∗ ; MA× ,D× ⊂ MA,D

hence, they are closable. 2 Proof. Take f ∈ D(MA,D ) and g ∈ D(MA× ,D× ). Then using twice *

MA,D f, gK =

∞ ∞ An,k Pk f, Pn gHn = An,k Pk f, Pn gHn n=0

=

n=0 k

k

∞

Pk f, A∗n,k Pn gHk k n=0

=

k

Pk f,

∞

A∗n,k Pn gHk

n=0

= f, MA× ,D× gK , with the fourth equality due to the fact that (A× , D × ) determines an operator. Therefore we get MA,D f, g = f, MA× ,D× g,

f ∈ D(MA,D ), g ∈ D(MA× ,D× )

which completes the proof of both inclusions.

As an immediate consequence of Proposition 3 we get the following. Corollary 4. If either ∗ (a) D(MA,D ) is a core 4 of MA × ,D × , or ∗ (b) D(MA× ,D× ) is a core of MA,D , then ∗ ∗ MA,D = MA × ,D × and MA× ,D × = MA,D . Consequently, both (a) and (b) must necessarily hold. 3. The following simple observation will be useful in the sequel. Fact 5. For closable operators A and B ¯ ⊂ D(B) ¯ and 1o if D(A) ⊂ D(B) and Af = Bf for f ∈ D(A), then D(A) ¯ ¯ ¯ Af = Bf for f ∈ D(A). If, in addition, D(A) is a core of B, then ¯ = D(B); ¯ D(A) o ¯ ¯ and Af ¯ = Bf ¯ for f ∈ D(A) ¯ = D(B), ¯ then D(A) is a 2 if D(A) = D(B) core of B. 4

D is a core of a closable operator A if A ⊂ A|D .

On Normal Extensions of Unbounded Operators

341

Our eﬀorts so far are surmounted by the result which follows. Let us remind ﬁrst that an operator N is said to be normal if D(N ) = D(N ∗ ) and N f = N ∗ f for f ∈ D(N ); N is called essentially normal if its closure N is normal. Theorem 6. Suppose both (A, D) and (A× , D) determine operators. If D(MA,D ) ∗ ∗ is a core of MA× ,D as well as that of MA,D (or, equivalently, that of MA × ,D ) and MA,D f = MA× ,D f ,

f ∈ D(MA,D ),

(3)

then MA,D is essentially normal. Proof. As D(MA,D ) is a core of MA× ,D and (3) holds, using Fact 5 we infer that D(MA,D ) = D(MA× ,D ) with (3) extended to this common domain. On the other ∗ hand, as D(MA,D ) is a core of MA,D Corollary 4 leads directly to normality of MA,D . 4. So as to utilize the above theorem we work out some conditions for the most worrying part of its assumptions to be satisﬁed. First we expand our considerations a bit to make them easier to comply with. Let B = (Bn )n be a ﬁnite or inﬁnite sequence of operators deﬁned on a common dense subspace D of a Hilbert space L, taking values in Hn respectively and such that Bn x2Hn < +∞, x ∈ D. (4) n

Then the operator CB from D into K = sense.

N n

Hn deﬁned by CB x = def

N n

Bn x makes

Proposition 7. Let CB be as above. Then 1o for every n and f ∈ K ∗ Pn f ∈ D(CB ) ⇐⇒ Pn f ∈ D(Bn∗ ); M # def ∗ 2o with DB = {f ∈ n Pn−1 (D(Bn∗ )) : n Bn Pn f converges weakly}, the equality ∗ ∗ DB = {f ∈ D(CB ) : Pn f ∈ D(CB ), n = 0, 1, . . .} holds and ∗ CB f= Bn∗ Pn f, f ∈ DB . n

M Proof. Part 1o is obvious. For 2o take x ∈ D, f ∈ n Pn−1 (D(Bn∗ )) and write Bn x, Pn f Hn = x, Bn∗ Pn f L . CB x, f K = #∞

n

n

∗ Because n=0 BM n Pn f converges weakly, the vector ∗ is such that f ∈ n Pn−1 D(CB ) then, due to 1o , ∗ f L = CB x, f K = x, CB

∗ ∗ f is in D(CB ). If f ∈ D(CB )

Bn x, Pn f Hn =

n

and f is in DB . The rest of the conclusion follows then.

x, Bn∗ f L

n

342

F.H. Szafraniec

∗ Corollary 8. If all the Bn ’s are zero but a ﬁnite number and f ∈ D(CB ) implies ∗ Pn f ∈ D(CB ) for every n, then : ∗ ∗ f= Bn∗ Pn f, f ∈ D(CB )= Pn−1 (D(Bn∗ )). CB n

n

Remark 9. If only two of the Bn ’s are diﬀerent from zero, say Bi and Bj , then ∗ ∗ ∗ f ∈ D(CB ) and Pi f ∈ D(CB ) implies Pj f ∈ D(CB ); this happens for instance when one of the operators is bounded. It can be extended properly to any ﬁnite number of operators. In order to adjust the above to the current situation make a shorthand notation def Ak = (Ai,k )∞ i=0 with D(Ak ) = Dk , for k = 0, 1, . . . , which apparently applies to L = Hk . If (A, D) determines an operator, each CAk is well deﬁned because (4) is automatically satisﬁed. We may think of the operator CAk as a column operator of the matrix A; it maps Dk into K. In the same way we deﬁne CA× replacing the matrix A by A× . k 2 takes Notice that, in the case of D ﬁn = D(MA,D ), the deﬁning formula * the very convenient form in terms of column operators ∞ MA,D = CAi Pi |D(MA,D ) (5) i=0

with the series converging on D(MA,D ) due to the fact that the corresponding sum is eﬀectively ﬁnite, that is it is so when applied to vectors in D(MA,D ). More precisely, for f ∈ Dﬁn each Pi f ∈ Di = D(An,i ) which means Pi f can be inserted 1 yielding in * ∞ An,i Pi f 2Hn < +∞ n=0 1 we have and this in turn means Pi f is in D(CAi ). Now starting from * ∞ ∞ +∞ > An,k Pk f 2Hn = An,k Pk f, An,l Pl f Hn n=0

=

k ∞

n=0 k,l

An,k Pk f, An,l Pl f Hn =

k,l n=0

=

∞ ∞ O O An,k Pk f, An,l Pl f K k,l n=0

n=0

Ck Pk f 2K

k

and this establishes (5). Based on the above we take a further step towards depicting the operator ∗ MA,D , in which the column operators play the essential rˆ ole. Let us remind that ∞ ∞ :∞ O def ∗ ∗ CAn ) = {f ∈ D(CA ) : CA f 2 < +∞}. D( n k n=0

k=0

n=0

On Normal Extensions of Unbounded Operators

343

Theorem 10. Suppose Dﬁn ⊂ D(MA,D ). Then O ∗ ∗ MA,D = CA . i i

Proof. The formula (5) results in ∗ MA,D =

∞

∗

CAi Pi |D(MA,D )

.

i=0

N∞ ∗ Take f ∈ D(MA,D ) = Dﬁn and g ∈ D( i=0 CA ). Then i ∞ ∞ ∞ ∞ O ∗ ∗ CAi Pi f, gK = CAi Pi f, gK = Pi f, CA g = f, CA gK . H i i i i=0

i=0

and, consequently, g is in D

i=0

#∞ i=0 ∞ O

CAi Pi |D(MA,D )

∗ CA ⊂ i

i=0

Notice that ∗ f, gK = f, MA,D gK = MA,D

∞

∗

i=0

=

∗ D(MA,D )

as well as

∗ MA,D .

(6)

i=0

f, CAk Pk gK , k ∗ f ∈ D(MA,D ), g ∈ D(MA,D ).

If g is such that Pk g = g for a temporarily ﬁxed k, then, because all such g’s ﬁll up the whole of Dk (notice that the assumption Dﬁn ⊂ D(MA,D ) has to be used ∗ here), we get immediately f ∈ D(CA ) and k ∗ ∗ f, gK = CA f, Pk gHk . MA,D k

Consequently, ∗ D(MA,D )⊂

:

(7)

∗ D(CA ). k

N As an arbitrary g ∈ Dﬁn = D(MA,D ) is of the form g = k Pk g the equality (7) implies O ∗ ∗ ∗ MA,D f, gK = CA f, P g = CA f, gK k H k k k k

k

k

the sum being ﬁnite. Therefore, O ∗ ∗ CA f, gK | ≤ MA,D f g, | k

∗ f ∈ D(MA,D ), g ∈ D(MA,D ),

k

N ∗ f are uniformly bounded, which results in f belonging to hence the sums k CA k N∞ ∗ ). This gives us the opposite to the inclusion (6) and the conclusion D( k=0 CA k follows. Corollary 11. With assumption of Corollary 8 to be satisﬁed :∞ : ∗ D(MA,D )= Pn−1 D(A∗n,k ). n=0

k

344

F.H. Szafraniec The twin Theorem 10 is as follows and the proof can repeated verbatim.

Theorem 12. Suppose Dﬁn ⊂ D(MA× ,D ). Then O ∗ ∗ CA MA × ,D = ×. k

k

∗ MA,D

Using the graph norm of leads straightforwardly to a characterization of Dﬁn to be a core of the adjoint to MA,D , given in terms of column operators. ∗ Corollary 13. Suppose Dﬁn ⊂ D(MA× ,D )∩D(MA,D ). Then Dﬁn is a core of MA,D if and only if the following condition holds O ∗ f ∈ D( CA × ), k

Pk f, gk Hk +

∞

k

∗ CA f, A∗k,n gk Hn = 0, n

gk ∈ Dk , k = 0, 1, . . .

n=0

implies f = 0.

Andˆ o’s construction in unbounded circumstances; suﬃciency 5. Having done all the preliminary work on matrix operators we state the working ∞ part of Andˆ o’s setting. Suppose we are given two sequences (Sn )∞ n=0 and (Dn )n=0 of closable operators acting in a Hilbert space H as well as a sequence D = (Dn )∞ n=0 of dense subspaces of H such that ∗ ) for n = 0, 1, . . . Dn ⊂ D(Sn ) ∩ D(Sn∗ ) ∩ D(Dn ) ∩ D(Dn+1

It becomes clear that for the two diagonal matrix ⎛ S0 D 1 0 0 ⎜ ⎜ 0 S1 D 2 0 def ⎜ N =⎜ ⎜ 0 0 S2 D 3 ⎝ .. .. .. . . .

(8)

⎞ .. ⎟ . ⎟ ⎟ .. ⎟ . ⎟ ⎠ .. .

the operators MN ,D and MN × ,D are determined and they act in K = H ⊕ H ⊕ · · · with domains D(MN ,D ) = D(MN × ,D ) = Dﬁn . def

Proposition 14. Suppose D0 = 0. Then the operator MN ,D satisﬁes (3) if and only if ∗ ∗ Sn f, Dn+1 gH = Dn+1 f, Sn+1 gH , ∗ Dn+1 f 2

+

Sn∗ f 2

f ∈ Dn , g ∈ Dn+1 ,

= Dn f + Sn f 2 , f ∈ Dn , 2

n = 0, 1, . . .

(∗)

∗ Moreover, if under the above circumstances Dﬁn is a core of MN ,D , then the

operator MN ,D is essentially normal.

On Normal Extensions of Unbounded Operators

345

Proof. The ‘if’ part of the ﬁrst conclusion is direct. For the ‘only if’ part we suggest to plug into the formal normality condition f = Pn f ﬁrst and then f = Pn f + Pn+1 f and consider it step by step. The additional conclusion follows from Theorem 6 if one notices that Dﬁn as equals D(MA× ,D ) is apparently a core of MN × ,D . Before we come to the main result of this paper recall an operator S in a Hilbert space H is called subnormal if there is another Hilbert space K, which contains isometrically H, and a normal operator N in it such that Sf = N f,

f ∈ D(S).

