Operator Theory in Inner Product Spaces

Operator Theory: Advances and Applications Vol. 175 Editor: I. Gohberg Editorial Office: School of Mathematical Science...

Author: Karl-Heinz Förster | Peter Jonas | Heinz Langer

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Operator Theory: Advances and Applications Vol. 175 Editor: I. Gohberg Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: D. Alpay (Beer-Sheva) J. Arazy (Haifa) A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Böttcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) R. E. Curto (Iowa City) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (College Station) A. Dijksma (Groningen) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) B. Gramsch (Mainz) J. A. Helton (La Jolla) M. A. Kaashoek (Amsterdam) H. G. Kaper (Argonne) Subseries: Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze Universität Potsdam Germany Sergio Albeverio Universität Bonn Germany

S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) B. Mityagin (Columbus) V. Olshevsky (Storrs) M. Putinar (Santa Barbara) L. Rodman (Williamsburg) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) I. M. Spitkovsky (Williamsburg) S. Treil (Providence) H. Upmeier (Marburg) S. M. Verduyn Lunel (Leiden) D. Voiculescu (Berkeley) D. Xia (Nashville) D. Yafaev (Rennes) Honorary and Advisory Editorial Board: C. Foias (Bloomington) P. R. Halmos (Santa Clara) T. Kailath (Stanford) H. Langer (Vienna) P. D. Lax (New York) M. S. Livsic (Beer Sheva) H. Widom (Santa Cruz) Michael Demuth Technische Universität Clausthal Germany Jerome A. Goldstein The University of Memphis, TN USA Nobuyuki Tose Keio University, Yokohama Japan

Operator Theory in Inner Product Spaces

Karl-Heinz Förster Peter Jonas Heinz Langer Carsten Trunk Editors Advances in Partial Differential Equations

Birkhäuser Basel . Boston . Berlin

Editors: Karl-Heinz Förster Peter Jonas Carsten Trunk Institut für Mathematik, MA 6-4 Technische Universität Berlin Straße des 17. Juni 136 D-10623 Berlin

Heinz Langer Mathematik Technische Universität Wien Wiedner Hauptstrasse 8–10/1411 A-1040 Wien e-mail: [email protected]

e-mail: [email protected] [email protected] [email protected]

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Bibliographic information published by Die Deutsche Bibliothek 'LH'HXWVFKH%LEOLRWKHNOLVWVWKLVSXEOLFDWLRQLQWKH'HXWVFKH1DWLRQDOELEOLRJUD¿HGHWDLOHG bibliographic data is available in the Internet at . ISBN 3978-3-7643-8268elston – Birkhäuser Berlin ISBN 978-3-7643-8269-8 Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the PDWHULDOLVFRQFHUQHGVSHFL¿FDOO\WKHULJKWVRIWUDQVODWLRQUHSULQWLQJUHXVHRI LOOXVWUDWLRQVUHFLWDWLRQEURDGFDVWLQJUHSURGXFWLRQRQPLFUR¿OPVRULQRWKHUZD\VDQG storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2007 Birkhäuser Verlag AG, 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 f Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN-10: 3-7643-8269-3 ISBN-13: 978-3-7643-8269-8

e-ISBN-10: 3-7643-8270-8 e-ISBN-13: 978-3-7643-8270-4

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Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

T.Ya. Azizov and L.I. Soukhotcheva Linear Operators in Almost Krein Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

J. Behrndt, A. Luger and C. Trunk Generalized Resolvents of a Class of Symmetric Operators in Krein Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

J. Behrndt, H. Neidhardt and J. Rehberg Block Operator Matrices, Optical Potentials, Trace Class Perturbations and Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

V. Derkach, S. Hassi and H. de Snoo Asymptotic Expansions of Generalized Nevanlinna Functions and their Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

A. Fleige A Necessary Aspect of the Generalized Beals Condition for the Riesz Basis Property of Indeﬁnite Sturm-Liouville Problems . . . . . .

89

K.-H. F¨ orster and B. Nagy On Reducible Nonmonic Matrix Polynomials with General and Nonnegative Coeﬃcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

S. Hassi, H. de Snoo and H. Winkler On Exceptional Extensions Close to the Generalized Friedrichs Extension of Symmetric Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 P. Jonas and H. Langer On the Spectrum of the Self-adjoint Extensions of a Nonnegative Linear Relation of Defect One in a Krein Space . . . . . . . . . . . . . . . . . . . . . . 121 M. Kaltenb¨ ack and H. Woracek Canonical Diﬀerential Equations of Hilbert-Schmidt Type . . . . . . . . . . . . 159 I. Karabash and A. Kostenko Spectral Analysis of Diﬀerential Operators with Indeﬁnite Weights and a Local Point Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

vi

Contents

C. Mehl and C. Trunk Normal Matrices in Degenerate Indeﬁnite Inner Product Spaces . . . . . . 193 V. Pivovarchik Symmetric Hermite-Biehler Polynomials with Defect . . . . . . . . . . . . . . . . . 211 L. Rodman A Note on Indeﬁnite Douglas’ Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225

A. Sandovici Some Basic Properties of Polynomials in a Linear Relation in Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Preface This volume contains papers written by the participants of the 4th Workshop on Operator Theory in Krein Spaces and Applications, which was held at the Technische Universit¨ at Berlin, Germany, December 17 to 19, 2004. The workshop covered topics from spectral, perturbation and extension theory of linear operators and relations in inner product spaces. They included spectral analysis of diﬀerential operators, the theory of generalized Nevanlinna functions and related classes of functions, spectral theory of matrix polynomials and problems from scattering theory. All these topics are reﬂected in the present volume. The workshop was attended by 58 participants from 12 countries. It is a pleasure to acknowledge the substantial ﬁnancial support received from the – Deutsche Forschungsgemeinschaft (DFG), ur Schl¨ ussel– DFG-Forschungszentrum MATHEON “Mathematik f¨ technologien”, – Institute of Mathematics of the Technische Universit¨ at Berlin. We would also like to thank Petra Grimberger for her great help. Last but not least, special thanks are due to Jussi Behrndt and Christian Mehl for their excellent work in the organisation of the workshop and the preparation of this volume. Without their assistance the workshop might not have taken place. The Editors

Operator Theory: Advances and Applications, Vol. 175, 1–11 c 2007 Birkh¨ auser Verlag Basel/Switzerland

Linear Operators in Almost Krein Spaces Tomas Ya. Azizov and Lioudmila I. Soukhotcheva Abstract. The aim of this paper is to study the completeness and basicity problems for selfadjoint operators of the class K(H) in almost Krein spaces and prove criteria for the basicity and completeness of root vectors of linear pencils. Mathematics Subject Classiﬁcation (2000). Primary 47B50; Secondary 46C50. Keywords. Krein space, operator pencil, completeness and basicity problem.

1. Introduction Let H be a Hilbert space, let A and B be compact operators and let A be additionally a positive operator. Consider the linear operator pencil L(λ) = A−1 − λ(I + B).

(1)

Such a pencil appears, for instance, if the following spectral problem in L2 (0, π) is considered: π ⎧ 2 ⎨ − d f + q(t)f (t) = λ(f (t) + K(t, s)f (s)ds) dt2 (2) 0 ⎩ f (0) = f (π) = 0. Assume q is a continuous real function with q(t) > −1, the kernel K(t, s) is symmetric and continuous on [0, π]×[0, π]. It remains to deﬁne A and B in the following way: d2 f A−1 f = − 2 + qf, dt dom A−1 = f ∈ L2 (0, π) | f, f absolutely continuous on (0, π), d2 f − 2 + qf ∈ L2 (0, π), f (0) = f (π) = 0 dt This research is supported partially by the grant RFBR 05-01-00203.

2

T.Ya. Azizov and L.I. Soukhotcheva π (Bf )(t) =

K(t, s)f (s)ds. 0

The completeness and basicity problems for the pencil (1) and for the operator H = A(I + B)

(3)

are closely related. The completeness problem for operators (3) under the assumption λ = 0 is not an eigenvalue of H (0 ∈ / σp (H)) was considered for the ﬁrst time by M.V. Keldysh (see, for instance, [6, Theorem 8.1]). The case when B is also selfadjoint and ker H = {0} was studied by I.Ts. Gokhberg and M.G. Krein in [6, p. 322]. There it was shown that there is a Riesz basis in H which consists of root vectors of H. The proof is based on the selfadjointness of the operator H with respect to the indeﬁnite inner product [·, ·] = ((I + B)·, ·),

(4)

which turns {H, [·, ·]} into a Pontryagin space. General criteria for the completeness and basicity of root vectors of selfadjoint operators in Pontryagin spaces can be found in [2], and, more generally, in [3, Theorem IV.2.12]. If ker(I + B) = 0, the inner product (4) is degenerate and, due to the compactness of the operator B, we have (a) the isotropic part H0 := {x ∈ H | [x, y] = 0, for all y ∈ H} of H is ﬁnite-dimensional, and = H/H0 is a Pontryagin space. (b) the factor-space H Indeﬁnite inner product spaces H with the properties (a) and (b) are called almost Pontryagin spaces. The operator H in (3) is selfadjoint in the degenerate almost Pontryagin space {H, [·, ·]}. First results about the completeness and basicity problems for compact selfadjoint operators in almost Pontryagin space were obtained in [1]. Recently, spectral properties of operators acting in almost Pontryagin and almost Krein spaces (for a deﬁnition of almost Krein spaces we refer to Deﬁnition 1 below) and their applications were studied in, for example, [4], [5], [7]–[13].1 The main aim of this paper is to study the completeness and basicity problems for selfadjoint operators of the class K(H) in almost Krein spaces (a deﬁnition of the class K(H) see below on p. 3). Moreover, we will prove criteria for the basicity and completeness of root vectors of the pencil (1). 1 The

authors thank Chr. Mehl (TU Berlin) for his help with the bibliography.

Linear Operators in Almost Krein Spaces

3

2. Main deﬁnitions We shortly recall some deﬁnitions and notions related to Krein spaces. For more details we refer to [3]. A linear space K equipped with an indeﬁnite inner product [·, ·] is called a Krein space if it admits a canonical decomposition K = K+ [+]K− , where K+ is [·, ·]-orthogonal to K− , and K± is a Hilbert space with respect to the scalar product ± [·, ·], respectively. Deﬁnition 1. A space K with an indeﬁnite inner product [·, ·] is called an almost Krein space if its isotropic part K0 is ﬁnite-dimensional (we do not exclude the case K0 = {0}) and the factor-space = K/K0 , K

ˆ x∈x [ˆ x, yˆ]= [x, y], xˆ, yˆ ∈ K, ˆ, y ∈ yˆ,

(5)

is a Krein space with respect to the naturally reduced indeﬁnite inner product [·, ·]. Let us note that each almost Pontryagin space is an almost Krein space. As in the Krein space case we say that a nonnegative/nonpositive subspace L± belongs to the class h± , if it is represented as a sum of a ﬁnite-dimensional neutral and a uniformly positive/negative subspace. Remark 2. From the deﬁnitions it follows immediately that nonnegative/nonpositive subspace L± in an almost Pontryagin space belongs to the class h± . All operators below are considered to be linear, everywhere deﬁned and bounded. An operator A in an almost Krein space is called selfadjoint, if [Ax, y] = [x, Ay] for all x, y ∈ K. By deﬁnition, a selfadjoint operator A belongs to the class H, if it has a maximal nonpositive and a maximal nonnegative invariant subspaces L± and each such a subspace belongs to h± , respectively. We say that a selfadjoint operator A belongs to K(H), if there is a selfadjoint operator B ∈ H, which commutes with A. Remark 3. Let K be an almost Pontryagin space. From Remark 2 and the deﬁnition of the class H it follows that I ∈ H. Hence each selfadjoint operator in an almost Pontryagin space belongs to the class K(H). Moreover, from Theorem 4 below we have that every selfadjoint operator in an almost Pontryagin space belongs to the class H.

4

T.Ya. Azizov and L.I. Soukhotcheva

3. Invariant dual pairs {L+ , L− } is said to be a dual pair, if L± is a nonnegative/nonpositive subspace and [x+ , x− ] = 0 for all x± ∈ L± . We prove an analog of a known result about invariant subspaces of operators of the class K(H) in a Krein space (see [3, § III.5]) for the case of almost Krein spaces. Theorem 4. Let A ∈ K(H) be a selfadjoint operator in an almost Krein space K and let B be a selfadjoint operator in H which commutes with A. Assume {L+ , L− } is an A-invariant dual pair (it is not excluded that L+ = L− = {0}). Then there exists an A-invariant maximal dual pair {L+ , L− } such that L± ⊂ L± . If {L+ , L− } is also B-invariant, then we can choose a maximal dual pair which is simultaneously A- and B-invariant. In particular, there exists an A-invariant maximal dual pair {L+ , L− } such that L± ∈ h± . Proof. Since the isotropic part K0 of K is a part of each maximal semideﬁnite subspace and {L+ + K0 , L− + K0 } is A-invariant, without loss of generality we assume that K0 ⊂ L+ ∩ L− . Consider the factor-space (5). Let us note that the isotropic part K0 of K is invariant under all selfadjoint operators in K, in particular, it is A- and B and B in K, generated by A and invariant. Therefore the selfadjoint operators A ∈ K(H). B, are well deﬁned. Since AB = B A and B ∈ H, we have A 0 Denote L± = L± /K . The dual pair {L+ , L− } is A-invariant. By [3, Corol lary III.5.13] and [3, Theorem III.1.13] there is an A-invariant maximal dual pair + − + − {L , L } in K. Then {L , L }, where L± = {x ∈ K | x ˆ ∈ L± }, is a desired A-invariant dual pair. If {L+ , L− } is an A- and B-invariant dual pair then by [3, Theorem III.5.12], [3, Theorem III.1.13] and the same arguments as above there is an A- and Binvariant maximal dual pair which is an extension of {L+ , L− }. In particular, if L+ = L− = {0}, then, by the above statement, there exists an A- and B-invariant maximal dual pair {L+ , L− }. B ∈ H implies L± ∈ h± . From Deﬁnition 1 it follows that a space K with an indeﬁnite inner product [·, ·] is an almost Krein space if and only if it admits a decomposition in a direct sum (6) K = K0 + K1 , 0 with ﬁnite-dimensional isotropic part K and a subspace K1 which is a Krein space with respect to [·, ·]. Let {L+ , L− } be an A-invariant maximal dual pair in K. Then L± = K0 +L± 1, + − ± = L ∩ K . The dual pair {L , L } is maximal in the Krein space K where L± 1 1. 1 1 1 − 0 0 ∩ L and let M ⊂ K be a subspace skew-linked with L . Then: Let L01 = L+ 1 1 1 1 1 − 0 K = K0 + L01 + L+ 1 + L1 + M1 .

(7)

Linear Operators in Almost Krein Spaces

5

Introduce in K a Hilbert scalar product (·, ·) such that all subspaces in (7) are ± orthogonal to each others with respect to (·, ·) and (·, ·)|L± 1 = ± [·, ·]|L1 . Then with respect to (7) the operator A has a triangular representation ⎡ ⎤ A00 A01 A02 A03 A04 ⎢ ⎥ ⎢ ⎥ ⎢ 0 A11 A12 A13 A14 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 A22 0 A24 ⎥ (8) A=⎢ 0 ⎥, ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 A33 A34 ⎥ ⎢ ⎥ ⎣ ⎦ 0 0 0 0 A44 where A0k , k = 0, 1, 2, 3, 4, are bounded operators, A11 and A∗44 are similar, and A22 and A33 are selfadjoint with respect to the introduced scalar product. We omit information about other components of (8) since it is not essential in this paper. Corollary 5. The spectrum of a selfadjoint operator A ∈ K(H) in an almost Krein space is real, except for at most ﬁnite number of non-real normal eigenvalues. There are not more than a ﬁnite number of real eigenvalues λ of A with degenerate ker(A − λ). In particular, the set of eigenvalues with nontrivial Jordan chains is ﬁnite. 4 Proof. The triangular representation (8) implies σ(A) = k=0 σ(Akk ). Since A22 and A33 are Hilbert space selfadjoint operators, the non-real spectrum of A coincides with the non-real spectra of A00 , A11 and A44 . Hence the statement about the non-real spectrum of A follows immediately from the ﬁnite dimensionality of K0 + L01 + M01 . Assume that A has an inﬁnite number of real eigenvalues λ such that ker(A− λ) is degenerate. Let L be the closed linear span of the isotropic parts of these kernels. Let B ∈ H be an operator which commutes with A. Then BL ⊂ L. This contradicts the assumption B ∈ H. The last statement of the corollary follows from the fact that kernels with non-trivial Jordan chains are degenerate.

4. Criteria for the completeness and basicity In the sequel we will need the following lemma. Lemma 6. Assume K is an almost Krein space and A ∈ K(H) is a selfadjoint operator in K with σ(A) = {0}. Then A is a nilpotent operator of ﬁnite rank. Proof. We consider the decomposition (7) of K and the corresponding decomposition (8) of A. Note that σ(A) = {0} implies σ(A00 ) = σ(A11 ) = σ(A22 ) = σ(A33 ) = σ(A44 ) = {0} and the selfadjointness of A22 and A33 implies A22 = 0 and A33 = 0.

6

T.Ya. Azizov and L.I. Soukhotcheva

Below we use the following notations: – Lλ (A) is the root subspace of A related to the eigenvalue λ : λ ∈ σp (A); – E(A) is the closed linear span (c.l.s. ) of the root vectors of A: E(A) = c.l.s. {Lλ (A) | λ ∈ σp (A)}; – E0 (A) = c.l.s. {ker(A − λ) | λ ∈ σp (A)}; – ER\0 (A) = c.l.s. {Lλ (A) | λ ∈ R \ {0}}. A vector system {ek } is called almost orthonormal if it is a union of two systems {ek } = {ek } ∪ {ek }, where {ek } is orthonormal: [ek , ek ] = 0, [ek , ej ] = δkj sign [ek , ek ], and {ek } is a ﬁnite system with [ek , ej ] = 0. Corollary 7. Let A ∈ K(H) be a compact selfadjoint operator in a Krein space K. Then E(A) = K if and only if L0 (A) ∩ ER\0 (A) = {0}. Moreover, if E(A) = K, there exists in K an almost orthonormal Riesz basis composed of root vectors of A. Proof. Indeed, let L = L0 (A)∩ER\0 (A). Then L is orthogonal to all root subspaces of A. If E(A) = K, we have the orthogonality of L to the whole Krein space which is non-degenerate. Hence L = {0}. Assume L = {0}. Let B ∈ H and AB = BA. Since ER\0 (A) is A- and Binvariant, the isotropic part ER\0 (A)0 of ER\0 (A) is also A- and B-invariant. Hence, taking into account that B ∈ H and ER\0 (A)0 is B-invariant neutral subspace, we obtain from Theorem 4 that dim ER\0 (A)0 < ∞. Since the root subspaces Lλ (A), λ = 0 are non-degenerate, we have ER\0 (A)0 ⊂ L0 (A), and hence ER\0 (A)0 ⊂ L, that is, ER\0 (A)0 = {0}. Therefore ER\0 (A) is a Krein space. The spectrum of the restriction of A to the orthogonal complement ER\0 (A)[⊥] to ER\0 (A) consists of a unique point λ = 0. Hence, it follows from Lemma 6 that L0 (A) = ER\0 (A)[⊥] . The latter implies E(A) = K. The existence of an almost orthonormal basis is proved in [3, Thm. IV.2.12]. Below we give necessary and suﬃcient conditions for the basicity and completeness of the set of root vectors of a compact selfadjoint operator in an almost Krein space. We show that in almost Krein spaces, in contrast to the Pontryagin or Krein space case, the completeness of the root vector system is not suﬃcient for its basicity. Theorem 8. Let A ∈ K(H) be a compact selfadjoint operator in an almost Krein space K and E(A) = K. Then the following assertions are equivalent: (i) K0 ∩ L0 (A) ∩ ER\0 (A) = {0}; (ii) L0 (A) ∩ ER\0 (A) = {0}; (iii) there exists in K an almost orthonormal Riesz basis consisting of root vectors of A. Proof. Corollary 5 implies that the non-real spectrum of A and the real nonzero eigenvalues λ ∈ σp (A) with degenerate kernels ker(A − λ) is a ﬁnite subset of the normal eigenvalues. Let P be the Riesz projector related to this part of σ(A). Since

Linear Operators in Almost Krein Spaces

7

dim P K < ∞ and P K is both A- and B-invariant, where B ∈ H commutes with A, we have that A|(I − P )K satisﬁes all conditions of the theorem and the basicity property for the root systems of A and A|(I − P )K is equivalent. Moreover, K0 ∩ L0 (A) ∩ ER\0 (A) = (I − P )K0 ∩ L0 (A|(I − P )K) ∩ ER\0 (A|(I − P )K) and L0 (A) ∩ ER\0 (A) = {0} ⇐⇒ L0 (A|(I − P )K) ∩ ER\0 (A|(I − P )K) = {0}. Hence without loss of generality we can assume P = 0, that is, both the set of the non-real eigenvalues and the set of nonzero real eigenvalues with degenerate kernels are empty. (i) ⇒ (ii). Let K0 ∩ L0 (A) ∩ ER\0 (A) = {0}. Then ER\0 (A) is a nondegenerate subspace. Indeed, let us suppose the opposite, i.e., that this subspace is degenerate and its isotropic part ER\0 (A)0 is nonzero. The subspace ER\0 (A)0 is ﬁnitedimensional and A-invariant. Therefore, there exists an eigenvalue λ0 and a cor0 responding eigenvector x0 ∈ ER\0 (A) . Since all kernels ker(A − λ) with λ = 0 are nondegenerate, we obtain λ0 = 0. So, x0 ∈ ker A, it is orthogonal to L0 (A) and thus is orthogonal to E(A). Hence, x0 ∈ K0 ∩ L0 (A) ∩ ER\0 (A), and x0 = 0. This contradicts the assumption x = 0. We have that ER\0 (A) is nondegenerate and therefore L0 (A) ∩ ER\0 (A) = {0}. (ii) ⇒ (i) is trivial. (ii) ⇒ (iii). According to the assumption all eigenvalues of A are real and all ker(A − λ), λ = 0, are nondegenerate. Hence the subspace ER\0 (A) is nondegenerate. Let us prove that this subspace is a Krein space. Really, it is B-invariant, where B ∈ H is an operator commuting with A. In each subspace ker(A − λ) − + − the operator B has an invariant maximal dual pair {L+ λ , Lλ }, and {L , L } with ± ± L = c.l.s. {Lλ | λ ∈ σ(A)}, is a B-invariant maximal dual pair in ER\0 (A). The assumption B ∈ H implies L± ∈ h± . Hence L+ + L− is an almost Krein space. By construction, the defect of this subspace in ER\0 (A) is ﬁnite. Hence ER\0 (A) is also an almost Krein space and, by assumption, it is nondegenerate. So, ER\0 (A) is a Krein space. Therefore the orthogonal decomposition K = ER\0 (A)[+]ER\0 (A)

[⊥]

holds. Since B ∈ H has in ER\0 (A) an invariant maximal dual pair, we have [⊥] B|ER\0 (A) ∈ H and B|ER\0 (A) ∈ H. Hence, A|ER\0 (A) ∈ K(H) and A|ER\0 (A) [⊥]

[⊥]

∈ K(H).

By Lemma 6, the subspace ER\0 (A) ⊃ L0 (A) coincides with L0 (A|ER\0 (A) [⊥] L0 (A). This implies ER\0 (A) ) = L0 (A) and K = ER\0 (A)[+]L0 (A).

[⊥]

)⊂ (9)

Since ER\0 (A) is a Krein space and A|ER\0 (A) ∈ K(H) it follows from [3, Theorem IV.2.12] that there exists in ER\0 (A) an orthonormal Riesz basis constructed from

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T.Ya. Azizov and L.I. Soukhotcheva

eigenvectors of A. If we add to this basis an arbitrary almost orthonormal Riesz basis composed of vectors of L0 (A), we obtain desired basis in K. (iii) ⇒ (ii). Assume there exists an almost orthonormal Riesz basis in K composed of root vectors of A. Let {ek } be the part of this basis contained in ER\0 (A). Using the same arguments as above we can assume, without loss of generality, that all vectors ek are deﬁnite andorthogonal to each other. If x0 ∈ L0 (A) ∩ ER\0 (A), then x0 ∈ c.l.s. {ek } : x0 = αk ek . On the other hand it is orthogonal to all these vectors. Hence, αk = [x0 , ek ] sign [ek , ek ] = 0, that is, x0 = 0. Remark 9. In the proof of Theorem 8 we have used the assumption that the root vector system of A is complete only in the proof of the implication (i) ⇒ (ii). We can rewrite the assumptions (i) and (ii) as 0 ∈ / σp (A|K0 ∩ ER\0 ) and 0∈ / σp (A|L0 (A) ∩ ER\0 ), respectively. Theorem 10. Let A ∈ K(H) be a compact selfadjoint operator in an almost Krein space K. Then {E(A) = K} ⇐⇒ {L0 (A) ∩ ER\0 (A) ⊂ K0 }. (10) Proof. {E(A) = K} =⇒ {L0 (A) ∩ ER\0 (A) ⊂ K0 }. Let x0 ∈ L0 (A) ∩ ER\0 (A). Since L0 (A) is orthogonal to c.l.s. {Lλ (A) | λ = 0}, the vector x0 is orthogonal to E(A), that is, x0 ∈ K0 . {E(A) = K} ⇐= {L0 (A) ∩ ER\0 (A) ⊂ K0 }. With respect to decomposition (6) the operator A admits the following matrix representation: ⎤ ⎡ A00 A01 ⎦. (11) A=⎣ 0 A11 induced by the operator A in the factor space K = Consider also the operator A 0 0 K/K . It follows from L0 (A) ∩ ER\0 (A) ⊂ K that L0 (A) ∩ ER\0 (A) = {0}. By a Riesz basis composed of root vectors of A. Since the Theorem 8, there is in K operators A and A11 are similar, there is in K1 a Riesz basis composed of root vectors of A11 . Hence, E(A11 ) = K1 . It remains to use that E(A) = K0 [+]E(A11 ) = K (see (11)).

5. Keldysh type operators Let H be a Hilbert space with the scalar product (·, ·), let A and B be compact selfadjoint operators acting in this space and A > 0. Consider an operator H as in (3) and introduce in H an inner product [·, ·] given by [·, ·] = ((I + B)·, ·). Denote K = {H, [·, ·]}. Then K is an almost Krein space, moreover, K is an almost Pontryagin space. The operator H is selfadjoint in K. Thus, we can apply to H Theorems 8 and 10.

Linear Operators in Almost Krein Spaces

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Theorem 11. Let an operator H as in (3) satisfy the above-mentioned properties. Then (a) E(H) = K; (b) There is an almost orthonormal Riesz basis in the almost Krein space K composed of root vectors of H if and only if one of the following equivalent assumptions holds. (b1) dim ker A(I + B) = dim ker(I + B)A; (b2) ker(I + B) ⊂ ran A. Proof. (a) The isotropic part K0 of the almost Krein space K coincides with ker(I + B). Since A > 0, we have L0 (H) = ker H = ker(I + B). Hence, L0 (H)∩ER\0 (H) ⊂ ker(I + B) = K0 . Now (a) follows directly from Theorem 10. (b) Let us note that the assumptions (b1) and (b2) are equivalent since the operator A has a trivial kernel. Suppose that there exists an almost orthonormal Riesz basis in K composed of root vectors of H. By Theorem 8(ii) we have K = ker H[+]ER\0 (H).

(12)

Since the operator H is diagonal with respect to (12) and the operator H|ER\0 (H) is a Pontryagin space selfadjoint, H|ER\0 (H) is similar to its Hilbert space adjoint. Hence H = A(I + B) and its Hilbert space adjoint H ∗ = (I + B)A are similar too. This implies (b1). Assume (b2). Consider the linear pencil L(λ) = A−1 − λ(I + B).

(13)

Since the set of root vectors of this pencil and of the operator H are the same, it is suﬃcient to check that there is in K an almost orthonormal Riesz basis composed of root vectors of the pencil (13). Consider the orthogonal decomposition of H: H = ker(I + B) ⊕ ran (I + B). −1

Let us rewrite the operator A to the decomposition (14):

(14)

and the pencil (13) in the matrix form with respect ⎡

A−1 = ⎣ ⎡ L(λ) = ⎣

C00 C10

C00

C01

C10

C11

⎤ ⎦,

C01

⎤

⎦. C11 − λ(I + B11 )

(15)

From (b2) it follows that C00 is a ﬁnite-dimensional operator and positive with respect to the Hilbert space scalar product, that C01 and C10 are bounded operators of ﬁnite ranks and that the operator C11 is positive with respect to the x0 Hilbert space scalar product and has a compact inverse. Hence, x = is an x1

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−1 C01 x1 , where x1 is eigenvector of (15), related to λ = 0, if and only if, x0 = −C00 an eigenvector of the pencil −1 C01 − λ(I + B11 ), L1 (λ) = C11 − C10 C00

(16)

corresponding to the same eigenvalue. As in the proof of Theorem 8 we assume without loss of generality that the root subspaces and kernels corresponding to −1 the same eigenvalue coincides. Since C11 − C10 C00 C01 is a Hilbert space positive operator in ran (I + B), its inverse is compact, and ker(I + B11 ) = {0}, the sets of eigenvectors of (16) coincide with the sets of eigenvectors of the operator −1 C01 )−1 (I + B11 ). H1 = (C11 − C10 C00

(17)

Consider ran (I + B) as a Pontryagin space K1 with indeﬁnite inner product [·, ·] = ((I + B11 )·, ·). Because the compact operator H1 is positive in K1 , the assumption (ii) of Theorem 8 is fulﬁlled. Hence, there is an almost orthonormal Riesz basis in K1 composed of eigenvectors of H1 . Taking into account the relation between eigenvectors of H1 and H and the equality ker(I + B) = ker H, we obtain the existence of an almost orthonormal Riesz basis in K composed of eigenvectors vectors of H. The latter is true if there are no associated vectors. In the general case, there is an almost orthonormal Riesz basis in K composed of root vectors of H.

References [1] Azizov T.Ya. On completely continuous operators that are selfadjoint with respect to a degenerate indeﬁnite metric, Matem. issled., 7 (1972), 4, 237–240. (Russ.) [2] Azizov T.Ya., Iokhvidov I.S. A criterion of the completeness and basicity of root vectors of a completely continuous J-selfadjoint operator in a Pontryagin space Πκ . Matem. issled., 6 (1971), 1, 158–161. (Russ.) [3] T.Ya. Azizov, I.S. Iokhvidov, Foundation of the theory of linear operators in spaces with an indeﬁnite metric, Nauka, Moscow, 1986 (Russ.); English transl.: Linear operators in spaces with an indeﬁnite metric, Wiley, New York, 1989. [4] P. Binding and R. Hryniv. Full and partial range completeness. Oper. Theory Adv. Appl. 130:121–133, 2002. [5] Vladimir Bolotnikov, Chi-Kwong Li, Patrick Meade, Christian Mehl, Leiba Rodman. Shells of matrices in indeﬁnite inner product spaces. Electron. J. Linear Algebra, 9: 67–92, 2002. [6] I.Ts. Gokhberg, M.G. Krein. Introduction to the theory of linear non-selfadjoint operators in a Hilbert space. Nauka, Moscow, 1965 (Russian). [7] M. Kaltenb¨ ack and H. Woracek. Selfadjoint extensions of symmetric operators in degenerated inner product spaces. Integral Equations Operator Theory, 28 (1997), 289–320. [8] P. Lancaster, A.S. Markus, and P. Zizler. The order of neutrality for linear operators on inner product spaces. LAA 259 (1997), 25–29.

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[9] H. Langer, R. Mennicken, and C. Tretter. A self-adjoint linear pencil Q − λP of ordinary diﬀerential operators. Methods Funct. anal. Topology, 2 (1996), 38–54. [10] A. Luger. A factorization of regular generalized Nevanlinna functions. Integral Equations Operator Theory 43: 326–345, 2002. [11] Christian Mehl, Leiba Rodman. Symmetric matrices with respect to sesquilinear forms. Linear Algebra Appl., 349: 55–75, 2002. [12] Christian Mehl, Andre C.M. Ran, Leiba Rodman. Semideﬁnite invariant subspaces: degenerate inner products, to appear in Oper. Theory Adv. Appl. [13] H. Woracek. Resolvent matrices in degenerate inner product spaces. Math. Nachr. 213 (2000), 155–175. Tomas Ya. Azizov and Lioudmila I. Soukhotcheva Department of Mathematics Voronezh State University Universitetskaya pl., 1 394022, Voronezh, Russia e-mail: [email protected] e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 175, 13–32 c 2007 Birkh¨ auser Verlag Basel/Switzerland

Generalized Resolvents of a Class of Symmetric Operators in Krein Spaces Jussi Behrndt, Annemarie Luger and Carsten Trunk Abstract. Let A be a closed symmetric operator of defect one in a Krein space K and assume that A possesses a self-adjoint extension in K which locally has the same spectral properties as a deﬁnitizable operator. We show that the Krein-Naimark formula establishes a bijective correspondence between the of A compressed resolvents of locally deﬁnitizable self-adjoint extensions A acting in Krein spaces K×H and a special subclass of meromorphic functions. Mathematics Subject Classiﬁcation (2000). Primary: 47B50; Secondary: 47B25. Keywords. Generalized resolvents, Krein-Naimark formula, self-adjoint extensions, locally deﬁnitizable operators, locally deﬁnitizable functions, boundary value spaces, Weyl functions, Krein spaces.