Theorem 15. Let S be a closable operator in a Hilbert space H and let two sequences ∞ (Sn )∞ n=0 and (Dn )n=0 of closable operators acting in a Hilbert space H as well as a sequence D = (Dn )∞ n=0 of dense subspaces of H satisfying (8) be given. Suppose S0 = S, D0 = 0 and D0 = D(S) def

as well as the condition (∗) holds. Then S is subnormal with the operator MN ,D being its normal extension in the space H ⊕ H ⊕ · · · if and only if the following condition is satisﬁed 5 : ∞ O ∗ f ∈ D( C(D ), k ,Sk ) k=0 ∗ ∗ ∗ Pn f, gn H + C(D f, Sn∗ gn H + C(D f, Dn+1 gn H = 0, n ,Sn ) n+1 ,Sn+1 )

gn ∈ Dn , n = 0, 1, . . . implies f = 0, def

with additional notation g−1 = 0. Remark 16. Remark 9, if applicable, can provide some help in further identiﬁcation ∗ of the domain of C(D . n ,Sn ) Proof of Theorem 15. Use Proposition 14 and Corollary 13.

Andˆ o’s construction in unbounded circumstances; necessity 6. Now we look upon the possibility of extorting conditions (∗) from the assumption that S has already a normal extension of any sort. In other words, we would like to know how much the limitations of Theorem 15 pose a challenge for subnormality of S. It follows directly from the deﬁnition of subnormality that D(S) ⊂ D(N ) ∩ H = D(N ∗ ) ∩ H ⊂ D(S ∗ ), 5

(9)

C(Dk ,Sk ) stands apparently for the (a bit overloaded in the present circumstances) hieroglyphics C(0,...,0,Dk ,Sk ,0,... ) with positioning of Dk , Sk easy to guess; it should not mean for the formalists B = (Dk , Sk ) as referring to the deﬁnition of CB , p. 341 – apology for this kind of perversity.

346

F.H. Szafraniec

with inclusions to be proper sometimes and S ∗ f ≤ Sf ,

f ∈ D(S),

(10)

the latter means hyponormality of S. Also as normal operators are closed, subnormal ones must necessarily be closable. The following result is a part of Theorem 37 in [10]. Theorem 17. An operator S in H with invariant domain D(S) is subnormal if and only if there exists a form ψ on N+ × D(S) which is positive deﬁnite 6 and ψ(m, n, f, g) = S m f, S n g,

m, n = 0, 1, . . . , f, g ∈ D(S).

As a consequence we get a way to transport subnormality from one space to another, which is an unbounded counterpart of Lemma 1 of [1]. Proposition 18. Let S be an operator in H with invariant domain D(S) and let V : H → H1 be a bounded operator such that V ∗ V S = S. If S is subnormal in H, then so is V SV ∗ in H1 provided it is densely deﬁned. Proof. Deﬁne a form ψ1 on N+ × D(V SV ∗ ) by ψ1 (m, n, x, y) = ψ(m, n, V ∗ x, V ∗ y) and remembering that (V SV ∗ )k = V S k V ∗ for k = 1, 2, . . . check that ψ1 is positive def deﬁnite and corresponds to the operator S1 = V SV ∗ as in Theorem 17. def

Suppose S is a subnormal operator in H. Let N be its normal extension in K (remember, H ⊂ K isometrically). Let us point out some relations between S ∗ and N ∗ . The ﬁrst of them is 7 N ∗ f − S ∗ f, gK = 0,

f ∈ D(N ∗ ) ∩ H, g ∈ H.

(α)

Indeed, N ∗ f, gK = f, N gK = f, SgH = S ∗ f, gH = S ∗ f, gK , ﬁrst for g ∈ D(S), then for all g ∈ H; all this makes sense as (9) holds. The second says f + N ∗ g − S ∗ g2K = f 2H + N ∗ g − S ∗ g2K ,

f ∈ H, g ∈ D(N ∗ ) ∩ H.

(β)

This is so because, due to (α), N ∗ g − S ∗ g ⊥ H. The third states N ∗ f − S ∗ f 2K = Sf 2H − S ∗ f 2H , 6 7

f ∈ D(S).

(γ)

For the relevant terminology and notation cf. [10]. Sometimes we emphasize the underlying space by adding a subscript to the inner product.

On Normal Extensions of Unbounded Operators

347

For this take f ∈ D(S), then, employing normality of N and using (α), one gets 8 N ∗ f − S ∗ f 2K = N ∗ f 2K − 2 ReN ∗ f, S ∗ f K + S ∗ f 2H = Nf 2K − 2 ReN ∗ f, S ∗ f K + S ∗ f 2H = Sf 2H − 2 ReS ∗ f, S ∗ f K + S ∗ f 2H = Sf 2H − S ∗ f 2H . Proposition 19. If (N ∗ − S ∗ )|D(S) is closable 9 , then there is a positive operator D in H having D(S) as its core and such that Df 2H = N ∗ f − S ∗ f 2K = Sf 2H − S ∗ f 2H ,

f ∈ D(S).

(11)

Proof. Set A = (N ∗ − S ∗ )|D(S) and consider it as an operator to K. A∗ A¯ is a positive and selfadjoint operator in H. Denote by D its positive square root. Then according to [13, Theorem 5.40] ¯ ¯ K , f ∈ D(A). ¯ D(D) = D(A), Df H = Af def

This implies in turn that D(S) is a core of D (use part 2o of Fact 5).

Suppose now that D(S) is invariant for S, that is SD(S) ⊂ D(S).

(12)

∗

Then D(S) + N D(S) is invariant for N . Indeed, because D(S) ⊂ D(N ) = D(N ∗ ), N (f + N ∗ g) = Sf + N N ∗ g = Sf + N ∗ N g = Sf + N ∗ Sg,

f, g ∈ D(S). ∗

Denote by K1 the closure in K of the linear space D(S) + N D(S). Then the operator N |D(S)+N ∗D(S) is subnormal in K1 . Due to (12), the second equality in (11) and (β), V : D(S) + (N ∗ − S ∗ )D(S) f + N ∗ g − S ∗ g → (f, Dg) ∈ H ⊕ H is a well deﬁned isometry which we extend to the whole of K1 . Assuming that D(S) is invariant for S ∗ as well, we get D(S) + N ∗ D(S) = D(S) + (N ∗ − S ∗ )D(S) which make the operator V N |D(S)+N ∗ D(S) V ∗ densely deﬁned in H ⊕ H; according to Lemma 19 it is subnormal. Then the rest of Andˆ o’s arguments can be done with due caution. The calculation which follows leads us to the matricial form of the operator V N |D(S)+N ∗D(S) V ∗ . Because V ∗ is a partial isometry with the initial space R(V ) = V K1 = {(f, Dg): f, g ∈ D(S)} we have R(V ) = R(D|D(S) ) = H ⊕ R(D) as D(S) is a core of D (Proposition 19). Therefore R(V )⊥ = 0 ⊕ N (D) and according to this for f, h ∈ D(S) we have the decomposition (f, h) = (f, Dg) ⊕ (0, h − Dg) with D2 g = Dh. 8

(13)

This gives an ad hoc argument for (10). Surprisingly we cannot prove it. If this is not because of our ignorance, it may be an interesting problem even in approximation theory in L2 -spaces over noncompact sets.

9

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F.H. Szafraniec

Notice that D(S) ⊂ D(D2 ), This stems from formula (11) and the fact that D(S) is a core of D (Proposition 19 again). Now, using (13), for f, h ∈ D(S) we get V N |D(S)+N ∗ D(S) V ∗ (f, h) = V N |D(S)+N ∗ D(S) V ∗ (f, Dg) = V N (f + (N ∗ − S ∗ )g) = V (Sf + (S ∗ S − SS ∗ )g + (N ∗ − S ∗ )Sg) = (Sf + D2 g, DSg) = (Sf + Dh, DSD−1 h) because, as we have already assumed, D(S) is invariant for S ∗ as well (which is used for the third equality) and, what has to be assumed furthermore, that D(S) is invariant 10 for D (which helps to make the calculation for the last equality 11 ). The latter also makes it sure the operator V N |D(S)+N ∗ D(S) V ∗ act in H ⊕ H as a matrix operator M(S,(D(S),D(S)) determined by the matrix S D|D(S) def . S= 0 DSD−1 |D(S) Notice that if D(S) is supposed to be invariant for S, S ∗ and D, as D is selfadjoint, D(S) ⊕ D(S) is invariant for the matrix operators M(S,(D(S),D(S)) and M(S × ,(D(S),D(S)) 12 . Nickname, for further references, the way we have passed from S to S Andˆ o’s procedure. 7. Now anticipating Andˆ o’s procedure to be a basic tool for the induction process we try to watch how it goes on. Suppose we have succeeded in constructing sequences (Sn )kn=0 and (Dn )k−1 n=0 of closable operators, S0 = S, D0 = 0, such that conditions (8) and (∗) are satisﬁed up to n = k − 1 with Dn = D(S) and the matrix operator determined by the operator matrix ⎛ ⎞ S0 D 1 0 · · · ··· 0 ⎜ ⎟ .. ⎜ 0 S1 D 2 0 . 0 ⎟ ⎜ ⎟ ⎜ ⎟ .. def ⎜ . 0 S2 D 3 0 ⎟ (14) Sk = ⎜ 0 ⎟ ⎜ . ⎟ .. .. .. .. ⎜ .. ⎟ . . . . 0 ⎟ ⎜ ⎝ 0 ··· ··· 0 Sk−1 Dk ⎠ 0 ··· ··· ··· 0 Sk is subnormal. If, in addition, we suppose all the Dn ’s are selfadjoint as well as D(S) is invariant for Sn , Sn∗ and Dn , n = 0, 1, . . . , k − 1 then D(S) ⊕ · · · ⊕ D(S), 10 11

Maybe this assumption is too rough. For a selfadjoint operator A in H its partial inverse A−1 here is understood as an operator + −1 f if f ∈ R(A) A|D(A)∩R(A) def , f ∈ R(A) ⊕ N (A), A−1 f = 0 if f ∈ R(A)⊥ = N (A)

which is in H. Thus A−1 A = P |D(A) and AA−1 = P |R(A)⊕N (A) , P being here a projection onto R(A). 12 In this case, because the matrix is ﬁnite, the deﬁnition of an operator determined by a matrix simpliﬁes and allows us here and there to shorten notation using in particular S for M(S ,(D(S),D(S))) and S× for M(S × ,(D(S),D(S))).