1. Introduction Let A be a densely deﬁned closed symmetric operator with defect one in a Hilbert space K and let {C, Γ0 , Γ1 } be a boundary value space for the adjoint operator A∗ . Let A0 be the self-adjoint extension A∗ ker Γ0 of A in K and denote the γ-ﬁeld and Weyl function corresponding to the boundary value space {C, Γ0 , Γ1 } by γ and M , respectively. Here M is a scalar Nevanlinna function, that is, it maps the upper half-plane C+ holomorphically into C+ ∪ R and is symmetric with respect to the real axis. It is well known that in this case the Krein-Naimark formula − λ)−1 |K = (A0 − λ)−1 − γ(λ) M (λ) + τ (λ) −1 γ(λ)∗ PK (A (1.1) establishes a bijective correspondence between the class of Nevanlinna functions τ (including the constant ∞) and the compressed resolvents of self-adjoint extensions of A in K × H, where H is a Hilbert space, cf. [22, 32]. The compressed resolvent A on the left-hand side of (1.1) is said to be a generalized resolvent of A. We note that if A has equal deﬁciency indices > 1 the generalized resolvents of A can still be described with formula (1.1), where the parameters τ are so-called Nevanlinna families, cf. [11, 23, 30, 31].

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Various generalizations of the Krein-Naimark formula in an indeﬁnite setting have been proved in the last decades. The case that A is a symmetric operator in a Pontryagin space K and H is a Hilbert space was investigated by M.G. Krein and H. Langer in [24]. Later V. Derkach considered both K and H to be Pontryagin or even Krein spaces, cf. [10]. In the general situation of Krein spaces K and H one obtains a correspondence between locally holomorphic relation-valued functions τ of A with a non-empty resolvent set. Under additional and self-adjoint extensions A assumptions other variants of (1.1) were proved in [6, 7, 8, 9, 10, 14, 27]. If, e.g., H is a Pontryagin space, then the parameters τ belong to the class of Nκ -families, a class of relation-valued functions which includes the generalized Nevanlinna functions. If H is a Krein space and the hermitian forms [A·, ·] and ·] both have ﬁnitely many negative squares, then τ belongs to a special subclass [A·, of the deﬁnitizable functions, cf. [7, 19]. It is the aim of this paper to prove a new variant of formula (1.1). Here we allow both K and H to be Krein spaces and we assume that A is of defect one and possesses a self-adjoint extension A0 in K which locally has the same spectral properties as a deﬁnitizable operator or relation, cf. [20, 28]. Under the assumption is also locally deﬁnitizable and that its sign types coincide “in essence” (i.e., that A with the exception of a discrete set, see Deﬁnition 2.6) with the sign types of A0 we prove in Theorem 3.2 that there exists a so-called locally deﬁnitizable function τ such that (1.1) holds. The proof is based on a coupling method developed in [11, §5] and a recent perturbation result from [4]. 2 One of the main diﬃculties here is to show that the symmetric relation A∩H possesses a self-adjoint extension in the Krein space H with a non-empty resolvent ∩ H2 in H such set and to choose a boundary value space for the adjoint of A that (1.1) holds with the corresponding Weyl function τ . In connection with a class of abstract λ-dependent boundary value problems the converse direction was already proved in [3], i.e., for a given locally deﬁnitizable function τ a self-adjoint of A in K × H such that (1.1) holds was constructed. extension A The paper is organized as follows: In Section 2 we recall the deﬁnitions and basic properties of locally deﬁnitizable self-adjoint operators and relations and the class of locally deﬁnitizable functions introduced and studied by P. Jonas, see, e.g., [20, 21]. The notion of dcompatibility of sign types of locally deﬁnitizable relations and functions is deﬁned in the end of Section 2.3. In the beginning of Section 3 we recall some basics on boundary value spaces and associated Weyl functions. Section 3.2 contains our main result. We prove in Theorem 3.2 that formula (1.1) establishes a bijective correspondence between an appropriate subclass of the locally deﬁnitizable functions and the compressed resolvents of locally deﬁnitizable of A in a Krein space K × H with K-minimal self-adjoint exit space extensions A spectral sign types d-compatible to those of A0 . Finally, in the end of Section 3.2, we formulate a variant of the Krein-Naimark formula for self-adjoint extensions of A in K and K × H, respectively, which locally have the same spectral A0 and A

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15

properties as self-adjoint operators or relations in Pontryagin spaces and functions τ from the local generalized Nevanlinna class.

2. Locally deﬁnitizable self-adjoint relations and locally deﬁnitizable functions 2.1. Notations and deﬁnitions Let (K, [·, ·]) be a separable Krein space with a corresponding fundamental symmetry J. 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 ). If K := K1 = K2 we simply write L(K). 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 to [15] and [16]. . and . The sum and the direct sum of subspaces in K2 are denoted by We deﬁne an indeﬁnite inner product on K2 by f g fˆ, gˆ := i [f, g ] − [f , g] , fˆ = , g ˆ = ∈ K2 . f g 0 −iJ ∈ L(K2 ) is a correspondThen (K2 , [[·, ·]]) is a Krein space and J = iJ 0 ing fundamental symmetry. If necessary we will indicate the underlying space by subscripts, e.g., [[·, ·]]K2 . Let A be a linear relation in K. The adjoint relation A+ ∈ C(K) is deﬁned as ˆ ∈ K2 | h, ˆ fˆ = 0 for all fˆ ∈ A , A+ := A[[⊥]] = h where A[[⊥]] denotes the orthogonal companion of A with respect to [[·, ·]]. A is said to be symmetric (self-adjoint) if A ⊂ A+ (resp. A = A+ ). Let S be a closed linear relation in K. The resolvent set ρ(S) of S ∈ C(K) −1 is the set of all λ ∈ C such that (S − λ) ∈ 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. A point λ ∈ C is called a point of regular type of S, λ ∈ r(S), if (S − λ)−1 is a bounded operator. We say that λ ∈ C belongs to the approximate point spectrum of S, denoted by σap (S), if there exists a sequence xynn ∈ S, n = 1, 2, . . . , such that xn = 1 and limn→∞ yn − λxn = 0. The extended approximate point spectrum σ ap (S) of S is deﬁned by σap (S) ∪ {∞} if 0 ∈ σap (S −1 ) σ ap (S) := . if 0 ∈ σap (S −1 ) σap (S) 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 self-adjoint relations, cf. [20] (for bounded self-adjoint operators see [29]). For

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equivalent descriptions of the spectra of positive and negative type we refer to [20, Theorem 3.18]. Deﬁnition 2.1. Let A0 be a self-adjoint relation in K. A point λ ∈ σap (A0 ) is to be of positive type (negative type) with respect to A0 , if for every sequence said xn yn ∈ A0 , 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 (A0 ), then ∞ is said to be of positive type (negative type) with respect to A0 if 0 is of positive type (resp. negative type) with respect to A−1 0 . We denote the set of all points of σ ap (A0 ) of positive type (negative type) by σ++ (A0 ) (resp. σ−− (A0 )). We remark that the self-adjointness of the relation A0 yields that the points of positive and negative type introduced in Deﬁnition 2.1 belong to R. An open subset ∆ of R is said to be of positive type (negative type) with (A0 ) is of positive type (resp. negative type) respect to A0 if each point λ ∈ ∆ ∩ σ with respect to A0 . An open subset ∆ of R is called of deﬁnite type with respect to A0 if it is either of positive or of negative type with respect to A0 . For each λ ∈ σ++ (A0 ) (σ−− (A0 )) there exists an open neighborhood Uλ in C such that (Uλ ∩ σ (A0 ) ∩ R) ⊂ σ++ (A0 ) (resp. (Uλ ∩ σ (A0 ) ∩ R) ⊂ σ−− (A0 )), Uλ \R ⊂ ρ(A0 ) and (A0 − λ)−1 ≤ M |Im λ|−1 holds for some M > 0 and all λ ∈ Uλ \R, cf. [1], [20] (and [29] for bounded operators). 2.2. Locally deﬁnitizable self-adjoint relations In this section we brieﬂy recall the notion of locally deﬁnitizable self-adjoint relations and intervals of type π+ and type π− from [20]. Let Ω be some 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. Deﬁnition 2.2. Let A0 be a self-adjoint 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 in Ω. The relation A0 is said to be deﬁnitizable over 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 A0 . (ii) For every ﬁnite union ∆ of open connected subsets of R, ∆ ⊂ Ω ∩ R, 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 holds for all λ ∈ U\R.

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By [20, Theorem 4.7] a self-adjoint relation A0 in K is deﬁnitizable over C if and only if A0 is deﬁnitizable, that is, the resolvent set of A0 is non-empty and there exists a rational function r = 0 with poles only in ρ(A0 ) such that r(A0 ) ∈ L(K) is a nonnegative operator in K, that is [r(A0 )x, x] ≥ 0 holds for all x ∈ K (see [28] and [16, §4 and §5]). Let A0 = A+ 0 be deﬁnitizable over Ω and let δ → E(δ) be the local spectral function of A0 on Ω ∩ R. Recall that E(δ) is deﬁned for all ﬁnite unions δ of connected subsets of Ω ∩ R, δ ⊂ Ω ∩ R, the endpoints of which belong to Ω ∩ R and are of deﬁnite type with respect to A0 (see [20, Section 3.4 and Remark 4.9]). With the help of the local spectral function E(·) the open subsets of deﬁnite type in Ω ∩ R can be characterized in the following way. An open subset ∆, ∆ ⊂ Ω ∩ R, is of positive type (negative type) with respect to A0 if and only if for every ﬁnite union δ of open connected subsets of ∆, δ ⊂ ∆, such that the boundary points of δ in R are of deﬁnite type with respect to A0 , the spectral subspace (E(δ)K, [·, ·]) (resp. (E(δ)K, −[·, ·])) is a Hilbert space (cf. [20, Theorem 3.18]). We say that an open subset ∆, ∆ ⊂ Ω ∩ R, is of type π+ (type π− ) with respect to A0 if for every ﬁnite union δ of open connected subsets of ∆, δ ⊂ ∆, such that the boundary points of δ in R are of deﬁnite type with respect to A0 the spectral subspace (E(δ)K, [·, ·]) is a Pontryagin space with ﬁnite rank of negativity (resp. positivity). We shall say that A0 is of type π+ over Ω (type π− over Ω) if Ω ∩ R is of type π+ (resp. type π− ) with respect to A0 and σ(A0 ) ∩ Ω\R consists of eigenvalues with ﬁnite algebraic multiplicity. We remark, that spectral points in sets of type π+ and type π− can also be characterized with the help of approximative eigensequences (see [1, 2]). 2.3. Matrix-valued locally deﬁnitizable functions In this section we recall the deﬁnition of matrix-valued locally deﬁnitizable functions from [21]. Although in the formulation of the main theorem in Section 3.2 below only scalar locally deﬁnitizable functions appear, matrix-valued functions will be used within the proof. 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 axis, 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 nonreal 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 ). The following deﬁnition of sets of positive and negative type with respect to matrix functions and Deﬁnition 2.4 below of locally deﬁnitizable matrix functions can be found in [21].

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Deﬁnition 2.3. 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 of negative type with respect to τ . Deﬁnition 2.4. 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 ﬁnite union ∆ of open connected subsets in R, ∆ ⊂ Ω ∩ R, there exist 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. The class of L(Cn )-valued deﬁnitizable functions in Ω will be denoted by Dn×n (Ω). In the case n = 1 we write D(Ω) instead of D1×1 (Ω) and we set where d∞

D(Ω) := D(Ω) ∪ {d∞ }, 0 denotes the relation c | c ∈ C ∈ C(C).

A function τ ∈ M n×n (C) which is deﬁnitizable in C is called deﬁnitizable, see [20]. We note that τ ∈ M n×n (C) is deﬁnitizable if and only if there exists a rational function g symmetric with respect to the real axis such that the poles of g belong to h(τ ) ∪ {∞} and gτ is the sum of a Nevanlinna function and a meromorphic function in C (cf. [20]). For a comprehensive study of deﬁnitizable functions we refer to the papers [18, 19] of P. Jonas. We mention only that the generalized Nevanlinna class is a subclass of the deﬁnitizable functions. Recall that a function τ ∈ M n×n (C) is said to be a generalized Nevanlinna function if the kernel Kτ , τ (λ) − τ (µ) , λ−µ has ﬁnitely many negative squares (see [25] and [26]). In [21] 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 τ = τ0 + τ(0) (2.1) n n of an L(C )-valued deﬁnitizable function τ0 and an L(C )-valued function τ(0) which is locally holomorphic on ∆. We say that a locally deﬁnitizable function τ ∈ Dn×n (Ω) is a generalized Nevanlinna function in Ω if the function τ0 in (2.1) can be chosen as a generalized Nevanlinna function. Kτ (λ, µ) =

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19

The class of L(Cn )-valued generalized Nevanlinna functions in Ω will be denoted by N n×n (Ω). In the case n = 1 we write N (Ω) instead of N 1×1 (Ω) and we set (Ω) := N (Ω) ∪ {d∞ }, N 0 | c ∈ C ∈ C(C). where d∞ denotes the relation c

The following theorem is a consequence of [21, Propositions 2.8 and 3.4]. It establishes a connection between self-adjoint relations which are locally deﬁnitizable (locally of type π+ ) and L(Cn )-valued locally deﬁnitizable functions (resp. local generalized Nevanlinna functions). Theorem 2.5. Let Ω be a domain as above and let A0 be a self-adjoint relation in the Krein space K which is deﬁnitizable over Ω. Let γ ∈ L(Cn , K) and S = S ∗ ∈ L(Cn ), ﬁx some point λ0 ∈ ρ(A0 ) ∩ Ω and deﬁne τ (λ) := S + γ + (λ − Re λ0 ) + (λ − λ0 )(λ − λ0 )(A0 − λ)−1 γ for all λ ∈ ρ(A0 ) ∩ Ω. Then the following holds. (i) The function τ is deﬁnitizable in Ω, τ ∈ Dn×n (Ω). (ii) If A0 is of type π+ over Ω, then τ belongs to N n×n (Ω). (iii) An open subset ∆ of Ω ∩ R which is of positive type (negative type) with respect to A0 is of positive type (resp. negative type) with respect to τ . In the sequel we shall often assume that the sign types of self-adjoint relations which are deﬁnitizable over Ω, and deﬁnitizable functions in Ω coincide outside of a discrete set in Ω ∩ R. A notion for this concept is introduced in the next deﬁnition, cf. [3, Deﬁnition 2.8]. Deﬁnition 2.6. Let A0 and A1 be self-adjoint relations which are deﬁnitizable over Ω and let τ and τ be L(Cn )-valued deﬁnitizable functions in Ω. We shall say that the sign types of A0 and A1 (A0 and τ , τ and τ) are d-compatible in Ω if for every 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 A0 and A1 (resp. A0 and τ , τ and τ) or of negative type with respect to A0 and A1 (resp. A0 and τ , τ and τ). If A0 is deﬁnitizable over Ω and the function τ ∈ Dn×n (Ω) is deﬁned as in Theorem 2.5, then obviously the sign types of A0 and τ are d-compatible in Ω. A typical nontrivial situation where d-compatibility of sign types of locally deﬁnitizable self-adjoint relations appears is shown in Theorem 2.7 below. For a proof we refer to [4, Theorem 3.2]. Theorem 2.7. Let A0 and A1 be self-adjoint relations in K, assume that the set ρ(A0 ) ∩ ρ(A1 ) ∩ Ω is non-empty and that A0 is deﬁnitizable over Ω. If (A1 − µ)−1 − (A0 − µ)−1 is a ﬁnite rank operator for some (and hence for all) µ ∈ ρ(A0 ) ∩ ρ(A1 ) ∩ Ω, then A1 is deﬁnitizable over Ω and the sign types of A0 and A1 are d-compatible in Ω.

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3. Generalized resolvents of a class of symmetric operators 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 be a closed symmetric relation in K. We say that A has 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 self-adjoint extension of A in K and that each self ˆ = m. adjoint extension Aˆ of A in K satisﬁes dim A/A We shall use the so-called boundary value spaces for the description of the self-adjoint extensions of closed symmetric relations in Krein spaces. The following deﬁnition is taken from [10]. 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 if+G is a2 Hilbert space and Γ0 A + Γ0 , Γ1 : A → G are mappings such that Γ := Γ1 : A → G is surjective, and the relation Γfˆ, Γˆ g G 2 = fˆ, gˆ K2 holds for all fˆ, gˆ ∈ A+ . In the following we recall some basic facts on boundary value spaces which can be found in, e.g., [8] and [10]. For the Hilbert space case we refer to [17], [12] and [13]. Let A be a closed symmetric relation in K. Then Nλ,A+ := ker(A+ − λ) = ran (A − λ)[⊥] denotes the defect subspace of A at the point λ ∈ r(A) and we set fλ fλ ∈ Nλ,A+ . Nˆλ,A+ := λfλ

ˆλ instead of Nλ,A+ and N ˆλ,A+ . When no confusion can arise we write Nλ and N If there exists a self-adjoint 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. [10]). 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 self-adjoint 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)

Generalized Resolvents of Symmetric Operators in Krein Spaces

21

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 (self-adjoint) extensions of A and the closed symmetric (resp. self-adjoint) 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 {G, Γ0 , Γ1 }. They 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. [10]). It follows that M (λ) = Re M (λ0 )+γ(λ0 )+ (λ − Re λ0 )

+ (λ − λ0 )(λ − λ0 )(A0 − λ)−1 γ(λ0 )

(3.5)

holds for a ﬁxed λ0 ∈ ρ(A0 ) and all λ ∈ ρ(A0 ). If, in addition, the condition K = clsp {Nλ | λ ∈ ρ(A0 )} is fulﬁlled, then it follows from (3.3) and (3.4) that the function M is strict, that is M (λ) − M (µ)∗ ker = {0} (3.6) λ−µ λ∈h(M)

holds for some (and hence for all) µ ∈ h(τ ). If Θ ∈ C(G) and AΘ is the corresponding extension of A (see (3.1)), then for every point λ ∈ ρ(A0 ) we have λ ∈ ρ(AΘ )

if and only if

0 ∈ ρ(Θ − M (λ)).

For λ ∈ ρ(AΘ ) ∩ ρ(A0 ) the well-known resolvent formula −1 (AΘ − λ)−1 = (A0 − λ)−1 + γ(λ) Θ − M (λ) γ(λ)+

(3.7)

(3.8)

holds (for a proof see, e.g., [10]). 3.2. A variant of the Krein-Naimark formula We choose a domain Ω as in the beginning of Section 2.2. Let A be a (not necessarily densely deﬁned) closed symmetric operator in the Krein space K, let {G, Γ0 , Γ1 } be a boundary value space for A+ and let H be a further Krein space. A self-adjoint of A in K × H is said to be an exit space extension of A and H is extension A

22

J. Behrndt, A. Luger and C. Trunk

of A is said to be K-minimal if called the exit space. The exit space extension A ρ(A) ∩ Ω is non-empty and − λ)−1 K | λ ∈ ρ(A) ∩Ω K × H = clsp K, (A holds. Note, that the deﬁnition of K-minimality depends on the domain Ω. The elements of K × H will be written in the form {k, h}, k ∈ K, h ∈ H. Let PK : K × H → H, {k, h} → k, be the projection onto the ﬁrst component of K × H. Then the compression − λ)−1 |K , PK (A

λ ∈ ρ(A),

to K is called a generalized resolvent of A. of the resolvent of A Theorem 3.2. Let A be a closed symmetric operator of defect one in the Krein space K and let {C, Γ0 , Γ1 }, A0 = ker Γ0 , be a boundary value space for A+ with corresponding γ-ﬁeld γ and Weyl function M . Assume that A0 is deﬁnitizable over Ω and that the condition K = clsp {Nλ,A+ | λ ∈ ρ(A0 ) ∩ Ω} is fulﬁlled. Then the following assertions hold. be a K-minimal self-adjoint exit space extension of A in K × H which (i) Let A and A0 are dis deﬁnitizable over Ω and assume that the sign types of A compatible in Ω. Then there exists a locally deﬁnitizable function τ ∈ D(Ω) such that the sign types of τ , A and A0 are d-compatible in Ω, ∩ ρ(A0 ) ∩ h(τ ) ∩ Ω ρ(A) is a subset of h((M + τ )−1 ) and the formula − λ)−1 |K = (A0 − λ)−1 − γ(λ) M (λ) + τ (λ) −1 γ(λ)+ PK (A

(3.9)

∩ ρ(A0 ) ∩ h(τ ) ∩ Ω. holds for all λ ∈ ρ(A) (ii) Let τ ∈ D(Ω) be a locally deﬁnitizable function such that M (µ) + τ (µ) = 0 for some µ ∈ Ω, assume that the sign types of τ and A0 are d-compatible in Ω and let Ω be a domain with the same properties as Ω, Ω ⊂ Ω. Then there exists a Krein space H and a K-minimal self-adjoint exit space extension A τ of A in K × H which is deﬁnitizable over Ω , such that the sign types of A, and A0 are d-compatible in Ω , ρ(A0 ) ∩ h(τ ) ∩ h (M + τ )−1 ∩ Ω and formula (3.9) holds for all points λ belonging to is a subset of ρ(A) ρ(A0 ) ∩ h(τ ) ∩ h((M + τ )−1 ) ∩ Ω . be a K-minimal selfProof. The proof of assertion (i) consists of four steps. Let A adjoint exit space extension of A in K × H which is deﬁnitizable over Ω such that and A0 are d-compatible in Ω. the sign types of A

Generalized Resolvents of Symmetric Operators in Krein Spaces

23

is a 1. In this ﬁrst step we prove assertion (i) for the case H = {0}. Here A canonical extension of A and therefore, by (3.1), there exists a self-adjoint constant τ ∈ R ∪ {d∞ } such that − λ)−1 = (A0 − λ)−1 − γ(λ) M (λ) + τ −1 γ(λ)+ (A coincides with the canonical self-adjoint extension A−τ holds (cf. (3.8)), that is, A of A and by (3.7) we have ρ(A−τ ) ∩ ρ(A0 ) ⊂ h((M + τ )−1 ). Here each point in Ω∩R is of positive as well as of negative type with respect to τ and hence assertion (i) follows. 2. In the following we assume H = {0}. Following the lines of [11, §5] we deﬁne in this step a symmetric relation T in H and a special boundary value space for the adjoint T + . Below we will deal with direct products of linear relations. The following notation will be used. If U is a relation in K and V is a relation in H we shall write U × V for the direct product of U and V which is a relation in K × H, {f1 , f2 } f1 f2 U ×V = ∈ U, ∈ V . f1 f2 {f1 , f2 } f f {f ,f } For the pair {f1 ,f2 } we shall also write {fˆ1 , fˆ2 }, where fˆ1 = f1 and fˆ2 = f2 . 1

1

2

The linear relations ∩ K2 = S := A

k {k, 0} ∈A k {k , 0}

∩ H2 = T := A

h {0, h} ∈ A h {0, h }

and

2

are closed and symmetric in K and H, respectively, and we have A ⊂ S. Let JK and JH be fundamental symmetries in the Krein spaces K and H, respectively, and choose JK 0 ∈ L(K × H) J := 0 JH ∩ K2 and as a fundamental symmetry in the Krein space K × H. Then JK S = J A ∩ H2 are symmetric relations in the Hilbert spaces (K, [JK ·, ·]) and JH T = J A (H, [JH ·, ·]), respectively. It follows from [11, §5] that the deﬁciency indices of JK S and −JH T coincide. As JK S is a symmetric extension of the symmetric operator JK A in the Hilbert space (K, [JK ·, ·]) the deﬁciency indices n± (JK S) of JK S are (1, 1) or (0, 0). The case n± (JK S) = 0 is impossible here as otherwise also the relation would coincide with JH T would be self-adjoint in (H, [JH ·, ·]) and therefore J A = S × T is by assumption a K-minimal exit space extension JK S × JH T . But as A of A we would obtain H = {0}, a contradiction.

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Therefore, it remains to consider the case n± (JK S) = 1. Then the operators A and S coincide. Let us show that A+ coincides with k {k, h} PK {k, h} {k, h} ∈ A = ∈ A . R= {k , h } k {k , h } PK {k , h } is self-adjoint we have In fact, as A g, PK {k , h } − g , PK {k, h} = {g, 0}, {k , h } − {g , 0}, {k, h} = 0 g ˆ ˆ ˆ = h . Hence A ⊂ R+ . Similarly it kˆ = k , h h} ∈ A, for all g ∈ A and {k, k h follows that R+ ⊂ A holds. Therefore A+ coincides with the closure of R and as A has ﬁnite defect and R is an extension of A we conclude A+ = R. Replacing PK by the projection PH onto the second component of K × H the same arguments show PH {k, h} {k, h} h {k, h} T+ = ∈ A = ∈ A . PH {k , h } {k , h } h {k , h } We deﬁne the mappings PK and PH by k {k, h} → A+ , PK : A → k {k , h } and → T +, PH : A

{k, h} h → . {k , h } h

In the sequel we denote the elements in A+ and T + by fˆ1 and fˆ2 , respectively. −1 −1 As the multivalued part of PH coincides with A it follows that Γ0 PK PH and −1 + Γ1 PK PH are operators. We deﬁne Γ0 , Γ1 : T → C by −1 ˆ Γ0 fˆ2 := −Γ0 PK PH f2 ,

−1 ˆ Γ1 fˆ2 := Γ1 PK PH f2 ,

fˆ2 ∈ T + ,

is self-adjoint, one veriﬁes that {C, Γ0 , Γ1 } is cf. [11]. Taking into account that A + a boundary value space for T . We set T0 := ker Γ0 .

(3.10)

if and only if fˆ1 − PK P −1 fˆ2 is An element {fˆ1 , fˆ2 } ∈ A+ × T + belongs to A H is the canonical self-adjoint extension of the symmetric contained in A. Therefore A relation A × T in K × H given by = {fˆ1 , fˆ2 } ∈ A+ × T + | Γ0 fˆ1 + Γ fˆ2 = Γ1 fˆ1 − Γ fˆ2 = 0 . A (3.11) 0 1 3. In order to show that T0 has a non-empty resolvent set we construct in this step an auxiliary self-adjoint extension Tα of T in H such that ρ(Tα ) ∩ Ω is non-empty and with the help of Theorem 2.7 we will show that Tα is deﬁnitizable over Ω and and A0 in Ω. that the sign types of Tα are d-compatible with the sign types of A

Generalized Resolvents of Symmetric Operators in Krein Spaces

25

It is easy to see that {C2 , Γ0 , Γ1 }, where Γ0 fˆ1 Γ1 fˆ1 ˆ ˆ { f , f } := Γ0 {fˆ1 , fˆ2 } := and Γ , fˆ1 ∈ A+ , fˆ2 ∈ T + , (3.12) 1 2 1 Γ0 fˆ2 Γ1 fˆ2 is a boundary value space for A+ × T + . Setting ⎛ ⎞ 0 0 −1 1 ⎜1 1 0 0 ⎟ 4 ⎟ W := ⎜ ⎝1 0 0 0⎠ ∈ L(C ) 0 0 0 1 and

&

0 Γ Γ1

'

Γ0 := W Γ1

(3.13)

0, Γ 1 } for A+ × T + . This follows, e.g., we obtain a boundary value space {C2 , Γ from the fact that W is unitary in the Krein space (C4 , [[·, ·]]C4 ), where 0 −iIC2 [[·, ·]]C4 := J ·, · , J = , (3.14) iIC2 0 (see Section 2.1). Here we have = ker Γ 0 . A

(3.15)

is by assumption deﬁnitizable over Ω it follows from Theorem 2.5 and (3.5) As A 0, Γ 1 } is a deﬁnitizable function ( corresponding to {C2 , Γ that the Weyl function M 2×2 ( ∈ D (Ω), and the sign types of M ( and A are d-compatible in Ω. in Ω, M It is not diﬃcult to verify that − λ)−1 |K = ker(T + − λ) = Nλ,T + , ran PH (A λ ∈ ρ(A), is an K-minimal exit space extension of A we have holds. Since A ∩Ω − λ)−1 |K | λ ∈ ρ(A) H = clsp ran PH (A ∩Ω = clsp Nλ,T + | λ ∈ ρ(A)

(3.16)

and from the assumption K = clsp {Nλ,A+ | λ ∈ ρ(A0 ) ∩ Ω} we obtain ∩Ω . K × H = clsp Nλ,A+ ×T + | λ ∈ ρ(A0 ) ∩ ρ(A) ( ∈ D2×2 (Ω) is strict (see (3.6)). We claim that This implies that the function M there exists α ∈ R such that the function ((λ) − 0 0 λ → M 0 α ∩ Ω. Indeed, let M ((λ) = (mij (λ))2 is invertible for some λ ∈ ρ(A) i,j=1 and suppose ∩ Ω and every α ∈ R we have that for all λ ∈ ρ(A) 0 0 ( = m11 (λ)(m22 (λ) − α) − m21 (λ)m12 (λ) = 0. det M (λ) − 0 α

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J. Behrndt, A. Luger and C. Trunk

This implies m11 (λ) = m12 (λ)m21 (λ) = 0 and since m12 and m21 are piecewise ( is symmetric with respect to the real axis meromorphic functions in Ω\R and M ∩ Ω, which contradicts the strictness we conclude m12 (λ) = m21 (λ) = 0, λ ∈ ρ(A) ( of M . It is straightforward to check that the matrix ⎛ ⎞ 0 0 1 0 ⎜ 0 −α α 1⎟ ⎟ ∈ L(C4 ) V := ⎜ (3.17) ⎝−1 0 0 1⎠ 0 −1 1 0 0 , Γ 1 } be the boundary value space is unitary in (C4 , [[·, ·]]C4 ), cf. (3.14). Let {C2 , Γ + + for A × T deﬁned by & ' & ' 0 0 Γ Γ Γ0 := V =VW , (3.18) Γ1 Γ1 Γ1 (see (3.13)). From

we obtain 0 {fˆ1 , fˆ2 } = Γ

⎛ 1 ⎜0 VW =⎜ ⎝0 0

0 −α 0 −1

0 0 1 0

⎞ 0 1⎟ ⎟ 0⎠ 0

Γ0 fˆ1 , Γ1 fˆ2 − αΓ0 fˆ2

and 1 {fˆ1 , fˆ2 } = Γ

fˆ1 ∈ A+ , fˆ2 ∈ T + ,

Γ1 fˆ1 , −Γ0 fˆ2

fˆ1 ∈ A+ , fˆ2 ∈ T + .

We denote the self-adjoint extension ker(Γ1 − αΓ0 ) ∈ C(H) of T in H by Tα . Then the self-adjoint extension ker Γ0 of A × T in K × H coincides with A0 × Tα . Since (3.18) and (3.17) imply 0 0 1 0 0 = ker Γ0 + Γ1 A0 × Tα = ker Γ 0 −α α 1 1 − 0 0 Γ 0 , = ker Γ 0 α belongs to the set we ﬁnd from (3.15), (3.2) and (3.7) that a point λ ∈ ρ(A) ρ(A0 × Tα ) if and only if 0 belongs to the resolvent set of 0 0 ( M (λ) − . 0 α

Generalized Resolvents of Symmetric Operators in Krein Spaces

27

∩ Ω, But we have chosen α such that this function is invertible for some λ ∈ ρ(A) therefore λ belongs to ρ(A0 × Tα ). In particular λ ∈ ρ(A0 ) ∩ ρ(Tα ) and ∩ Ω = ∅. ρ(Tα ) ∩ ρ(A0 ) ∩ ρ(A) and A0 × Tα are As A × T is a symmetric relation of defect two and A self-adjoint extensions of A × T in K × H we have − λ)−1 − ((A0 × Tα ) − λ)−1 ≤ 2 dim ran (A ∩ ρ(A0 ) ∩ ρ(Tα ) ∩ Ω. Since A is deﬁnitizable over Ω we obtain from for all λ ∈ ρ(A) Theorem 2.7 that also the self-adjoint relation A0 × Tα is deﬁnitizable over Ω and and the sign types of A0 × Tα are d-compatible in Ω. that the sign types of A It is a simple consequence from Deﬁnition 2.1 that σ++ (A0 × Tα ) ∩ σap (Tα ) ⊂ σ++ (Tα ) and

σ−− (A0 × Tα ) ∩ σap (Tα ) ⊂ σ−− (Tα ) holds. Hence, real points from σ++ (A0 × Tα ) (σ−− (A0 × Tα )) belong to ρ(Tα ) or to σ++ (Tα ) (resp. σ−− (Tα )). Therefore Tα is deﬁnitizable over Ω and the sign types of Tα in Ω are d-compatible with the sign types of A0 × Tα and, hence, with the and A0 in Ω. sign types of A

4. In this step we show that also T0 in (3.10) has a non-empty resolvent set and that formula (3.9) holds with the Weyl function τ corresponding to the boundary value space {C, Γ0 , Γ1 }. Moreover, we show that τ is locally deﬁnitizable and that in Ω. its sign types are d-compatible with the sign types of A0 and A It is straightforward to verify that {C, Γ1 − αΓ0 , −Γ0 } is a boundary value space for T + and we have Tα = ker(Γ1 − αΓ0 ) and T0 = ker(−Γ0 ). The corresponding Weyl function τα is deﬁned for all λ ∈ ρ(Tα ). As Tα is deﬁnitizable over Ω the function τα belongs to the class D(Ω) and the sign types of τα are and A0 in Ω (cf. Theorem 2.5 and (3.5) d-compatible with the sign types of Tα , A or [3, Proposition 3.2]). Relation (3.16) implies that τα is strict and in particular τα is not identically equal to zero. Then, by (3.8), for λ ∈ ρ(Tα ) ∩ h(τα−1 ) we have 1 γ (λ)+ , (T0 − λ)−1 = (Tα − λ)−1 − γα (λ) τα (λ) α where γα is the γ-ﬁeld of the boundary value space {C, Γ1 − αΓ0 , −Γ0 }. Therefore the set ρ(Tα )∩ρ(T0 )∩Ω is non-empty and by Theorem 2.7 the self-adjoint relation T0 is deﬁnitizable over Ω and the sign types of T0 and Tα are d-compatible in Ω. The Weyl function τ corresponding to the boundary value space {C, Γ0 , Γ1 } satisﬁes τ (λ) = −τα (λ)−1 + α,

λ ∈ h(τα−1 ) ∩ ρ(Tα ).

and is holomorphic on ρ(T0 ). It follows from Theorem 2.5 and (3.5) that τ belongs to the class D(Ω) and that its sign types are d-compatible with the sign types of T0 and Tα and hence

28

J. Behrndt, A. Luger and C. Trunk

and A0 . The γ-ﬁeld corresponding to {C, Γ0 , Γ1 } will also with the sign types of A be denoted by γ . Since A0 and T0 are both deﬁnitizable over Ω the set ρ(A0 )∩ρ(T0 )∩Ω is nonempty. The γ-ﬁeld γ and the Weyl function M corresponding to the boundary value space {C2 , Γ0 , Γ1 } deﬁned in (3.12) are given by γ(λ) 0 λ → γ (λ) = (3.19) , λ ∈ ρ(A0 ) ∩ ρ(T0 ) ∩ Ω, 0 γ (λ) and λ → M (λ) =

M (λ) 0

0 , τ (λ)

λ ∈ ρ(A0 ) ∩ ρ(T0 ) ∩ Ω,

(3.20)

respectively. The relation

{u, −u} 2) Θ := u, v ∈ C ∈ C(C {v, v}

(3.21)

is self-adjoint and the corresponding self-adjoint extension of A × T is given by −1 Γ0 + + ˆ ˆ ˆ ˆ ˆ ˆ Θ = { f , f } ∈ A × T | Γ + Γ = Γ − Γ = 0 (3.22) f f f f 1 2 0 1 2 1 1 2 0 1 Γ1 (see (3.11)). and coincides with A if and only if By (3.7) a point λ ∈ ρ(A0 × T0 ) belongs to ρ(A) 0 ∈ ρ(Θ − M (λ)). ∩ Ω = ρ(A0 ) ∩ h(τ ) ∩ ρ(A) ∩Ω Hence, for λ ∈ ρ(A0 × T0 ) ∩ ρ(A) −1 {v − M (λ)u, v + τ (λ)u} = Θ − M (λ) u, v ∈ C {u, −u} is an operator. Therefore (M (λ) + τ (λ))u = 0 implies u = 0 and we conclude that ∩ ρ(A0 ) ∩ h(τ ) ∩ Ω is a subset of h((M + τ )−1 ). Setting x = v − M (λ)u the set ρ(A) and y = v + τ (λ)u we obtain −1 −1 u = − M (λ) + τ (λ) x + M (λ) + τ (λ) y ∩ Ω. This implies for λ ∈ ρ(A0 × T0 ) ∩ ρ(A) −1 −(M (λ) + τ (λ))−1 Θ − M (λ) = (M (λ) + τ (λ))−1

(M (λ) + τ (λ))−1 . −(M (λ) + τ (λ))−1

∩ Ω the relation For all λ ∈ ρ(A0 × T0 ) ∩ ρ(A) − λ)−1 = (A0 × T0 ) − λ −1 + γ (λ) Θ − M (λ) −1 γ (λ)+ (A

(3.23)

(3.24)

holds (cf. (3.8)) and it follows from (3.24), (3.19) and (3.23) that the formula − λ)−1 |K = (A0 − λ)−1 − γ(λ) M (λ) + τ (λ) −1 γ(λ)+ PK (A holds. This completes the proof of assertion (i). 5. Assertion (ii) was already proved in [3] in a slightly diﬀerent form. For the convenience of the reader we sketch the proof.

Generalized Resolvents of Symmetric Operators in Krein Spaces

29

If τ is identically equal to a real constant, then A−τ := ker(Γ1 + τ Γ0 ) is a canonical self-adjoint extension of A. As the Weyl function M corresponding to A+ and {C, Γ0 , Γ1 } is strict we obtain ρ(A−τ ) ∩ Ω = ∅ and Theorem 2.7 implies that A−τ is deﬁnitizable over Ω and that the sign types of A0 , A−τ and τ ∈ R are d-compatible. By (3.8) −1 γ(λ)+ (A−τ − λ)−1 = (A0 − λ)−1 − γ(λ) M (λ) + τ holds for all λ ∈ ρ(A0 ) ∩ ((M + τ )−1 ). In the case τ = d∞ = 0c | c ∈ C we have A−τ = A0 . Assume now that τ ∈ D(Ω) is not equal to a constant and let Ω be a domain with the same properties as Ω, Ω ⊂ Ω. With the help of [21, Theorem 3.8] it was shown in [3, Theorem 3.3] that there exists a Krein space H, a closed symmetric operator T of defect one in H and a boundary value space {C, Γ0 , Γ1 } for T + such that T0 := ker Γ0 is deﬁnitizable over Ω , the sign types of τ and T0 are d-compatible and τ coincides with the Weyl function corresponding to {C, Γ0 , Γ1 } on Ω ∩ ρ(T0 ). Moreover the condition H = clsp γ (λ) | λ ∈ ρ(T0 ) ∩ Ω (3.25) is fulﬁlled. We choose the boundary value space {C2 , Γ0 , Γ1 } for A+ × T + as in (3.12) with γ-ﬁeld and Weyl function given by (3.19) and (3.20), respectively. The Then self-adjoint extension corresponding to Θ in (3.21) via (3.1) is denoted by A. has the form (3.22) and the relation (3.24) holds for all λ ∈ Ω which belong to A ρ(A0 × T0 ) and fulﬁl 0 ∈ ρ(Θ − M (λ)). From (3.23) we conclude ρ(A0 ) ∩ h(τ ) ∩ h (M + τ )−1 ∩ Ω ⊂ ρ(A) and (3.24) implies that the formula (3.9) holds. Since the minimality condition is a K-minimal exit space extension (3.25) is fulﬁlled it follows from (3.24) that A of A. As A0 × T0 is deﬁnitizable over Ω the relation (3.24) and Theorem 2.7 A0 and τ are is also deﬁnitizable over Ω and the sign types of A, imply that A d-compatible. The next theorem is a variant of the Krein-Naimark formula for the case that are locally of type π+ and τ is a local generalized Nevanlinna function. A0 and A The proof of Theorem 3.3 below is essentially the same as the proof of Theorem 3.2. Instead of the result on ﬁnite rank perturbations of locally deﬁnitizable self-adjoint relations from [4], cf. Theorem 2.7, one has to use [5, Theorem 2.4] on the stability of self-adjoint operators and relations locally of type π+ under compact perturbations in resolvent sense. We leave the details to the reader. Theorem 3.3. Let A be a closed symmetric operator of defect one in the Krein space K and let {C, Γ0 , Γ1 } be a boundary value space for A+ with corresponding γ-ﬁeld γ and Weyl function M . Assume that A0 = ker Γ0 is of type π+ over Ω and that the condition K = clsp {Nλ,A+ | λ ∈ ρ(A0 ) ∩ Ω} is fulﬁlled.

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J. Behrndt, A. Luger and C. Trunk

Then the following assertions hold: of A in K × H which (i) For every K-minimal self-adjoint exit space extension A is of type π+ over Ω there exists a function τ ∈ N (Ω) such that ∩ ρ(A0 ) ∩ h(τ ) ∩ Ω ρ(A) is a subset of h((M + τ )−1 ) and the formula − λ)−1 |K = (A0 − λ)−1 − γ(λ) M (λ) + τ (λ) −1 γ(λ)+ PK (A

(3.26)

∩ ρ(A0 ) ∩ h(τ ) ∩ Ω. holds for all λ ∈ ρ(A) (Ω) be a local generalized Nevanlinna function such that M (µ) + (ii) Let τ ∈ N τ (µ) = 0 for some µ ∈ Ω and let Ω be a domain with the same properties as Ω, Ω ⊂ Ω. Then there exists a Krein space H and a K-minimal self-adjoint of A in K × H which is of type π+ over Ω , such that exit space extension A ρ(A0 ) ∩ h(τ ) ∩ h (M + τ )−1 ∩ Ω and formula (3.26) holds for all points λ belonging to is a subset of ρ(A) ρ(A0 ) ∩ h(τ ) ∩ h((M + τ )−1 ) ∩ Ω.

References [1] T.Ya. Azizov, J. Behrndt, P. Jonas, C. Trunk: Spectral Points of Type π+ and Type π− for Closed Linear Relations in Krein Spaces, submitted. [2] T.Ya. Azizov, P. Jonas, C. Trunk: Spectral Points of Type π+ and Type π− of Selfadjoint Operators in Krein Spaces, J. Funct. Anal. 226 (2005), 114–137. [3] J. Behrndt: A Class of Abstract Boundary Value Problems with Locally Deﬁnitizable Functions in the Boundary Condition, Operator Theory: Advances and Applications 163 (2005), 55–73. [4] J. Behrndt: Finite Rank Perturbations of Locally Deﬁnitizable Operators in Krein Spaces, to appear in J. Operator Theory. [5] J. Behrndt, P. Jonas: On Compact Perturbations of Locally Deﬁnitizable Selfadjoint Relations in Krein Spaces, Integral Equations Operator Theory 52 (2005), 17–44. [6] J. Behrndt, H.C. Kreusler: Boundary Relations and Generalized Resolvents of Symmetric Relations in Krein Spaces, submitted. [7] J. Behrndt, C. Trunk: On Generalized Resolvents of Symmetric Operators of Defect One with Finitely many Negative Squares, Proceedings of the Algorithmic Information Theory Conference, Vaasan Yliop. Julk. Selvityksi¨a Rap., 124, Vaasan Yliopisto, Vaasa, (2005), 21–30. [8] 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. [9] V.A. Derkach: On Krein Space Symmetric Linear Relations with Gaps, Methods of Funct. Anal. Topology 4 (1998), 16–40.

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[10] V.A. Derkach: On Generalized Resolvents of Hermitian Relations in Krein Spaces, J. Math. Sci. (New York) 97 (1999), 4420–4460. [11] 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. [12] 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. [13] 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. [14] V.A. Derkach, M.M. Malamud: On some Classes of Holomorphic Operator Functions with Nonnegative Imaginary Part, Operator Theory, Operator Algebras and related Topics (Timi¸soara, 1996), Theta Found., Bucharest (1997), 113–147. [15] A. Dijksma, H.S.V. de Snoo: Symmetric and Selfadjoint Relations in Krein Spaces I, Operator Theory: Advances and Applications 24, Birkh¨ auser Verlag Basel (1987), 145–166. [16] A. Dijksma, H.S.V. de Snoo: Symmetric and Selfadjoint Relations in Krein Spaces II, Ann. Acad. Sci. Fenn. Math. 12, (1987), 199–216. [17] V.I. Gorbachuk, M.L. Gorbachuk: Boundary Value Problems for Operator Diﬀerential Equations, Kluwer Academic Publishers, Dordrecht (1991). [18] P. Jonas: A Class of Operator-valued Meromorphic Functions on the Unit Disc, Ann. Acad. Sci. Fenn. Math. 17 (1992), 257–284. [19] P. Jonas: Operator Representations of Deﬁnitizable Functions, Ann. Acad. Sci. Fenn. Math. 25 (2000), 41–72. [20] P. Jonas: On Locally Deﬁnite Operators in Krein Spaces, in: Spectral Theory and Applications, Theta Foundation 2003, 95–127. [21] P. Jonas: On Operator Representations of Locally Deﬁnitizable Functions, Operator Theory: Advances and Applications 162, Birkh¨ auser Verlag Basel (2005), 165–190. [22] M.G. Krein: On Hermitian Operators with Defect-indices equal to Unity, Dokl. Akad. Nauk SSSR, 43 (1944), 339–342. [23] M.G. Krein: On the Resolvents of an Hermitian Operator with Defect-index (m, m), Dokl. Akad. Nauk SSSR, 52 (1946), 657–660. [24] M.G. Krein, H. Langer: On Defect Subspaces and Generalized Resolvents of Hermitian Operators in Pontryagin Spaces, Funktsional. Anal. i Prilozhen. 5 No. 2 (1971) 59-71; 5 No. 3 (1971) 54-69 (Russian); English transl.: Funct. Anal. Appl. 5 (1971/1972), 139–146, 217–228. ¨ [25] M.G. Krein, H. Langer: Uber einige Fortsetzungsprobleme, die eng mit der Theorie angen. I. Einige Funktionenhermitescher Operatoren im Raume Πκ zusammenh¨ klassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187–236. [26] M.G. Krein, 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. [27] H. Langer: Verallgemeinerte Resolventen eines J-nichtnegativen Operators mit endlichem Defekt, J. Funct. Anal. 8 (1971), 287–320.

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[28] 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. [29] H. Langer, A. Markus, V. Matsaev: Locally Deﬁnite Operators in Indeﬁnite Inner Product Spaces, Math. Ann. 308 (1997), 405–424. [30] H. Langer, B. Textorius: On Generalized Resolvents and Q-functions of Symmetric Linear Relations (Subspaces) in Hilbert Space, Paciﬁc J. Math. 72 (1977), 135–165. [31] M.M. Malamud: On a Formula for the Generalized Resolvents of a Non-densely deﬁned Hermitian Operator (Russian), Ukrain. Mat. Zh. 44 (1992), 1658–1688; translation in Ukrainian Math. J. 44 (1993), 1522–1547. [32] M.A. Naimark: On Spectral Functions of a Symmetric Operator, Izv. Akad. Nauk SSSR, Ser. Matem. 7 (1943), 373–375. 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] Annemarie Luger Institut f¨ ur Analysis und Scientiﬁc Computing Technische Universit¨ at Wien Wiedner Hauptstraße 8-10 A-1040 Wien, Austria e-mail: [email protected] Carsten Trunk Institut f¨ ur Mathematik, MA 6-3 Technische Universit¨ at Berlin Straße des 17. Juni 136 D-10623 Berlin, Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 175, 33–49 c 2007 Birkh¨ auser Verlag Basel/Switzerland

Block Operator Matrices, Optical Potentials, Trace Class Perturbations and Scattering Jussi Behrndt, Hagen Neidhardt and Joachim Rehberg Abstract. For an operator-valued block-matrix model, which is called in quantum physics a Feshbach decomposition, a scattering theory is considered. Under trace class perturbations the channel scattering matrices are calculated. Using Feshbach’s optical potential it is shown that for a given spectral parameter the channel scattering matrices can be recovered either from a dissipative or from a Lax-Phillips scattering theory. Mathematics Subject Classiﬁcation (2000). Primary 47A40; Secondary 47A55, 47B44. Keywords. Feshbach decomposition, optical potential, Lax-Phillips scattering theory, dissipative scattering theory, scattering matrix, characteristic function, dissipative operators.

1. Introduction Let L and L0 be self-adjoint operators in a separable Hilbert space L and denote by P ac (L0 ) the orthogonal projection onto the absolutely continuous subspace Lac (L0 ) of L0 . The pair {L, L0 } of self-adjoint operators is said to perform a scattering system if the wave operators W± (L, L0 ), W± (L, L0 ) := s − lim eitL e−itL0 P ac (L0 ), t→±∞

(1.1)

exist, cf. [5]. If the wave operators exist, then they are isometries from the absolutely continuous subspace Lac (L0 ) into the absolutely continuous subspace Lac (L), i.e., ran (W± (L, L0 )) ⊆ Lac (L). The scattering system {L, L0 } is called complete if the ranges of the wave operators W± (L, L0 ) coincide with Lac (L), cf. [5]. The operator S(L, L0 ) := W+ (L, L0 )∗ W− (L, L0 ) is called the scattering operator of the scattering system {L, L0 }. The scattering operator regarded as an operator in Lac (L0 ) commutes with Lac 0 . If the scattering

34

J. Behrndt, H. Neidhardt and J. Rehberg

system {L, L0} is complete, then S(L, L0 ) is a unitary operator in Lac (L0 ). In physical applications L0 is usually called the unperturbed or free Hamiltonian while L is called the perturbed or full Hamiltonian. Since S(L, L0 ) commutes with the free Hamiltonian L0 the scattering operator is unitarily equivalent to a multiplication operator induced by a family {S(λ)}λ∈R of unitary operators in the spectral representation of L0 . This family is called the scattering matrix of the complete scattering system {L, L0 } and is the most important quantity in the analysis of scattering processes. In this paper we investigate the special case that the Hilbert space L splits into two subspaces H1 and H2 , H1 L= ⊕ , H2 and the unperturbed Hamiltonian L0 is of the form L0 =

H1 0

0 H2

H1 H1 : ⊕ −→ ⊕ . H2 H2

(1.2)

In physics the subspaces Hj and the self-adjoint operators Hj , j = 1, 2, are often called scattering channels and channel Hamiltonians, respectively. With respect to the decomposition (1.2) one introduces the channel wave operators W± (L, Hj ) := s − lim eitL Jj e−itHj P ac (Hj ) t→±∞

where Jj : Hj −→ L is the natural embedding operator. Introducing the channel scattering operators Sij = W+ (L, Hi )∗ W− (L, Hj ) : Hj −→ Hi ,

i, j = 1, 2,

one obtains a channel decomposition of the scattering operator S11 S12 S(L, L0 ) = . S21 S22

(1.3)

In physics the decomposition (1.2) is often motivated either by the exclusive interest to scattering data in a certain channel or by the limited measuring process which allows to measure the scattering data only of a certain channel, say H1 . ac Thus, let us assume that only the channel scattering operator S11 : Hac 1 −→ H1 in the scattering channel H1 is known. This gives rise to the following problem: Is it possible to replace the full Hamiltonian L by an eﬀective one H acting only in H1 such that the scattering operator of the scattering system {H, H1 } coincides with S11 ? Since S11 is a contraction, in general, this implies that either the scattering system {H, H1 } cannot be complete or H is not self-adjoint. The problem has a solution within the scope of dissipative scattering systems developed in [18, 19, 20] for pairs {H, H1 } of dissipative and self-adjoint operators

Block Operator Matrices and Scattering

35

in some separable Hilbert space. For such pairs the wave operators W±D (H, H1 ) are deﬁned by ∗

W+D (H, H1 ) := s − lim eitH e−itH1 P ac (H1 ) t→+∞

and W−D (H, H1 ) := s − lim e−itH eitH1 P ac (H1 ), t→+∞

and the notion of completeness is generalized, cf. [18, 19, 20]. The scattering operator of a dissipative scattering system {H, H1 } is deﬁned by SD := W+D (H, H1 )∗ W−D (H, H1 ). It turns out that SD is a contraction acting on the absolutely continuous subspace Hac 1 of H1 which commutes with H1 . In [17, 18] it was shown that for any self-adjoint operator H1 in H1 and any contraction SD acting on the absolutely continuous subspace Hac 1 and commuting with H1 there is a maximal dissipative operator H on H1 such that {H, H1 } performs a complete scattering system with scattering operator given by SD . In particular, this holds for the self-adjoint operator H1 and the channel scattering operator S11 . That means, there is a maximal dissipative operator H on H1 such that the channel scattering operator S11 is the scattering operator of the complete dissipative scattering system {H, H1 }. Hence, roughly speaking, the scattering operator S11 can be always viewed as the scattering operator of a suitable chosen dissipative scattering system on H1 . The disadvantage of this fact is that H is not known explicitly. Another approach to this problem was suggested by Feshbach in [10, 11], see also [6, 9]. He proposes a concrete dissipative perturbation V1 of the channel Hamiltonian H1 , called “optical potential”, such that the scattering operator S1 of the dissipative scattering system {H1 + V1 , H1 } approximates S11 with a certain accuracy. To explain this approach in more detail let us assume that the full Hamiltonian L is obtained from L0 by an additive perturbation, L = L0 + V , where V is given by H1 H1 0 G ⊕ ⊕ . −→ (1.4) : V = G∗ 0 H2 H2 Introducing the “optical potential” V1 (λ) := −G(H2 − λ − i0)−1 G∗ ,

λ ∈ R,

(1.5)

it was shown in [8, Theorem 4.4.4] that under strong assumptions indeed the scattering operator S1 [λ] of the (in general dissipative) scattering system {H1 (λ), H1 }, H1 (λ) := H1 + V1 (λ),

λ ∈ R,

(1.6)

coincides with the scattering operator S11 with an error of second order in the coupling constant. We show that Feshbach’s proposal can be made precise in another sense. Note ﬁrst that the decomposition (1.2) leads not only to the decomposition (1.3)

36

J. Behrndt, H. Neidhardt and J. Rehberg

of the scattering operator S but also to a decomposition of the scattering matrix {S(µ)}µ∈R , S11 (µ) S12 (µ) S(µ) := , S21 (µ) S22 (µ) where {Sij (µ)}µ∈R are called the channel scattering matrices. Denoting the scattering matrix of the dissipative scattering system {H1 (λ), H1 } by {S1 [λ](µ)}µ∈R we prove that S11 (λ) = S1 [λ](λ)

(1.7)

holds for a.e. λ ∈ R. This shows, that Feshbach’s proposal gives in fact a good approximation of the channel scattering matrix {S11 (µ)}µ∈R in a neighborhood of the chosen spectral parameter λ of the optical potential V1 (λ). Moreover, Feshbach’s proposal implies a second problem. Similarly to the optical potential V1 (λ) in the ﬁrst channel H1 one can introduce an optical potential V2 (λ) in the second channel, V2 (λ) := −G∗ (H1 − λ − i0)−1 G,

λ ∈ R,

(1.8)

and deﬁne a perturbed operator H2 (λ), H2 (λ) := H2 + V2 (λ),

λ ∈ R,

(1.9)

in H2 . We show below that the characteristic function Θ2 [λ](ξ), ξ ∈ C− , of the dissipative operator H2 (λ) and the scattering matrix {S11 (λ)}λ∈R are related by S11 (λ) = Θ2 [λ](λ)∗

(1.10)

for a.e. λ ∈ R. By [1]–[4] the last relation also yields that the scattering matrix {S11 (λ)}λ∈R can be regarded as the scattering matrix SLP [λ](µ) of a Lax-Phillips scattering system at the point λ. Below we restrict ourself to a complete scattering system {L,L0 }, L = L0 + V , where the perturbation V is a self-adjoint trace class operator. The assumption that V is a trace class operator is made for simplicity. Indeed, it would be suﬃcient to assume that the resolvent diﬀerence (L − z)−p − (L0 − z)−p is nuclear for a certain p ∈ N or, more generally, that the conditions of the so-called “stationary” scattering theory are satisﬁed, cf. [5, Section 14]. However, we emphasize that in contrast to [8] the smallness of the perturbation V is not assumed. Following the lines of [5] we show in Section 2 how the scattering matrix of the scattering system {L, L0 } can be calculated. Under the additional assumptions (1.2) and (1.4) we ﬁnd in Section 3 the channel scattering matrices {Sij (λ)}λ∈R . In Section 4 we prove relation (1.7). Section 5 is devoted to the proof of (1.10). Moreover, the Lax-Phillips scattering theory for which {Θ2 [λ](µ)∗ }µ∈R is the scattering matrix is indicated.

Block Operator Matrices and Scattering

37

2. Scattering matrix In this section we brieﬂy recall the notion of the scattering matrix {S(λ)}λ∈R of a scattering system {L, L0}, where it is assumed that the unperturbed operator L0 is self-adjoint in the separable Hilbert space L and the perturbed operator L diﬀers from L0 by a self-adjoint trace class operator V ∈ B1 (L), V = V ∗ ∈ B1 (L).

L = L0 + V,

(2.1)

Let E0 (·) be the spectral measure of L0 and denote by B(R) the set of all Borel subsets of the real axis R. Without loss of generality we assume throughout the paper that the condition L = clospan{E0 (∆)ran (|V |) : ∆ ∈ B(R)} ∗

(2.2)

is satisﬁed, where |V | := (V V ) . By Theorem X.4.4 of [13] the scattering system {L, L0 } is complete, that is, the ranges of the wave operators W± (L, L0 ) in (1.1) coincide with the absolutely continuous subspace Lac (L) of L. The operator V admits the representation 1/2

V = |V |1/2 C|V |1/2 ,

|V | = (V ∗ V )1/2 ,

C = sgn(V ),

(2.3)

where |V |1/2 belongs to the Hilbert-Schmidt class B2 (L) and sgn(·) is the signum function. By Proposition 3.14 of [5] the limits |V |1/2 (L − λ ± i0)−1 |V |1/2 = lim |V |1/2 (L − λ ± i)−1 |V |1/2 →+0

(2.4)

exist in B2 (L) for a.e. λ ∈ R. The same holds for the limits |V |1/2 (L0 − λ ± i0)−1 |V |1/2 . Moreover by Proposition 3.13 of [5] the derivative |V |1/2 E0 (dλ)|V |1/2 ≥0 dλ exists in B1 (L) for a.e. λ ∈ R. We set Qλ := clo ran (M0 (λ)) ⊆ L. M0 (λ) :=

(2.5)

By {Q(λ)}λ∈R we denote the family of orthogonal projections from L onto Qλ . One veriﬁes that {Q(λ)}λ∈R is measurable. Let us consider the standard Hilbert space L2 (R, dλ, L). On L2 (R, dλ, L) we introduce the projection Q (Qf )(λ) := Q(λ)f (λ),

λ ∈ R,

f ∈ L2 (R, dλ, L),

and set Q = ran (Q). Further, in L2 (R, dλ, L) we deﬁne the multiplication operator ML by λf (λ), λ ∈ R, (ML f )(λ) := dom (ML ) := f ∈ L2 (R, dλ, L) : λf (λ) ∈ L2 (R, dλ, L) . Obviously, the multiplication operator ML and the projection Q commute. We set MQ := ML dom (ML ) ∩ Q.

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J. Behrndt, H. Neidhardt and J. Rehberg

From Section 4.5 of [5] one gets that the absolutely continuous part Lac of the perturbed operator L and the operator MQ are unitarily equivalent. In the following we denote the subspace Q by L2 (R, dλ, Qλ ) which can be regarded as the direct integral of the family of subspaces {Qλ }λ∈R with respect to the Lebesgue measure dλ on R, cf. [5]. Since the scattering operator S = W+ (L, L0 )∗ W− (L, L0 ) acts on Lac (L0 ) and commutes with Lac 0 there is a measurable family {S(λ)}λ∈R of operators S(λ) : Qλ −→ Qλ such that S is unitarily equivalent to the multiplication operator (MQ (S)f )(λ) dom (MQ (S))

:= :=

S(λ)f (λ), L2 (R, dλ, Qλ ).

The family {S(λ)}λ∈R is called the scattering matrix of the scattering system {L, L0 }. Since the scattering system {L, L0 } is complete the operator S(λ) is unitary on Qλ for a.e. λ ∈ R. The following representation theorem of the scattering matrix is a consequence of Corollary 18.9 of [5], see also [5, Section 18.2.2]. Theorem 2.1. Let L, L0 and V be self-adjoint operators in L as in (2.1). Then {L, L0 } is a complete scattering system and the corresponding scattering matrix matrix {S(λ)}λ∈R admits the representation 1/2 1/2 S(λ) = IQ λ − 2πi M0 (λ) C − C|V |1/2 (L − λ − i0)−1 |V |1/2 C M0 (λ)

for a.e. λ ∈ R.

3. Channel scattering matrices Let us now assume that the Hilbert space L is the orthogonal sum of two subspaces H1 and H2 , L = H1 ⊕ H2 , that L0 is a diagonal block operator matrix of the form L0 =

H1 0

0 H2

H1 H1 : ⊕ −→ ⊕ , H2 H2

(3.1)

cf. (1.2), where H1 and H2 are self-adjoint operators in H1 and H2 and that V ∈ B1 (L) is a self-adjoint trace class operator of the form V =

0 G∗

H1 H1 G : ⊕ −→ ⊕ , 0 H2 H2

(3.2)

see (1.4). The operator G : H2 −→ H1 describes the interaction between the channels. Since V is a trace class operator we have G ∈ B1 (H2 , H1 ).

Block Operator Matrices and Scattering

39

The perturbed or full Hamiltonian L has the form H1 H1 H1 G : ⊕ −→ ⊕ . L := L0 + V = ∗ G H2 H2 H2

(3.3)

The following lemma is known as the Feshbach decomposition in physics, cf. [10, 11]. We use the notation H1 (z) = H1 + V1 (z)

and

z ∈ C\R,

(3.4)

V2 (z) = −G∗ (H1 − z)−1 G,

(3.5)

H2 (z) = H2 + V2 (z),

where V1 (z) = −G(H2 − z)−1 G∗

and

see (1.6), (1.9), (1.5) and (1.8). Lemma 3.1. Let L, H1 (z) and H2 (z), z ∈ C\R, be given by (3.3) and (3.4), respectively. Then we have z ∈ res(Hi (z)), i = 1, 2, for all z ∈ C\R and &

(L − z)

−1

=

(H1 (z) − z)−1

−(H1 − z)−1 G(H2 (z) − z)−1

−(H2 (z) − z)−1 G∗ (H1 − z)−1

(H2 (z) − z)−1

'

.

(3.6)

Proof. From Im (H1 (z) − z)h1 , h1 = Im zh1 2 + Im z(H2 − z)−1 G∗ h1 2 , z ∈ C\R, h1 ∈ H1 , we conclude that (H1 (z)−z)−1 is a bounded everywhere deﬁned operator for all z ∈ C\R. Analogously one veriﬁes that (H2 (z) − z)−1 is a bounded everywhere deﬁned operator for all z ∈ C\R. A straightforward computation shows (L − z)−1 & ' (H1 − z)−1 + (H1 − z)−1 G(H2 (z) − z)−1 G∗ (H1 − z)−1 −(H1 − z)−1 G(H2 (z) − z)−1 = −(H2 (z) − z)−1 G∗ (H1 − z)−1 (H2 (z) − z)−1

for z ∈ C\R. From the identity −1 ∗ −1 G = G∗ I − (H1 − z)−1 G(H2 − z)−1 G∗ I − G∗ (H1 − z)−1 G(H2 − z)−1 we obtain (H1 − z)−1 + (H1 − z)−1 G(H2 (z) − z)−1 G∗ (H1 − z)−1 −1 ∗ = (H1 − z)−1 I + G(H2 − z)−1 I − G∗ (H1 − z)−1 G(H2 − z)−1 G (H1 − z)−1 = (H1 − z)−1 I + G(H2 − z)−1 G∗ (H1 (z) − z)−1 = (H1 (z) − z)−1 for all z ∈ C\R, which proves (3.6).

In the next lemma we calculate the limit |V |1/2 (L − λ − i0)−1 |V |1/2 , λ ∈ R, cf. (2.4). Here and in the following it is convenient to use the functions N1 (z) := |G∗ |1/2 (H1 − z)−1 |G∗ |1/2 , N2 (z) := |G|1/2 (H2 − z)−1 |G|1/2 ,

z ∈ C\R,

(3.7)

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J. Behrndt, H. Neidhardt and J. Rehberg

and

F1 (z) := |G∗ |1/2 (H1 (z) − z)−1 |G∗ |1/2 , F2 (z) := |G|1/2 (H2 (z) − z)−1 |G|1/2 ,

z ∈ C\R.

(3.8)

Lemma 3.2. Let V ∈ B1 (L) be given by (3.2) with G ∈ B1 (H2 , H1 ) and let U be a partial isometry such that G = U |G|. Then the limits Ni (λ) := lim Ni (λ + i) →+0

and

Fi (λ) := lim Fi (λ + i), i = 1, 2, →+0

exist in B2 (Hi ) for a.e. λ ∈ R and the representation F1 (λ) −N1 (λ)U F2 (λ) 1/2 −1 1/2 = |V | (L − λ − i0) |V | F2 (λ) −F2 (λ)U ∗ N1 (λ)

(3.9)

(3.10)

holds for a.e. λ ∈ R. Proof. By |G|1/2 ∈ B2 (H2 ) and |G∗ |1/2 ∈ B2 (H1 ) the existence of the limits Ni (λ) in (3.9) for a.e. λ ∈ R follows from Proposition 3.13 of [5]. Using the representations F1 (z) = |G∗ |1/2 P1 (L − z)−1 H1 |G∗ |1/2 and F2 (z) = |G|1/2 P2 (L − z)−1 H2 |G|1/2 , z ∈ C\R, which follow from (3.6), and taking into account [5, Proposition 3.13] we again obtain the existence of Fi (λ), i = 1, 2, for a.e. λ ∈ R. It is easy to see that ∗ 1/2 |G | 0 1/2 ∗ 1/4 |V | = (V V ) = (3.11) 0 |G|1/2 holds. Let U be a partial isometry from H2 into H1 such that G = U |G|. Making use of the factorizations G = |G∗ |1/2 U |G|1/2

and G∗ = |G|1/2 U ∗ |G∗ |1/2 , −1

the block matrix representation of (L − z) veriﬁes (3.10).