On Normal Extensions of Unbounded Operators k + 1-times, is invariant for S k and ⎛ 0 ⎜ .. ⎜ . × S× k Sk − SkSk = ⎜ ⎝ 0 0

S ∗k . Moreover, cf. footnote 12, ⎞ ··· 0 0 ⎟ . .. ⎟ · · · .. . ⎟. ⎠ ··· 0 0 2 ∗ ∗ · · · 0 D k + Sk Sk − Sk Sk

After applying Andˆ o’s procedure to S k , we come to a subnormal operator S k Dk+1 |D(S k ) def ˜ S k+1 = 0 Dk+1 S k D −1 k+1 |D(S k )

349

(15)

(16)

in Hk+1 ⊕ Hk+1 , where Dk+1 determined as in Proposition 19 as a (k + 1) × (k + 1) matrix operator, due to (15), has the only non-zero entry in its downright corner; the latter is the operator Dk+1 satisfying Dk+1 f 2 + Sk∗ f 2 = Dk f 2 + Sk f 2 for f ∈ D(S). Because of this, when one opens up the blocks in (16) to (k+1)×(k+ 1) (still operator) matrices, it turns out that there is a surplus of rows composed totaly of zero entries. Thus the operator V k+1 : Hk+1 ⊕ Hk+1 (f0 , . . . , fk+1 , g0 , . . . , gk+1 ) → (f0 , . . . , fk+1 , gk+1 ) ∈ Hk+2 , crossing out needless coordinates makes 13 the operator 14 S k C(0,...,0,Dk+1 |D(S) ) def ∗ ˜ S k+1 = V k+1 S k+1 V k+1 = −1 0 Dk+1 Sk Dk+1 |D(S) subnormal in Hk+2 . The only missing thing here is the requirement D(S) to be invariant for Dk+1 , which has to be kept going. Summary. If S is subnormal operator in H, then there are two sequences (Sn )∞ n=0 and (Dn )∞ n=0 having common domain D(S), S0 = S, D0 = 0, Dn ’s being positive, and such that (∗) holds provided all those ‘ifs’ concerning invariance of D(S) and the assumption of closability as in Proposition 19 on every stage can be released. If this happens then each of the operators (14) is a subnormal extension of S. This gives us a starting point to go backwards but for that we need the other assumptions of Theorem 15 to be satisﬁed but this is another story. Let us mention that for the famous creation operator, which is subnormal, all the Sn ’s are equal to D while all the Dn ’s are equal to the identity operator I. Another ﬂicker of hope is in some room left by not requiring the operators Dn to be positive (in the suﬃciency part). This a bit optimistic remark brings us on to the ﬁnal question which follows. Question. Though there is a lot of unbounded subnormal operators for which all the invariance assumptions are satisﬁed, a gap between the class of operators satisfying (∗) and all the subnormal ones under the present circumstances has been opened. So how wide is it and is it really the case? In other words, does this 13 14

˜k+1 . ˜k+1 = S This is due to Proposition 18 as V ∗k+1 V k+1 S Notice at the upright corner the column operator appears again.

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gap come from the fact that our way of realizing Andˆ o’s programme is not tuned ﬁnely enough or it is a result of intrinsic inﬂexibility of Andˆ o’s method itself? Acknowlegement Thanks are due to Jan Stochel for his criticism eﬀecting the ﬁnal version of this paper.

References [1] T. Andˆ o, Matrices of normal extensions of subnormal operators, Acta Sci. Math. (Szeged), 24 (1963), 91–96. [2] Yu.A. Berezanskii, Eigenfunction expansions for selfadjoint operators, Naukova Dumka, Kiev, 1965. [3] E. Bishop, Spectral theory of operators on a Banach space, Trans. Amer. Math. Soc., 86 (1957), 414–445. ´ [4] C. Foia¸s, D´ecomposition en op´erateurs et vecteurs propres. I. Etudes de ces d´ecompositions et leurs rapports avec les prolongements des op´erateurs, Rev. Roumaine Math. Pures Appl., 7 (1962), 241–282. [5] A.S. Holevo, Probabilistic and statistical aspects of quantum theory, North-Holland, Amsterdam – New York – Oxford, 1982. [6] J. Janas and J. Stochel, Selfadjoint operators with ﬁnite rows, Ann. Polon. Math., 66 (1997), 155–172. [7] J. Stochel and F.H. Szafraniec, On normal extensions of unbounded operators. I, J. Operator Theory, 14 (1985), 31–55. [8] , On normal extensions of unbounded operators. II, Acta Sci. Math. (Szeged), 53 (1989), 153–177. , On normal extensions of unbounded operators. III. Spectral properties, Publ. [9] RIMS, Kyoto Univ., 25 (1989), 105–139. [10] , The complex moment problem and subnormality: a polar decomposition approach, J. Funct. Anal., 159 (1998), 432–491. [11] M.H. Stone, Linear transformations in Hilbert space, Amer. Math. Soc., Providence, R.I, 1932. [12] F.H. Szafraniec, Subnormality in the quantum harmonic oscillator, Commun. Math. Phys., 210 (2000), 323–334. [13] J. Weidmann, Linear operators in Hilbert spaces, Springer-Verlag, New York, Heidelberg, Berlin, 1980. Franciszek Hugon Szafraniec Instytut Matematyki Uniwersytet Jagiello´ nski ul. Reymonta 4 PL-30059 Krak´ ow e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 351–363 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Directing Mappings in Kre˘ın Spaces Bj¨orn Textorius To Heinz Langer on the occasion of his retirement.

Abstract. M.G. Kre˘ın’s method of directing mappings is extended to quasidensely deﬁned symmetric operators with selfadjoint deﬁnitizable extensions in some Kre˘ın space, by direct use of the spectral function of the extension, thus avoiding a reduction to the semideﬁnite situation. Mathematics Subject Classiﬁcation (2000). Primary 47B50, 46C20 ; Secondary 34B09. Keywords. Kre˘ın space, Pontrjagin space, deﬁnitizable operator, directing mapping.

1. Introduction The method of directing functionals or directing mappings is often useful in order to prove spectral representations for operators and functions. Originally, this method was developed in [3] for a linear space L with a semideﬁnite inner product, a symmetric operator S in L with quasidense domain and a ﬁnite system of directing functionals; it is explained, e.g., in M.A. Naimark’s book [11]. Generalizations to an inﬁnite number of directing functionals or a directing mapping for a symmetric linear operator were given in [7]; for a symmetric linear relation in [9], [10], and analogous results for isometric operators have been formulated in [1]. It is the aim of this note to show that this method can also be generalized to deﬁnitizable operators in an indeﬁnite inner product space (L, [·, ·]). First results in this direction were discussed by M.G. Kre˘ın and H. Langer more than 30 years ago. There the main idea was to use a new semideﬁnite inner product (f, g) := [p(S)f, g] on D(S π ), where π is the degree of the deﬁnitizing polynomial p of S, or (f, g) := [q(S)f, q(S)g] on D(S ρ ), ρ = deg q, if the deﬁnitizing polynomial p is of the form p = q q¯ and to apply the results concerning the case of a semideﬁnite inner product to the restrictions of S in these spaces. There arise, however, some technical diﬃculties and restrictions. It has to be shown that, e.g.,

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the domain D(S π+1 ) is quasidense in D(S π ) with respect to the inner product (·, ·). For diﬀerential operators the domains of these powers are often not so easy to deal with. Several years ago Heinz Langer and the author began to study a way to avoid this diﬃculty. Instead of reducing the problem to the semideﬁnite situation, the idea is to assume the existence of a deﬁnitizable extension S of S in some

and make use of the spectral function E

of S directly. It is then Kre˘ın space K necessary to change the deﬁnition of a directing mapping slightly. While in the original deﬁnition the equation (S − z)f = g is considered for ﬁxed g, f at single points z, we replace f by a holomorphic function f (z) in some domain Dg ⊂ C and with values in L, and suppose that the corresponding equation (S − z)f (z) = g(z) has a holomorphic solution f (z) if and only if Φz g(z) = 0 (z ∈ Dg ). In the case of a semideﬁnite inner product, this assumption follows from from the original condition. It should be mentioned, however, that in applications where S is, e.g., an ordinary diﬀerential operator, this in some sense stronger condition dealing with these holomorphic functions can as easily be veriﬁed as the original one. The results of this note can be applied to obtain representations of kernels and functions having a ﬁnite number of negative squares and to ordinary diﬀerential problems with an indeﬁnite weight function. I thank a referee for valuable remarks.

2. Deﬁnitizable operators Let A be a deﬁnitizable selfadjoint operator in a Kre˘ın space K with deﬁnitizing polynomial p. Recall that this means that the resolvent set ρ(A) is not empty and that [p(A)x, x] ≥ 0 for all x ∈ D(Aπ ), where π denotes the degree of the polynomial p. The non-real spectrum σ0 (A) of the operator A consists of a ﬁnite number of eigenvalues of ﬁnite order, located symmetrically with respect to the real axis. Moreover, A has a spectral function E deﬁned on the semiring RA of all intervals of R with endpoints not in the set c(A) of critical points of A, and their complements in R. The spectral function can be

for which ∆

∩ R ∈ RA extended in a natural way to not necessarily real sets ∆, by putting

:= E(∆

∩ R) + E(∆) Eλj ,

λj ∈∆∩σ 0 (A)

where Eλ denotes the Riesz projection corresponding to the (non-real) eigenvalue

A . Furthermore, as in [6] and [8], λ of A. This extended semiring is denoted by R c(A) denotes the set of ﬁnite critical points of A: c(A) = c(A) ∩ R, and we put σ /0 (A) := c(A) ∪ σ0 (A). A critical point is either regular or singular, see, e.g., [8].

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353

Lemma 2.1. Let α ∈ R be a real critical point, and let ∆ ∈ RA be bounded and such that ∆ ∩ c(A) = {α}. If q is a nonnegative integer such that [(A − α)q f, f ] ≥ 0 for all f ∈ E(∆)K, then for all these elements f and z ∈ ρ(A) we have E D q−1 −1 −1 −k−1 k (λ − z) + (z − α) (λ − α) E(dλ)f (A − z) f = ∆

−

q−1

k=0

(z − α)−k−1 Ak f − (z − α)−q−1 N f,

(2.1)

k=0

where N is a bounded selfadjoint operator in K with the properties N 2 = 0, JN = 0,

A such that α ∈ ∆ , and the integral exists in and N E(∆ ) = 0 for all sets ∆ ∈ R the strong operator topology as an improper one with respect to the singularity of E at α. This singularity is, however, such that ∆ (λ − α)q E(dλ) converges. Proof. Without loss of generality we assume that α = 0, σ

0 (A) = {α} and that A is bounded: σ(A) ⊂ ∆. The identity λq z −q (λ − z)−1 = (λ − z)−1 + z −1 + λz −2 + · · · + λq−1 z −q implies (A − z)−1 = z −q Aq (A − z)−1 − z −1 − z −2 A − · · · − z −q Aq−1 .

(2.2)

According to [6] and [8], the representation z −q Aq (A − z)−1 E(∆) = z −q λq (λ − z)−1 E(dλ) − z −q−1 N

(2.3)

∆

holds. The relations (2.2), (2.3) evidently give (2.1).

3. Preliminaries on integration Let H, K be two Kre˘ın spaces, Ψ : z → Ψz a function deﬁned and holomorphic on some set σ ⊂ C and with values in L(H, K), and let A ∈ L(K) be a selfadjoint operator in K such that σ(A) ⊂ σ. If CA is a Jordan contour in σ surrounding σ(A), symmetric with respect to the real axis, we deﬁne the bounded linear operator ΨA ∈ L(H, K) by 1 ΨA := − (A − z)−1 Ψz dz. (3.1) 2πi CA If Θ. is a function with the same properties as Ψ. , then the following formula is easy to verify for arbitrary functions a, b, holomorphic on σ and with values in H: 6 7 P P 1 1 −1 −1 − (A − z) Ψz a(z)dz, − (A − z) Θz b(z)dz 2πi CA 2πi CA P 1 [(A − z)−1 Ψz a(z), Θz¯b(¯ z )]dz. =− 2πi CA

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This formula does still hold if the operator is deﬁnitizable and σ(A) is not necessarily surrounded by CA , but CA intersects R nontangentially and only in points which do not belong to c(A) ∪ σp (A). In this case the integrals are to be understood as principal values in these points of R. In the following such integrals are denoted by . Now let H be a Hilbert space. By V we denote the following class of L(H)valued measures V : 1. V is deﬁned on the semiring RV of bounded intervals of R whose closures do not contain the points of a ﬁnite exceptional set s(V ) = {α1 , . . . , αl } (depending on V ). 2. V is either nonnegative or nonpositive on each interval in R \ {α1 , . . . , αl }, which does not contain points from a given ﬁnite set (depending on V ). 3. For each α ∈ s(V ) there exists an integer q ≥ 0 such that the integral α+ (λ − α)q V (dλ) exists if > 0 is so small that [α − , α + ] ∩ s(V ) = {α}. α− In the sequel we have to deal with integrals of holomorphic H-valued functions on R with respect to the operator measure V (dλ). For an interval ∆ such that ∆ ∩ s(V ) = ∅ these integrals are deﬁned in [7]. Now assume that the interval ∆ contains one exceptional point α in its interior ∆i , that it has positive distance α+ from all the other exceptional points of V , and that the integral α− (λ−α)q V (dλ) exists. Let φ, ψ be holomorphic, H-valued functions on ∆, and assume that ∆ is so small that on ∆ the expansions φ(λ) =

∞

φµ (λ − α)µ ,

µ=0

ψ(λ) =

∞

ψν (λ − α)ν

ν=0

hold. Then an improper integral with respect to the singularity of V at α is deﬁned by means of the following regularization: (V (dλ)φ(λ), ψ(λ))(α;q) ∆ , + q−1 (λ − α)l (l) (V (dλ)φ(λ), ψ(λ)) − := (V (dλ)φ(t), ψ(t)) |t=α l! ∆ l=0 (V (dλ)φµ , ψν )λµ+ν . (3.2) = µ,ν≥0; µ+ν≥q

∆

If, e.g., dimH = 1 the second expression makes sense in the usual way. It then reads as q−1 (λ − α)l (l) . V (dλ) φ(λ)ψ(λ) − (φ(t)ψ(t)) |t=α l! ∆ l=0

For the integral deﬁned above we need a kind of Stieltjes-Lifshits inversion formula.