(3.12)

in Lemma 3.1 and relation (3.11) one

We note that if U is a partial isometry such that G = U |G| holds and C is deﬁned by 0 U , (3.13) C := U∗ 0 then the operator V in (3.2) can be written in the form |V |1/2 C|V |1/2 , cf. (2.3). Let E1 (·) and E2 (·) be the spectral measures of H1 and H2 , respectively. The operator function M0 (·) from (2.5) here admits the representation M1 (λ) 0 M0 (λ) = (3.14) 0 M2 (λ) for a.e λ ∈ R, where the derivatives |G∗ |1/2 E1 (dλ)|G∗ |1/2 |G|1/2 E2 (dλ)|G|1/2 and M2 (λ) = dλ dλ exist in B1 (H1 ) and B1 (H2 ) for a.e. λ ∈ R, respectively. Setting Qj,λ := clo ran (Mj (λ)) , j = 1, 2, M1 (λ) =

(3.15)

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41

and Qλ := Q1,λ ⊕ Q2,λ for a.e. λ ∈ R we obtain the decomposition

(3.16)

L2 (R, dλ, Qλ ) = L2 (R, dλ, Q1,λ ) ⊕ L2 (R, dλ, Q2,λ ), cf. Section 2. From (2.2) the conditions H1 = clospan{E1 (∆)ran (|G∗ |) : ∆ ∈ B(R)}, H2 = clospan{E2 (∆)ran (|G|) : ∆ ∈ B(R)}

(3.17)

follow. Moreover, the converse is also true, that is, condition (3.17) implies (2.2). Hence, without loss of generality we assume that condition (3.17) is satisﬁed. Therefore the reduced multiplication operators MQj , MQj := MHj dom (MHj ) ∩ L2 (R, dλ, Qj,λ ), where

(MHj f )(λ) := λf (λ), λ ∈ R, dom (MHj ) := f ∈ L2 (R, dλ, Hj ) : λf (λ) ∈ L2 (R, dλ, Hj ) . are unitary equivalent to the absolutely continuous parts Hjac of the operators Hj , j = 1, 2. With respect to the decomposition (3.16) the scattering matrix {S(λ)}λ∈R admits the decomposition Q1,λ Q1,λ S11 (λ) S12 (λ) S(λ) = (3.18) : ⊕ −→ ⊕ S21 (λ) S22 (λ) Q2,λ Q2,λ for a.e. λ ∈ R. The entries {Sij (λ)}λ∈R , i, j = 1, 2, are called channel scattering matrices. We note that the multiplication operators induced by the channel scattering matrices are unitary equivalent to the channel scattering operators Sij = Pi SPj , i, j = 1, 2, where Pi is the orthogonal projection in L onto the subspace Hj and S is the scattering operator of the complete scattering system {L, L0}. In the next proposition we give a more explicit description of the channel scattering matrices Sij (λ). The proof is an immediate consequence of Theorem 2.1, Lemma 3.2 and relations (3.14), (3.15) and (3.13). Proposition 3.3. Let L0 , V and L be given in accordance with (3.1), (3.2) and (3.3), respectively. Then the scattering matrix {S(λ)}λ∈R of the complete scattering system {L, L0} admits the representation (3.18) with entries Sij (λ) given by S11 (λ) = IQ1,λ + 2πiM1 (λ)1/2 U F2 (λ)U ∗ M1 (λ)1/2 , S12 (λ) = −2πiM1 (λ)1/2 {U + U F2 (λ)U ∗ N1 (λ)U }M2 (λ)1/2 , S21 (λ) = −2πiM2 (λ)1/2 {U ∗ + U ∗ N1 (λ)U F1 (λ)U ∗ }M1 (λ)1/2 , S22 (λ) = IQ2,λ + 2πiM2 (λ)1/2 U ∗ F1 (λ)U M2 (λ)1/2 . for a.e. λ ∈ R.

(3.19)

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4. Dissipative channel scattering In this section we consider the (dissipative) scattering system {H1 (λ), H1 } for a.e. λ ∈ R, where H1 (λ) = H1 + V1 (λ), (4.1) is deﬁned for a.e. λ ∈ R, and H1 is the self-adjoint operator in H1 from (3.1). The limit V1 (λ) = lim→+0 V1 (λ + i) (see Lemma 4.1) is called the optical potential of the channel H1 . We recall that a linear operator T in a Hilbert space is said to be dissipative if Im (T f, f ) ≤ 0, f ∈ dom (T ), and T is called maximal dissipative if T is dissipative and does not admit a proper dissipative extension. In Theorem 4.4 below we establish a connection between the scattering matrices corresponding to the scattering systems {H1 (λ), H1 } and the channel scattering matrix S11 (λ) from (3.18) and (3.19). Lemma 4.1. Let V1 (z) = −G(H2 − z)−1 G∗ , z ∈ C\R, be deﬁned by (3.5) with G ∈ B1 (H2 , H1 ). Then the limit V1 (λ) = lim→+0 V1 (λ + i) exists in B1 (H1 ) and V1 (λ) is dissipative for a.e. λ ∈ R. Proof. Using the factorizations (3.12) of G and G∗ we ﬁnd V1 (z) = −|G∗ |1/2 U N2 (z)U ∗ |G∗ |1/2 ,

z ∈ C\R,

(4.2)

where N2 (z) is given by (3.7). According to Lemma 3.2 the limit lim→+0 N2 (λ+i) exists in B2 (H1 ) and since |G∗ |1/2 ∈ B2 (H1 ) we conclude that the limit V1 (λ) = lim V1 (λ + i) = lim −|G∗ |1/2 U N2 (λ + i)U ∗ |G∗ |1/2 →+0

→+0

exists in B1 (H1 ) for a.e. λ ∈ R. It is not diﬃcult to see that Im V1 (z) ≤ 0 for z ∈ C+ and therefore also the limit V1 (λ) is dissipative for a.e. λ ∈ R. It follows from Lemma 4.1 that for a.e. λ ∈ R the operator H1 (λ) = H1 +V1 (λ) is maximal dissipative and therefore {H1 (λ), H1 } is a dissipative scattering system in the sense of [19, 20]. By Theorem 4.3 of [20] the corresponding wave operators ∗

W+D (H1 (λ), H1 ) = s − lim eitH1 (λ) e−itH1 P ac (H1 ) t→+∞

and

W−D (H1 (λ), H1 ) = s − lim e−itH1 (λ) eitH1 P ac (H1 ) t→+∞

exist and are complete which yields that {H1 (λ), H1 } performs a complete dissipative scattering system for a.e. λ ∈ R, see [19] for details. The associated scattering operators are deﬁned by SD [λ] := W+D (H1 (λ), H1 )∗ W−D (H1 (λ), H1 ) and act on the absolutely continuous subspaces Hac 1 (H1 ). Since SD [λ] commutes with H1 the scattering operator is unitary equivalent to a multiplication operator in the spectral representation L2 (R, dλ, Q1,µ ) of H1 induced by a family of contractions {SD [λ](µ)}µ∈R . The family {SD [λ](µ)}µ∈R is called the scattering matrix of the complete dissipative scattering system {H1 (λ), H1 }.

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43

Using the fact that every maximal dissipative operator admits a self-adjoint dilation, i.e., there exists a self-adjoint operator in a (in general) larger Hilbert space such that its compressed resolvent coincides with the resolvents of the maximal dissipative operator for all z ∈ C+ , cf. [7, Section 7], see also [12], one concludes from Proposition 3.14 of [5] that the limit F1 [λ](µ) = lim F1 [λ](µ + i) →+0

exist in B1 (H1 ) for a.e. µ ∈ R, where F1 [λ](z) := |G∗ |1/2 (H1 (λ) − z)−1 |G∗ |1/2 ,

z ∈ C+ ,

is deﬁned for a.e. λ ∈ R. The next proposition is a direct consequence of Theorem 2.2 of [16], see also [15]. Proposition 4.2. Let G ∈ B1 (H2 , H1 ) and H1 (λ) be given by (4.1). Then for a.e. λ ∈ R the scattering matrix {SD [λ](µ)}µ∈R of the complete dissipative scattering system {H1 (λ), H1 } admits the representation SD [λ](µ) = IQ 1,µ + 2πiM1 (µ)1/2 U N2 (λ) + N2 (λ)U ∗ F1 [λ](µ)U N2 (λ) U ∗ M1 (µ)1/2

for a.e. (µ, λ) ∈ R2 with respect to the Lebesgue measure in R2 . In the next lemma we show that the limit F1 [λ](λ), F1 [λ](λ) = lim F1 [λ](λ + i) = lim |G∗ |1/2 (H1 (λ) − λ − i)−1 |G∗ |1/2 , →+0

→+0

(4.3)

exist in B2 (H1 ) for a.e. λ ∈ R. Lemma 4.3. Let L0 , V and L be given by (3.1), (3.2) and (3.3), respectively, with G ∈ B1 (H2 , H1 ). Further, let F1 (λ) be as in Lemma 3.2 and let H1 (λ) be deﬁned by (4.1). Then the limit F1 [λ](λ) in (4.3) exists in B2 (H1 ) for a.e. λ ∈ R and the relation F1 [λ](λ) = F1 (λ) (4.4) holds for a.e. λ ∈ R. Proof. We have

F1 (z) − F1 [λ](z) = |G∗ |1/2 (H1 (z) − z)−1 − (H1 (λ) − z)−1 |G∗ |1/2 = |G∗ |1/2 (H1 (z) − z)−1 (V1 (λ) − V1 (z))(H1 (λ) − z)−1 |G∗ |1/2 .

(4.5)

From (4.2) we obtain

V1 (λ) − V1 (z) = |G∗ |1/2 U N2 (z) − N2 (λ) U ∗ |G∗ |1/2

and inserting this expression into (4.5) and using the deﬁnitions of F1 (z) in (3.8) and F1 [λ](z) yields F1 (z) − F1 [λ](z) = F1 (z)U (N2 (z) − N2 (λ))U ∗ F1 [λ](z). Hence

F1 (z) = {IH1 + F1 (z)U (N2 (z) − N2 (λ))U ∗ } F1 [λ](z)

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J. Behrndt, H. Neidhardt and J. Rehberg

and for z = λ + i, > 0 suﬃciently small, the operator {IH1 + F1 (z)U (N2 (z) − N2 (λ))U ∗ } is invertible. Therefore we conclude −1

{IH1 + F1 (λ + i)U (N2 (λ + i) − N2 (λ))U ∗ }

F1 (λ + i) = F1 [λ](λ + i).

From this representation we get the existence of F1 [λ](λ) in B2 (H1 ) and the equality (4.4) for a.e. λ ∈ R. The next theorem is the main result of this section. We show how the channel scattering matrix S11 (λ) of the scattering system {L, L0 } is connected with the scattering matrices SD [λ](µ) of the dissipative scattering systems {H1 (λ), H1 }. Theorem 4.4. Let {L, L0 } be the scattering system from Section 3, where L0 , V and L are given by (3.1), (3.2) and (3.3), respectively, and G ∈ B1 (H2 , H1 ). Further, let {Sij (λ)}, i, j = 1, 2, be the corresponding scattering matrix from (3.18) and let SD [λ](µ) be the scattering matrices of the dissipative scattering systems {H1 (λ), H1 }. Then the scattering matrix SD [λ](λ) exists for a.e λ ∈ R and satisﬁes the relation SD [λ](λ) = S11 (λ) for a.e. λ ∈ R. Proof. From Proposition 4.2 and Lemma 4.3 we obtain that SD [λ](λ) exists for a.e λ ∈ R and has the form SD [λ](µ) = IQ 1,µ + 2πiM1 (µ)1/2 U N2 (λ) + N2 (λ)U ∗ F1 (λ)U N2 (λ) U ∗ M1 (µ)1/2 (4.6)

A similar calculation as in the proof of Lemma 3.1 shows F2 (z) = N2 (z) + N2 (z)U ∗ F1 (z)U N2 (z),

z ∈ C+ .

If z tends to λ ∈ R, then we get F2 (λ) = N2 (λ) + N2 (λ)U ∗ F1 (λ)U N2 (λ)

(4.7)

for a.e. λ ∈ R. Inserting (4.7) into (4.6) we obtain SD [λ](λ) = IQ1,λ + 2πiM1 (λ)1/2 U F2 (λ)U ∗ M1 (λ)1/2 and by Proposition 3.3 this coincides with S11 (λ) for a.e. λ ∈ R.

5. Lax-Phillips channel scattering Similarly to Lemma 4.1 one veriﬁes that V2 (λ) = lim→+0 V2 (λ + i) exists in B1 (H2 ) for a.e. λ ∈ R. The limit V2 (λ), which is called the optical potential of the channel H2 , is dissipative for a.e. λ ∈ R. The optical potential deﬁnes the maximal dissipative operator H2 (λ) := H2 + V2 (λ)

Block Operator Matrices and Scattering

45

for a.e. λ ∈ R. The operator H2 (λ) decomposes for a.e. λ ∈ R into a self-adjoint part and a completely non-self-adjoint part. Let Θ2 [λ](ξ), ξ ∈ C− , be the characteristic function, cf. [12], of the completely non-self-adjoint part of H2 (λ). We are going to verify S11 (λ) = Θ2 [λ](λ)∗ for a.e. λ ∈ R which shows that for a.e. λ ∈ R the scattering matrix S11 (λ) can be regarded as the result of a certain Lax-Phillips scattering theory, cf. [1, 2, 3, 4, 14]. There is an orthogonal decomposition self H2 = Hcns 2,λ ⊕ H2,λ self for a.e. λ ∈ R such that Hcns 2,λ and H2,λ reduce H2 (λ) into a completely non-selfadjoint operator H2cns (λ) and a self-adjoint operator H2self (λ),

H2 (λ) = H2cns (λ) ⊕ H2self (λ). Taking into account Proposition 3.14 of [5] we get that m (V2 (λ)) = −π|G|1/2 U ∗ M1 (λ)U |G|1/2 for a.e. λ ∈ R. Let us introduce the operator ) α(λ) := 2πM1 (λ) U |G|1/2 .

(5.1)

Notice that clo{ran (α(λ))} = Q1,λ for a.e. λ ∈ R. With the completely non-self-adjoint part H cns (λ) one associates the characteristic function Θ2 [λ](·) : Q1,λ −→ Q1,λ deﬁned by Θ2 [λ](ξ) := IQ1,λ − iα(λ)(H2 (λ)∗ − ξ)−1 α(λ)∗ , ξ ∈ C− . The characteristic function is a contraction-valued holomorphic function in C− . From [12, Section V.2] we get that the boundary values Θ2 [λ](µ) := s − lim Θ2 [λ](µ − i) →+0

exist for a.e. µ ∈ R. Theorem 5.1. Let L0 , V and L be given by (3.1), (3.2) and (3.3). If the condition G ∈ B1 (H2 , H1 ) is satisﬁed, then the limit Θ2 [λ](λ), Θ2 [λ](λ) := s − lim Θ2 [λ](λ − i) →+0

exists for a.e. λ ∈ R and the relation S11 (λ) = Θ2 [λ](λ)∗ holds for a.e. λ ∈ R. Proof. We set Θ∗2 [λ](ξ) := Θ2 [λ](ξ)∗ = IQ1,λ + iα(λ)(H2 (λ) − ξ)−1 α(λ)∗ ξ ∈ C+ . Using (5.1) we get

) ) Θ∗2 [λ](ξ) = IQ1,λ + 2πi M1 (λ) U F2 [λ](ξ)U ∗ M1 (λ)

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J. Behrndt, H. Neidhardt and J. Rehberg

for a.e. λ ∈ R, where F2 [λ](ξ) := |G|1/2 (H2 (λ) − ξ)−1 |G|1/2 ,

ξ ∈ C+ .

Similar to the proof of Lemma 4.3 one veriﬁes that the limit F2 [λ](λ) F2 [λ](λ) = lim F2 [λ](λ + i) →+0

exist in B2 (H2 ) for a.e. λ ∈ R and satisﬁes the relation F2 [λ](λ) = F2 (λ). Hence the limit Θ∗2 [λ](λ) = s − lim→+0 Θ2 [λ](λ − i)∗ exists for a.e. λ ∈ R and the relation ) ) Θ∗2 [λ](λ) = IQ1,λ + 2πi M1 (λ) U F2 [λ](λ)U ∗ M1 (λ) holds for a.e λ ∈ R. From (3.19) we obtain that S11 (λ) = Θ∗2 [λ](λ) for a.e. λ ∈ R. Since the limit Θ∗2 [λ](λ) exists for a.e. λ ∈ R one concludes that Θ2 [λ](λ) := s − lim Θ2 [λ](λ − i) →+0

exists for a.e. λ ∈ R and Θ2 [λ](λ)∗ = Θ∗2 [λ](λ) is valid. This completes the proof Theorem 5.1. The last theorem admits an interpretation of the scattering matrix S11 (λ) as the result of a Lax-Phillips scattering. Indeed, let us introduce the minimal self-adjoint dilation K2 (λ) of the maximal dissipative operator H2 (λ). We set K2,λ = D−,λ ⊕ H2 ⊕ D+,λ , where D±,λ := L2 (R± , dx, Q1,λ ). Further, we deﬁne ⎛

⎞ ⎛ ⎞ d f− f− −i dx K2 (λ) ⎝ f ⎠ := ⎝ e (H2 (λ))f − 12 α(λ)∗ [f+ (0) + f− (0)] ⎠ d f+ −i dx f+

for elements of the domain

⎫ ⎧⎛ ⎞ f ∈ dom (H2 (λ)) ⎬ ⎨ f− dom (K2 (λ)) := ⎝ f ⎠ : f± ∈ W 1,2 (R± , dx, Q1,λ ) . ⎭ ⎩ f+ f+ (0) − f− (0) = −iα(λ)f

The operator K2 (λ) is self-adjoint and is a minimal self-adjoint dilation of the maximal dissipative operator H2 (λ), that is, K

(H2 (λ) − z)−1 = PH22,λ (K2 (λ) − z)−1 H2 for z ∈ C+ and

K2,λ = clospan EK2 (λ) (∆)H2 : ∆ ∈ B(R) ,

where EK2 (λ) (·) is the spectral measure of K2 (λ). It turns out that D±,λ are incoming and outgoing subspaces with respect to K2 (λ), i.e., e−itK2 (λ) D+,λ ⊆ D+,λ ,

t ≥ 0,

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and e−itK2 (λ) D−,λ ⊆ D−,λ ,

t ≤ 0.

However, we remark that the completeness condition K2,λ = clospan e−itK2 (λ) D±,λ : t ∈ R

(5.2)

is in general not satisﬁed. Condition (5.2) holds if and only if the maximal dissipative operator H2 (λ) is completely non-selfadjoint and H2 is singular, that means, the absolutely continuous part H2ac of H2 is trivial. On the subspace Dλ , Dλ = D−,λ ⊕ D+,λ = L2 (R, dx, Q1,λ ) ⊆ K2,λ , let us deﬁne the operator K0 (λ), d g(x), dom (K0 (λ)) := W 1,2 (R, dx, Q1,λ ). dx The self-adjoint operator K0 (λ) generates the shift group, i.e, (K0 (λ)g)(x) := −i

(e−itK0 (λ) g)(x) = g(x − t),

g ∈ Dλ .

Using the Fourier transform F : L2 (R, dx, Q1,λ ) −→ L2 (R, dµ, Q1,λ ), 1 dx e−iµx f (x), (F f )(µ) = √ 2π R the operator K0 (λ) transforms into the multiplication operator on the Hilbert space L2 (R, dµ, Q1,λ ). Furthermore, one has e−itK2 (λ) D+,λ = e−itK0 (λ) D+,λ ,

t ≥ 0,

e−itK2 (λ) D−,λ = e−itK0 (λ) D−,λ ,

t ≤ 0.

and The last properties yield the existence of the Lax-Phillips wave operators W±LP [λ] := s − lim eitK2 (λ) J± (λ)e−itK0 (λ) , t→±∞

cf. [5, 14] where J± (λ) : D±,λ −→ K2,λ is the natural embedding operator. The Lax-Phillips scattering operator SLP (λ) is deﬁned by SLP [λ] := W+LP [λ]∗ W−LP [λ], cf. [5, 14]. With respect to the spectral representation L2 (R, dµ, Q1,λ ) the LaxPhillips scattering matrix {SLP [λ](µ)}µ∈R coincides with {Θ2 [λ](µ)∗ }µ∈R , see [1, 2, 3, 4]. Hence the scattering matrix {S11 (λ)}λ∈R can be regarded as the result of a Lax-Phillips scattering for a.e. λ ∈ R. Acknowledgement The support of the work by DFG, Grant 1480/2, is gratefully acknowledged. The authors thank Prof. P. Exner for discussion and literature hints.

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References [1] Adamjan, V.M., Arov, D.Z.: Unitary couplings of semi-unitary operators, Akad. Nauk Armjan. SSR Dokl. 43, no. 5, 257–263 (1966). [2] Adamjan, V.M., Arov, D.Z.: Unitary couplings of semi-unitary operators, Mat. Issled. 1, vyp. 2, 3–64 (1966). [3] Adamjan, V.M., Arov, D.Z.: On scattering operators and contraction semigroups in Hilbert space, Dokl. Akad. Nauk SSSR 165, 9–12 (1965). [4] Adamjan, V.M., Arov, D.Z.: On a class of scattering operators and characteristic operator-functions of contractions. Dokl. Akad. Nauk SSSR 160, 9–12 (1965). [5] Baumg¨ artel, H., Wollenberg, M.: Mathematical scattering theory, Akademie-Verlag, Berlin, 1983. [6] Davies, E.B.: Two-channel Hamiltonians and the optical model of nuclear scattering, Ann. Inst. H. Poincar´e Sect. A (N.S.) 29, no. 4, 395–413 (1979). [7] Davies, E.B.: Quantum theory of open systems, Academic Press, London-New York, 1976. [8] Exner, P.: Open quantum systems and Feynman integrals, D. Reidel Publishing Co., Dordrecht, 1985. ´ [9] Exner, P., Ulehla, I.: On the optical approximation in two-channel systems, J. Math. Phys. 24, no. 6, 1542–1547 (1983). [10] Feshbach, H.: A uniﬁed theory of nuclear reactions. II, Ann. Physics 19, 287–313 (1962). [11] Feshbach, H.: Uniﬁed theory of nuclear reactions, Ann. Physics 5, 357–390 (1958). [12] Foias, C., Sz.-Nagy, B.: Harmonic analysis of operators on Hilbert space, NorthHolland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akad´emiai Kiad´ o, Budapest 1970. [13] Kato, T.: Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York. [14] Lax, P.D., Phillips, R.S.: Scattering theory, Academic Press, New York-London, 1967. [15] Neidhardt, H.: Scattering matrix and spectral shift of the nuclear dissipative scattering theory. II, J. Operator Theory 19, no. 1, 43–62 (1988). [16] Neidhardt, H.: Scattering matrix and spectral shift of the nuclear dissipative scattering theory, In “Operators in indeﬁnite metric spaces, scattering theory and other topics” (Bucharest, 1985), 237–250, Oper. Theory Adv. Appl., 24, Birkh¨ auser, Basel, 1987. [17] Neidhardt, H.: On the inverse problem of a dissipative scattering theory. I, In “Advances in invariant subspaces and other results of operator theory” (Timi¸soara and Herculane, 1984), 223–238, Oper. Theory Adv. Appl., 17, Birkh¨ auser, Basel, 1986. [18] Neidhardt, H.: Eine mathematische Streutheorie f¨ ur maximal dissipative Operatoren, Report MATH, 86-3. Akademie der Wissenschaften der DDR, Institut f¨ ur Mathematik, Berlin, 1986. [19] Neidhardt, H.: A nuclear dissipative scattering theory, J. Operator Theory 14, no. 1, 57–66 (1985).

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[20] Neidhardt, H.: A dissipative scattering theory, In “Spectral theory of linear operators and related topics” (Timi¸soara/Herculane, 1983), 197–212, Oper. Theory Adv. Appl., 14, Birkh¨ auser, Basel, 1984. Jussi Behrndt Technische Universit¨ at Berlin Institut f¨ ur Mathematik Straße des 17. Juni 136 D–10623 Berlin, Germany e-mail: [email protected] Hagen Neidhardt Weierstraß-Institut f¨ ur Angewandte Analysis und Stochastik Mohrenstr. 39 D–10117 Berlin, Germany e-mail: [email protected] Joachim Rehberg Weierstraß-Institut f¨ ur Angewandte Analysis und Stochastik Mohrenstr. 39 D–10117 Berlin, Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 175, 51–88 c 2007 Birkh¨ auser Verlag Basel/Switzerland

Asymptotic Expansions of Generalized Nevanlinna Functions and their Spectral Properties Vladimir Derkach, Seppo Hassi and Henk de Snoo Abstract. Asymptotic expansions of generalized Nevanlinna functions Q are investigated by means of a factorization model involving a part of the generalized zeros and poles of nonpositive type of the function Q. The main results in this paper arise from the explicit construction of maximal Jordan chains in the root subspace R∞ (SF ) of the so-called generalized Friedrichs extension. A classiﬁcation of maximal Jordan chains is introduced and studied in analytical terms by establishing the connections to the appropriate asymptotic expansions. This approach results in various new analytic characterizations of the spectral properties of selfadjoint relations in Pontryagin spaces and, conversely, translates analytic and asymptotic properties of generalized Nevanlinna functions into the spectral theoretical properties of self-adjoint relations in Pontryagin spaces. Mathematics Subject Classiﬁcation (2000). Primary 46C20, 47A06, 47B50; Secondary 47A10, 47A11, 47B25. Keywords. Generalized Nevanlinna function, asymptotic expansion, Pontryagin space, symmetric operator, selfadjoint extension, operator model, factorization, generalized Friedrichs extension.

1. Introduction Let Nκ be the class of generalized Nevanlinna functions, i.e., meromorphic functions on C \ R with Q(¯ z ) = Q(z) and such that the kernel NQ (z, λ) =

Q(z) − Q(λ) , ¯ z−λ

z, λ ∈ ρ(Q),

¯ z = λ,

The research was supported by the Academy of Finland (project 212150) and the Research Institute for Technology at the University of Vaasa.

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V.A. Derkach, S. Hassi and H.S.V. de Snoo

has κ negative squares on the domain of holomorphy ρ(Q) of Q, see [20]. If the function Q ∈ Nκ belongs to the subclass Nκ,−2n , n ∈ N, (see [6]) then it admits the following asymptotic expansion 2n+1 - sj−1 1 Q(z) = γ − + o , z. →∞, (1.1) zj z 2n+1 j=1 →∞ means that z tends to ∞ nontangentially (0 < ε < where γ, sj ∈ R and z. arg z < π − ε). Asymptotic expansions for Q ∈ Nκ of the form (1.1) (with γ = 0) were introduced in [21]. They naturally appear, for instance, in the indeﬁnite moment problem considered in [22]. The expansion (1.1) is equivalent to the following operator representation of the function Q ∈ Nκ,−2n : Q(z) = γ + [(A − z)−1 ω, ω],

(1.2)

where ω ∈ dom A and A is a selfadjoint operator in a Pontryagin space H; see [21, Satz 1.10] and Corollary 3.4 below. The representation (1.2) can be taken to be minimal in the sense that ω is a cyclic vector for A, i.e., n

H = span { (A − z)−1 ω : z ∈ ρ(A) }, in which case the negative index sq− (H) of H is equal to κ. The representation (1.2) shows that ∞ is a generalized zero of the function Q(z) − γ, or equivalently, that ∞ is a generalized pole of the function Q∞ (z) = −1/(Q(z) − γ). This means that the underlying symmetric operator S is nondensely deﬁned in H with dom S = { f ∈ dom A : [f, ω] = 0 }

(1.3)

and that ({0} × span {ω}) SF = S +

(1.4) is a selfadjoint extensions of S in H with ∞ ∈ σp (SF ). Here + stands for the componentwise sum in the Cartesian product H × H. In other words, the extension SF is multivalued and, in fact, can be interpreted as the generalized Friedrichs extension of S, see [5] and the references therein. It follows from (1.1) and (1.2) that s0 = [ω, ω] ∈ R. If κ > 0 then it is possible that s0 ≤ 0, in which case ∞ is a generalized pole of nonpositive type (GPNT) of the function Q∞ , cf. [23]. More precisely, if ∞ is a GPNT of Q∞ with multiplicity κ∞ := κ∞ (Q∞ ) (see (2.2) below for the deﬁnition), then in (1.1) one automatically has s0 = · · · = sj = 0,

for every j < 2κ∞ − 2.

Furthermore, if m is the ﬁrst nonnegative index in (1.1) such that sm = 0 (if it exists), then, equivalently, the function Q∞ admits an asymptotic expansion of the form Q∞ (z) = pm+1 z m+1 + · · · + p2+1 z 2+1 + o z 2+1 , z. →∞, (1.5)

Asymptotic expansions

53

where pm+1 = 1/sm , pi ∈ R, i = m + 1, . . . , 2 + 1, and the integers m, n, and are connected by = m − n with m ≥ 2; see Theorem 5.4 below for further details. It turns out that (1.5) holds for some ≤ 0 if and only if ∞ is a regular critical point of SF , or equivalently, if and only if the corresponding root subspace R∞ (SF ) = { h ∈ H : {0, h} ∈ SFk for some k ∈ N } of the generalized Friedrichs extensions SF in (1.4) is nondegenerate. In this case the GPNT ∞ of Q∞ as well as the corresponding root subspace R∞ (SF ) are shortly called regular. On the other hand, if ∞ is a singular critical point of SF , then in (1.5) > 0 and, moreover, the minimal integer such that the expansion (1.5) exists coincides with the dimension κ0∞ of the isotropic subspace of the root subspace R∞ (SF ), see Theorem 5.6. In this case the GPNT ∞ of Q∞ and the corresponding root subspace R∞ (SF ) are shortly called singular with the index of singularity κ0∞ . The above-mentioned results reﬂect the close connections between the asymptotic expansions (1.1), (1.5), and the root subspace R∞ (SF ) of SF . The given assertions are examples of the results in the present paper which have been derived by means of the factorization model of the function Q∞ recently constructed by the authors in [9]. This model is based on the following “proper” factorization of the function Q∞ ∈ Nκ : Q∞ (z) = q(z)q (z)Q0 (z),

(1.6)

where q is a (monic) polynomial, q (z) = q(¯ z ), and Q0 ∈ Nκ such that κ∞ (Q0 ) = 0

and κ = κ − deg q,

see Lemma 4.3 below. Such a factorization for Q∞ is in general not unique, but the factorization model based on such a factorization carries the complete information about the root subspace R∞ (SF ) of SF . A major part of the results presented in this paper is associated with the structure of the root subspace R∞ (SF ) of SF in a model space and the various connections to the asymptotic expansions (1.1) and (1.5). By using the factorization model based on a proper factorization (1.6) of Q∞ maximal Jordan chains in R∞ (SF ) are constructed in explicit terms. Their construction leads to three different types of maximal Jordan chains in R∞ (SF ). Each of these three types of maximal Jordan chains admits its own characteristic features, reﬂecting various properties of the root subspace R∞ (SF ). The construction shows explicitly, for instance, when the root subspace R∞ (SF ) is regular and when it is singular. The length of the maximal Jordan chain as well as the signature of the root subspace R∞ (SF ) can be easily read oﬀ from their construction. In the case that the root subspace R∞ (SF ) is regular, the three types of maximal Jordan chain can be characterized by their length. The ﬁrst type of maximal Jordan chain is of length 2k+1, where k = deg q = κ∞ (Q∞ ), and the second and third type of maximal Jordan chains are of length 2k and 2k − 1, respectively. The classiﬁcation of these maximal Jordan chains remains the same in the case when the root subspace R∞ (SF ) is

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V.A. Derkach, S. Hassi and H.S.V. de Snoo

singular. In that case the index of singularity κ0∞ as introduced above enters to the formulas, while the diﬀerence κ− (R∞ (SF )) − κ+ (R∞ (SF )) of the negative and the positive index of R∞ (SF ) remains unaltered, see Theorem 4.12. All of these facts can be translated into the analytical properties of the functions Q∞ and Q = γ − 1/Q∞ via the asymptotic expansions (1.1) and (1.5), and conversely. The classiﬁcation of maximal Jordan chains in R∞ (SF ) motivates an analogous classiﬁcation of generalized zeros and poles of nonpositive type of the function Q ∈ Nκ , which turns out to be connected with the characterization of the multiplicities of GZNT and GPNT of the function Q due to H. Langer in [24]; see Subsection 3.2 for the deﬁnitions of generalized zeros and poles of types (T1)– (T3). This induces a classiﬁcation for the asymptotic expansions for the functions Q and Q∞ ; see Theorems 5.3 and 5.4. Some further characterizations of the three diﬀerent types of generalized zeros and poles are obtained by means of the factorized integral representations of the functions Q and Q∞ , which are based on their canonical factorizations, see [11]; for deﬁnitions, see Subsection 2.1, cf. also [5]. In particular, Theorem 6.1 and Theorem 6.3 extend some earlier results by the authors in [6] (where κ = 1) and in [8], from the regular case to the singular case in an explicit manner involving the index of singularity κ0∞ , which is characterized in Theorem 5.6 below. The construction of the maximal Jordan chains in R∞ (SF ) using the factorization model for Q∞ in (1.6) is carried out in Section 4. The most careful treatment of the model is required in the construction of maximal Jordan chains which are of the third type (T3). The reason is that the factorization of Q∞ does not produce a minimal model for the function Q∞ directly. In the minimal factorization model the maximal Jordan chains of type (T3) are roughly speaking the shortest ones, cf. (4.21), (4.24), (4.28); see also Theorem 5.3 and Theorem 6.3. The results in Lemma 4.11 and part (iii) of Theorem 4.12 characterize maximal Jordan chains of type (T3). In this case the underlying symmetric relation S(Q) is multivalued (before the auxiliary part of the space is factored out). This statement is true more generally: for an arbitrary Nκ -function Q the occurrence of generalized zeros and poles of type (T3) in R ∪ {∞} is an indication that point spectrum σp (S(Q)) of S(Q) is nonempty, see Lemma 6.4 below, which by part (i) of Theorem 4.6 is equivalent to S(Q) being not simple. In fact, the existence of maximal Jordan chains of type (T3) or, equivalently, the existence of GZNT and GPNT of type (T3) can be used to give criteria for minimality of various factorization models for Nκ -functions, see Propositions 6.6 and 6.7 below. The topics considered in this paper have connections to some other recent studies involving asymptotic expansions of Nκ -functions, see in particular [6], [8], [9], [12], [13], [15], and their canonical factorization, see, e.g., [3], [5], [7], [11], [14]. For instance, in [13] the authors investigate the subclass of Nκ -functions with κ = κ∞ (Q) and extend some results, e.g., from [6], [8]. General operator models based on the canonical factorization of Nκ -functions have been introduced in [3]; for another model not using the canonical factorization of Q, see [18]. The construction of a minimal canonical factorization model by using reproducing kernel Pontryagin

Asymptotic expansions

55

space methods has been recently worked out in [12], cf. also [3, Theorem 4.1]. Some of the results in the present paper can be naturally augmented by the results which can be found from [15], where characteristic properties of the generalized zeros and poles of Nκ -functions have been studied with the aid of their operator representations. The present paper forms a continuation of the paper [9], where the details concerning the construction of the announced factorization model can be found. Some basic deﬁnitions and concepts which will be used throughout the paper are given in Section 2. In Section 3 some additions concerning the subclasses Nκ,− as introduced in [6] are given, including a proof for [6, Proposition 6.2] as announced in that paper, cf. Theorem 3.3 below; see also Theorem 5.4 for an extension of these results. Asymptotic expansions are introduced in Section 3 and a classiﬁcation of generalized zeros and poles is given. In Section 4 the main ingredients concerning the factorization model are given and the construction of maximal Jordan chains in R∞ (SF ) is carried out. The connection between the properties of the root subspace R∞ (SF ) and the asymptotic expansions of the form (1.1) and (1.5) is investigated in Section 5. Finally, in Section 6 the classiﬁcation of GZNT and GPNT is connected with factorized integral representations of the functions Q and Q∞ (z). In this section also the generalized zeros and poles of nonpositive type of Nκ -functions which belong to R are brieﬂy treated and some consequences as announced above are established.

2. Preliminaries 2.1. Canonical factorization of Q ∈ Nκ The notions of generalized poles and generalized zeros of nonpositive type were introduced in [23]. The following deﬁnitions are based on [24]. A point α ∈ R is called a generalized pole of nonpositive type (GPNT) of the function Q ∈ Nκ with multiplicity κα (= κα (Q)) if −∞ < lim (z − α)2κα +1 Q(z) ≤ 0, z. →α

0 < lim (z − α)2κα −1 Q(z) ≤ ∞. z. →α

(2.1)

Similarly, the point ∞ is called a generalized pole of nonpositive type (GPNT) of Q with multiplicity κ∞ (= κ∞ (Q)) if 0 ≤ lim

Q(z)

z. →∞ z 2κ∞ +1

< ∞,

−∞ ≤ lim

Q(z)

z. →∞ z 2κ∞ −1

< 0.

(2.2)

A point β ∈ R is called a generalized zero of nonpositive type (GZNT) of the function Q ∈ Nκ if β is a generalized pole of nonpositive type of the function −1/Q. The multiplicity πβ (= πβ (Q)) of the GZNT β of Q can be characterized by the inequalities: 0 < lim

z. →β

Q(z) ≤ ∞, (z − β)2πβ +1

−∞ < lim

z. →β

Q(z) ≤ 0. (z − β)2πβ −1

(2.3)

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V.A. Derkach, S. Hassi and H.S.V. de Snoo

Similarly, the point ∞ is called a generalized zero of nonpositive type (GZNT) of Q with multiplicity π∞ (= π∞ (Q)) if −∞ ≤ lim z 2π∞ +1 Q(z) < 0, z. →∞

0 ≤ lim z 2π∞ −1 Q(z) < ∞. z. →∞

(2.4)

It was shown in [23] that for Q ∈ Nκ the total number (counting multiplicities) of poles (zeros) in C+ and generalized poles (zeros) of nonpositive type in R ∪ {∞} is equal to κ. Let α1 , . . . , αl (β1 , . . . , βm ) be all the generalized poles (zeros) of nonpositive type in R and the poles (zeros) in C+ with multiplicities κ1 , . . . , κl (π1 , . . . , πm ). Then the function Q admits a canonical factorization of the form p (2.5) Q(z) = r(z)r (z)Q00 (z), Q00 ∈ N0 , r = , q / / πj and q(z) = lj=1 (z − αj )κj are relatively prime where p(z) = m j=1 (z − βj ) polynomials of degree κ−π∞ (Q) and κ−κ∞ (Q), respectively; see [11], [5]. It follows from (2.5) that the function Q admits the (factorized) integral representation 1 t p Q(z) = r(z)r (z) a + bz + − (2.6) dρ(t) , r = , 2 t−z 1+t q R where a ∈ R, b ≥ 0, and ρ(t) is a nondecreasing function satisfying the integrability condition dρ(t) < ∞. (2.7) 2 R t +1 2.2. The subclasses Nκ,1 and Nκ,0 A function Q ∈ Nκ is said to belong to the subclass Nκ,1 , if ∞ Q (z) |Im Q (iy) | = 0 and dy < ∞, lim z. →∞ z y η with η > 0 large enough. Similarly Q ∈ Nκ is said to belong to the subclass Nκ,0 , if lim

z. →∞

Q(z) = 0 and lim sup |z Im Q(z)| < ∞, z z. →∞

see [5]. In the following theorems the subclasses Nκ,1 and Nκ,0 are characterized both in terms of the integral representation (2.6) and in terms of operator representations of the form (1.2). Let Et be a spectral function of a selfadjoint operator A in a Pontryagin space H, 0 see [1]. Denote by H := H (A), ∈ N, the set of all elements h ∈ H such that ∆ |t| d[Et h, h] < ∞ for some neighborhood ∆ of ±∞. Moreover, let H− (A), ∈ N, be the corresponding dual spaces. Here, for instance, H−1 (A) can be identiﬁed as the set of all generalized elements obtained by com0 pleting H with respect to the inner product ∆ (1 + |t|)−1 d[Et h, h] < ∞ with some from H neighborhood ∆ of ±∞. The operator A admits a natural continuation A into H−1 , see [5] for further details. The classes Nκ,1 and Nκ,0 are characterized in the following two theorems, see [5].

Asymptotic expansions

57

Theorem 2.1. ([5]) For Q ∈ Nκ the following statements are equivalent: (i) Q belongs to Nκ,1 ; − z)−1 ω, ω], z ∈ ρ(A), for some selfadjoint operator A in a (ii) Q(z) = γ + [(A Pontryagin space H, a cyclic vector ω ∈ H−1 , and γ ∈ R; (iii) Q has the integral representation (2.6) with deg q − deg p = π∞ (Q) > 0, or with deg p = deg q (π∞ (Q) = 0), b = 0, and (1 + |t|)−1 dρ(t) < ∞. (2.8) R

Theorem 2.2. ([5]) For Q ∈ Nκ the following statements are equivalent: (i) Q belongs to Nκ,0 ; (ii) Q (z) = γ + O (1/z), z. →∞; (iii) Q(z) = γ + [(A − z)−1 ω, ω], z ∈ ρ(A), for some selfadjoint operator A in a Pontryagin space H, a cyclic vector ω ∈ H, and γ ∈ R; (iv) Q has the integral representation (2.6) with deg q − deg p = π∞ (Q) > 0, or with deg p = deg q (π∞ (Q) = 0), b = 0, and dρ(t) < ∞. (2.9) R

Remark 2.3. If Q ∈ Nκ,0 , then the operator representation of Q in part (iii) of Theorem 2.2 implies that lim −z(Q(z) − γ) = [ω, ω].

z. →∞

Hence, the statement (ii) in Theorem 2.2 can be strengthened in the sense that for every function Q ∈ Nκ,0 there are real numbers γ and s0 , such that 1 s0 Q (z) = γ − +o , z. →∞. (2.10) z z

3. Asymptotic expansions of generalized Nevanlinna functions Asymptotic expansions of generalized Nevanlinna functions (as in (2.10)) can be used for studying operator and spectral theoretical properties of selfadjoint extensions of symmetric operators in Pontryagin and Hilbert spaces, see [20], [16]. In this section a subdivision of the class Nκ of generalized Nevanlinna functions is given along the lines of [16], [6]. Moreover, a classiﬁcation for generalized zeros of nonpositive type is introduced and interpreted via asymptotic expansions. 3.1. The subclasses Nκ,− of generalized Nevanlinna functions Deﬁnition 3.1. A function Q ∈ Nκ is said to belong to the subclass Nκ,−2n , n ∈ N, if there are real numbers γ and s0 , . . . , s2n−1 such that the function ⎛ ⎞ 2n sj−1 ⎠ Q(z) = z 2n ⎝Q(z) − γ + (3.1) zj j=1

58

V.A. Derkach, S. Hassi and H.S.V. de Snoo

is O(1/z) as z. →∞. Moreover, Q ∈ Nκ is said to belong to the subclass Nκ,−2n+1 in (3.1) belongs to Nκ ,1 for some κ ∈ N. if the function Q The next lemma clariﬁes the above deﬁnition of the subclasses Nκ,− , ∈ N. Lemma 3.2. If the function Q belongs to the subclass Nκ,−2n (Nκ,−2n+1 ) for some in (3.1) belongs to the subclass Nκ ,0 (resp. Nκ ,1 ) with n ∈ N, then the function Q κ ≤ κ. Moreover, the following inclusions are satisﬁed · · · ⊂ Nκ,−2n−1 ⊂ Nκ,−2n ⊂ Nκ,−2n+1 ⊂ · · · ⊂ Nκ,0 ⊂ Nκ,1 .

(3.2)

in (3.1) in the form Q(z) Proof. Rewrite the expression for the function Q = 2n z Q(z). Then Q(z) as a sum of two generalized Nevanlinna functions is also a is a generalized generalized Nevanlinna function, and therefore, in view of (2.5), Q ≤ κ(Q) is Nevanlinna function, too. Next it is shown that the inequality κ(Q) satisﬁed. First observe that the condition Q(z) = O(1) and hence, in particular, the condition Q(z) = O(1/z) as z. →∞ implies that = 0, κ∞ (Q)

(3.3)

= κα (Q) for every α = 0, ∞, while for α = 0 one cf. (2.2), (2.4). Clearly, κα (Q) derives from (2.1) the estimate ≤ κ0 (Q). κ0 (Q)

(3.4)

≤ κ(Q). Now by Therefore, one can conclude from (3.3) and (3.4) that κ(Q) ∈ Nκ ,0 Theorem 2.2 the condition Q(z) = O(1/z), z. →∞, is equivalent to Q with κ ≤ κ, which proves the ﬁrst statement for the subclasses Nκ,−2n . If Q ∈ ≤ κ(Q), one actually has Q ∈ Nκ ,1 = O(1) and since κ(Q) Nκ,−2n+1 , then Q(z) for κ ≤ κ. Since Nκ,0 ⊂ Nκ,1 the inclusions Nκ,−2n ⊂ Nκ,−2n+1 , n ∈ N, follow from the ﬁrst part of the lemma. Now let Q ∈ Nκ,−2n−1 . Then by deﬁnition + zs2n + s2n+1 ∈ Nκ ,1 , z 2 Q(z)

(3.5)

is as in (3.1) and κ ≤ κ. It is clear from (3.5) (see Theorem 2.1) that where Q Q(z) = O(1/z) as z. →∞. Hence, Q ∈ Nκ,−2n and this proves the remaining inclusions in (3.2). The subclasses Nκ,− , ∈ N, are now characterized by means of the operator and the integral representation of Q in (1.2) and (2.6), respectively. Theorem 3.3. For Q ∈ Nκ the following statements are equivalent: (i) Q ∈ Nκ,− , ∈ N; (ii) Q(z) = γ + [(A − z)−1 ω, ω], z ∈ ρ(A), for some selfadjoint operator A in a Pontryagin space H, a cyclic vector ω ∈ H , and γ ∈ R;

Asymptotic expansions

59

(iii) Q has an integral representation (2.6) with π∞ (Q) = deg q − deg p ≥ 0 (and b = 0 if π∞ (Q) = 0), such that (1 + |t|)−2π∞ dρ(t) < ∞. (3.6) R

Proof. (i) ⇒ (iii) Let Q ∈ Nκ,− , where is either 2n or 2n − 1, n ∈ N. In view of (3.2) and Theorems 2.1, 2.2 one has π∞ (Q) = deg q− deg p ≥ 0, and if π∞ (Q) = 0 in (3.1) then b = 0 and (2.8) or (2.9) is satisﬁed. By Lemma 3.2 the function Q admits the factorization belongs to Nκ ,2n− with κ ≤ κ. Hence, Q 1 t p2 Q = r(z) r (z) − a + bz + (3.7) d ρ(t), r = , 2 t − z 1 + t q2 R cf. (2.5). Moreover, the where p2 and q2 are the polynomials associated to Q, inequality π∞ (Q) = deg q2 − deg p2 ≥ 0 holds by Theorems 2.1 and 2.2. On the admits also the representation other hand, it follows from (2.6) and (3.1) that Q 1 t − (3.8) Q(z) = z 2n r(z)r (z) a + bz + dρ(t) + p1 (z), t − z 1 + t2 R where p1 is a polynomial with deg p1 ≤ 2n. An application of the generalized Stieltjes inversion formula (see [19]) shows that the measures d ρ(t) in (3.7) and dρ(t) in (3.8) are connected by ρ(t) = t2n |r(t)|2 dρ(t). | r (t)|2 d

(3.9)

∈ Nκ ,1 \ Nκ ,0 so that = 2n − 1, then deg p2 = deg q2 and d ρ(t) Therefore, if Q satisﬁes the condition (2.8) in Theorem 2.1. The condition (3.6) follows now from ∈ Nκ ,0 so that = 2n, then either deg p2 = deg q2 in which case d ρ(t) (3.9). If Q satisﬁes the condition (2.9) in Theorem 2.2, or π∞ (Q) = deg q2 − deg p2 > 0 in which case d ρ(t) satisﬁes the condition (2.7). In both cases | r (t)|2 d ρ(t) < ∞ for M > 0 large enough. |t|>M

Hence, again the condition (3.6) follows from (3.9). (ii) ⇔ (iii) Let Et be the spectral function of a selfadjoint operator A in the minimal representation (1.2) of Q. It follows from (1.2), (2.6), and the generalized Stieltjes inversion formula that d[Et ω, ω] = |r(t)|2 dρ(t),

t ∈ ∆,

in some neighborhood ∆ of ±∞. This implies that (1 + |t|)−2π∞ dρ(t) < ∞ if and only if (1 + |t|) d[Et ω, ω] < ∞, R

∆

i.e., ω ∈ H , which proves the equivalence of (ii) and (iii).

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(ii) ⇒ (i) First consider the case = 2n. Then ω ∈ H means that ω ∈ dom An . in (3.1) by setting Deﬁne the function Q sj = [Aj ω, ω],

sn+j = [Aj ω, An ω],

j = 0, . . . , n.

(3.10)

admits the operator representaThen a straightforward calculation shows that Q tion Q(z) = [(A − z)−1 ω , ω ], ω = An ω ∈ H. (3.11) Therefore, Q(z) = O(1/z) and Q ∈ Nκ,− . Now let = 2n − 1. Then ω ∈ H means that ω := An ω ∈ H−1 . Hence , ω ] is well deﬁned. Moreover, by deﬁning s0 , . . . , s2n−2 as in (3.10) s2n−1 := [Aω in (3.1) admits the operator representation it follows that the function Q − z)−1 ω , ω ], ω = An ω ∈ H−1 . Q(z) = [(A (3.12) ∈ Nκ ,1 for some κ ∈ N and thus Q ∈ Nκ,− . This Hence, by Theorem 2.1 Q completes the proof. In the case of even indices = 2n the equivalence of (i) and (ii) in Theorem 3.3 coincides with the following result of M.G. Kre˘ın and H. Langer, see [21, Satz 1.10]. Corollary 3.4. ([21]) The function Q ∈ Nκ admits an operator representation Q(z) = γ + [(A − z)−1 ω, ω] with γ ∈ R and ω ∈ H2n (= dom An ) if and only if there are real numbers γ and s0 , . . . , s2n , such that 2n+1 - sj−1 1 Q(z) = γ − +o , z. →∞. (3.13) zj z 2n+1 j=1 In this case the numbers s0 , . . . , s2n are given by (3.10). Proof. The proof of Theorem 3.3 shows that the condition ω ∈ dom An is equiv alent to the operator representation (3.11) of the function Q(z) in (3.1). Now by applying (2.10) in Remark 2.3 to the function Q(z) in (3.11) and taking into account (3.1) the equivalence to the expansion (3.13) follows. The criterion of M.G. Kre˘ın and H. Langer formulated in Corollary 3.4 does not hold in the case of an odd index = 2n − 1. However, it is clear that if ω ∈ H2n−1 then the analog of the expansion (3.13) exists. Corollary 3.5. If the function Q ∈ Nκ admits an operator representation Q(z) = γ + [(A − z)−1 ω, ω] with γ ∈ R and ω ∈ H2n−1 (= dom |A|n−1/2 ) then there are real numbers γ and s0 , . . . , s2n , such that 2n 1 sj−1 + o , z. →∞. (3.14) Q(z) = γ − zj z 2n j=1 Proof. Since ω ∈ H2n−1 the operator representation (3.12) in the proof of Theo rem 3.3 shows that Q(z) = o(1). The expansion (3.14) for the function Q follows in (3.1). now from the deﬁnition of Q

Asymptotic expansions

61

It is emphasized that the existence of the expansion (3.14) does not imply that ω ∈ H2n−1 or, equivalently, that ω := An ω belongs to H−1 . In this case , ω ] need not be deﬁned and hence it cannot coincide with the coeﬃcient [Aω s2n−1 in (3.14). 3.2. A classiﬁcation of generalized zeros of nonpositive type For what follows it will be useful to give a classiﬁcation for generalized zeros and poles of nonpositive type of a function Q ∈ Nκ . Let ∞ be a GZNT of Q with multiplicity π∞ > 0. It follows from (2.4) that precisely one of the following three cases can occur: 2π∞ +1 2π∞ −1 (T1) −s2π∞ := limz. Q(z) < 0, limz. Q(z) = 0; →∞ z →∞ z 2π∞ +1 2π∞ −1 Q(z) = ∞, limz. Q(z) = 0; (T2) limz. →∞ z →∞ z 2π∞ +1 2π∞ −1 (T3) limz. Q(z) = ∞, −s2π∞ −2 := limz. Q(z) > 0. →∞ z →∞ z In these cases ∞ is said to be a generalized zero of type (T1), (T2), or (T3), respectively; the shorter notations GZNT1, GZNT2, and GZNT3 are used accordingly. The corresponding classiﬁcation for a ﬁnite generalized zero β ∈ R of Q is deﬁned analogously: Q(z) Q(z) (T1) limz. > 0, limz. = 0; →β →β (z − β)2πβ +1 (z − β)2πβ −1 Q(z) Q(z) = ∞, limz. = 0; (T2) limz. →β →β 2π +1 β (z − β) (z − β)2πβ −1 Q(z) Q(z) = ∞, limz. < 0. (T3) limz. →β →β 2π +1 β (z − β) (z − β)2πβ −1 A generalized pole of nonpositive type β ∈ R ∪ {∞} of Q is said to be of type (T1), (T2), or (T3), if β is a generalized zero of nonpositive type of the function −1/Q which is of type (T1), (T2), or (T3), respectively. To give some immediate implications of the above classiﬁcation consider the generalized zero ∞ of Q. If it is of the ﬁrst type, then it follows from (T1) that Q ∈ Nκ,−2π∞ . Moreover, Q has the following asymptotic expansion: 1 s2π∞ (3.15) Q(z) = − 2π∞ +1 + o , z. →∞, s2π∞ > 0. z z 2π∞ +1 If the generalized zero ∞ of Q is of type (T3), then Q ∈ Nκ,−2π∞ −2 and Q has the following asymptotic expansion 1 s2π −2 + o (3.16) Q(z) = − 2π∞ , z. →∞, s2π∞ −2 < 0. z ∞ −1 z 2π∞ −1 In the case that the generalized zero ∞ is of type (T2) there are two possibilities: either Q belongs to Nκ,−2π∞ , in which case both of the moments s2π∞ −1 and s2π∞ are ﬁnite and Q has the asymptotic expansion 1 s2π∞ s2π∞ −1 (3.17) , z. →∞, s2π∞ −1 = 0, Q(z) = − 2π∞ − 2π∞ +1 + o z z z 2π∞ +1

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or Q belongs to Nκ,−2(π∞ −1) \ Nκ,−2π∞ and it has the asymptotic expansion 1 s2π∞ −1 Q(z) = − 2π∞ + o , z. →∞, (3.18) z z 2π∞ or 1 Q(z) = o , z. →∞. (3.19) z 2π∞ −1 Observe, that the expansions (3.17) and (3.18) are also special cases of the expansion (3.19). Hence, if ∞ is a generalized zero of type (T2), then Q has an expansion of the form (3.16), but now with s2π∞ −2 = 0; however, Q does not have an expansion of the form (3.15). Similar observations remain true for generalized zeros β ∈ R and poles α ∈ R ∪ {∞}. For instance, to get the analogous expansions for a generalized zero β ∈ R apply the transform −Q(1/(z − β)) to the expansions in (3.15)–(3.19); cf. also [15]. The role of the above classiﬁcation for generalized zeros and poles of nonpositive type will be described in detail in Sections 4–6.

4. An operator model for the generalized Friedrichs extension 4.1. Boundary triplets and Weyl functions The construction of the model uses the notion of a boundary triplet in a Pontryagin space setting. Let H be a Pontryagin space with negative index κ, let S be a closed symmetric relation in H with defect numbers (n, n), and let S ∗ be the adjoint of S. A triplet Π = {Cn , Γ0 , Γ1 } is said to be a boundary triplet for S ∗ , if the following two conditions are satisﬁed: (i) the mapping Γ : f → {Γ0 f, Γ1 f} from S ∗ to Cn ⊕ Cn is surjective; (ii) the abstract Green’s identity [f , g] − [f, g ] = (Γ1 f, Γ0 g) − (Γ0 f, Γ1 g)

(4.1)

holds for all f = {f, f }, g = {g, g } ∈ S ∗ , see, e.g., [2], [10]. It is easily seen that A0 = ker Γ0 and A1 = ker Γ1 are selfadjoint extensions of S. Associated to every boundary triplet there is the Weyl function Q deﬁned by Q(z)Γ0 fz = Γ1 fz , z ∈ ρ(A0 ), z , and Nz = ker (S ∗ − z) denotes the defect subspace of where fz := {fz , zfz } ∈ N S at z ∈ C. It follows from (4.1) that the Weyl function Q is also a Q-function of the pair {S, A0 } in the sense of Kre˘ın and Langer, see [20]. If S is simple, so that H = span {Nz : z ∈ ρ(A0 )}, then the Weyl function Q belongs to the class Nκ , otherwise Q ∈ Nκ with κ ≤ κ. Moreover, if S is simple and H is a selfadjoint extension of S in H, then the point spectrum of H is also simple, that is, every eigenspace of H is one-dimensional, and if α ∈ R ∪ {∞}, then the root subspace at α is at most 2κ + 1-dimensional.

Asymptotic expansions

63

In the case where S is given by (1.3) one can deﬁne a boundary triplet for S ∗ as follows. Proposition 4.1. (cf. [5]) Let A be a selfadjoint operator in a Pontryagin space H and let the restriction S of A be deﬁned by (1.3) with ω ∈ H. Then the adjoint S ∗ of S in H is of the form S ∗ = {{f, Af + cω} : f ∈ dom A, c ∈ C } ∞ ∗ and a boundary triplet Π∞ = {C, Γ∞ 0 , Γ1 } for S is determined by

Γ∞ 0 f = [f, ω],

Γ∞ 1 f = c,

f = {f, Af + cω} ∈ S ∗ .

The corresponding Weyl function Q∞ is given by Q∞ (z) = −

1 , [(A − z)−1 ω, ω]

z ∈ ρ(A).

(4.2)

4.2. The model operator S(Q∞ ) corresponding to a proper factorization Operator models for generalized Nevanlinna functions whose only generalized pole of nonpositive type is at ∞ have been constructed in [6] and [14]. Such functions admit a canonical factorization of the form Q∞ (z) = q(z)q (z)Q0 (z),

(4.3)

where Q0 ∈ N0 , q(z) = z + qk−1 z + · · · + q0 is a polynomial, and q (z) = q(¯ z ). In general, models which are based on the canonical factorization of Q ∈ Nκ are not necessarily minimal, i.e., the underlying model operator S(Q∞ ) need not be simple and it can even be a symmetric relation (multivalued operator). However, with the canonical factorization the nonsimple part of S(Q∞ ) can be easily identiﬁed and factored out to produce a simple symmetric operator from S(Q∞ ), cf. [3]. The model constructed for S(Q∞ ) in [6] uses an orthogonal coupling of two symmetric operators. In [9] this model was adapted to the case where the function Q0 is allowed to be a generalized Nevanlinna function, too. In this case the situation becomes more involved and, in general, one cannot represent S(Q∞ ) as an orthogonal sum of a simple symmetric operator and a selfadjoint relation. However, such a simple orthogonal decomposition for S(Q∞ ) can still be obtained if the factorization (4.3) of Q∞ is proper. This concept is deﬁned as follows. k

k−1

Deﬁnition 4.2. ([9]) The factorization Q∞ (z) = q(z)q (z)Q0 (z) is said to be proper if q is a divisor of degree κ∞ (Q∞ ) > 0 of the polynomial q in the canonical factorization of the function Q∞ , cf. (2.5). Clearly, proper factorizations of Q∞ ∈ Nκ always exist, but they are not unique if q has more than one zero and κ∞ (Q∞ ) < κ. Proper factorizations Q∞ = qq Q0 can be characterized also without using the canonical factorization of Q∞ . Lemma 4.3. Let Q∞ ∈ Nκ have a factorization of the form Q∞ (z) = q(z)q (z)Q0 (z),

deg q = k ≥ 1,

(4.4)

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where q(z) is a monic polynomial, and let α ∈ σ(q) be a zero of q with multiplicity kα . Then the following statements are equivalent: (i) the factorization (4.4) of Q∞ is proper; (ii) the multiplicities κ∞ (Q∞ ) and πα (Q∞ ) satisfy the following relations: κ∞ (Q∞ ) = deg q and πα (Q∞ ) ≥ kα for all α ∈ σ(q);

(4.5)

(iii) κ∞ (Q0 ) and κ(Q∞ ) = κ satisfy the following identities κ∞ (Q0 ) = 0 and κ(Q∞ ) = deg q + κ(Q0 ).

(4.6)

Proof. (i) ⇔ (ii) In a proper factorization (4.4) κ∞ (Q∞ ) = deg q and clearly the inequalities in (4.5) just mean that q divides the polynomial q in the canonical factorization of Q∞ . (i) ⇒ (iii) If the factorization (4.4) is proper, then in the canonical factorization of the function Q0 the numerator q0 and denominator p0 (= p) of the corresponding rational factor r0 are of the same degree κ(Q0 ), and this implies (4.6). (iii) ⇒ (i) It follows from the second equality in (4.6) that q and the polynomial p0 in the canonical factorization of the function Q0 are relatively prime and, therefore, q is a factor of the polynomial q in the canonical factorization of Q∞ . Moreover, π∞ (Q0 ) = 0. Now the assumption κ∞ (Q0 ) = 0 implies that κ∞ (Q∞ ) = deg q. The construction of factorization models is now brieﬂy described. Let q be a polynomial as in (4.4) of degree k = deg q. Deﬁne the k × k matrices Bq and Cq by ⎛ ⎞ ⎞ ⎛ q1 . . . qk−1 1 0 1 ... 0 ⎜ .. ⎟ ⎟ ⎜ .. . .. .. ⎜ . 1 0⎟ .. . . . 0 ⎟ ⎜ ⎟ , Cq = ⎜ Bq = ⎜ ⎟, ⎜ .. ⎟ ⎝ 0 . 0 ... 1 ⎠ ⎝qk−1 . . . .⎠ .. −q0 −q1 . . . −qk−1 1 0 ... 0 so that σ(Cq ) = σ(q). Moreover, let Hq be a 2k-dimensional Pontryagin space deﬁned by 0 Bq . (Ck ⊕ Ck , B·, ·), B = Bq 0 A general factorization model for functions Q∞ of the form (4.4) was constructed in [9] and can be applied, in particular, for proper factorizations of Q∞ . Theorem 4.4. (cf. [9]) Let Q∞ ∈ Nκ be a generalized Nevanlinna function and let Q∞ (λ) = q(λ)q (λ)Q0 (λ),

(4.7)

be a proper factorization of Q∞ , where q is a monic polynomial of degree k = deg q ≥ 1. Let S0 be a closed symmetric relation in a Pontryagin space H0 with the boundary triplet Π0 = {C, Γ00 , Γ01 } whose Weyl function is Q0 .

Asymptotic expansions

65

Then: (i) the function Q0 in (4.7) belongs to the class Nκ−k ; (ii) the linear relation ⎧ ⎧⎛ ⎞ ⎛ ⎞⎫ ⎪ f0 ⎬ f0 = {f0 , f0 } ∈ S0∗ , ⎨ ⎨ f0 ⎠ ⎝ ⎠ ⎝ f C f : S(Q∞ ) = , f1 = Γ01 f0 , q ⎭ ⎪ 0 ⎩ ⎩ f Cq f + Γ0 f0 ek f1 = 0

⎫ ⎪ ⎬ ⎪ ⎭

(4.8)

is closed and symmetric in H := H0 ⊕ Hq and has defect numbers (1, 1); (iii) the adjoint S(Q∞ )∗ of S(Q∞ ) is given by ⎫ ⎧ ⎧⎛ ⎞ ⎛ ⎞⎫ f0 ⎬ f0 = {f0 , f0 } ∈ S0∗ , ⎬ ⎨ ⎨ f0 ⎝ f ⎠ , ⎝ Cq f + ϕe k ⎠ : S(Q∞ )∗ = ; f1 = Γ01 f0 , ⎭ ⎭ ⎩⎩ 0 f Cq f + Γ0 f0 ek ϕ ∈C (iv) a boundary triplet Π = {C, Γ0 , Γ1 } for S(Q∞ )∗ is determined by Γ0 (f0 ⊕ F ) = f1 ,

Γ1 (f0 ⊕ F) = ϕ,

f0 ⊕ F ∈ S(Q∞ )∗ ;

(4.9)

(v) the corresponding Weyl function coincides with Q∞ . Proof. Since the factorization (4.7) is proper the statement (i) is immediate from the equality (4.6) in Lemma 4.3. All the other statements are contained in [9]. In fact, the statement (iv) in Theorem 4.4 can be obtained directly also from Proposition 4.1, since S(Q∞ ) in (4.8) is a restriction of the selfadjoint relation ⎫ ⎧ ⎧⎛ ⎞ ⎛ ⎞⎫ f0 ⎬ ⎨ ⎨ f0 ∗ ⎬ ⎠ : f0 = {f0 , f0 } ∈ S0 , ⎝f ⎠,⎝ Cq f A(Q∞ ) = (4.10) ⎭ ⎭ ⎩⎩ f1 = Γ01 f0 f Cq f + Γ00 f0 ek to the subspace H ω0 , where ω0 = col(0, ek , 0); compare (1.3). The generalized Friedrichs extension of S(Q∞ ) is given by ⎧ ⎧⎛ ⎞ ⎛ ⎫ ⎞⎫ ⎪ f0 ⎨ ⎨ f0 ⎬ ⎬ f0 = {f0 , f0 } ∈ S0∗ , ⎪ ⎝ f ⎠ , ⎝ Cq f + ϕe k ⎠ : SF (Q∞ ) = . (4.11) f1 = Γ01 f0 , ⎪ ⎪ ⎭ 0 ⎩ ⎩ f ⎭ Cq f + Γ0 f0 ek f1 = 0, ϕ ∈C According to (4.2) in Proposition 4.1 the Weyl function Q∞ (z) corresponding to the boundary triplet (4.9) is of the form Q∞ (z) = −

1 . [(A(Q∞ ) − z)−1 ω0 , ω0 ]

Thus the function Q = −1/Q∞ has the representation Q(z) = [(A(Q∞ ) − z)−1 ω, ω],

(4.12)

which is, however, not necessarily minimal, since mul S and ker (S − α), α ∈ σ(q), can be nontrivial. The following lemma describes these subspaces.