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Lemma 3.1. Let V, α, q, ∆, φ, ψ be as before and assume that C∆ is a Jordan contour, symmetric with respect to R with ∆ in its interior domain and all other exceptional points of V in its exterior domain. Assume that φ and ψ are holomorphic on C∆ and its interior and that V is continuous on the boundary of ∆. Then, for z ∈ /∆ E , P + D q−1 1 −1 −k−1 k (λ − z) + z )) dz (z − α) (λ − α) (V (dλ)φ(z), ψ(¯ − 2πi C∆ ∆ k=0 = (V (dλ)φ(λ), ψ(λ))(α; q) . (3.3) ∆

Proof. Without loss of generality we assume α = 0. The left-hand side of (3.3) then equals . P 1 q −q −1 − λ z (λ − z) (V (dλ)φ(z), ψ(¯ z )) dz 2πi C∆ ∆ + , P ∞ 1 λq z −q (λ − z)−1 (V (dλ)φµ , ψν )z µ+ν dz =− 2πi C∆ ∆ µ,ν=0 P ∞ 1 q −q −1 µ+ν = − λ z (λ − z) z (V (dλ)φµ , ψν ) dz 2πi C∆ ∆ µ,ν=0 ⎛ ⎞ ⎝ = λµ+ν (V (dλ)φµ , ψν ⎠ , ∆

µ,ν≥0,µ+ν≥q

which, according to (3.2), equals the right-hand side of (3.3). In the last step we used the formula µ+ν P 1 µ+ν ≥ q λ , λq z −q (λ − z)−1 z µ+ν dz = − 0 µ+ν < q 2πi C∆

which holds if λ is in the domain surrounded by C∆ .

4. Directing mappings Let L be a linear space equipped with an inner product [·, ·], that is a possibly degenerate Hermitian sesquilinear form. We always assume that L is decompos·

·

able, that is there are subspaces L+ , L− , L0 such that L = L+ [+]L− [+]L0 , where [f, f ] ≥ 0 (≤ 0) if 0 = f ∈ L+ (L− ), and L0 = L ∩ L[⊥] is the isotropic part of ·

L. Here [+] denotes a direct sum of linear manifolds, which are orthogonal with respect to [·, ·], and [⊥] denotes the orthogonal complement with respect to [·, ·]. 1 Completing L+ (L− ) with respect to the norms f = [f, f ] 2 if f ∈ L+ and 1 f = (−[f, f ]) 2 if f ∈ L− and forming the orthogonal sum K of these completions, then K becomes in a natural way a Kre˘ın space containing the factor space

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L = L/L0 as a dense subspace. In the sequel, if L is a decomposable inner product space, we always ﬁx one decomposition and everything is to be understood with respect to this decomposition. Recall that L is always decomposable in each of the following two cases: (a) L is a Hilbert space with respect to some inner product (·, ·) and the indeﬁnite inner product [·, ·] is continuous in this Hilbert space. (b) The inner product [·, ·] has only a ﬁnite number κ, 0 ≤ κ < ∞ of negative squares. Let S be a symmetric linear operator in L with a quasidense domain D(S) ⊂ L. Here symmetric means that [Sf, g] = [f, Sg] for all f, g ∈ D(S), and a set D ∈ L is said to be quasidense in L if for each f ∈ L there exists a sequence (fn ) ⊂ D such that fn − f → 0 if n → ∞. Then S generates a symmetric linear operator S in K as follows: and Sf := Sf * , f ∈ D(S). = D(S) D(S) It is easy to see that this deﬁnition of S is correct. If f, f1 ∈ D(S) and f = f1 , * − Sf Q1 , g] = [Sf − Sf1 , g] = [f − f1 , Sg] = 0 for all g ∈ D(S). But the set then [Sf * = Sf Q1 . of all g ∈ K where g ∈ D(S) is dense in K, hence Sf In the following we shall always assume that Sˆ has a selfadjoint deﬁnitizable ˜ ⊃ K. It is well known (see, e.g., [7]) that in extension S˜ in some Kre˘ın space K particular this holds if the inner product [·, ·] on L has a ﬁnite number of negative squares. It also holds if the Hermitian sesquilinear form [Sf, g] on D(S) has a ﬁnite number of negative squares, and for at least one λ0 ∈ C the range R(S − λ0 ) is closed in K (see, e.g., [2]). A function f : z → f (z) with values in L is called holomorphic on Df if f : z → f(z) with values in K is holomorphic. Let again S be a symmetric linear operator in the decomposable inner product space L, and let H be a Hilbert space. A mapping Φ from L × C into H Φ : (f ; z) → Φz f, f ∈ L, z ∈ C, is called a directing mapping for S if it has the following properties: 1. For each z ∈ C the mapping f → Φz f, f ∈ L is linear. 2. For each f ∈ L the mapping z → Φz f, z ∈ C is holomorphic on C. 3. If the function z → h(z) with values in L is holomorphic on Dh ⊂ C, then Φz h(z) = 0, z ∈ Dh , if and only if there exists a holomorphic function z → g(z) on Dh with values in D(S) such that (S − z)g(z) = h(z). (Γ) 4. For each compact set Γ ⊂ C there is a mapping Ψ(Γ) : (x; z) → Ψz x from H × Γ into L such that (Γ) (a) for each z ∈ Γ the mapping z → Ψz x, x ∈ H, is linear and for each (Γ) x ∈ H the mapping z → Ψz x is holomorphic on the interior Γi of Γ;

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(Γ)

(b) Φz Ψz x = x, x ∈ H, z ∈ Γ; (Γ) (c) if z ∈ Γ is ﬁxed we have Ψz x ≤ C(z; Γ)x with some constant C(z; Γ) depending on z and Γ. Recall that in the original deﬁnition of a directing functional or a directing mapping (see [7]), concerning the case of a nonnegative semideﬁnite inner product [·, ·] on L the following condition (30 ) was imposed instead of (3): 30 . For z ∈ C and h ∈ L we have Φz h = 0 if and only if there exists an element g ∈ D(S) such that Sg − zg = h. Evidently, if we assume (30 ), then for a function h(z) as in (3) there exists a function g(·) on Dh which solves (S − z)g(z) = h(z) on Dh . However, it is not clear whether g(·) is holomorphic on Dh . The following lemma implies that this is the case if the inner product [·, ·] is nonnegative semideﬁnite on L, i.e., in this case condition (30 ) implies condition (3). Lemma 4.1. Assume that the mapping Φ has the properties (1), (2), (30 ). Then the condition (3) is satisﬁed in each of the following two cases: (a) The inner product [·, ·] on L is nonnegative semideﬁnite.

⊃ K and for each (b) S has a deﬁnitizable extension S in some Kre˘ın space K function z → h(z) which is holomorphic on Dh ⊂ C and such that Φz h(z) = 0, z ∈ Dh , the function z → g(z) on Dh deﬁned by the solutions of the equation (S − z)g(z) = h(z) (which exist according to (30 )) is locally bounded on Dh . Proof. If the inner product [·, ·] on L is nonnegative, then the symmetric operator S on the Hilbert space K has at least one selfadjoint extension S in some Hilbert space

⊃ K, which can also be regarded as a deﬁnitizable operator. In both cases we can K

⊃ K. Let z → h(z) consider a deﬁnitizable extension S of S in some Kre˘ın space K be as in condition (3), and let z → g(z) on Dh be a solution of the equation Q = h(z), Q (S − z)g(z) = h(z) (which exists according to (30 )). Then (S − z)g(z) hence also Q = h(z) Q (S − z)g(z) (4.1) Q Q = (S − z)−1 h(z). ˜ then g(z) and if z ∈ / σp (S),

we consider the holomorphic function g : On the set Dh \σ(S) Q g (z) := (S − z)−1 h(z).

(4.2)

Q and from (4.1) (S − z)g(z) Q = h(z). Q It Then we have (S − z) g(z) = h(z), Q

follows that g(z) is holomorphic on Dh \σ(S). / Let Γ = (α, β) be a real interval such that Γ ⊂ Dh . Assume that α, β ∈ ˜ ∪ σp (S) and consider again the function g(z) given by (4.2) for z = z¯, z ∈ / c(S) Q is bounded if z approaches α or β σp (S). Then the function z → (S − z)−1 h(z) nontangentially.

358

B. Textorius Indeed, we have, e.g.,

Q (S − z)−1 h(z) Q Q − h(α) Q + ((S − z)−1 − (S − α)−1 )h(α) Q + (S − α)−1 h(α) = (S − z)−1 (h(z) Q + g(α) Q Q − h(α))(z Q = (z − α)(S − z)−1 (h(z) − α)−1 + (z − α)(S − z)−1 g(α) and this remains bounded if z → α nontangentially as |z − α|(S − z)−1 has this property. We now choose a closed Jordan contour CΓ surrounding Γ, intersecting the real axis in α and β nontangentially and such that the non-real spectrum of S is outside of CΓ . Consider in the domain surrounded by CΓ the function g0 P 1 Q g0 (z) := (ζ − z)−1 (S − ζ)−1 h(ζ)dζ. 2πi CΓ It is holomorphic, and we have P P 1 1 −1 Q Q Q h /0 (4.3)

h(ζ)dζ = h(z)+ (ζ−z)−1 h(ζ)dζ+ (S−ζ) (S−z) g0 (z) = 2πi CΓ 2πi CΓ /0 ∈ E(Γ) /0 ∈ K, deﬁned by the last integral. Evidently, h

K,

with some element h

where by E we denote the spectral function of S. It follows from (4.3) and (4.1) that for all z in the interior of CΓ Q =h /0 . (S − z)(g0 (z) − g(z)) (4.4)

Consider an interval Γ0 ⊂ Γ, Γ0 ∈ RS such that all the points of Γ0 ∩ σ(S) are of either positive or negative type (see [6], [8]). Then (4.4) implies % % %

[E(dλ) h0 , h0 ] %% % % %<∞ % Γ0 |λ − z|2 %

0 )

0 ) for all z ∈ Γ0 , hence [E(Γ h0 , h0 ] = 0 and E(Γ h0 = 0. From (4.4) follows Q g(z) = g0 (z) on Γ0 , hence g(z) is holomorphic there and the ﬁrst statement of the lemma is proved. In the second situation the above reasoning implies that h0 belongs to the

∩ linear span of the algebraic eigenspaces of S corresponding to eigenvalues in c(S)

Γ. Then, if h0 has a non-zero component in the algebraic eigenspace corresponding

∩ Γ, we ﬁnd to λ0 ∈ c(S) h0 = (S − z)−1

n0 j=0

(j)

h0 + ··· (λ0 − z)j+1

(n ) for some n0 ≥ 0, h0 0 = 0, where . . . denotes terms which are holomorphic at λ0 . Q is unbounded at λ0 , which is impossiBut then (4.4) implies that g0 (z)− g(z) Q are bounded there. It follows that Q ble, since g0 (z) and g(z) h0 = 0, so g0 (z) = g(z).