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Lemma 4.5. ([9]) Under the assumptions of Theorem 4.4 let S0 be a simple closed symmetric operator in the Pontryagin space H0 and let A0i = ker Γ0i (⊃ S0 ), i = 0, 1. Then: (i) mul S(Q∞ ) is nontrivial if and only if mul A01 is nontrivial and in this case mul S(Q∞ ) = { (g, 0, Γ00 gek ) : g = {0, g} ∈ A01 };

(4.13)

(ii) if mul S(Q∞ ) is nontrivial, then it is spanned by a positive vector; (iii) if mul A00 is nontrivial, then it is spanned by a positive vector; (iv) σp (S(Q∞ )) = σp (A00 ) ∩ σ(q ) and for α ∈ σp (A00 ) ∩ σ(q ) one has ker (S(Q∞ ) − α) = { (g0 , Γ01 g0 Λ|λ=α , 0) : g0 ∈ ker (A00 − α) },

(4.14)

where Λ = (1, λ, . . . , λk−1 ), λ ∈ C; (v) if ker (S(Q∞ ) − α) or, equivalently, ker (A00 − α) is nontrivial, then it is spanned by a positive vector. It follows from (ii) and (iv) that the linear relation S(Q∞ ) can be decomposed into a direct sum of an operator S with an empty point spectrum and a selfadjoint part in a Hilbert space which is the sum of mul S(Q∞ ) and ker (S(Q∞ ) − α), α ∈ σp (A00 ) ∩ σ(q ). The next theorem shows that the reduced operator S is simple. Theorem 4.6. Let the assumptions of Theorem 4.4 and Lemma 4.5 be satisﬁed and let S(Q∞ ), A(Q∞ ), and SF (Q∞ ) be given by (4.8), (4.10), and (4.11), respectively. Then: (i) S(Q∞ ) is simple if and only if σp (S(Q∞ )) = ∅. In this case the linear relations S = S(Q∞ ), A = A(Q∞ ), and SF = SF (Q∞ ) satisfy the equalities (1.3) and (1.4) with ω = ω0 and the operator representation (4.12) of Q = −1/Q∞ is minimal. (ii) If S(Q∞ ) is not simple, then the subspace H = span { mul S(Q∞ ), ker (S(Q∞ ) − α) : α ∈ σp (A00 ) ∩ σ(q) }

(4.15)

is positive and reducing for S(Q∞ ). The simple part of S(Q∞ ) coincides with the restriction S of S(Q∞ ) to H := H H . The compressions S , A , and SF of S(Q∞ ), A(Q∞ ), and SF (Q∞ ) to the subspace H satisfy the equalities (1.3) and (1.4), with ω ∈ H given by ω0 , if k > 1, ω= (g, −1/Γ0 g, Γ0 g) , if k = 1, and the function Q = −1/Q∞ admits the minimal representation Q(λ) = −1/Q∞ (λ) = [(A − λ)−1 ω, ω].

Asymptotic expansions

67

4.3. The root subspace of the generalized Friedrichs extension at ∞ Let the assumptions of Theorem 4.6 be satisﬁed and let the linear relations S(Q∞ ), A(Q∞ ), SF (Q∞ ) and S , A , SF be as deﬁned in that theorem. Since S is a simple symmetric operator in the Pontryagin space H its selfadjoint extension SF has a simple point spectrum. In particular, the multivalued part mul SF of SF is at most one-dimensional. The corresponding root subspace R∞ (SF ) = span { g ∈ H : {0, g} ∈ SF , for some k ∈ N } k

is at most 2κ+1-dimensional. The root subspace R∞ (SF ) is spanned by the vectors ωj which form a maximal Jordan chain in R∞ (SF ): R∞ (SF ) = span { ωj ∈ H : {ωj−1 , ωj } ∈ SF ,

j = 0, . . . , ν − 1 },

dim R∞ (SF )

where ν = and ω−1 = 0. The main properties of the root subspace R∞ (SF ) of the selfadjoint extension SF of S are given in Lemmas 4.7–4.11 below. The proofs of these lemmas are constructive, since they are based on the factorization model for S(Q∞ ) given in Theorem 4.4, cf. also [9]. As a consequence a detailed description for the structure of the corresponding Jordan chains in the factorization model is obtained. It turns out that three diﬀerent types of maximal Jordan chains can appear in this model. In each of these cases maximal Jordan chains in R∞ (SF ) can be regular or singular and the signature of R∞ (SF ) has its own speciﬁc nature in each case. The ﬁrst lemma concerns the dimension and nondegeneracy of the root subspace R∞ (SF ); further equivalent conditions and descriptions (without a speciﬁc model space) for arbitrary generalized poles and zeros of Q ∈ Nκ with direct (nonconstructive) proofs can be found from [15]. Lemma 4.7. Let Q∞ ∈ Nκ , let κ∞ (Q∞ ) = deg q(= k) > 0, and let S , SF , and ω ∈ H, [ω, ω] ≤ 0, be as in Theorem 4.6. Then: (i) dim R∞ (SF ) ≥ deg q; ν−1 ν (ii) dim R∞ (SF ) is equal to ν if and only if ω ∈ dom SF \ dom SF ; (iii) R∞ (SF ) is a regular subspace of dimension ν if and only if ω ∈ dom SF (iv)

R∞ (SF )

ν−1

\ dom SF

ν

and [ω, ων−1 ] = 0 for {ω, ων−1 } ∈ SF

ν−1

;

is a singular subspace of dimension ν if and only if

ω ∈ dom SF

ν−1

\ dom SF

ν

and [ω, ων−1 ] = 0 for {ω, ων−1 } ∈ SF

ν−1

.

(4.16)

Proof. (i) Consider a proper factorization (4.7) for Q∞ and the model operator S(Q∞ ) constructed in Theorem 4.4 and denote by A0 = ker Γ0 the selfadjoint extension SF (Q∞ ) in (4.11) of S(Q∞ ) in (4.8). Clearly, the vectors ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 0 w0 = ⎝ek ⎠ , . . . , wk−2 = ⎝e2 ⎠ , wk−1 = ⎝e1 ⎠ (4.17) 0 0 0 form a Jordan chain in R∞ (A0 ), that is, w0 ∈ mul A0 and {wj−1 , wj } ∈ A0 for j = 1, . . . , k − 1. If S(Q∞ ) is simple then A0 = SF and this proves (i). In the case

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when S(Q∞ ) is not simple, but mul S(Q∞ ) is trivial the vectors w0 , . . . , wk−1 still belong to R∞ (SF ) since for all α ∈ σp (A00 ) ∩ σ(q ) these vectors are orthogonal to the eigenspaces ker (S(Q∞ ) − α) which were described in (4.14). Assume that mul S(Q∞ ) is not trivial. Again it follows from Lemma 4.5 that the vectors w0 , . . . , wk−1 are orthogonal to ker (S(Q∞ ) − α) for all α ∈ σp (A00 ) ∩ σ(q ). Moreover, by using (4.13) in Lemma 4.5 it is seen that the vectors w0 , . . . , wk−2 are orthogonal to mul S(Q∞ ). Since mul S(Q∞ )[⊥]ker (S(Q∞ ) − α) for all α ∈ σp (A00 ) ∩ σ(q ), it is easy to check that the projections wj = P wj of wj , j = 1, . . . , k − 1, to the subspace H = H H , where H is given by (4.15), take the form ⎞ ⎛ g)g −(Γ00 ⎠, w0 = w0 , . . . , wk−2 = wk−2 , wk−1 =⎝ (4.18) e1 0 2 −|Γ0 g| ek where g ∈ mul A01 , [g, g] = 1, g = {0, g} ∈ A01 , and Γ00 g = 0. Therefore, the sequence {w0 , . . . , wk−1 } forms a Jordan chain in R∞ (SF ) and hence again dim R∞ (SF ) ≥ k. (ii) This statement holds true since SF has a simple spectrum. (iii)&(iv) Let ω0 , . . . , ων−1 be a maximal chain in R∞ (SF ). Since {0, ω0 } ∈ SF one obtains immediately [ω0 , ωj ] = 0 for all j < ν − 1. Therefore, the subspace R∞ (SF ) is regular if and only if [ω0 , ων−1 ] = 0. This proves (iii) and (iv). The explicit construction of a maximal Jordan chain {ω0 , . . . , ων−1 } spanning R∞ (SF ) in the factorization model will be carried out by a continuation of the chains (4.17) and (4.18), which correspond to the cases mul S(Q∞ ) = {0} and mul S(Q∞ ) = {0}, respectively. Observe, that by Lemma 4.5 mul S(Q∞ ) is nontrivial if and only if mul A01 is nontrivial. Moreover, since S0 is simple at most one of the selfadjoint extensions A00 or A01 of S0 can be multivalued. Case I: mul S(Q∞ ) = {0}. The proof of Lemma 4.7 shows that {ω0 , . . . , ων−1 } can be constructed as a continuation of the chain (4.17) if ν > k with ωj = wj , j = 0, . . . , k − 1. The formula (4.11) implies that the vector ωk should be of the form ωk = (g0 , f, f) , where g0 = {0, g0 } ∈ S0∗ ,

Γ01 g0 = 1,

f = Cq e1 + ϕe k,

By choosing ϕe k = −Cq e1 one obtains ⎞ ⎛ g0 ωk = ⎝ 0 ⎠ , g0 = {0, g0} ∈ S0∗ , Γ00 g0 ek

f = Γ00 g0 ek .

Γ01 g0 = 1.

(4.19)

Here g0 = 0 if and only if ωk = 0. Therefore, one can continue the chain (4.17) if and only if mul S0∗ is nontrivial. This proves the following Lemma 4.8. If mul S0∗ = {0}, then R∞ (SF ) is isotropic and dim R∞ (SF ) = κ0 (R∞ (SF )) = k.

Asymptotic expansions

69

Now assume that mul S0∗ is nontrivial, so that one can continue the chain (4.17) by a vector ωk = 0 of the form (4.19). Here two further diﬀerent cases can occur: either mul A00 is nontrivial, in which case mul A00 = mul S0∗ , or mul A00 = {0}. These two cases give rise to two diﬀerent types of maximal Jordan chains in R∞ (SF ). Lemma 4.9. Let mul A00 be nontrivial. Then R∞ (SF ) is regular if and only if ν = 2k + 1. If R∞ (SF ) is singular then k + 1 ≤ ν ≤ 2k and κ0 (R∞ (SF )) = 2k + 1 − ν,

κ− (R∞ (SF )) = ν − k − 1,

κ+ (R∞ (SF )) = ν − k. (4.20)

Proof. Since mul A00 = {0} one has Γ00 g0 = 0. It follows from (4.11) and (4.19) that the chain {ωk , . . . , ων−1 } can be taken to be of the form ⎞ ⎞ ⎛ ⎞ ⎛ ⎛ g1 gν−k−1 g0 ⎠, 0 ωk = ⎝ 0 ⎠ , ωk+1 = ⎝ 0 ⎠ , . . . , ων−1 = ⎝ (4.21) 0 f(1) f(ν−k−1) with {gj−1 , gj } ∈ A01 and ek−j+1 + f(j) = a

j−2 -

cj,i ek−i ,

cj,i ∈ C,

j = 1, . . . , ν − k − 1,

i=0

where a := Γ00 {g0 , g1 } = [ωk−1 , ωk+1 ] = [ωk , ωk ] = [g0 , g0 ] > 0, since the vector g0 ∈ mul A00 is positive in view of Lemma 4.5. There are two reasons for the chain to break: either gν−k−1 ∈ dom A01 , or one has (f(ν−k−1) )1 = 0 in which case ν = 2k + 1. Since [ων−1 , ω0 ] = (fν−k−1 )1 , the subspace R∞ (A0 ) is regular if and only if ν = 2k + 1. Observe, that the maximal chain {ω0 , . . . , ων−1 } in R∞ (SF ) is also a maximal chain in R∞ (SF ), because the vectors ωj ∈ R∞ (SF ), j = 0, . . . , ν − 1, are orthogonal to the eigenspaces ker (S(Q∞ ) − α) for α ∈ σp (A00 ) ∩ σ(q). If R∞ (SF ) is singular then k + 1 ≤ ν ≤ 2k. The isotropic subspace of R∞ (SF ) is spanned by the vectors ωj , j = 0, . . . , 2k − ν, and [ω2k−ν+1 , ων−1 ] = [ωk , ωk ] = [g0 , g0 ] = a > 0. Therefore, the Gram matrix of the chain ω0 , . . . , ων−1 is a Hankel matrix of the form ⎛ ⎞ 0 a 0 0 . ⎠, G= with G0 = ⎝ (4.22) .. 0 G0 a ∗ where the left upper corner of G is the matrix 02k+1−ν . The other two equalities in (4.20) are implied by the structure of G0 in (4.22), since a > 0.

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{0}. Then R∞ (SF ) is Lemma 4.10. Let mul A00 = mul A01 = {0} and let mul S0∗ = regular if and only if ν = 2k. If R∞ (SF ) is singular then k + 1 ≤ ν ≤ 2k − 1 and κ0 (R∞ (SF )) = 2k − ν,

κ− (R∞ (SF )) = κ+ (R∞ (SF )) = ν − k.

(4.23)

Proof. By assumptions Γ00 g0 = 0. In this case the chain {ωk , . . . , ων−1 } takes the form ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ g1 gν−k−1 g0 ⎠, 0 ωk = ⎝ 0 ⎠ , ωk+1 = ⎝ 0 ⎠ , . . . , ων−1 = ⎝ (4.24) a ek f(1) f(ν−k−1) where a := Γ00 g0 = 0, {gj−1 , gj } ∈ A01 , and f(j) are given by ek−j + f(j) = a

j−1 -

cj,i ek−i ,

cj,i ∈ C,

j = 1, . . . , ν − k − 1.

(4.25)

i=0

As in Lemma 4.9 it is seen from (4.24) that the subspace R∞ (SF ) is regular if and only if ν = 2k. If R∞ (SF ) is singular then k + 1 ≤ ν ≤ 2k − 1. The isotropic subspace of R∞ (SF ) is spanned by the vectors ωj , j = 0, . . . , 2k − 1 − ν, and [ω2k−ν , ων−1 ] = [ωk , ωk−1 ] = a = 0. Therefore, the Gram matrix of the chain ω0 , . . . , ων−1 takes the form (4.22), where the left upper corner of G is the matrix 02k−ν . The equalities in (4.23) are implied by the structure of G. Case II: mul S(Q∞ ) = {0}. The proof of Lemma 4.7 shows that a maximal Jordan chain {ω0 , . . . , ων−1 } in R∞ (SF ) can now be constructed as a continuation of the chain (4.18) with ωj = wj , j = 0, . . . , k − 1. Lemma 4.11. Let mul S(Q∞ ) = {0}. Then R∞ (SF ) is regular if and only if ν = 2k − 1. If R∞ (SF ) is singular then 1 < k ≤ ν ≤ 2k − 2 and κ0 (R∞ (SF )) = 2k − 1 − ν,

κ− (R∞ (SF )) = ν − k + 1,

κ+ (R∞ (SF )) = ν − k. (4.26)

Proof. If k = 1 then ω0 = (−(Γ00 g)g, 1, −|Γ00 g|2 ) and it follows from (4.11) that the chain (4.18) cannot be continued, in which case ν = k = 1 and [ω0 , ω0 ] = g|2 < 0. −|Γ00 Now let k > 1 and consider the continuation of the chain (4.18). According to (4.11) the condition {ωk−1 , ωk } ∈ A0 for some ωk means that for some vector g1 one has h := {−(Γ00 g)g, g1 } ∈ S0∗ , Γ01 h = 1, (4.27) where g = {0, g} ∈ A01 , [g, g] = 1, Γ00 g = 0. Observe, that here the conditions 0 Γ1 h = 1 and [g, g] = 1 are equivalent, since by Green’s identity (4.1) −Γ00 g[g, g] = Γ01 gΓ00 h − Γ00 gΓ01 h = −Γ00 gΓ01 h.

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71

Observe also, that g ∈ dom A01 , since g ∈ mul A01 and the vector g is positive. Moreover, it follows from (4.11) that the continuation of the chain (4.18) can be taken to be of the form ⎞ ⎞ ⎛ ⎛ g1 gν−k ωk = ⎝ 0 ⎠ , . . . , ων−1 = ⎝ 0 ⎠ , (4.28) f(1) f(ν−k) where {gj−1 , gj } ∈ A01 for j = 2, . . . , ν − k and f(j) , j = 1, . . . , ν − k, are given by (4.25), where a := −|Γ00 g|2 = [ωk−1 , ωk−1 ] < 0. By (4.13) the vector ωj is orthogonal to mul S(Q∞ ) for all j = k, . . . , ν − 2. In addition, one can select gν−k in (4.28) such that [gν−k , g] = 0 and then also ων−1 is orthogonal to mul S(Q∞ ). Moreover, the vectors ωj , j = k, . . . , ν − 1, are orthogonal to the eigenspaces ker (S(Q∞ ) − α), α ∈ σp (A00 ) ∩ σ(q). Hence (4.18) together with (4.28) forms a maximal Jordan chain in R∞ (SF ). Moreover, the subspace R∞ (SF ) is regular if and only if ν = 2k − 1. If R∞ (SF ) is singular then k > 1 and k ≤ ν ≤ 2k −2, see (4.27). The isotropic subspace of R∞ (SF ) is spanned by the vectors ωj , j = 0, . . . , 2k − ν − 2, and [ω2k−ν−1 , ων−1 ] = [ωk−1 , ωk−1 ] = a < 0. Therefore, the Gram matrix of the chain ω0 , . . . , ων−1 is a Hankel matrix of the form (4.22), where the left upper corner of G is the matrix 02k−1−ν and a < 0. This proves the equalities (4.26). The above considerations show that three diﬀerent types of maximal Jordan chains in R∞ (SF ) can occur, cf. (4.21), (4.24), and (4.28). The longest Jordan chains appear in the ﬁrst case, where mul A00 = {0}, see (4.21), while the shortest Jordan chains appear in the third case, where mul A01 = {0}, or equivalently, mul S(Q∞ ) = {0}, see (4.28). The main properties associated with each of these maximal Jordan chains in Lemmas 4.8–4.11 are collected in the next theorem. Theorem 4.12. Let Q∞ ∈ Nκ , κ∞ (Q∞ ) = k > 0, ν = dim (R∞ (SF )), and let S , SF be as in Theorem 4.6. Then one of the following three cases occurs: (i) If mul A00 = {0}, then k + 1 ≤ ν ≤ 2k + 1, κ− (R∞ (SF )) = κ+ (R∞ (SF )) − 1, and κ0 (R∞ (SF )) = 2k + 1 − ν. Moreover, R∞ (SF ) is singular if and only if ν ≤ 2k. (ii) If mul A00 = mul A01 = {0}, then k ≤ ν ≤ 2k, κ− (R∞ (SF )) = κ+ (R∞ (SF )), and κ0 (R∞ (SF )) = 2k − ν. Moreover, R∞ (SF ) is singular if and only if ν ≤ 2k − 1. (iii) If mul A01 = {0}, then k ≤ ν ≤ 2k − 1, κ− (R∞ (SF )) = κ+ (R∞ (SF )) + 1, and κ0 (R∞ (SF )) = 2k − 1 − ν. Moreover, R∞ (SF ) is singular if and only if ν ≤ 2k − 2 and k > 1. Proof. It is enough to prove the statement (ii). For this one combines the results in Lemma 4.8 and Lemma 4.10. Indeed, if mul S0∗ = {0} then by Lemma 4.8 one has ν = dim R∞ (SF ) = k ≤ 2k − 1 and κ0 (R∞ (SF )) = k = 2k − ν.

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5. Analytic characterizations of the root subspace at ∞ Let Q∞ ∈ Nκ have a minimal operator representation (1.2) with γ = 0, where A is a selfadjoint operator in H, and let S and SF be deﬁned by (1.3) and (1.4), respectively. In Section 4 one such minimal operator representation was constructed in Theorem 4.6 by using the factorization (4.2) of Q∞ in Theorem 4.4. By unitary equivalence general statements concerning S, A, and SF in a minimal representation of the function Q∞ can be obtained by considering the corresponding objects S , A , and SF in the model of Theorem 4.6. In this section the results concerning the root subspace of the generalized Friedrichs extension SF in Section 4 are connected with the asymptotic expansions in Section 3. In particular, the connection between the classiﬁcation of generalized poles and zeros introduced in Subsection 3.2 and the three diﬀerent types of maximal Jordan chains in Section 4 is explained. 5.1. The root subspace of the generalized Friedrichs extension at ∞ and operator representations In order to establish the connection between the Jordan chains in the root subspace R∞ (SF ) of the generalized Friedrichs extension SF and the asymptotic expansions in Section 3 the following lemma will be useful. Lemma 5.1. Let A be a selfadjoint operator in a Pontryagin H, let ω ∈ H, and let S and SF be deﬁned by (1.3) and (1.4). Then: (i) dom S n = { f ∈ dom An : [Aj f, ω] = 0 for all j < n }, n ∈ N; (ii) ω ∈ dom S n if and only if ω ∈ dom SFn , n ∈ N; (iii) if ω0 , ω1 , . . . , ωn with ω0 = ω is a Jordan chain in R∞ (SF ) such that [ωi , ωj ] = 0

for all

i + j < l ≤ 2n,

l ∈ N,

(5.1)

i + j ≤ l ≤ 2n.

(5.2)

then [S i ω, S j ω] = [ωi , ωj ]

for all

Proof. (i) If f ∈ dom S n , then the deﬁnition of S in (1.3) shows that f ∈ dom An and [Aj f, ω] = 0 for all j < n, since Aj f ∈ dom S for all j < n and ω is orthogonal to dom S. Conversely, assume that f ∈ dom An and [Aj f, ω] = 0 for all j < n. Then the deﬁnition (1.3) shows that Aj f ∈ dom S for all j < n. Hence f ∈ dom S n . (ii) Note that the condition ω ∈ dom SFn means that there is a chain of vectors ω0 , ω1 , . . . , ωn , ω0 = ω, such that {ωj−1 , ωj } ∈ SF for j ≤ n. According to (1.4) this is equivalent to ωj = Sωj−1 + cj ω

for some cj ∈ C,

j = 1, . . . , n.

(5.3)

Hence, if ω ∈ dom SFn , then ωj ∈ dom S and ωj−1 ∈ dom S 2 for all j < n, and this leads to ω ∈ dom S n . The reverse implication is also clear from (5.3).

Asymptotic expansions

73

(iii) It follows from (5.1) and (5.3) with l ≥ 1 that [ω0 , ω1 ] = [ω0 , Sω0 ]. Now, if [S i ω0 , S j ω0 ] = [ωi , ωj ] = 0 for all i+j < m ≤ l, then one obtains from (5.3) that [ωi+1 , ωj ] = [S i+1 ω0 +

i -

ci+1−α S α ω0 , S j ω0 +

α=0

= [S

i+1

j−1 -

cj−β S β ω0 ]

β=0

j

ω0 , S ω0 ],

which completes the proof.

The statements (iii) and (iv) in Lemma 4.7 can now be reformulated in terms of the moments sj = [Aj ω, ω] of the operator A, cf. also [15]. Proposition 5.2. Let S, SF , ω ∈ H, and [ω, ω] ≤ 0, be as in (1.3) and (1.4), in a minimal representation of Q∞ ∈ Nκ with κ∞ (Q∞ ) = k > 0. Then: (i) R∞ (SF ) is a regular subspace of dimension ν if and only if ω ∈ dom Aν−1 , and sν−1 = 0, sj = 0 for all j < ν − 1;

(5.4)

(ii) R∞ (SF ) is a singular subspace of dimension ν if and only if ω ∈ dom Aν−1 \ dom Aν , and sj = 0 for all j ≤ ν − 1.

(5.5)

Proof. (i) Assume that R∞ (SF ) is a regular subspace of dimension ν. Then by Lemma 4.7 ω ∈ dom SFν−1 \dom SFν and by Lemma 5.1 ω ∈ dom S ν−1 ⊂ dom Aν−1 . This implies that sj = [Aj ω, ω] = [S j ω, ω] = 0 for j < ν − 1. Moreover, if {ω, ων−1 } ∈ SFν−1 then sν−1 = [Aν−1 ω, ω] = [S ν−1 ω, ω] = [ων−1 , ω] = 0, so that (5.4) follows. Conversely, if (5.4) holds, then ω ∈ dom SFν−1 by Lemma 5.1. Moreover, for {ω, ων−1 } ∈ SFν−1 one has [ων−1 , ω] = [Aν−1 ω, ω] = sν−1 = 0, so that ω ∈ dom SFν . Hence, R∞ (SF ) is a regular subspace of dimension ν. (ii) Assume that R∞ (SF ) is a singular subspace of dimension ν. Then it follows from Lemma 4.7 and Lemma 5.1 that ω ∈ dom S ν−1 ⊂ dom Aν−1 and [Aj ω, ω] = [S j ω, ω] = 0 for all j ≤ ν − 1. If ω ∈ dom Aν , then by Lemma 5.1 ω ∈ dom SFν , a contradiction to (4.16). Thus, ω ∈ dom Aν and (5.5) follows. Conversely, assume that (5.5) holds. Then one has ω ∈ dom SFν−1 \ dom SFν by Lemma 5.1. Moreover, by (5.2) one has [ων−1 , ω] = [Aν−1 ω, ω] = sν−1 = 0. Hence, (4.16) holds and R∞ (SF ) is a singular subspace of dimension ν.

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5.2. Asymptotic expansions and the classiﬁcation for the generalized zero (pole) ∞ in the model space The classiﬁcation of generalized poles of Q∞ or, equivalently, of the generalized zeros of Q = −1/Q∞ in Subsection 3.2 is now connected with the maximal Jordan chains constructed in Section 4. For this purpose observe that the assumptions of Theorem 4.12 can be expressed in terms of the Weyl function Q0 (λ) of S0 in the following equivalent form: mul A00 = {0}

⇔

mul A01 = {0}

⇔

Q0 (z) = 0, z. →∞ z lim zQ0 (z) = ∞. lim

z. →∞

(5.6) (5.7)

Now one can reformulate Theorem 4.12 in the form which makes clear the connection with the classiﬁcation of generalized zeros and poles of nonpositive type introduced in Section 3. Theorem 5.3. Let Q ∈ Nκ have a minimal representation (1.2) and let S and SF be deﬁned by (1.3) and (1.4). Let ∞ be a generalized zero of negative type of Q with multiplicity π∞ (Q) = k > 0 and let the root subspace R∞ (SF ) be of dimension ν. Then R∞ (SF ) is regular if and only if Q has an asymptotic expansion of the form 1 sν−1 s2ν−2 Q(z) = − ν − · · · − 2ν−1 + o (5.8) , z. →∞, sν−1 = 0. z z z 2ν−1 Moreover, precisely one of the following three cases occurs: (i) If ∞ is a GZNT1, then k+1 ≤ ν ≤ 2k+1 and Q has the asymptotic expansion s2k s2ν−2 1 Q(z) = − 2k+1 − · · · − 2ν−1 + o , z. →∞, (5.9) z z z 2ν−1 where s2k > 0. In this case κ− (R∞ (SF )) = κ+ (R∞ (SF )) − 1, κ0 (R∞ (SF )) = 2k + 1 − ν, and R∞ (SF ) is singular if and only if ν ≤ 2k. (ii) If ∞ is a GZNT2, then k ≤ ν ≤ 2k and for ν ≥ k + 1 Q has the asymptotic expansion 1 s2ν−2 s2k−1 Q(z) = − 2k − · · · − 2ν−1 + o , z. →∞, (5.10) z z z 2ν−1 where s2k−1 = 0, and for ν = k Q has the asymptotic expansion 1 Q(z) = o , z. →∞. z 2ν−1

(5.11)

In this case κ− (R∞ (SF )) = κ+ (R∞ (SF )), κ0 (R∞ (SF )) = 2k−ν, and R∞ (SF ) is singular if and only if ν ≤ 2k − 1. (iii) If ∞ is a GZNT3, then k ≤ ν ≤ 2k − 1 and Q has the asymptotic expansion 1 s2ν−2 s2k−2 Q(z) = − 2k−1 − · · · − 2ν−1 + o , z. →∞, (5.12) z z z 2ν−1

Asymptotic expansions

75

where s2k−2 < 0. In this case κ− (R∞ (SF )) = κ+ (R∞ (SF ))+1, κ0 (R∞ (SF )) = 2k − 1 − ν, and R∞ (SF ) is singular if and only if ν ≤ 2k − 2 and k > 1. Proof. Since the root subspace R∞ (SF ) is of dimension ν it follows from Proposition 5.2 that ω ∈ dom Aν−1 . By Theorem 3.3 this means that Q ∈ Nκ,−2(ν−1) . Now Corollary 3.4 and (5.4) show that in the regular case the asymptotic expansion is of the form (5.8). If the root subspace R∞ (SF ) is singular then Proposition 5.2 and Theorem 3.3 yield ω ∈ dom Aν−1 \ dom Aν and Q ∈ Nκ,−2(ν−1) \ Nκ,−2ν . Now consider the classiﬁcation given in Subsection 3.1. (i) Assume that ∞ is a GZNT1 of Q. Then by (T1) the following limit exists: lim z 2k+1 Q(z) < 0.

z. →∞

It follows from the factorization (4.7) that Q0 (z) Q∞ (z) 1 = lim 2k+1 = − lim 2k+1 > 0. z z z Q(z) z. →∞ z. →∞ z. →∞ lim

According to (5.6) this means that mul A00 = {0}. The asymptotic expansion (5.9) is implied by (3.15) and Corollary 3.4. The remaining statements are obtained from part (i) of Theorem 4.12. (ii) Assume that ∞ is a GZNT2 of Q. Then by (T2) one has lim z 2k+1 Q(z) = ∞,

lim z 2k−1 Q(z) = 0.

z. →∞

z. →∞

It follows from the factorization (4.7) that lim

z. →∞

Q0 (z) 1 = − lim 2k+1 = 0, z z. →∞ z Q(z)

lim zQ0 (z) = − lim

z. →∞

z. →∞

1 = ∞. z 2k−1 Q(z)

According to (5.6) and (5.7) this means that mul A00 = mul A01 = {0}. The asymptotic expansion (5.10) is implied by (3.17), (3.18), and Corollary 3.4, while the expansion (5.11) is obtained from (3.18), (3.19). The remaining statements are obtained from part (ii) of Theorem 4.12. (iii) Finally, assume that ∞ is a GZNT3 of Q. Then by (T3) the following limit exists: lim z 2k−1 Q(z) > 0. z. →∞

It follows from (4.7) that lim zQ0 (z) = − lim

z. →∞

1

z. →∞ z 2k−1 Q(z)

< 0,

and in view of (5.7) this means that mul A01 = {0}. The asymptotic expansion (5.12) is implied by (3.16) and Corollary 3.4, and the remaining statements are obtained from part (iii) of Theorem 4.12.

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The characterizations in Theorem 5.3 can be translated also for the generalized poles of nonpositive type of the function Q∞ by means of the following theorem. In fact, this result is an extension of [6, Theorem 5.2] and can be seen to augment also the result stated in Theorem 3.3. Theorem 5.4. Let Q ∈ Nκ , Q = 0, with limz. →∞ Q(z) = 0 and let Q∞ = −1/Q. Then the following statements are equivalent: (i) Q ∈ Nκ,−2n \ Nκ,−2n−2 and m ≥ 0 is the maximal integer such that sj = 0 for all j ≤ m − 1(≤ 2n); (ii) Q(z) = [(A − z)−1 ω, ω], z ∈ ρ(A), for some selfadjoint operator A in a Pontryagin space H and a cyclic vector ω ∈ dom An \ dom An+1 satisfying [Aj ω, Ai ω] = 0 for all i + j ≤ m − 1(≤ 2n), i, j ≤ n; (iii) Q has an asymptotic expansion of the form 1 s2n sm Q(z) = − m+1 − · · · − 2n+1 + o , z. →∞, (5.13) z z z 2n+1 where m(≤ 2n + 1) and n are maximal nonnegative integers, such that (5.13) holds; (iv) Q∞ = −1/Q has an asymptotic expansion of the form Q∞ (z) = pm+1 z m+1 + · · · + p2+1 z 2+1 + o z 2+1 , z. →∞, (5.14) where pm+1 = 0 if m ≥ 2 and ∈ Z (with 2 ≤ m + 1) is minimal such that (5.14) holds. In this case the integers m, n ≥ 0 and are connected by = m − n. Moreover, in (5.13) sm = 0 if and only if pm+1 = 0 in (5.14), in which case pm+1 = 1/sm and m ≤ 2n or, equivalently, 2 ≤ m. Proof. (i) ⇔ (ii) This equivalence follows from Theorem 3.3 and the formulas (3.10) for the moments sj , j ≤ 2n. (i) ⇔ (iii) If (i) holds then by Corollary 3.4 Q has an asymptotic expansion of the form (5.13) and maximality of n in this expansion follows from the assumption Q ∈ Nκ,−2n+2 . The converse statement is also clear. (iii) ⇔ (iv) Assume that Q satisﬁes (5.13). If m ≤ 2n then sm = 0. Otherwise m = 2n + 1 and the expansion (5.13) reduces to 1 Q(z) = o , z. →∞. (5.15) z 2n+1 In the case that m ≤ 2n, sm = 0 and the expansion (5.13) can be rewritten in the form s2n 1 m+1 , z. →∞. (5.16) Q(z) = −sm − · · · − 2n−m + o z z z 2n−m Since Q∞ = −1/Q, by inverting the expansion (5.16) one concludes that the expansion (5.13) for Q with sm = 0 is equivalent for Q∞ to admit an expansion

Asymptotic expansions

77

of the form

Q∞ (z) = pm+1 z m+1 + · · · + p2(m−n)+1 z 2(m−n)+1 + o z 2(m−n)+1 ,

z. →∞,

where pm+1 = 1/sm = 0. This means that Q∞ has an asymptotic expansion of the form (5.14), where the integer = m − n, 2 < m + 1, is minimal if n is maximal, and conversely. Hence the equivalence of (iii) and (iv) is shown in the case that m ≤ 2n. Next consider the case that m = 2n + 1. Then Q satisﬁes (5.15) and this is an expansion of the form (5.13) with the maximal integers n ≥ 0 and m = 2n + 1. Since Q∞ = −1/Q, one concludes that Q∞ (z) = o(z 2n+3 ),

z. →∞,

so that Q∞ has an expansion of the form (5.14) with := n+1 > 0 and m+1 = 2. Moreover, here = n+1 is the minimal integer, such that Q∞ admits an expansion of the form (5.14). Observe, that since = n + 1 is minimal, the maximal m in (5.13) is equal to m = 2n + 1, so that the equality = m − n holds also in this case. In particular, (5.13) and (5.14) are still equivalent if m = 2n + 1, in which case one can take m + 1 = 2. To establish the classiﬁcation of the asymptotic expansions for the function Q∞ along the lines of Theorem 5.3 it is enough to consider expansions of the form (5.14) with = m − n ≥ 0 (so that in (5.13) n ≤ m) and then apply the last statement of Theorem 5.3. In this case (5.14) takes the form Q∞ (z) = P (z) + o z 2+1 , z. →∞, where P (z) := pm+1 z m+1 + · · · + p2+1 z 2+1 ,

≥ 0,

is a real polynomial of degree deg P = m + 1, whose leading coeﬃcient is given by pm+1 = 1/sm if deg P > 2. If m + 1 = 2, then one can take P = 0. 5.3. Asymptotic expansions and the index of singularity As another consequence of Theorem 5.4 some characterizations for the index of singularity of the generalized pole (zero) of nonpositive type of Q∞ (of Q = −1/Q∞) at ∞ are given. The motivation for this notion is given in the end of this section. Deﬁnition 5.5. Let ∞ be a generalized pole of Q∞ (zero of Q) of nonpositive type of order ν (= dim R∞ (SF )) and let Hν = (si+j )ν−1 i,j=0 be the ν × ν Hankel matrix which is determined by the ﬁnite moments sj , 0 ≤ j ≤ 2ν − 2, of Q. Then κ0∞ = dim (ker Hν ) is called the index of singularity of Q∞ at ∞. Theorem 5.6. Let n and m be maximal nonnegative integers with n ≤ m, such that Q ∈ Nκ,−2n and sj = 0 for all j ≤ m − 1(≤ 2n), and let S, A, SF , ω be from the minimal operator representation (1.2) of Q.

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Then the following assertions are equivalent: (i) ∞ is a generalized zero of nonpositive type of Q with the index of singularity equal to κ0∞ ; (ii) κ0∞ = m − n(≥ 0); (iii) Q∞ = −1/Q admits an asymptotic expansion of the form 0

Q∞ (z) = Pn+1 (z) + o(z 2κ∞ +1 ),

z. →∞,

(5.17)

where Pn+1 is a polynomial of degree n+1 ≥ and ≥ 0 is the minimal integer such that (5.17) holds; (iv) κ0∞ is equal to the dimension of the isotropic subspace of R∞ (SF ). 2κ0∞

κ0∞

Proof. (i) ⇔ (ii) It follows from Theorem 5.3 and the maximality of n and m, n ≤ m, that n = ν − 1, where ν = dim R∞ (SF ), cf. [15, Theorem 5.2]. Since clearly dim (ker Hν ) = m − n(≥ 0) the equivalence of (i) and (ii) is shown. (ii) ⇔ (iii) This follows immediately from Theorem 5.4 with = m − n = κ0∞ ≥ 0. Here minimality of κ0∞ ≥ 0 in (5.17) is equivalent to the maximality of n (= m − κ0∞ ) in (5.13). (ii) ⇔ (iv) To prove this equivalence the indices m and n are calculated in each of the cases (i)–(iii) in Theorem 5.3. In the case (i) one obtains from (5.9) m = 2k, n = ν − 1. Hence κ0∞ = m − n = 2k − ν + 1 = κ0 (R∞ (SF )). Similarly in the case (ii) for both expansions (5.10) and (5.11) one has m = 2k − 1, n = ν − 1. Thus κ0∞ = m − n = 2k − ν = κ0 (R∞ (SF )). Finally in the case (iii) m = 2k − 2, n = ν − 1, and κ0∞ = m − n = 2k − 1 − ν = κ0 (R∞ (SF )).

This completes the proof.

The equivalence of (i) and (iv) in Theorem 5.6 is also a direct consequence of [15, Corollary 4.4]. From Theorem 5.6 one obtains immediately the following characterization for the regularity of a critical point. Corollary 5.7. ([6, Theorem 4.1], [17, Proposition 1.6]) The root subspace R∞ (SF ) is nondegenerate (equivalently ∞ is a regular critical point of SF ) if and only if Q∞ admits the representation Q∞ (z) = P (z) + Q0 (z),

where Q0 (z) = o(z),

z. →∞.

Proof. This is immediate from the equivalence of (iii) and (iv) in Theorem 5.6.

The index κ0∞ measures the degree of singularity of the singular critical point ∞ of SF . The characterization of κ0∞ via the asymptotic expansion (5.17) in Theorem 5.6 is particularly appealing: it extends the result stated in Corollary 5.7 to the case of singular critical points in an explicit manner.

Asymptotic expansions

79

6. Spectral characterizations via the underlying Weyl functions In this section the structure of the underlying root subspace corresponding to the three diﬀerent types of maximal Jordan chains constructed in Section 4 is studied by means of the factorized integral representations of the underlying Weyl functions. First detailed results are presented for the point ∞. Then the case of ﬁnite generalized zeros and poles of nonpositive type is treated brieﬂy. Furthermore, it is shown how the classiﬁcation of all generalized zeros and poles of Q ∈ Nκ belonging to R ∪ {∞} can be applied in establishing analytic criteria for the minimality of (not necessarily canonical) factorization models of Nκ -functions. 6.1. The classiﬁcation of generalized zeros and poles at ∞ The canonical factorization of generalized Nevanlinna functions in (1.6) implies that Q(z) = −1/Q∞ (z) has the following integral representation: 1 t p(z) p (z) − Q(z) = −1/Q∞ (z) = a + bz + dρ(t) , (6.1) q(z) q (z) t − z t2 + 1 R where p and q are as in (2.5), a ∈ R, b ≥ 0, and ρ(t) satisﬁes dρ(t) < ∞. (6.2) 2+1 t R 0 0 In the case that R dρ(t) < ∞, denote a0 = a − R t/(t2 + 1) dρ(t) ∈ R. In the next theorem the classiﬁcation of the Jordan chains is characterized via the spectral properties of the function Q using the factorized integral representation (6.1). Theorem 6.1. Let Q∞ ∈ Nκ , let k = κ∞ (Q∞ ) > 0, and let Q(z) = −1/Q∞ (z) have the factorized integral representation (6.1). Then: (i) the GZNT ∞ of Q is regular and of type (T1) if and only if dρ(t) > 0, and (1 + |t|)2k dρ(t) < ∞; b = a0 = 0,

(6.3)

(ii) the GZNT ∞ of Q is regular and of type (T2) if and only if b = 0, a0 = 0, and (1 + |t|)2(k−1) dρ(t) < ∞;

(6.4)

(iii) the GZNT ∞ of Q is regular and of type (T3) if and only if b > 0 and (1 + |t|)2(k−2) dρ(t) < ∞;

(6.5)

R

R

R

R

(iv) the GZNT ∞ of Q is singular and of type (T1) with the index of singularity κ0∞ (> 0) if and only if b = a0 = 0 and 0 0 (1 + |t|)2(k−κ∞ ) dρ(t) < ∞, (1 + |t|)2(k−κ∞ +1) dρ(t) = ∞; (6.6) R

R

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(v) the GZNT ∞ of Q is singular of type (T2) with the index of singularity (0 <)κ0∞ < k if and only if b = 0, a0 = 0, and 0 0 (1 + |t|)2(k−1−κ∞ ) dρ(t) < ∞, (1 + |t|)2(k−κ∞ ) dρ(t) = ∞, (6.7) R

R

κ0∞

= k(> 0) if and only if b = 0 and and with the index of singularity dρ(t) = ∞; (6.8) R

(vi) the GZNT ∞ of Q is singular and of type (T3) with the index of singularity κ0∞ (> 0) if and only if b > 0 and 0 0 (1 + |t|)2(k−2−κ∞ ) dρ(t) < ∞, (1 + |t|)2(k−1−κ∞ ) dρ(t) = ∞. (6.9) R

R

Proof. By Proposition 5.2 the root subspace R∞ (SF ) is regular (singular) of dimension ν if and only if (5.4) holds (respectively (5.5) holds). According to Theorem 3.3 the condition ω ∈ dom Aν−1 is equivalent to the condition (1 + |t|)2(ν−k−1) dρ(t) < ∞. (6.10) R

This leads to the integrability conditions in (6.3)–(6.5) in the regular case and to the integrability conditions (6.7)–(6.9) in the singular case, see Theorem 4.12. It follows from the expansions (5.9)–(5.12) of Q in Theorem 5.3 and the factorized integral representation of Q in (6.1) that sj = 0 for j < 2k − 2,

s2k−2 (Q) = −b,

and moreover, that dρ(t) < ∞ and b = 0, if R

then s2k−1 = −a0 ,

and if

R

(6.11)

(6.12)

dρ(t) < ∞ and b = a0 = 0,

then s2k =

dρ(t).

(6.13)

R

All the statements of the theorem can now be obtained from Theorem 5.3 by combining Proposition 5.2 with (6.10)–(6.13). Observe that the regular cases (i)–(iii) are obtained from the singular cases (iv)–(vi) by taking κ0∞ = 0 and excluding the second condition in (6.6), (6.7), and (6.9), respectively. All of the conditions in Theorem 6.1 are based on the canonical factorization of Q = rr Q00 in (6.1) involving the ordinary Nevanlinna function Q00 ∈ N0 in (2.5). In the factorization model of Theorem 4.4 the factor Q0 belongs to the class Nκ−k , where k = κ∞ (Q∞ ). The classiﬁcation of Jordan chains was described in Theorem 4.12 by means of the selfadjoint extensions A00 and A01 of S0 in the Pontryagin space H0 whose negative index is equal to κ(Q0 ) = κ(Q∞ ) − κ∞ (Q∞ ). Analogous descriptions remain true also for the canonical factorization of Q∞ .

Asymptotic expansions

81

Proposition 6.2. Let Q = rr Q00 be the canonical factorization of Q ∈ Nκ in (6.1) with k = π∞ (Q) > 0, let S00 be a simple symmetric operator in a Hilbert space 00 H00 with a boundary triplet Π00 = {C, Γ00 0 , Γ1 } whose Weyl function is equal to 00 00 00 Q00 , and let A0 = ker Γ0 and A1 = ker Γ00 1 be the corresponding selfadjoint extensions of S00 in H00 . Then: (i) the GZNT ∞ of Q is of type (T1) if and only if mul A00 1 = {0}; 00 (ii) the GZNT ∞ of Q is of type (T2) if and only if mul A00 0 = mul A1 = {0}; 00 (iii) the GZNT ∞ of Q is of type (T3) if and only if mul A0 = {0}. Proof. The identities (6.11) and (6.12) concern the function0Q00 . The conditions which describe GZNT ∞ of type (T1) are b = a0 = 0 and R dρ(t) < ∞. These conditions are equivalent to − limz. →∞ zQ00 (z) < ∞, which holds if and only if 1 − lim > 0 ⇔ mul A00 1 = {0}, z. →∞ zQ00 (z) where the last equivalence follows from the simplicity of the operator 0 S00 in H00 . Similarly, for type (T2) one 0has the conditions b = 0 and a0 = 0 if R dρ(t) < ∞, or the conditions b = 0 and R dρ(t) = ∞. These conditions are equivalent to Q00 (z) 00 = 0, lim zQ00 = ∞ ⇔ mul A00 0 = mul A1 = {0}. z z. →∞ z. →∞ Finally, for type (T3) one has the condition b > 0 which is equivalent to lim

Q00 (z) >0 z which completes the proof. lim

z. →∞

⇔

mul A00 0 = {0},

The spectral theoretic characterization in Theorem 6.1 was based on the canonical factorization of the function Q = −1/Q∞ . Since ∞ is a GPNT of the function Q∞ it is natural to translate this result for the canonical factorization of Q∞ : 1 t q(z) q (z) − Q∞ (z) = (6.14) a∞ + b ∞ z + dσ∞ (t) , p(z) p (z) t − z t2 + 1 R where p, q are as0in (6.1), a∞ ∈ R, b∞ ≥ 0, and σ∞ (t)0satisﬁes the analog of (6.2). In the case that R dσ∞ (t) < ∞, denote γ∞ = a∞ − R t/(t2 + 1) dσ∞ (t) ∈ R. Theorem 6.3. Let Q∞ ∈ Nκ with k = κ∞ (Q∞ ) > 0 have the factorized integral representation (6.14). Then: (i) the GPNT ∞ of Q∞ is regular and of type (T1) if and only if b∞ > 0 and (1 + |t|)2(k−1) dσ∞ (t) < ∞; (6.15) R

(ii) the GPNT ∞ of Q∞ is regular and of type (T2) if and only if b∞ = 0, γ∞ = 0, and the integrability condition (6.15) is satisﬁed; (iii) the 0 GPNT ∞ of Q∞ is regular and of type (T3) if and only if b∞ = γ∞ = 0, R dρ(t) > 0, and the integrability condition (6.15) is satisﬁed;

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(iv) the GPNT ∞ of Q∞ is singular and of type (T1) with the index of singularity κ0∞ (> 0) if and only if b∞ > 0 and 0 0 (1 + |t|)2(k−1−κ∞ ) dσ∞ (t) < ∞, (1 + |t|)2(k−κ∞ ) dσ∞ (t) = ∞; (6.16) R

R

(v) the GPNT ∞ of Q∞ is singular and of type (T2) with the index of singularity (0 <)κ0∞ < k if and only if b∞ = 0, γ∞ = 0, and the integrability 0 conditions (6.16) are satisﬁed, 0 and with the index of singularity κ∞ = k(> 0) if and only if b∞ = 0 and R dσ∞ (t) = ∞; (vi) the GPNT ∞ of Q∞ is singular and of type (T3) with the index of singularity κ0∞ (> 0) if and only if b∞ = γ∞ = 0, and the integrability conditions (6.16) are satisﬁed. Proof. In each case the conditions concerning the parameters b∞ , a∞ , and γ∞ are immediate from Proposition 6.2. It remains to establish the integrability conditions for σ∞ (t). First observe that the integrability conditions for ρ(t) in Theorem 6.1 concern the function Q00 in (2.5), while the integrability conditions for σ∞ (t) concern the function −1/Q00 . Now if the GPNT ∞ is of type (T1) then due to the conditions b = a0 = 0 the measure dρ(t) has two more ﬁnite moments than the measure dσ∞ (t). Therefore, the integrability conditions in (6.3) and (6.6) are equivalent to those in (6.15) and (6.16), respectively; cf. [16, Theorem 4.2]. Similarly, due to b∞ = γ∞ = 0, the integrability conditions in (6.5) and (6.9) are equivalent to those in (6.15) and (6.16), respectively. Moreover, by [16, Theorem 4.2] the integrability conditions in (6.4) and (6.7) are equivalent to those in (6.15) and (6.16), respectively, while the 0 conditions b = 0 and (6.8) are clearly equivalent to the conditions b∞ = 0 and R dσ∞ (t) = ∞. In the case that κ(Q∞ ) = κ∞ (Q∞ ) = 1 the result in Theorem 6.3 simpliﬁes to [8, Theorem 4.1]. Observe also that if the function σ∞ (t) has a compact support in 0R, then (6.16) shows that ∞ cannot be a singular critical point of SF . Moreover, if R dσ∞ (t) = 0 and κ∞ (Q∞ ) > 0, then only the cases (i) and (ii) in Theorem 6.3 can occur. 6.2. The classiﬁcation of generalized zeros and poles in R The classiﬁcation of generalized zeros and poles of nonpositive type has been studied in detail at the point z = ∞. Similar results hold true also for ﬁnite generalized zeros and poles of nonpositive type of Q ∈ Nκ which belong to R. Here some main characterizations for the classiﬁcation of ﬁnite generalized zeros β ∈ R and ﬁnite generalized poles α ∈ R of Q are presented. First the classiﬁcation of zeros and poles is characterized by means of the canonical factorization of Q ∈ Nκ . Lemma 6.4. Let Q ∈ Nκ , κ > 0, be factorized as in (6.1). Then the types (T1)– (T3) of a generalized zero β ∈ R of Q are characterized as follows:

Asymptotic expansions

83

(i) the GZNT β ∈ R of Q is of type (T1) if and only if β ∈ σp (A00 1 ); 00 (ii) the GZNT β ∈ R of Q is of type (T2) if and only if β ∈ σp (A00 0 ) ∪ σp (A1 ); 00 (iii) the GZNT β ∈ R of Q is of type (T3) if and only if β ∈ σp (A0 ). Moreover, the types (T1)–(T3) of a generalized pole α ∈ R of Q are characterized as follows: (iv) the GPNT α ∈ R of Q is of type (T1) if and only if α ∈ σp (A00 0 ); 00 (v) the GPNT α ∈ R of Q is of type (T2) if and only if α ∈ σp (A00 0 ) ∪ σp (A1 ); 00 (vi) the GPNT α ∈ R of Q is of type (T3) if and only if α ∈ σp (A1 ). Proof. (i)–(iii) In view of the canonical factorization of Q in (6.1) one obtains the following representation for the limits in (2.3): lim

z. →β

and if

0

Q(z) = lim (z − β)Q00 (z) = −(ρ(β+) − ρ(β−)) (≤ 0) (z − β)2πβ −1 z. →β

dρ(t) R (t−β)2

< ∞ and limz. →β Q00 (z) = 0, then

lim

z. →β

Q(z) Q00 (z) =b+ = lim 2π +1 β (z − β) z. →β (z − β)

R

dρ(t) (> 0), (t − β)2

(6.17)

(6.18)

and otherwise the limit in (6.18) is not ﬁnite. Observe that the limit in (6.17) is negative if and only if β ∈ σp (A00 0 ). The limit in (6.18) is ﬁnite if and only if for the function −1/Q00 the limit in (6.17) is negative, which is equivalent to β ∈ σp (A00 1 ). The statements (i)–(iii) are now obvious from the deﬁning properties of the classiﬁcations given in Subsection 3.2. (iv)–(vi) Apply the characterizations of generalized zeros in the ﬁrst part of the lemma to the function −1/Q. Next some characteristic properties of the underlying root subspaces associated with the classiﬁcation of generalized zeros β ∈ R (generalized poles α ∈ R) are established in a minimal representation of Q (of −1/Q, respectively). Let S(Q) be a simple symmetric operator in a Pontryagin space H, let {Γ0 , Γ1 , H} be a boundary triplet for S ∗ such that the corresponding Weyl function is the given Nκ -function Q, and let A(Q) = ker Γ0 and A(−1/Q) = ker Γ1 . Moreover, let Rα (A(Q)) and Rβ (A(−1/Q)) be the root subspaces of the selfadjoint extension A(Q) and A(−1/Q) of S(Q) associated with the generalized pole α and the generalized zero β ∈ R of Q, respectively. The following result is analogous to Theorem 4.12. For simplicity, the classiﬁcation of a GPNT α ∈ R and a GZNT β ∈ R of Q is characterized here by using the signature of the corresponding root subspace. Here the following notations will be used: να := dim Rα (A(Q)), νβ := dim Rβ (A(−1/Q)), κα ± := κ± (Rα (A(Q))), κα 0 := κ0 (Rα (A(Q))),

κβ± := κ± (Rβ (A(−1/Q))), κβ0 := κ± (Rβ (A(−1/Q))).

Proposition 6.5. With the notations given above the following assertions hold for a GPNT α ∈ R and a GZNT β ∈ R of Q ∈ Nκ :

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V.A. Derkach, S. Hassi and H.S.V. de Snoo

α (i) α is of type (T1) if and only if κα − = κ+ −1, in this case κα +1 ≤ να ≤ 2κα +1 α and κ0 = 2κα + 1 − να ; α (ii) α is of type (T2) if and only if κα − = κ+ , in this case κα ≤ να ≤ 2κα and α κ0 = 2κα − να ; α (iii) α is of type (T3) if and only if κα − = κ+ + 1, in this case κα ≤ να ≤ 2κα − 1 α and κ0 = 2κα − 1 − να ; (iv) β is of type (T1) if and only if κβ− = κβ+ −1, in this case πβ +1 ≤ νβ ≤ 2πβ +1 and ν0β = 2πβ + 1 − νβ ; (v) β is of type (T2) if and only if κβ− = κβ+ , in this case πβ ≤ νβ ≤ 2πβ and ν0β = 2πβ − νβ ; (vi) β is of type (T3) if and only if κβ− = κβ+ + 1, in this case πβ ≤ νβ ≤ 2πβ − 1 and ν0β = 2πβ − 1 − νβ .

Proof. The statements (i)–(iii) are obtained from Theorem 4.12 by considering the transform Q∞ (z) := −Q(α + 1/z). Namely, the root subspace Rα (A(Q)) associated with the generalized pole α of Q coincides with the root subspace R∞ (SF ) associated with the generalized pole ∞ of the function Q∞ . The statements (iv)–(vi) follow by applying the results in (i)–(iii) to the function −1/Q. The classiﬁcation for a GZNT β ∈ R and a GPNT α of Q can be characterized also via asymptotic expansions of the function Q in a neighborhood of these points. The deﬁning properties of the types (T1)–(T3) are reﬂected in these expansions in a similar manner as in Theorems 5.3 and 5.4 above. Such expansions have been studied in [15] and, for instance, the analog of Theorem 5.3 for a GZNT β ∈ R of Q can be easily derived from [15, Theorem 5.2] and for a GPNT α of Q form [15, Theorem 5.4]. Moreover, these expansions can be characterized also via the canonical factorization of Q along the lines of Theorems 6.1 and 6.3 by using Lemma 6.4; see also [15, Section 6]. A detailed formulation of these results is left for the reader. 6.3. Analytic criteria for the minimality of (localized) factorization models The classiﬁcation of generalized zeros and poles of nonpositive type of Q ∈ Nκ can be used to give analytic criteria for general (localized) factorizations models based on (proper) factorizations of Q of the form 0 , Q(z) = r(z) r (z)Q

r =

p , q

(6.19)

where p and q are divisors of the polynomials p and q in the canonical factorization of Q in (6.1), with multiplicity of a zero equal to its original multiplicity. Such factorization models can be used, for instance, in studying local spectral properties of the function Q ∈ Nκ , along the lines carried out in the previous sections of the present paper at ∞ with the aid of the model in Theorem 4.6 for proper factorizations of Q at ∞.

Asymptotic expansions

85

The basic observation here is that according to Lemma 4.11 the occurrence of a maximal Jordan chain of type (T3) means that mul S(Q∞ ) = {0}. Therefore, the existence of such a Jordan chain is connected with the non-minimality of the factorization model of Q. The next result is formulated for the canonical factorization model as constructed in [3, Theorem 3.3]; by unitary equivalence the result holds for all other canonical factorization models, too; cf. [12]. Proposition 6.6. The canonical factorization model constructed for Q ∈ Nκ in [3, Theorem 3.3] is simple if and only if all generalized zeros β ∈ R ∪ {∞} and generalized poles α ∈ R ∪ {∞} of Q of nonpositive type are either of type (T1) or of type (T2). Proof. By Lemma 6.4 a real GZNT β (a real GPNT α) of Q is not of type (T3) 00 if and only if β ∈ σp (A00 0 ) (respectively α ∈ σp (A1 )). Moreover, by Proposition 6.2 the GZNT β = ∞ (the GPNT α = ∞) is not of type (T3) if and 00 only if mul A00 0 = {0} (respectively, mul A1 = {0}). Now, according to [3, Theorem 4.1], these conditions characterize the simplicity of the corresponding factorization model for Q. These observations lead to the following analytic characterization for the simplicity of the factorization model in Theorem 4.4 ([9, Theorem 4.2]) which is based on a proper factorization of Q∞ at ∞ (see Deﬁnition 4.2). Proposition 6.7. Let Q∞ ∈ Nκ with k = κ∞ (Q∞ ) > 0 and let Q∞ = qq Q0 be a proper factorization of Q∞ with some monic polynomial q, deg q = k. Moreover, let the symmetric operator S0 be simple in the Pontryagin space H0 (see Lemma 4.5). Then the factorization model for Q∞ in Theorem 4.4 is minimal if and only if the following two conditions are satisﬁed: (1) α = ∞ is not a GPNT of Q∞ of type (T3); (2) all the real zeros of q as GZNT of Q∞ are either of the type (T1) or (T2). Proof. By Theorem 4.6 the symmetric relation S(Q∞ ) is simple if and only if it has an empty point spectrum, or equivalently, the subspace H in (4.15) is trivial. In view of Lemma 4.5 the last condition is equivalent to mul A01 = {0} and σp (A00 ) ∩ σ(q) = ∅.

(6.20)

By Theorem 4.12 the ﬁrst condition in (6.20) is equivalent to the property formulated in part (1). The condition α ∈ σp (A00 ) ∩ σ(q) means that q(α) = 0 and lim (z − α)Q0 (z) < 0,

z. →α

(6.21)

since by Lemma 4.5 ker (A00 −α) is spanned by a positive vector, cf. [9, Lemma 2.3]. Hence, if the multiplicity of α as a root of q is κα , then Q∞ (z) lim < 0, (6.22) z. →α (z − α)2κα −1 and thus πα (Q∞ ) = κα and α is of type (T3). Conversely, (6.22) implies (6.21) with κα = πα (Q∞ ). This completes the proof.

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Since the polynomial q can be selected to be an arbitrary divisor of degree k = κ∞ (Q∞ ) of the polynomial q in the canonical factorization of Q∞ , the condition (2) in Proposition 6.7 can be satisﬁed if, for instance, the total number (counting multiplicities) of all generalized zeros in C+ and all generalized zeros of nonpositive type in R of Q∞ which are of type (T1) or (T2) is at least equal to κ∞ (Q∞ ). Therefore, the factorization model in Theorem 4.4 can be minimal, while the canonical factorization model of Q∞ need not be minimal. Remark 6.8. Similar facts can be derived for other (localized) factorization models which are build on some (proper) factorization of Q ∈ Nκ , cf. (6.19). The construction of such (localized) factorization models can be based on the (orthogonal) coupling of two minimal models (a method studied in another context in [4]), one of which is a ﬁnite-dimensional Pontryagin space model for the rational 2 × 2-matrix function 0 r(z) R(z) = , (6.23) r (z) 0 whose reproducing kernel space model has been identiﬁed in matrix terms with Bezoutians and companion operators in [3, Proposition 3.3]. The other is a minimal 0 0 . The model for the product Q = rr Q Pontryagin space model for the factor Q in (6.19) is obtained from the orthogonal sum of the models for R and Q0 as a straightforward extension of the model constructed in [3, Theorem 3.3], simply by 0 to be a generalized Nevanlinna function, too. This procedure allowing the factor Q can also be described in pure function theoretic terms: perform suitable “block 0 , cf. [3, ⊕Q 0 to get the product Q = rr Q transforms” to the orthogonal sum R Section 3]. Such factorization models are not minimal in general. Non-minimality of 0 may cancel such models reﬂects the fact that some poles and zeros of rr and Q each others when these functions are multiplied. The simplest example here is the function Q(z) = −z, which belongs to N1 and whose canonical factorization is given by 1 2 − Q(z) = −z = z , z where Q0 (z) = −1/z ∈ N0 , cf. [9, Section 3]. Hence a model which is built on the canonical factorization of Q(z) does not produce a minimal model for Q directly. The general characterization for minimality of canonical factorization models was established in [3, Theorem 4.1]. This result is equivalent to the analytic criterion given in Proposition 6.6: Q does not have any generalized zeros or poles in R∪{∞} which are of type (T3); cf. also [12, Theorem 4.4]. Finally, it is noted that the construction of a minimal model in the case of the canonical factorization of Q ∈ Nκ has been recently studied in [12] by using reproducing kernel Pontryagin spaces. In particular, in that paper a detailed analysis concerning the reproducing kernel space model for the matrix function R in (6.23) has been carried out.

Asymptotic expansions

87

References [1] T.Ya. Azizov and I.S. Iokhvidov, Foundations of the theory of linear operators in spaces with an indeﬁnite metric, Nauka, Moscow, 1986 (English translation: Wiley, New York, 1989). [2] V.A. Derkach, “On generalized resolvents of Hermitian relations”, J. Math. Sciences, 97 (1999), 4420–4460. [3] V.A. Derkach and S. Hassi, “A reproducing kernel space model for Nκ -functions”, Proc. Amer. Math. Soc., 131 (2003), 3795–3806. [4] 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. [5] V. Derkach, S. Hassi, and H.S.V. de Snoo, “Operator models associated with Kac subclasses of generalized Nevanlinna functions”, Methods of Functional Analysis and Topology, 5 (1999), 65–87. [6] V.A. Derkach, S. Hassi, and H.S.V. de Snoo, “Generalized Nevanlinna functions with polynomial asymptotic behaviour and regular perturbations”, Oper. Theory Adv. Appl., 122 (2001), 169–189. [7] V.A. Derkach, S. Hassi, and H.S.V. de Snoo, “Operator models associated with singular perturbations”, Methods of Functional Analysis and Topology, 7 (2001), 1–21. [8] V.A. Derkach, S. Hassi, and H.S.V. de Snoo, “Rank one perturbations in a Pontryagin space with one negative square”, J. Funct. Anal., 188 (2002), 317–349. [9] V.A. Derkach, S. Hassi, and H.S.V. de Snoo, “A factorization model for the generalized Friedrichs extension in a Pontryagin space”, Oper. Theory Adv. Appl., 162 (2005), 117–133. [10] V. Derkach and M. Malamud, “The extension theory of hermitian operators and the moment problem”, J. Math. Sciences, 73 (1995), 141–242. [11] A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, “A factorization result for generalized Nevanlinna functions of the class Nκ ”, Integral Equations and Operator Theory, 36 (2000), 121–125. [12] A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, “Minimal realizations of scalar generalized Nevanlinna functions related to their basic factorization”, Oper. Theory Adv. Appl., 154 (2004), 69–90. [13] A. Dijksma, H. Langer, and Yu. Shondin, “Rank one perturbations at inﬁnite coupling in Pontryagin spaces”, J. Funct. Anal., 209 (2004), 206–246. [14] A. Dijksma, H. Langer, Yu.G. Shondin, and C. Zeinstra, “Self-adjoint operators with inner singularities and Pontryagin spaces”, Oper. Theory Adv. Appl., 117 (2000), 105–176. [15] S. Hassi and A. Luger, “Generalized zeros and poles of Nκ -functions: on the underlying spectral structure”, Methods of Functional Analysis and Topology, 12 no. 2 (2006), 131–150. [16] S. Hassi, H.S.V. de Snoo, and A.D.I. Willemsma, “Smooth rank one perturbations of selfadjoint operators”, Proc. Amer. Math. Soc., 126 (1998), 2663–2675.

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[17] P. Jonas, “Operator representations for deﬁnitizable functions”, Annales Academiae Scientiarum Fennicae, Mathematica, 25 (2000) 41–72. [18] P. Jonas, H. Langer, and B. Textorius, “Models and unitary equivalence of cyclic selfadjoint operators in Pontryagin spaces”, Oper. Theory Adv. Appl., 59 (1992), 252–284. [19] I.S. Kac and M.G. Kre˘ın, “R-functions – analytic functions mapping the upper halfplane into itself”, Supplement to the Russian edition of F.V. Atkinson, Discrete and continuous boundary problems, Mir, Moscow 1968 (Russian) (English translation: Amer. Math. Soc. Transl. Ser. 2, 103 (1974), 1–18). ¨ [20] M.G. Kre˘ın and H. Langer, “Uber die Q-function eines π-hermiteschen Operators im Raume Πκ ”, Acta. Sci. Math. (Szeged), 34 (1973), 191–230. ¨ [21] M.G. Kre˘ın and H. Langer, “Uber einige Fortsetzungsprobleme, die eng mit der angen. 1. Einige FunkTheorie hermitescher Operatoren im Raume Πκ zusammenh¨ tionenklassen und ihre Darstellungen”, Math. Nachr., 77 (1977), 187–236. [22] M.G. Kre˘ın and 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 (I)”, Beitr¨age Anal., 14 (1979), 25–40. [23] M.G. Kre˘ın and 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. [24] H. Langer, “A characterization of generalized zeros of negative type of functions of the class Nκ ”, Oper. Theory Adv. Appl., 17 (1986), 201–212. Vladimir Derkach Department of Mathematical Analysis Donetsk National University Universitetskaya str. 24 83055 Donetsk, 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. 175, 89–94 c 2007 Birkh¨ auser Verlag Basel/Switzerland

A Necessary Aspect of the Generalized Beals Condition for the Riesz Basis Property of Indeﬁnite Sturm-Liouville Problems Andreas Fleige Abstract. For the Sturm-Liouville eigenvalue problem −f = λrf on [−1, 1] with Dirichlet boundary conditions and with an indeﬁnite weight function r changing it’s sign at 0 we discuss the question whether the eigenfunctions form a Riesz basis of the Hilbert space L2|r| [−1, 1]. In the nineties the suﬃcient so called generalized one hand Beals condition was found for this Riesz basis property. Now using a new criterion of Parfyonov we show that already the old approach gives rise to a necessary and suﬃcient condition for the Riesz basis property under certain additional assumptions. Mathematics Subject Classiﬁcation (2000). Primary 34B4; Secondary 34L10. Keywords. Indeﬁnite Sturm-Liouville problem, Riesz basis, deﬁnitizable operator.