Directing Mappings in Kre˘ın Spaces

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∩ Dh can be treated in a similar way, and the lemma The non-real points of σ0 (S) is proved. (Γ) Let Γ ∈ C be a closed disc around zero. By Ψ we denote the mapping in z (Γ) L(H, K) generated by Ψz , that is Q (Γ) (Γ) x = Ψ Ψ z x, x ∈ H. z · is holomorphic on Γi and it can also be considered as a function with Then Ψ

For A ∈ L(K)

the operator Ψ (Γ) is deﬁned according to (3.1). values in L(H, K). A (Γ)

Lemma 4.2. Let the closed disc Γ ⊂ C around zero be such that / σ0 (S) ⊂ Γi , i and let ∆ be a simply connected subset of Γ , symmetric with respect to R with smooth boundary ∂∆ which intersects the real axis nontangentially and is such that

∪ c(S))

= ∅. Denote by S ∆ the restriction of S to E(∆)K.

(∂∆) ∩ (σp (S) Then, for f ∈ L P 1

(Γ) Φz f dz. E(∆) f = − (S − z)−1 Ψ ˜∆ S 2πi ∂∆

Proof. For any g ∈ K 7 P 6 P 1 1

(Γ) Φz f dz, (S − ζ)−1 g dζ¯ E(∆) f + (S − z)−1 Ψ − ˜∆ S 2πi ∂∆ 2πi ∂∆ 7 P 6 P 1 1 −1

(Γ) f + (S ∆ − z)−1 Ψ Φ f dz, ( S − ζ) g dζ¯ =− z ∆ z 2πi ∂∆ 2πi ∂∆ P 1 (Γ)

f − Ψ ¯)−1 g dz. = z Φz f, (S∆ − z 2πi ∂∆ (Γ)

Furthermore, Φz (f −Ψz Φz f ) = 0, hence there exists a holomorphic solution (Γ) Q = g(z) ∈ L of the equation (S − z)g(z) = f − Ψz Φz f , therefore (S − z)g(z) (Γ) f − Ψz Φz f , the last integral becomes P 1 Q g]dz = 0, [g(z), 2πi ∂∆

and the lemma is proved. The proof of the following lemma is similar.

Lemma 4.3. Let Γ, Γ be discs around zero, both containing some set ∆ as in Lemma 4.2. If Ψ, Ψ are mappings with the properties (4), then

) Q )(Γ (Γ) = (Ψ Ψ

∆ . S

S ∆

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B. Textorius (Γ )

Proof. If z ∈ Γi ∩ (Γ )i , then for x ∈ H we have Φz (Ψz x − (Ψ )z x) = 0, hence (Γ) (Γ ) Ψz x − (Ψ )z x = (S − z)g(z) with some function g holomorphic in ∆ and with

it follows values in D(S). For arbitrary g ∈ K P P (Γ ) 1 1 (Γ) −1 Q Q g]dz = 0,

[Ψ x − (Ψ )S˜∆ x, (S − z) g]dz = − [g(z), − 2πi ∂∆ S˜∆ 2πi δ∆ (Γ)

and the lemma is proved.

5. The main theorem Theorem 5.1. Let L be a linear space with an inner product [·, ·] which is decomposable, let some decomposition be ﬁxed, and let S be a symmetric linear operator in L with a quasidense domain D, such that its corresponding operator S in K has

⊃ K. Assume also that at least one deﬁnitizable extension in some Kre˘ın space K S has a directing mapping Φ with values in some Hilbert space H. Then, for each deﬁnitizable selfadjoint extension S of S there exist operators Ai,ν = A∗i,ν ∈ L(H), ν = 0, 1, . . . , qi , i = 1, 2, . . . , l, Ai,qi ≥ 0, Bj,ν ∈ L(H), ν = 0, 1 . . . , rj , j = 1, 2 . . . , n, ∗ Ck,ν = Ck,ν ∈ L(H), Ck,pk = 0; ν = 0, 1, . . . , pk ; pk ≥ 1, k = 0, 1, . . . , m, points β1 , . . . , βn ∈ C + , an L(H)-valued measure V ∈ V with s(V ) = {α1 , . . . , αl }, points γ1 , . . . , γm ∈ R\s(V ) where V ({γ1 , . . . , γm }) = 0, such that for arbitrarily chosen intervals ∆i ∈ RV around αi which satisfy the condition ({γ1 , . . . , γm } ∪ s(V )) ∩ ∆i = {αi }, i = 1, 2, . . . , l, the representation qi l (V (dλ)Φλ f, Φλ g)(αi ;qi ) + (Ai,ν Φt f, Φt g)(ν) |t=αi [f, g] = i=1

∆i

ν=1

(V (dλ)Φλ f, Φλ g)

+ R\∪li=1 ∆i n rj

+

∗ (Bj,ν Φt f, Φt g)(ν )|t=βj + (Bj,ν Φt f, Φt g)(ν) |t=βj

j=0 ν=0 m pj

+

(Ck,ν Φt f, Φt g)(ν) |t=γk

(5.1)

k=1 ν=0

holds, if at least one of the elements f, g belongs to D(S), the other one being

arbitrary in L. If the degree of the deﬁnitizing polynomial of S is zero (that is K is a Hilbert space) or even, then the representation holds for all f, g ∈ L.

Directing Mappings in Kre˘ın Spaces

361

The representation can be chosen such that s(V ) is the set of singular critical

γ1 , . . . , γm are the regular critical points of S which belong to σp (S)

points of S, and are such that S has a Jordan chain of length > 1 in γj , and β1 , . . . , βn are the eigenvalues of S˜ in C + . If in particular the inner product [·, ·] has a ﬁnite number κ of negative squares and the numbers qi , i = 1, 2, . . . , l are chosen minimal: qi = 2κi , κi ∈ {0, 12 , 1, . . . }, then the representation can be chosen so that V with possible exception of κ ≤ κ points δ1 , . . . , δκ ∈ supp V is nonnegative and

κ

κ− (V ({δν }) +

ν=1

where

κi +

i=1

⎛ ⎜ ⎜ Bj = ⎜ ⎝

and

l

⎛

n

rank Bj +

j=1

m

rank Ck ≤ κ, ⎞

Bj,rj Bj,rj −1 .. .

Bj,rj .. .

... ... .. .

Bj,0

Bj,1

. . . Bj,rj

Ck,qk ⎜ .. Ck = ⎝ . Ck,0

0

0 .. . Ck,1

(5.2)

k=1

... .. .

0 0 .. .

0 .. .

⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎠.

. . . Ck,pk

Remark 5.2. If l = 0, m = 0 etc. in the representation (5.1) it means that the corresponding sums are 0. Proof. Let S˜ be a deﬁnitizable selfadjoint extension of Sˆ in some Kre˘ın space ˜ ⊃ K. If α is a ﬁnite critical point of S and ∆ is a bounded interval around α K

then, for some

= {α}, and the endpoints of ∆ are not in σp (S), such that ∆ ∩ c(S) q

nonnegative integer q, [(S − α) f , f ] ≥ 0 for all f ∈ E(∆)K (see [8]). According to Lemma 2.1, for f, g ∈ L and with suitably chosen Γ and C∆ , we then have 7 6 P P 1 ˜∆ − z)−1 Ψ (Γ) Φz f dz,− 1 (Γ) Φz gdz

(S˜∆ − z)−1 Ψ ( S [E(∆) fˆ,ˆ g] = − ˜∆ ˜∆ S S 2πi C∆ 2πi C∆ 7 P 6 −1 1 (Γ) Φz¯g dz (Γ) Φz f, Ψ S˜∆ − z (5.3) Ψ =− ˜∆ ˜∆ S S 2πi C∆ 7 P 6 1 (λ − α)q−1 ˜ 1 1 (Γ) Φz f, Ψ (Γ) Φz¯g dz + + ···+ E(dλ) Ψ =− ˜∆ S˜∆ S 2πi C∆ ∆ λ − z z − α (z − α)q D E P q−1 S˜∆ S˜∆ 1 N 1 (Γ) (Γ) Φz f, Ψ Φz¯g dz. + Ψ + ···+ + + ˜∆ ˜∆ S S 2πi C∆ z − α (z − α)2 (z − α)q (z − α)q+1 If we put ˜ (Γ) , (Γ) )∗ E(dλ) Ψ V (dλ) := (Ψ ˜ ˜ S S ∆

∆

(5.4)

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B. Textorius

then, according to (3.3), the ﬁrst integral on the right-hand side equals (V (dλ)Φλ f, Φλ g)(α;q) . ∆

With α = αi , ∆ = ∆i , q = qi , i = 1, 2, . . . , l this gives the ﬁrst sum in (5.1) and with ∗ ˜ν ∗ Ai,ν := (Ψ ˜ ) S∆ i Ψ S ˜ , ν = 0, . . . , qi − 1; Ai,qi := (ΨS ˜ ) N ΨS ˜ , S (Γ)

∆i

(Γ)

(Γ)

∆i

∆i

(Γ)

∆i

i = 1, 2, . . . , l, the second integral in (5.3) gives the second sum in (5.1). The other terms in (5.1) follow in the same way from (5.3), if only dλ has a positive distance from all the critical points. The terms in (5.1) corresponding to the non-real points β1 , . . . βn follow in the same way from the Laurent expansion of (S˜ − z)−1 around βk , k = 1, 2, . . . , n. In the middle integral in (5.1) over R\ ∪li=1 ∆i the measure V (dλ) is again deﬁned by (5.4), and this integral exists in the case that ∞ is a critical point ⊂ D(S).

If ∞ is of S˜ if at least one of the elements f or g belongs to D(S)

The statement not a critical point of S˜ the integral exists for all f, g ∈ D(S). V ({γ1 , . . . , γm }) = 0 follows by continuity. For the statement in the case that the inner product [·, ·] has a ﬁnite number κ of negative squares we observe, e.g., (see [4], [5]) that the dimension of the algebraic eigenspace corresponding to an eigenvalue βj in C + , j = 1, . . . , n, coincides with the dimension of the range of the matrix ⎛ ⎞ Ej,sj 0 ... 0 ⎜ Ej,sj −1 Ej,sj ... 0 ⎟ ⎜ ⎟ ⎜ .. .. .. .. ⎟ , ⎝ . . . . ⎠ Ej,2 Ej,sj −1 Ej,sj Ej,1 where Ej,ν are the coeﬃcients of the Laurent expansion at z = βj of the resolvent: (S˜ − z)−1 =

Ej,sj Ej,1 + ··· , + ···+ (βj − z)sj βj − z

and (Γ) )∗ Ej,ν Ψ

j ⊂ C +, (Γ) , βj ∈ ∆ Bj,ν := (Ψ ˜ ) ˜ S S

∆ j

∆ j

∩∆

j = {βj }. where σ(S) On the other hand, this dimension equals the number of negative squares of [·, ·] on the eigenspace. We also observe that for i = 1, . . . , l, κi at most equals the number of negative

κ (see Deﬁnition 2.2 and Folgerung 3.11 in [6]). The other

i )Π squares of [·, ·] on E(∆ terms arise in the same way, thus the estimate (5.2) holds.

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References [1] M.B. Bekker, Isometric operators with directing mappings. Current analysis and its applications 218, 3–9, “Naukova Dumka”, Kiev, 1989 (Russian). ´ [2] B. Curgus, H. Langer, Kre˘ın space approach to symmetric ordinary diﬀerential operators with an indeﬁnite weight function. J. Diﬀerential Equations 79 (1989), no. 1, 31–61. [3] M.G. Kre˘ın, On hermitian operators with directing functionals. Sb. Trud. Inst. Mat. Kiev 10 (1948), 83–106 (Ukrainian). ¨ [4] M.G. Kre˘ın, H. Langer, Uber die Q-Funktion eines π-hermiteschen Operators im Raume Πκ . Acta Sci. Math. (Szeged) 34 (1973), 191–230. [5] M.G. Kre˘ın, H. Langer, Some propositions on analytic matrix functions related to the theory of operators in the space Πκ . Acta Sci. Math. (Szeged) 43 (1981), 181–205. [6] H. Langer, Zur Spektraltheorie linearer Operatoren in J-R¨ aumen und einige Anwenat dungen auf die Schar L(λ) = λ2 I + λB + C. Habilitationsschrift, Techn. Universit¨ Dresden, 1965. ¨ [7] H. Langer, Uber die Methode der richtenden Funktionale von M.G.Kre˘ın. Acta Math. Acad. Sci. Hungar. 21 (1970), 207–224. [8] H. Langer, Spectral functions of deﬁnitizable operators in Kre˘ın spaces. Functional analysis, Proc. Dubrovnik 1981, Lecture Notes in Math. 948, pp 1–46, SpringerVerlag, Berlin, 1982. [9] H. Langer, B. Textorius, Spectral functions of a symmetric linear relation with a directing mapping. Proc. Roy. Soc. Edinburgh 81A (1978), 237–246. [10] H. Langer, B. Textorius, Spectral functions of a symmetric linear relation with a directing mapping, I. Proc Roy. Soc. Edinburgh 97A (1984), 165–176. [11] M.A. Naimark, Linear diﬀerential operators. Ungar, New York 1968. Bj¨ orn Textorius Link¨ oping University SE 58183 Link¨ oping Sweden e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 163, 365–379 c 2005 Birkh¨ auser Verlag Basel/Switzerland

The Extension Problem for Positive Deﬁnite Functions. A Short Historical Survey Zolt´an Sasv´ari Dedicated to Heinz Langer on the occasion of his retirement

Abstract. The aim of the present paper is to give a short historical survey on the extension problem for positive deﬁnite functions. Mathematics Subject Classiﬁcation (2000). Primary 43A35, 47A57, 34A55; Secondary 42A82, 46C20, 34B20. Keywords. continuation, positive deﬁnite functions, functions with negative squares.

1. Introduction Acknowledgments are usually placed at the end of a paper. In this case, however, I would like to use the ﬁrst page to thank Heinz for drawing my attention to the extension problem, for posing many interesting questions and answering many of my questions. He and M.G. Krein studied many interpolation, moment and continuation problems. In the present paper I try to give a short historical survey only in a special area of these continuation problems. I will consider only functions, not kernels. These functions will be complex-valued (i.e., not operator- or matrix-valued) and in most cases deﬁned on the real line. However, I will also treat functions with negative squares, as a natural generalization of positive deﬁnite functions. Even in this special area, the survey is not complete, the choice of the topics was also inﬂuenced by the author’s research interests.

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Z. Sasv´ ari

2. The emergence of the concept of positive deﬁnite functions The history of positive deﬁnite functions started in 1907 with Carath´eodory’s paper [10]. In this paper he studied functions of the form ∞ 1+ (ak + ibk ) · z k (ak , bk ∈ R) k=1

analytic in the unit disc and having positive real part. He characterized these functions by the property that for each n = 1, 2, . . . the point (a1 , b1 , . . . , an , bn ) ∈ R2n should lie in the smallest convex set containing the points 2 · (cos ϕ, sin ϕ, cos 2ϕ, sin 2ϕ, . . . , cos nϕ, sin nϕ),

0 ≤ ϕ < 2π.

In 1911 Toeplitz [56] noticed that Carath´eodory’s condition is equivalent to the inequality n dk−l ck c¯l ≥ 0 (n = 1, 2, . . . ; ck ∈ C) (2.1) k,l=1

where d0 = 2, dk = ak − ibk , d−k = dk . In the same year Herglotz [17] solved the so-called trigonometric momentproblem. He proved that condition (2.1) of Toeplitz holds if and only if there exists a non-negative measure µ on [0, 2π) such that 2π dn = eint dµ(t) (n = 0, ±1, ±2, . . .). 0

Note that if such a measure exists, then it is unique. Inspired by the work of Carath´eodory and Toeplitz, Mathias [41] introduced in 1923 the notion of a positive deﬁnite function on R. He called a complex-valued function f on R positive deﬁnite if f (−x) = f (x) (x ∈ R) and

n

f (xi − xj )ci c¯j ≥ 0

(2.2) (2.3)

i,j=1

for every choice of x1 , . . . , xn ∈ R and c1 , . . . , cn ∈ C. 1 He showed that a continuous function f ∈ L1 (R) is positive deﬁnite if and only if ∞ f (t) · eitx dt ≥ 0 (x ∈ R). −∞

This condition can be used for example to show that the functions 2 1 and max(0, 1 − |x|) x → e−|x| , e−x , x2 + 1 1 It is interesting to note that condition (2.2) remains part of the deﬁnition of a positive deﬁnite function until 1933, when F. Riesz [47] points out that it follows easily from (2.3).

The Extension Problem

367

are positive deﬁnite. Among the few examples, where the validity of (2.3) can easily be checked are the functions x → eixy (x, y ∈ R). Indeed, % %2 % n % n % % ei(xj −xk )y cj ck = %% eixj y cj %% ≥ 0. %j=1 % j,k=1 In 1932 Bochner [6] proved that these functions are the building stones of positive deﬁnite functions in the following sense: A continuous, complex-valued function f on R is positive deﬁnite if and only if there exists a non-negative ﬁnite measure µ on R such that ∞ f (x) = eixt dµ(t). −∞

If such a measure exists, then it is unique. In 1933 Bochner [8] generalized this result to functions of several real variables. Another important theorem of the same period is due to Stone [54, 55]: for each t ∈ R let Ut be a unitary operator in a Hilbert space H such that Ut+s = Ut Us and t −→ (v, Ut w) is continuous for all v, w ∈ H. Then there exists a so-called partition of unity E(λ) such that ∞ Ut = eitλ dE(λ). −∞

About the same time both F. Riesz [47] and Bochner [7] pointed out that Bochner’s theorem can be used to prove Stone’s result. Their proofs use the fact that for any v ∈ H the function t → (v, Ut v) is positive deﬁnite: ⎛ ⎞ n n n (Uxj −xk v, v)cj ck = ⎝ cj Uxj v, ck Uxk v ⎠ ≥ 0. j,k=1

j=1

k=1

Another application of positive deﬁnite functions is in the theory of random processes. Let Z = {Z(x) : x ∈ R} be a random process, where the random variables Z(x) are real-valued and are deﬁned on a common probability space. The process Z is said to be second-order stationary, or simply stationary, if second moments exist and the expectation E(Z(x)) and the covariance cov(Z(x), Z(x+h)) do not depend on x. We may then deﬁne the covariance function C(h) = cov(Z(x), Z(x + h)),

h ∈ R.

Khintchin [23] used Bochner’s theorem to show that a continuous real-valued function C is the covariance function of a continuous stationary random process if and only if there exists a non-negative measure µ on R such that ∞ C(h) = cos ht dµ(t), −∞

i.e., if C is a real-valued positive deﬁnite function.

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Z. Sasv´ ari

In 1959 M.G. Krein [27] (see also [26, 19, 20, 21]) generalized the concept of positive deﬁniteness in the following way: A complex-valued Hermitian function f on R (that is, f (−x) = f (x) for all x ∈ R) is said to have k negative squares if the Hermitian matrix n

A = (f (xi − xj ))i,j=1 has at most k negative eigenvalues (counted with their multiplicities) for any choice of n and x1 , . . . , xn ∈ R, and for some choice of n and x1 , . . . , xn the matrix A has exactly k negative eigenvalues. We denote by Pk (R) the set of all functions on R with k negative squares. Note that P0 (R) is the set of positive deﬁnite functions. We list a few examples: x → xk ex + (−x)k e−x ∈ Pk (R) x → (−1)k x2k ∈ Pk (R), x → (−1)k+1 x2k ∈ Pk+1 (R). More generally: if a ∈ (2k − 2, 2k] (k > 0) then x → (−1)k |x|a ∈ Pk (R). In [27] M.G. Krein proved that every continuous function f ∈ Pk (R) is deﬁnitizable in the following sense: there exists a polynomial Q of degree k such that the inequality ∞ ∞ d d h(y) Q −i h(x) dy dx ≥ 0 f (x − y)Q −i dy dx −∞ −∞ holds for every inﬁnitely diﬀerentiable function h with compact support. He obtained the integral representation ∞ itx e − S(x, t) f (x) = p(x) + dµ(t) |Q0 (t)|2 −∞ where p is a Hermitian solution of the diﬀerential equation d d Q −i p(x) = 0 Q −i Q(t) = Q(t¯) . dx dx Q0 is a polynomial that is obtained by deleting the non-real zeros of Q, S is a regularizing correction compensating for the real zeros of Q, and µ is a nonnegative measure satisfying ∞ 1 dµ(t) < ∞ 2 m −∞ (1 + t ) where m denotes the degree of Q0 . See the Notes to Section 6 in [50] for more historical remarks.

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3. Results on the existence of extensions having certain properties Now we turn to the extension problem that was ﬁrst explicitly posed by M.G. Krein [24] in 1940, though, as we will see, Carath´eodory has proved some results in the discrete case as early as 1911. A complex-valued function f deﬁned on the ﬁnite interval (−2a, 2a), a > 0, is called positive deﬁnite, if the inequality n

f (xj − xk )cj ck ≥ 0

j,k=1

holds, whenever cj ∈ C and xj ∈ (−a, a). More generally, let V be a symmetric subset of a group G containing the identity of G. A complex-valued function f on V is called positive deﬁnite if the inequality n f (x−1 i xj )ci cj ≥ 0 i,j=1

holds for all ﬁnite systems c1 , . . . , cn of complex numbers and elements x1 , . . . , xn of V with x−1 i xj ∈ V (i, j = 1, . . . , n). We will denote by P (V ) the set of all positive deﬁnite functions on V . Can every function f ∈ P (V ) be extended to a function in P (G)? The case where G is the group Z of integers and V has the form V = Vn = {k ∈ Z : |k| ≤ n} with some non-negative integer n has been investigated by Carath´eodory [11, 10]. As we have seen in the previous section, he proved that the real part of an analytic function f (z) = 1 +

∞

(ak + ibk )z k

(|z| < 1)

k=1

is positive if and only if (a1 , b1 , . . . , an , bn ) ∈ Qn for all n where Qn denotes the smallest convex set containing the points 2 · (cos ϕ, sin ϕ, . . . , cos nϕ, sin nϕ),

0 ≤ ϕ < 2π.

Carath´eodory calls the point (a1 , b1 , . . . , an , bn ) the nth geometric representative of f and proves: An arbitrary point of Qn is the nth geometric representative of at least one function f (as above). Moreover, for the points on the surface of Qn this function f is uniquely determined.2 2 He

also proved that in this case f is a rational function with at most n zeros.

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Z. Sasv´ ari

In [11] he shows, of course using a diﬀerent terminology, that the points of Qn can be identiﬁed with what we denoted by P (Vn ). Hence, we can reformulate the ﬁrst statement as: Theorem 3.1 (C. Carath´eodory). Every positive deﬁnite function on Vn = {k ∈ Z : |k| ≤ n} can be extended to a positive deﬁnite function on Z. This statement can be proved as follows. Suppose that f is positive deﬁnite on Vn and set f (n + 1) := z, f (−n − 1) := z where z is a complex number. Then f is positive deﬁnite on Vn+1 if and only if % % % f (0) f (1) ... f (n) z % % % % f (1) f (0) . . . f (n − 1) f (n) %% % % % .. D(z) := % % ≥ 0. . % % % f (n) f (n − 1) . . . f (0) f (1) %% % % z f (n) ... f (1) f (0) % It can be shown that the inequality above holds if and only if |z − z0 | ≤ r with some z0 ∈ C and r ≥ 0. Thus, f can be extended to Vn+1 in at least one way. The continuous analogue of Carath´eodory’s result has been given by Krein. Theorem 3.2 (M.G. Krein, 1940). Any continuous positive deﬁnite function f on (−A, A) can be extended to a positive deﬁnite function on R. In [24] Krein gives the following sketch of the proof. Let LA be the linear space of all functions ϕ of the form ϕ(t) = ck eixk t (ﬁnite sum) k

where −A < xk < A and ck ∈ C. On LA we deﬁne a linear functional Φ by ck f (xk ). Φ(ϕ) = k

Since f is positive deﬁnite this functional is non-negative (i.e., Φ(ϕ) ≥ 0 if ϕ ≥ 0) and hence it can be extended to a non-negative linear functional on L∞ . Setting F (x) := Φ(eixt ), −∞ < x < ∞ we have f (x) = F (x) (−A < x < A) and ⎛% %2 ⎞ % % % % ⎟ ⎜ F (xj − xk )cj ck = Φ ⎝%% cj eixj t %% ⎠ ≥ 0. % j % j,k In [46] D.A. Raikov gave another proof of Theorem 3.2.

The Extension Problem

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Krein used the continuity of f to show that Φ is non-negative. It was A.P. Artjomenko [3] who pointed out that the continuity assumption can be dropped. Theorem 3.3 (A.P. Artjomenko, 1941). Any positive deﬁnite function on (−A, A) can be extended to a positive deﬁnite function on R. About one decade later Calder´ on and Pepinsky [9] showed that in higher dimensions the extension is not always possible. Theorem 3.4 (A. Calder´ on, R. Pepinsky, 1952). There exist positive deﬁnite functions deﬁned on rectangles in Zn (n ≥ 2) that cannot be extended to positive deﬁnite functions deﬁned on all of Zn . The results of Calder´ on and Pepinsky were rediscovered and extended to Rn by Rudin [48]. Theorem 3.5 (W. Rudin, 1963). There exist continuous positive deﬁnite functions deﬁned on rectangles in Rn (n ≥ 2) that cannot be extended to positive functions deﬁned on all of Rn . The proofs of these nonexistence results use the fact that there exist positive polynomials of two real variables that cannot be represented as a ﬁnite sum of squares of polynomials with real coeﬃcients. D. Hilbert showed in 1888 [18] that there exist non-negative homogeneous polynomials in three variables which are not the sum of squares of homogeneous polynomials. The ﬁrst simple explicit example has been found by T.S. Motzkin [43]. A very simple example for R2 is the polynomial F (s, t) = s2 t2 (s2 + t2 − 1) + 1 (see [4]). Rudin [49] has also shown that positive deﬁnite extensions do exist in Rn if V is a ball and if the functions are isotropic.3 Theorem 3.6 (W. Rudin, 1970). Continuous, isotropic positive deﬁnite functions deﬁned on a ball {x ∈ Rn : x < r} can be extended to positive deﬁnite functions deﬁned on all of Rn . In higher dimensions (e.g., analysis of geostatistical data) random processes are often assumed to be both stationary and isotropic (i.e., the covariance function is isotropic). Covariance models ﬁtted to observed data frequently involve a socalled nugget eﬀect, that is a discontinuity at the origin 0 ∈ Rn : c if t = 0 C0 (t) = 0 if t = 0 If C1 is a continuous isotropic covariance function, then C = C0 + C1 3 A function f deﬁned on a subset of Rn is called isotropic if f (x) = ϕ(x) where ϕ is a realvalued function on [0, ∞) and · denotes the Euclidean norm.

372

Z. Sasv´ ari

is an isotropic covariance function with a nugget eﬀect. It was conjectured by I.J. Schoenberg [53] in 1938 that all isotropic covariance functions have this form if n ≥ 2. A positive answer was given by T. Gneiting, Z. Sasv´ ari [14]: Theorem 3.7 (T. Gneiting, Z. Sasv´ari, 1999). Any (Lebesgue) measurable, isotropic covariance function on a ball of Rn can be extended to a covariance function C on Rn . If n > 1 then C has the form C = C0 + C1 with some continuous covariance function C1 . For n = 1, the second statement is not true: the indicator function of the rational numbers is positive deﬁnite and isotropic but nowhere continuous. One of the basic problems of statistical estimation is to guess an unknown probability measure, given certain observations. In general there are inﬁnitely many measures consistent with the data. The question, which of these is the best is an important one. The principle that nature favors the states of largest entropy was applied with success by physicists in statistical mechanics. For a probability measure µ with density p we deﬁne the entropy of µ by 1 ∞ log p(x) K(µ) := dx − log 4. π −∞ 1 + x2 The next result was proved in [12]: Theorem 3.8 (J. Chover, 1961). Let f be a positive deﬁnite function on (−A, A) satisfying f (0) = 1. Suppose that f (0) is ﬁnite and that there is at least one probability measure µ with density such that ∞ (i) f (x) = eitx dµ(t), −A < x < A −∞

(ii) K(µ) > −∞. Then there exists a unique measure µ satisfying (i) and (ii) for which K(µ) is maximal. We refer to [38] for more information on the maximum entropy principle in connection with the trigonometric moment problem. This principle has also been intensively studied in connection with matrix completion problems. A partial matrix is an array in which some entries are exactly speciﬁed and the remaining ones are thought of as free variables. In a typical matrix completion problem, one asks for conditions under which the unspeciﬁed entries can be chosen so that the resulting ordinary matrix is of a desired type (e.g., positive deﬁnite like in the sketched proof of Theorem 3.1). There is a vast literature on matrix completion problems, we mention here the paper [16]. We close this section with results stating that certain local properties of positive deﬁnite functions imply global ones. The ﬁrst one is due to A.P. Artjomenko and follows immediately from the inequality |f (x) − f (y)|2 ≤ 2f (0) · [f (0) − Re f (y − x)] satisﬁed by an arbitrary positive deﬁnite function f .

(x, y ∈ R)

The Extension Problem

373

Theorem 3.9. If the real part of a function f ∈ P (R) is continuous at 0 then f is uniformly continuous on R. A positive deﬁnite function f is said to be analytic if there exists a complexvalued function θ of one complex variable, which is holomorphic in a circle around 0 and f (t) = θ(t) for all real t in this circle. The next result deals with analyticity and has been proved independently by P. L´evy [40] and D.A. Raikov [45]. Theorem 3.10. If f is an analytic positive deﬁnite function then there exist αf , βf ∈ (0, ∞] such that f extends to a function which is holomorphic in the strip { z ∈ C : −αf < Im z < βf } and such that αf and βf are maximal with this property. We refer to Section 1.12 of [5] for more details. As a corollary we obtain that an analytic positive deﬁnite function on (−A, A) has exactly one positive deﬁnite extension (which is analytic). The next theorem deals with diﬀerentiability. Unfortunately we cannot assign exact priorities to it. Theorem 3.11. If for some positive integer k the real part of f is 2k-times diﬀerentiable at 0 then f is 2k-times continuously diﬀerentiable on R. The question, whether (Lebesgue) measurable positive deﬁnite functions have measurable positive deﬁnite extensions has been posed by M.G. Krein [25, 28]. An aﬃrmative answer was given by the author [51].4 Theorem 3.12 (Z. Sasv´ari, 1986). If f is measurable on (−A, A) then f is measurable on R. Combining this with Theorem 3.3 we see that any measurable positive deﬁnite function on (−A, A) can be extended to a measurable positive deﬁnite function on R.

4. Description of all extensions In this section we deal with the following problem which was ﬁrst investigated by M.G. Krein [24] in 1940 and which plays an important role in his method for solving inverse spectral problems [29, 30, 31]. Let f be a continuous positive deﬁnite function f on [−2a, 2a]. Describe all positive ˜ = f (t), t ∈ [−2a, 2a]. deﬁnite functions f˜ on R satisfying f(t) We will call f˜ an extension or continuation of f . 4 A personal remark: I learned Krein’s problem from Heinz, who informed Krein in a letter about my solution. As a young mathematician I was very glad when Heinz showed me a part of Krein’s answer in which he congratulated me.

374

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There are two possibilities: 1. There exists exactly one extension. This is the case for example if f is analytic or f (0) = 1 and |f (t0 )| = 1 for some t0 ∈ [−2a, 2a], t0 = 0. 2. There are inﬁnitely many extensions (the so-called indeterminate case). In [24] Krein gave several criteria for uniqueness of the extension by employing methods reminiscent of those used in the classical moment problems (see [1] or [52]). We mention also E. Akutowicz [2] and Devinatz [13] who gave criteria for uniqueness in terms of the self-adjointness of certain operators in Hilbert spaces. The classiﬁcation of all possible extensions in the indeterminate case is of the same nature as that in the classical moment problems. An essential role is played by the description of all selfadjoint extensions of a given Hermitian operator with defect (1, 1). In the papers [32, 33, 34, 35, 36] and in the unpublished manuscript [37] M.G. Krein and H. Langer formulate and study indeﬁnite analogues of interpolation, moment, and continuation problems. Some of their results were new even when restricted to the positive deﬁnite case. To formulate the indeﬁnite continuation problem denote by Pk,a the set of all continuous, Hermitian functions f on the interval [−2a, 2a] having k negative squares, i.e., such that the maximum of the numbers of the negative eigenvalues, counted with multiplicities, of the matrices (f (ti − tj ))ni,j ,

n ∈ N, t1 , . . . , tn ∈ (−a, a)

is equal to k. With this notation the indeﬁnite continuation problem can be formulated as follows: Given a function f ∈ Pk,a ﬁnd all functions f˜ ∈ Pk (R) such that f˜(t) = f (t), t ∈ [−2a, 2a]. An even continuous function C deﬁned on [−2a, 2a] is said to have an accelerant H if (i) H(t) = −C (t) exists for t = 0, (ii) H(t) is absolutely integrable over [−2a, 2a], and (iii) C (0+) < 0. With the accelerant H we associate the operator H in L2 [0, 2a], deﬁned by 2a (H ϕ) (t) = H(t − s) ϕ(s) ds (0 ≤ t ≤ 2a). 0

We are now able to formulate one of the numerous results of Krein and Langer [36]: Theorem 4.1 (M.G. Krein, H. Langer, 1985). Assume that f ∈ Pk,a has an accelerant and that −1 does not belong to the spectrum of the integral operator H. If f has more than one continuation then there exist four entire functions wjk (z) = wjk (a, z), such that the equality ∞ w11 (z)T (z) + w12 (z) , Im z > γ eizt f˜(t) dt = i w21 (z)T (z) + w22 (z) 0

The Extension Problem

375

for some γ ≥ 0 establishes a bijective correspondence between all continuations f˜ ∈ Pk (R) and all functions T of the form T ≡ ∞ or ∞ tz + 1 T (z) = α + βz + dσ(t), z ∈ C \ R −∞ t − z where α ∈ R, β ≥ 0 and σ is a non-negative ﬁnite measure.5 The method of proof is based on the extension theory of symmetric operators in a Pontryagin space (see [34] and the references therein). The matrix function w11 (·; z) w12 (·; z) W(·; z) = w21 (·; z) w22 (·; z) called the resolvent matrix of this continuation problem, is shown to satisfy the canonical system dW(a; z) J = zW(a; z)H(a) (4.1) da and the initial condition 1 0 . (4.2) W(0; z) = −z 1 0 −1 , and the Hamiltonian H(a) is a continuous 2 × 2-matrix Here J = 1 0 function. The paper [39] contains a detailed study of the special case f (t) = 1 − |t|. This function is positive deﬁnite on [−2a, 2a] if a ≤ 1 and it has one negative square if a > 1. The resolvent matrix is given by (1−(a−1)z 2 ) sin az−az cos az W (a, z) =

sin az−z cos az (a−1)z z cos az a−1

and the Hamiltonian is

H(a) =

z2

(a − 1)z sin az + cos az

(a − 1)2 0

0 1 (a−1)2

.

The mentioned papers of Krein and Langer contain a wealth of interesting results and their connections to other parts of mathematics. To give a detailed survey would require a book rather than a paper. M. Kaltenb¨ ack and H. Woracek [22] gave a description of all extensions with k negative squares of a continuous function f ∈ Pk,a , without posing additional assumptions on f . 5 T is a so-called Nevanlinna function: it is holomorphic in the upper half-plane and has a nonnegative imaginary part there.

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They proved that there are the following three cases: 1. f has inﬁnitely many extensions with k negative squares. 2. f has a unique extension with k negative squares and no other extension with a ﬁnite number of negative squares. 3. f has a unique extension with k negative squares and there exists a positive integer ∆(f ) such that the function f has no extensions with k negative squares, where k < k < k + ∆(f ), and inﬁnitely many extensions with k ≥ k + ∆(f ) negative squares. At last, we mention that the results in [36] have been successfully applied in [42] (see also [15]) to characterize a class of stationary processes that have polynomial covariance functions on an interval.

References [1] N.I. Akhiezer, The classical moment problem. Edinburgh: Oliver and Boyd 1965. [2] E.J. Akutowicz, On extrapolating a positive deﬁnite function from a ﬁnite interval. Math. Scand. 7 (1959), 157–169. [3] A.P. Artjomenko, Hermitian positive functions and positive functionals. (Russian), Dissertation, Odessa State University (1941). Published in Teor. Funkci˘ı, Funkcional. Anal. i Priloˇ z en. 41, (1983) 1–16; 42 (1984), 1–21. [4] C. Berg, J.P.R. Christensen, P. Ressel, Harmonic Analysis on Semigroups. Theory of Positive Deﬁnite and Related Functions. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag 1984. [5] T.M. Bisgaard, Z. Sasv´ ari, Characteristic Functions and Moment Sequences. Positive Deﬁniteness in Probability. Huntigton, NY: Nova Science Publishers 2000. [6] S. Bochner, Vorlesungen u ¨ber Fouriersche Integrale. Leipzig: Akademische Verlagsgesellschaft 1932. [7] S. Bochner, Spektralzerlegung linearer Scharen unit¨ arer Operatoren. Sitzungsber. Preuss. Akad. Wiss. phys.-math. (1933), 371–376. [8] S. Bochner, Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse. Math. Ann. 108 (1933), 378–410. [9] A. Calder´ on, R. Pepinsky, On the phases of Fourier coeﬃcients for positive real periodic functions. Computing Methods and the Phase Problem in X-Ray Crystal Analysis, published by The X-Ray Crystal Analysis Laboratory, Department of Physics, The Pennsylvania State College (1952), 339–348. ¨ [10] C. Carath´eodory, Uber den Variabilit¨ atsbereich der Koeﬃzienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann. 64 (1907), 95–115. ¨ [11] C. Carath´eodory, Uber den Variabilit¨ atsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo, 32 (1911), 193–217. [12] J. Chover, On normalized entropy and the extensions of a positive-deﬁnite function. J. Math. Mech. 10 (1961), 927–945.

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[13] A. Devinatz, On the extensions of positive deﬁnite functions. Acta Math. 102 (1959), 109–134. [14] T. Gneiting, Z. Sasv´ari, The Characterization Problem for Isotropic Covariance Functions Math. Geology 31(1) (1999), 105–111. [15] T. Gneiting, Z. Sasv´ari, M. Schlather, Analogies and correspondences between variograms and covariance functions. Adv. Appl. Probab. 33 (2001), 617–630. [16] I. Gohberg, M.A. Kaashoek, H.J. Woerdemann, A Maximum Entropy Principle in the General Framework of the Band Method. J. Funct. Anal. 95 (1991), 231–254. ¨ [17] G. Herglotz, Uber Potenzreihen mit positivem, reellen Teil im Einheitskreis. Leipziger Berichte, math.-phys. Kl. 63 (1911), 501–511. ¨ [18] D. Hilbert, Uber die Darstellung deﬁniter Formen als Summe von Formenquadraten. Math. Ann. 32 (1988), 342–350. [19] I.S. Iohvidov, Unitary and selfadjoint operators in spaces with an indeﬁnite metric. (Russian) Dissertation, Odessa (1950). [20] I.S. Iohvidov, On the theory of indeﬁnite Toeplitz forms. (Russian) Dokl. Akad. Nauk SSSR 101(2) (1955), 213–216. [21] I.S. Iohvidov, M.G. Krein, Spectral theory of operators in spaces with an indeﬁnite metric II. (Russian) Trudy Moskov. Mat. Ob˘s˘c. 8 (1959), 413–496. English translation: Amer. Math. Soc. Translations 2(3), (1963), 283–373. [22] M. Kaltenb¨ ack, H. Woracek, On extensions of hermitian functions with a ﬁnite number of negative squares. J. Operator Theory 40 (1998), 147–183. [23] A. Khintchin, Korrelationstheorie der station¨ aren stochastischen Prozesse. Math. Ann. 109 (1934), 604–615. [24] M.G. Krein, Sur le probl`eme du prolongement des fonctions hermitiennes positives et continues. Dokl. Akad. Nauk SSSR 26 (1940), 17–22. [25] M.G. Krein, On the representation of functions by Fourier-Stieltjes integrals. (Russian) U˘cenije Zapiski Kuibishevskogo Gosud. Pedag. i U˘citelskogo Inst. 7 (1943), 123–148. [26] M.G. Krein, Screw lines in inﬁnite-dimensional Lobachevski space and the Lorentz transformation. (Russian) Usp. Mat. Nauk 3(3) (1948), 158–160. [27] M.G. Krein, On the integral representation of a continuous Hermitian-indeﬁnite function with a ﬁnite number of negative squares. (Russian) Dokl. Akad. Nauk SSSR, 125(1) (1959), 31–34. [28] M.G. Krein, On measurable Hermitian-positive functions. (Russian) Mat. Zametki 23 (1978), 79–89. English translation: Math. Notes 23 (1978), 45–50. [29] M.G. Krein, Solution of the inverse Sturm–Liouville problem. Dokl. Akad. Nauk. SSSR 76:1 (1951), 21–24. [30] M.G. Krein, On the transition function of the one-dimensional second order boundary value problem. Dokl. Akad. Nauk. SSSR88:3 (1953), 405–408. [31] M.G. Krein, On the determination of the potential of a particle by its S-function. Dokl. Akad. Nauk. SSSR 105:3 (1955), 433–436. [32] M.G. Krein, H. Langer, On the indeﬁnite power moment problem. Soviet Math. Dokl. 17 (1976), 90–93.

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¨ [33] M.G. Krein, H. Langer, Uber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Πκ zusammenh¨ angen. I. Einige Funktionenklassen und ihre Darstellungen. Math. Nachr. 77 (1977), 187–236. ¨ [34] M.G. Krein, H. Langer, Uber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Πk zusammenh¨ angen. II. Verallgemeinerte Resolventen, u-Resolventen und ganze Operatoren. J. Funct. Anal. 30 (1978), 390–447. [35] M.G. Krein, H. Langer, On some extension problems which are closely related with the theory of hermitian operators in a space πk . III. Indeﬁnite analogues of the Hamburger and Stieltjes problems. Part (I). Beitr¨ age zur Analysis 14 (1979), 25– 40. [36] M.G. Krein, H. Langer, On some continuation problems which are closely related to the theory of operators in spaces Πk . IV: Continuous analogues of orthogonal polynomials on the unit circle with respect to an indeﬁnite weight and related coninuation problems for some classes of functions. J. Operator Theory 13 (1985), 299–417. [37] M.G. Krein, H. Langer, Continuation of Hermitian positive deﬁnite functions and related questions. (Manuscript) [38] H.J. Landau, Maximum Entropy and the Moment Problem, Bull. Amer. Math. Soc. 16(1), (1987), 47–77. [39] H. Langer, M. Langer, Z. Sasv´ ari, Continuations of Hermitian Indeﬁnite Functions and Corresponding Canonical Systems: An Example. Methods. of Funct. Anal. and Topology, 10(1) 2004, 39–53. [40] P. L´evy, Th´ eorie de l’addition des variables al´eatoires. Paris: Gauthier-Villars, 1937. ¨ [41] M. Mathias, Uber positive Fourier-Integrale. Math. Zeitschrift 16 (1923), 103–125. [42] S. Mitra, T. Gneiting, Z. Sasv´ari, Polynomial covariance functions on intervals. Bernoulli 9(2) 2003, 229–241. [43] T.S. Motzkin, The arithmetic-geometric inequality. Published in: Inequalities (Ed. by O. Shisha). New York-London: Academic Press 1967. [44] G. P´ olya, Remarks on characteristic functions. Proc. First Berkeley Conf. on Math. Stat. and Prob. Berkeley: Univ. of Calif. Press (1949), 115–123. [45] D.A. Raikov, On the decomposition of Gauss and Poisson laws. (Russian) Izv. Akad. Nauk. SSSR, ser. math. 2 (1937), 91–124. [46] D.A. Raikov, Sur les fonctions positivement d´eﬁnies. Dokl. Akad. Nauk SSSR, 26 (1940), 860–865. ¨ [47] F. Riesz, Uber S¨ atze von Stone und Bochner. Acta Sci. Math. 6 (1932–1934), 184– 198. [48] W. Rudin, The extension problem for positive-deﬁnite functions. Illinois J. Math. 7 (1963), 532–539. [49] W. Rudin, An extension theorem for positive-deﬁnite functions. Duke Math. J. 37 (1970), 49–53. [50] Z. Sasv´ ari, Positive Deﬁnite and Deﬁnitizable Functions. Berlin: Akademie Verlag 1994. [51] Z. Sasv´ ari, The extension problem for measurable positive deﬁnite functions. Math. Zeitschrift 191 (1986), 475–478.

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[52] J. Shohat, J. Tamarkin, The problem of moments. Math. Surveys No. 1, Providence, R. I.: Amer. Math. Soc. 1943. [53] I.J. Schoenberg, Metric spaces and competely monotone functions. Ann. of Math. 39(4) (1938), 811–841. [54] M.H. Stone, Linear transformations in Hilbert space. III. Operational methods and group theory. Proc. Nat. Acad. Sci. U.S.A. 16 (1930), 172–175. [55] M.H. Stone, On one-parameter unitary groups in Hilbert space. Ann. of Math. 33(2) (1932) 643–648. ¨ [56] O. Toeplitz, Uber die Fourier’sche Entwickelung positiver Funktionen. Rend. Circ. Mat. Palermo 32 (1911), 191–192. Zolt´ an Sasv´ ari Mommsenstr. 13 D-01062 Dresden Germany e-mail: [email protected]