1. Introduction We consider the regular indeﬁnite Sturm-Liouville eigenvalue problem −f = λrf

on [−1, 1],

f (−1) = f (1) = 0

(1.1)

where the real weight function r ∈ L [−1, 1] has a single turning point, i.e., sign change, at 0. More precisely we assume 1

r(x) < 0 a.e. on [−1, 0),

r(x) > 0 a.e. on (0, 1]

(1.2)

It is well known that this eigenvalue problem has only real and simple eigenvalues ([5], [6], [7]). During the last two decades the question was intensively studied whether the eigenfunctions form a Riesz basis of the Hilbert space L2|r| [−1, 1] with the inner product 1 f g¯|r|dx (f, g ∈ L2|r| [−1, 1]). (f, g) := −1

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(e.g., [2], [5], [12], [6], [7], [13], [8], [1], [10]). This question is motivated by the fact that in case of a positive weight function r the eigenfunctions even form an orthonormal basis. However in the indeﬁnite situation it took some time to ﬁnd suﬃcient conditions for this Riesz basis property of the eigenvalue problem ([2], [5], [12], [6], [7], [13], [10]). In particular starting from a paper of Beals [2] a technique was developed which led to quite general suﬃcient conditions in the late eighties and nineties ([5], [6], [7], [13]). For this reason in [7] the author called his result the generalized one hand Beals condition. In our days in [4] this technique was still improved and applied to similar problems with λ-dependent boundary conditions. On the other hand there was a competition to ﬁnd counterexamples, i.e., weight functions r such that the Riesz Basis property fails to hold. After in [13] Volkmer had proved the existence of such counterexamples the ﬁrst concrete counterexample was presented in [8] and then in [1] counterexamples with continuously diﬀerentiable weight functions were given. Moreover, in [3] it was shown that counterexamples also depend on the type of the boundary conditions. Recently Parfyonov found a connection of the “positive” and the “negative” results in [10] where he gave a suﬃcient and necessary condition for the Riesz Basis property in the special situation of an odd weight function. We shall see here that already the previous approach using the generalized one hand Beals condition was not too far away from a necessary condition. More precisely we shall show that in the case of an odd weight function and under certain additional assumptions the generalized one hand Beals condition is equivalent to the condition of Parfyonov and hence to the Riesz Basis property of the eigenvalue problem. Moreover, this also leads to a suﬃcient condition of Parfyonov’s type for non odd weight functions. However, the question remains open whether in this general case we also get a necessary and suﬃcient condition for the Riesz Basis property in this way. In [5] it was detected that the Riesz Basis property is closely connected to the critical points of a certain deﬁnitizable operator. Therefore also here we shall make use of Langer’s theory of deﬁnitizable operators in Krein spaces outlined in [9].

2. Two known conditions for the Riesz basis property For the real function r ∈ L1 [−1, 1] with (1.2) the space L2r [−1, 1] of all (equivalence 01 classes of) measurable functions f on [−1, 1] with −1 |f |2 |r|dx < ∞ is a Krein space with the indeﬁnite inner product 1 [f, g] := f g¯rdx (f, g ∈ L2r [−1, 1]). −1

The operator J deﬁned by Jf := sgn(r)f (f ∈ L2r [−1, 1]) is a fundamental symmetry and induces the Hilbert space inner product 1 f g¯|r|dx (f, g ∈ L2r [−1, 1]). (f, g) = [Jf, g] = −1

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91

It is well known ([5], [6], [7]) that the operator A deﬁned by 1 f ∈ L2r [−1, 1], dom(A) : = {f ∈ L2r [−1, 1] | f, f abs. cont., r f (−1) = f (1) = 0}, 1 Af : = − f (f ∈ dom(A)) r is selfadjoint, nonnegative, boundedly invertible and hence deﬁnitizable in the Krein space (L2r [−1, 1], [·, ·]). Moreover the spectrum of A consists of a sequence of real and simple eigenvalues accumulating only at +∞ and −∞ and therefore ∞ is the only critical point of A. According to [7], Sections 3.2 and 3.3, we say that r satisﬁes the generalized right-hand Beals condition if there exist numbers µ > 0, 0 < ξ < min(1, µ1 ) and a continuously diﬀerentiable real function h on [0, ξ] such that h(t) =

r(t) a.e. on (0, ξ], µ · r(µt)

h(0) = 1.

(2.1)

Now we recall [7], Theorem 3.7: Theorem 2.1. If r satisﬁes the generalized right-hand Beals condition then ∞ is not a singular critical point of A. By [5], Proposition 4.1, the non-singularity of the critical point ∞ of A implies that the Hilbert space (L2r [−1, 1], (·, ·)) has a Riesz basis consisting of eigenfunctions of the eigenvalue problem (1.1), i.e., the Riesz basis property. Note that [7] includes a similar result for an analogues generalized left-hand Beals condition. Moreover we want to mention that there is a similar result in [13], Corollary 2.7. Now for µ > 0 we put µt t 1 r(x)dx = µr(µx)dx (t ∈ [0, ]). Rµ (t) := µ 0 0 and in view of [10] we say that r satisﬁes the Parfyonov condition if there exists a number µ ∈ (0, 1) such that for all ∈ (0, 1) 1 Rµ () ≤ R1 (). (2.2) 2 Here we recall a part of [10], Theorem 6: Theorem 2.2 (Parfyonov). Assume that the function r is odd. Then r satisﬁes the Parfyonov condition if and only if the eigenvalue problem (1.1) has the Riesz basis property.

3. A connection of the two conditions Throughout this section we additionally assume that the weight function r is continuously diﬀerentiable on (0, ξ] for a number 0 < ξ ≤ 1. Under some additional

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assumptions we shall show that the Parfyonov condition implies the generalized right-hand Beals condition. To this end for µ ∈ (0, 1) we consider the function hµ (t) :=

r(t) µr(µt)

(t ∈ (0, ξ])

with the derivative hµ (t) =

r(µt)r (t) − µr (µt)r(t) µr(µt)2

(t ∈ (0, ξ]).

Then if r satisﬁes the Parfyonov condition with µ ∈ (0, 1) it follows from (2.2) by l’Hˆ opital’s rule that 2 ≤ lim

t0

R1 (t) R (t) r(t) = lim 1 = lim = lim hµ (t) Rµ (t) t0 Rµ (t) t0 µr(µt) t0

(3.1)

if the last limit exists. Therefore if r is continuously diﬀerentiable on [0, ξ] with r(0) = 0 then so is the function hµ with hµ (0) ≥ 2 and hence (2.1) is satisﬁed with h = hµ . However, since hµ (0) = µ1 = 1 this is a trivial case of Theorem 3.1. Assume that r is continuously diﬀerentiable on (0, ξ] for a number 0 < ξ ≤ 1 and r is odd, i.e., r(−t) = −r(t) (t ∈ (0, 1]). Moreover assume that for all µ ∈ (0, 1) the limits hµ,0 := lim

t0

r(t) , µr(µt)

r(µt)r (t) − µr (µt)r(t) t0 µr(µt)2

hµ,0 := lim

(3.2)

exist. Then the following statements are equivalent: (i) The function r satisﬁes the generalized right-hand Beals condition. (ii) ∞ is not a singular critical point of A. (iii) The eigenvalue problem (1.1) has the Riesz basis property. (iv) The function r satisﬁes the Parfyonov condition. (v) For at least one µ ∈ (0, 1) we have hµ,0 = 1. Proof. The proofs of the implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) were already mentioned in section 2. The implication (iv) ⇒ (v) follows from (3.1). For the implication (v) ⇒ (i) we must show that the continuous function (t ∈ (0, ξ]) hµ (t) h(t) := hµ,0 (t = 0) is continuously diﬀerentiable. By the mean value theorem for each t ∈ (0, ξ] there is a number ξt ∈ (0, t) such that h(t) − h(0) = hµ (ξt ) −→ hµ,0 (t −→ 0). t Therefore h is diﬀerentiable at 0 and the derivative is continuous in 0.

Now from Theorem 3.1 we easily obtain the following suﬃcient condition of Parfyonov’s type without the restriction to odd weight functions. Note that this was already shown in [11].

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93

Corollary 3.2. Assume that r is continuously diﬀerentiable on (0, ξ] for a number 0 < ξ ≤ 1 and for all µ ∈ (0, 1) the limits (3.2) exist. Then the eigenvalue problem (1.1) has the Riesz basis property if the function r satisﬁes the Parfyonov condition. Proof. Consider the “odd extension” r˜ of r restricted to (0, 1] r(t) (t ∈ (0, 1]) r˜(t) := −r(−t) (t ∈ [−1, 0)). Then r˜ satisﬁes the Parfyonov condition and by Theorem 3.1 also the generalized right-hand Beals condition. Since this is then also true for r the Riesz basis property follows from Theorem 2.1. It is well known that the eigenvalue problem (1.1) has the Riesz basis property if r ∈ C 2 [0, ξ] with r(0) = 0 and r (0) = 0 (e.g., [7], Corollary 3.8). This also serves as an example for the existence of the limits (3.2) since in this case we have by l’Hospital’s rule t0

hµ,0

r (t)

1 (= 1), µ2 r (t) r (µt) r (0) r (0) − µh − . = lim hµ (t) = lim (t) = 2 µ 2 t0 t0 2µ r (µt) 2r (µt) 2µ r (0) 2µr (0)

hµ,0 = lim hµ (t) = lim

t0 µ2 r (µt)

=

References [1] N.L. Abasheeva, S.G. Pyatkov, Counterexamples in indeﬁnite Sturm-Liouville problems. Siberian Adv. Math. 7, No. 4 (1997), 1–8. [2] R. Beals, Indeﬁnite Sturm-Liouville problems and half range completeness. J. Diﬀerential Equations 56 (1985), 391–407. [3] P. Binding, B. Curgus, A counterexample in Sturm-Liouville completeness theory. Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 244–248. [4] P. Binding, B. Curgus, Riesz Bases of Root Vectors of Indeﬁnite Sturm-Liouville Problems with Eigenparameter Dependent Boundary Conditions, I. To appear. [5] 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. [6] A. Fleige, The “turning point condition” of Beals for indeﬁnite Sturm-Liouville problems. Math. Nachr. 172 (1995), 109–112. [7] A. Fleige, Spectral Theory of Indeﬁnite Krein-Feller Diﬀerential Operators. Mathematical Research 98. Akademie Verlag, Berlin, 1996. [8] A. Fleige, A counterexample to completeness properties for indeﬁnite Sturm-Liouville problems. Math. Nachr. 190 (1998), 123–128. [9] H. Langer, Spectral functions of deﬁnitizable operators in Krein spaces In: D. Butkovic, H. Kraljevic and S. Kurepa (eds.): Functional analysis. Conf. held at Dubrovnik, Yugoslavia, November 2–14, 1981. Lecture Notes in Mathematics, Vol. 948, Springer-Verlag, Berlin, Heidelberg, New York, 1982, 1–46.

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[10] A.I. Parfyonov, On an Embedding Criterion for Interpolation Spaces and Application to Indeﬁnite Spectral Problems. Siberian Mathematical Journal, Vol. 44, No. 4 (2003), 638–644. [11] A.I. Parfyonov, On Curgus’ condition in indeﬁnite Sturm-Liouville problems. Matem. Trudy., Vol. 7, No. 1 (2004), 153–188. (In Russian, translated in Sib. Adv. Math.) [12] S.G. Pyatkov, Elliptic eigenvalue problems with an indeﬁnite weight function. Siberian Adv. Math. 4, No. 2 (1994), 87–121. [13] H. Volkmer, Sturm-Liouville problems with indeﬁnite weights and Everitt’s inequality. Proc. Roy. Soc. Edinburgh Sect. A 126, No. 5, (1996), 1097–1112. Andreas Fleige Baroper Schulstraße 27a D-44225 Dortmund, Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 175, 95–109 c 2007 Birkh¨ auser Verlag Basel/Switzerland

On Reducible Nonmonic Matrix Polynomials with General and Nonnegative Coeﬃcients K.-H. F¨orster and B. Nagy Abstract. We consider nonmonic quadratic polynomials acting on a general or on a ﬁnite-dimensional linear space as a continuation of our work in [7,8]. Conditions are given for the existence of right roots, if the coeﬃcient operators have lower block triangular representations. In the ﬁnite-dimensional case we consider (in a certain sense, entrywise) nonnegative coeﬃcient matrices in the general (reducible) case, and extend several earlier results from the case of irreducible coeﬃcients. In particular, we generalize results of Gail, Hantler and Taylor [9]. We show that our general methods are suﬃciently strong to prove a remarkable result by Butler, Johnson and Wolkowicz [3], proved there by ingenious ad hoc methods. Mathematics Subject Classiﬁcation (2000). Primary 15A22; Secondary 15A48, 47A56, 15A18. Keywords. Reducible matrix polynomials, block triangular operator coeﬃcients, (entrywise) nonnegative matrix coeﬃcients, nonnegative matrix roots.

1. Introduction We consider quadratic (in general nonmonic) polynomials Q(·) of the form λ2 M + λL + K. In Section 2 we study such polynomials when the coeﬃcients M , L and K are lower block triangular linear maps (with the same block size). Such polynomials can have (right) roots, but none of these roots need to be lower block triangular; see Example 2.2. In Theorem 2.6 we give a necessary and suﬃcient condition for the existence of a lower block triangular (of the same block size as the coeﬃcients) right root of this matrix polynomial. This work was completed with partial support of the Hungarian National Science Grant OTKA No T-047276 and partial support of the DAAD, the Technical University of Berlin and the Budapest University of Technology and Economics.

96

K.-H. F¨ orster and B. Nagy The main topic of this paper is the matrix polynomial Q(λ) = λIn×n − (λ2 A + λB + C)

with entrywise nonnegative n × n-matrices A, B and C. This class of matrix polynomials play an important role in the theory of quasi-birth-and-death Markov chains (see [2, 5.6]) and has been studied very intensively; for a rather extensive review on this topic see [2] and [14]. In [7] the spectral properties of such matrix polynomials were studied. In the case when S = A + B + C is irreducible a complete set of necessary and suﬃcient conditions for the existence of a nonnegative right root W of Q(·) were given and the spectral properties of the matrices W and V such that Q(λ) = (V − λA)(λIn×n − W ) and W ≥ 0n×n were studied; see [7, Section 4]. Results in this direction for the case when S is reducible can be found in [2, 4.7], [9, Theorem 5] and [3, Theorem 2.3]. In Section 3 we study systematically the existence of nonnegative right roots of Q(·) when the coeﬃcients are nonnegative and reducible. In contrast to the example in Section 2 mentioned above here Q(·) has nonnegative lower triangular right roots if and only if it has nonnegative right roots (Theorem 3.1). With the help of the Frobenius normal form of S and the function 0 ≤ ρ → spectral radius of S(ρ) = ρ2 A + ρB + C we give several suﬃcient conditions for the existence of nonnegative roots of Q(·), and we generalize the results mentioned above. In particular, we give a completely diﬀerent proof of [3, Theorem 2.3]. Our terminology is mostly traditional. For standard facts in the theory of linear bounded operators in Banach spaces we refer to [4], concerning (matrix or operator) polynomials to [10] or [15], concerning nonnegative matrices to [11], [1] or [16].

2. Roots of polynomials with lower triangular block operator coeﬃcients Let E be a (ﬁnite- or inﬁnite-dimensional complex) vector space. We consider the polynomial Q(λ) = λ2 M + λL + K, λ ∈ C, (2.1) where M , L and K are linear maps in E. We assume that E = E1 × E2 and that the coeﬃcients have lower block triangular representations; for example M11 0 M= , (2.2) M21 M22 where Mrs is a linear map form Es into Er for 1 ≤ s ≤ r ≤ 2 and 0 is the zero map from E2 into E1 .

On Reducible Nonmonic Matrix Polynomials

97

Then Q(λ) has the lower block triangular representation 0 Q11 (λ) . Q(λ) = Q21 (λ) Q22 (λ)

(2.3)

Note that if the dimension of the vector space E is ﬁnite, and the Q s are matrices, then the form (2.3) shows that Q(·) is reducible in the sense of [11].Note also that the irreducible case (for nonnegative coeﬃcients) was treated in [7, Section 4]. Assume that Q(·) has a right root W , i.e., W is a linear map in E such that Q(W ) = M W 2 + LW + K = 0, and let V be a linear map in E such that Q(λ) = (λM − V )(λIE − W ), where IE denotes the identity operator in E. If V and W have the block representation V11 V12 W11 V = and W = V21 V22 W21

(2.4)

W12 W22

,

(2.5)

direct computations show that (2.4) implies the following system of matrix equations M11 W12 + V12 = 0 V11 W12 + V12 W22 = 0, (2.6) V21 W11 + V22 W21 = K21 , M21 W11 + V21 + M22 W21 = −L21 . From the ﬁrst two equations in (2.6) it follows immediately: If W is lower block triangular, then V is lower block triangular (in this case the second equation is automatically fulﬁlled), the diagonal polynomials Qrr (·) have the right roots Wrr , and Qrr (λ) = (λMrr − Vrr )(λIEr − Wrr ) for r = 1, 2. Let the linear map X of E1 into E2 be a solution of the equation V22 X − M22 XW11 = K21 + (L21 + M21 W11 )W11 ,

(2.7)

and deﬁne W21 := X

and

V21 := −L21 − M21 W11 − M22 X.

(2.8)

Then the last two equations of (2.6) are fulﬁlled. On the other hand, from the last two equations of (2.6) it follows: if we deﬁne X := W21 , then (2.7) and (2.8) are satisﬁed. Therefore we have proved that 1. implies 2. in the following proposition. Proposition 2.1. With the assumptions, deﬁnitions and notations above the following assertions are equivalent: 1. The polynomial Q(·) has a lower block triangular right root (of the same block size as the coeﬃcients). 2. For r = 1, 2 the polynomial Qrr (·) has a right root Wrr , and equation (2.7) has a solution X, which is a linear map of E1 into E2 , where V22 satisﬁes Q22 (λ) = (λM22 − V22 )(λIE2 − W22 ).

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Proof. 2. implies 1.: Let Qrr (λ) = (λMrr − Vrr )(λIEr − Wrr ) for r = 1, 2, let X be a solution of (2.7), and W21 and V21 be deﬁned as in (2.8). Then these maps are the lower left blocks, and Wrr and Vrr are the diagonal blocks of a lower block triangular right root W of Q(·) and of a lower block diagonal V , respectively, such that Q(λ) = (λM − V )(λIE − W ). The next example exhibits a reducible matrix polynomial which has right roots, but none of its right roots is lower triangular. Example 2.2. Consider the matrix polynomial 0 0 0 0 + Q(λ) = λ2 M + K = λ2 1 0 −2 −2 0 0 0 0 1 1 = λ − λI2×2 − . 1 0 −1 −1 1 1 Let Q(λ) = (λM − V )(λI − W ) with 2 × 2-matrices V and W = (wrs ). Then 0 0 and V = −M W = − w11 w12 0 0 0 0 , =K =VW =− 2 + w12 w21 w12 (w11 + w22 ) −2 −2 w11 and this implies w12 = 0. Remark. Assume that the coeﬃcients M, L, K of Q(·) are lower triangular, that the pair (V, W ) factorizes Q(·) (as in (2.4)), and let (V0 , W0 ) denote the pair obtained by substituting the blocks V12 , W12 in (2.5) by zero blocks. We have seen that if the pair (V0 , W0 ) also factorizes Q(·), then the diagonal blocks (Vrr , Wrr ) factorize the corresponding diagonal block Qrr (·). We show that if the diagonal blocks (Vrr , Wrr ) factorize the corresponding diagonal block Qrr (·) and the block M21 is an injective operator, then (V0 , W0 ) = (V, W ), hence it factorizes Q(·). Indeed, if the diagonal blocks factorize as well as the pair (V, W ), then we have Vjj Wjj = Kjj , Mjj Wjj + Vjj = −Ljj (j = 1, 2), Vj1 W1j + Vj2 W2j = Kjj ,

Mj1 W1j + Mj2 W2j + Vjj = −Ljj

(j = 1, 2).

Hence we obtain M21 W12 = 0, and the injectivity of M21 implies W12 = 0. It means that W = W0 , and we have seen that then V12 = 0 follows. Hence (V0 , W0 ) = (V, W ) factorizes the polynomial Q(·). Corollary 2.3. Let the coeﬃcients of Q(·) be block diagonal (i.e., M12 = L12 = K12 = 0 and M21 = L21 = K21 = 0). Then Q(·) has a (block diagonal) right root if and only if all diagonal entries of Q(·) have right roots. Assume that V22 is invertible. Then equation (2.7) is equivalent to X − V22 −1 M22 XW11 = V22 −1 (K21 + (L21 + M21 W11 )W11 ).

(2.9)

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Let L(E1 , E2 ) denote the vector space of all linear maps from E1 into E2 . We deﬁne the linear map 3(V22 −1 M22 , W11 ) : L(E1 , E2 ) → L(E1 , E2 ), with X → V22 −1 M22 XW11 . (2.10) Corollary 2.4. Assume that Qrr (λ) = (λMrr −Vrr )(λIEr −Wrr ) for r = 1, 2. If V22 and IL(E1 ,E2 ) − 3(V22 −1 M22 , W11 ) are invertible, then Q(·) has a lower triangular right root. Now let E1 and E2 be Banach spaces, E their Banach product space and the coeﬃcients of Q(·) and their entries in the representation (2.2) be linear and bounded operators between the corresponding Banach spaces. By σ(T ) we denote the spectrum of a bounded linear operator T in a Banach space. Then we have: Corollary 2.5. Assume that Qrr (λ) = (λMrr − Vrr )(λIEr − Wrr ) for r = 1, 2. If −1 −1 M22 ) · σ(W11 ) = {λ · µ : λ ∈ σ(V22 M22 ) and µ ∈ σ(W11 )}, 1∈ / σ(V22

(2.11)

then Q(·) has a lower triangular right root. The proof follows from the last corollary and the equality σ(3(T, S)) = σ(T ) · σ(S), see [20, Theorem]. The next theorem follows from Proposition 2.1 by constructing inductively the blocks in the subdiagonals of W and V . Theorem 2.6. Let the polynomial Q(·) be as in (2.1), E = E1 × E2 × · · · × Em , and assume that the coeﬃcients M , L and K have lower block triangular representations, for example ⎛ ⎞ M11 0 0 ··· 0 ⎜ ⎟ .. ⎜ M21 ⎟ M22 0 . 0 ⎜ ⎟ ⎜ ⎟, .. .. .. .. M = ⎜ .. ⎟ . . . . ⎜ . ⎟ ⎝ Mm−1,1 · · · ⎠ · · · Mm−1,m−1 0 ··· ··· ··· Mmm Mm1 where Mpq are linear maps form Eq into Ep for 1 ≤ q ≤ p ≤ m. Then the following assertions are equivalent: 1. The polynomial Q(·) has a right root which is lower block triangular (of the same block size as the coeﬃcients). 2. For r = 1,2, . . . ,m the polynomial Qrr (·) has a right root Wrr and for 1 ≤ q < p ≤ m the equation Mpr Wrq )Wqq − Vpr Wrq (2.12) Vpp X − Mpp XWqq = Kpq + (Lpq + q≤r
q
has a solution X (of the same size as Mpq ) which is a linear map from Eq into Ep . Here Vpp is a linear map in Ep such that Qpp (λ) = (λMpp − Vpp )(λIEp − Wpp ).

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For the proof we note the following: Let Qrr (λ) = (λMrr − Vrr )(λIEr − Wrr ) for r = 1, 2, . . . , m, and let Xpq be a solution of (2.12) for 1 ≤ q < p ≤ m. Deﬁne for 1 ≤ q < p ≤ m and Vpq := −Lpq − Mpr Wrq . (2.13) Wpq := Xpq q≤r≤p

Then the ﬁrst of these maps are the lower left blocks, and Wrr are the diagonal blocks of a lower block triangular right root W of Q(·). The second of these maps are the lower left blocks, and Vrr are the diagonal blocks of a lower block triangular V such that Q(λ) = (λM − V )(λIE − W ). We can generalize Corollaries 2.3–2.4 in an obvious way to the more general case of the last theorem.

3. Nonnegative roots of nonmonic matrix polynomials with nonnegative coeﬃcients In this section we consider the matrix polynomial Q(λ) = λIn×n − (λ2 A + λB + C)

(3.1)

with entrywise nonnegative n × n-matrices A,B and C. Here and in the remainder In×n and 0n×n denote the n × n identity matrix and the n × n zero matrix, respectively; sometimes we write for short I and 0 for an appropriate (square) identity matrix and a (not necessarily square) zero matrix, respectively. As mentioned in the introduction, we are interested in the existence of nonnegative right roots of Q(·) in the case when S = A + B + C is reducible; for the simpler case when S is irreducible see [7, Sections 3 and 4]. The nonnegative matrix S is cogredient to its (lower triangular) Frobenius normal form (see [18, Section 2]), i.e., there exists a permutation matrix P such that ⎞ ⎛ S11 0 0 ··· 0 ⎟ ⎜ .. ⎟ ⎜ S21 S22 0 . 0 ⎟ ⎜ t ⎟ ⎜ . . . .. .. (3.2) P SP = ⎜ . .. .. ⎟ . . . ⎟ ⎜ ⎠ ⎝ Sm−1,1 · · · · · · Sm−1,m−1 0 Sm1 ··· ··· ··· Smm where the diagonal blocks Srr are irreducible square matrices. For our purpose we can and will assume, that S is in its Frobenius form. Since 0n×n ≤ A ≤ S, and a corresponding inequality for the other coeﬃcients B and C holds, all coeﬃcients and Q(·) are lower block triangular matrices of the same block size as S. If W is a right root of Q(·), then there exists a (unique) matrix V such that Q(λ) = (V − λA)(λIn×n − W ),

λ ∈ C,

(3.3)

On Reducible Nonmonic Matrix Polynomials

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and (3.3) is equivalent to V = In×n − B − AW

and C = V W

(3.4)

In the next two theorems it is not essential that (3.2) is the Frobenius normal form of S; only the lower block triangular form of S is needed, and the fact that the nonnegative coeﬃcients A, B and C have a corresponding lower block triangular form. If Q(·) has a nonnegative right root, we know from [7, Proposition 3.1] that Q(·) has a smallest (= minimal) nonnegative right root. Theorem 3.1. Let Q(λ) = λIn×n −(λ2 A+λB+C) with nonnegative n × n-matrices A, B and C and let S = A + B + C be in lower block triangular form (3.2). Then the following assertions are equivalent: 1. Q(·) has a nonnegative right root. 2. Q(·) has a nonnegative right root which is lower block triangular (of the same block sizes as S); in particular, the smallest nonnegative right root of Q(·) is lower block triangular. 3. For r = 1, 2, . . . , m the polynomial Qrr (·) has a nonnegative right root Wrr and for 1 ≤ q < p ≤ m the equation ⎛ ⎞ Vpp X − App XWqq = Cpq + ⎝Bpq + Apr Wrq ⎠ Wqq − Vpr Wrq (3.5) q≤r
q
has a nonnegative matrix X (of the same size as Spq ) as a solution. Here Vpp is a Z-matrix such that Qpp (λ) = (Vpp − λApp )(λI − Wpp ). Proof. 1. implies 2.: From [7, Proposition 3.1] we know that the ﬁxed point iteration Wl+1 = AWl2 + BWl + C

with

0n×n ≤ W0 ≤ C,

l = 0, 1, . . . , converges to the smallest nonnegative right root of Q(·). In our case W0 , and then all matrices of the sequence (Wl ) are lower block triangular (of the same block sizes as S). Therefore its limit is also lower block triangular (of the same block sizes as S). That 2. implies 1. is clear. For the equivalence of 2. and 3. apply Theorem 2.5 (note the sign change for V between the ﬁrst factors in (2.4) and (3.3)!) and results of [7, Section 3]. Note that in the case of Theorem 3.1 the matrices Vpq for 1 ≤ q < p ≤ m deﬁned in (2.13) are given by Apr Wrq − App X. (3.6) Vpq = −Bpq − q≤r
Therefore they are nonpositive, and the block matrix V is a Z-matrix (see (3.4)). In the following r(T ) denotes the spectral radius of the square matrix T , and S(λ) denotes the polynomial λ2 A + λB + C. Thus Q(λ) = λI − S(λ).

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Theorem 3.2. Let Q(λ) = λIn×n −(λ2 A+λB+C) with nonnegative n × n-matrices A, B and C and let S = A + B + C be in the lower block triangular form (3.2). Assume that for k = 1, 2, . . . , m there exists a positive ρk such that r(Skk (ρk )) = r( ρk 2 Akk + ρk Bkk + Ckk ) < ρk ,

(3.7)

ρ1 ≤ ρ2 ≤ · · · ≤ ρm .

(3.8)

and Then Q(·) has a nonnegative right root; more precisely, there exist lower block triangular matrices W and V (with the same block sizes as S) such that Q(λ) = (λM − V )(λIn×n − W ), where W is a nonnegative matrix and V is a nonsingular M -matrix. Proof. By [7, Theorem 3.4] there exists for k = 1, 2, . . . , m a nonnegative matrix Wkk with r(Wkk ) < ρk and a nonsingular M -matrix Vkk with r(Vkk −1 Akk ) < 1/ρk such that Qkk (λ) = λI − (λ2 Akk + λBkk + Ckk ) = (λMkk − Vkk )(λIkk − Wkk ). This and (3.8) imply that r(3(Vpp −1 App , Wqq )) < 1 for 1 ≤ q < p ≤ m, see the remark to the proof of Corollary 2.4. The right-hand side of equation (3.5) is a nonnegative matrix. Using the Neumann series expansion of the inverse of I − 3(Vpp −1 App , Wqq ) we obtain that for 1 ≤ q < p ≤ m the unique solution of (3.5) is a nonnegative matrix. Theorem 3.1 completes the proof of the ﬁrst assertion. Note that the diagonal blocks of V are nonsingular M -matrices, therefore they have nonnegative inverses. Further the left lower blocks of V are nonpositive (see (3.6)). Therefore V = I − B − AW is a nonsingular Z-matrix with a nonnegative inverse, thus it is a M -matrix (see [1, (N38 ), p. 137]). By Theorem 5.1 of [8] it is possible to generalize both theorems above to polynomials with coeﬃcients which are more general than matrices, for example operators in ordered Banach spaces. Note that the assumptions of Theorem 3.2 about the spectral radii do not imply that r( ρ2 A + ρB + C ) < ρ for some positive ρ (see [7, Example 3.3]). Therefore one cannot directly apply [7, Theorem 3.4]. The following example shows that the above theorem does not hold if we assume only (3.7), but not (3.8). Example 3.3. Let a be a positive number and deﬁne 0 0 0 0 1 0 Q(λ) = λI2×2 − λ2 +λ + . 0 a a 0 1 0 Then equation (3.5) becomes (1 − a)x = 1 + a. Theorem 3.1 shows that Q(·) has no nonnegative (triangular) right root for a > 1. For a > 1 and positive ρ1 and ρ2 1 = r(S11 (ρ1 )) < ρ1

and aρ2 2 = r(S22 (ρ2 )) < ρ2

imply ρ1 > 1 > 1/a > ρ2 . As a preparation for the next theorem we introduce the following notation. Let Q(λ) = λIn×n − (λ2 A + λB + C) with nonnegative n × n-matrices A, B and

On Reducible Nonmonic Matrix Polynomials

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C and let S = A + B + C be in (lower triangular) Frobenius normal form (3.2). Assume that r(S(ˆ ρ)) = r( ρˆ2 A + ρˆB + C ) ≤ ρˆ for some positive ρˆ.

(3.9)

Since for k = 1, 2, . . . , m the matrix Skk (1) is irreducible, the function ]0, ∞[→ [0, ∞[, ρ → r(Skk (ρ)) is real analytic; see [7, Section 3]. By r (Skk (·)) we denote its derivative. Now we deﬁne the following sets: M := {k ∈ {1, 2, . . . , m} : r(Skk (ˆ ρ)) = ρˆ}, M− := {k ∈ M : r (Skk (ˆ ρ)) < 1}, M1 := {k ∈ M : r (Skk (ˆ ρ)) = 1}

and

M+ := {k ∈ M : r (Skk (ˆ ρ)) > 1}. For k = 1, 2, . . . , m we denote by dk the index of phase imprimitivity (see [7]) of the graph G(Ckk ]Bkk [Akk ). We recall (see [7, Proposition 4.5]): dk = 0 implies that for a positive ρ with r(Skk (ρ)) = ρ and for a θ ∈ [0, 2π[ the complex number 2π 2π ρeiθ is an eigenvalue of Qkk (·) if and only if θ ∈ {0, 2π dk , 2 · dk , . . . , (dk − 1) dk }. The following theorem generalizes [9, Theorem 5] for quadratic matrix polynomials considered in this paper. Theorem 3.4. Let Q(λ) = λIn×n −(λ2 A+λB+C) with nonnegative n × n-matrices A, B and C and let S = A+B +C be in its Frobenius normal form (3.2). Assume that there exists a positive ρˆ such that r(S(ˆ ρ)) = r( ρˆ2 A + ρˆB + C ) ≤ ρˆ.

(3.10)

With the notation above the following assertions hold: 1. The set of the eigenvalues of Q(·) is the whole plane C if and only if there exists a k ∈ M such that r(Skk (ρ)) = ρ for all nonnegative ρ. 2. Assume that the set of the eigenvalues of Q(·) is not the whole plane C. Then (a) Q(·) has exactly n− k∈M− ∪ M1 dk eigenvalues (counting algebraic mul∈ C : |λ| < ρˆ }. tiplicities) in the open disc Dρˆ = {λ (b) Q(·) has exactly k∈M− ∪ M+ dk + 2 k∈M1 dk eigenvalues (counting algebraic multiplicities) on the circle Tρˆ = {λ ∈ C : |λ| = ρˆ }. Proof. The polynomial Q(·) is block triangular, therefore the set of its eigenvalues is the union of the sets of the eigenvalues of its diagonal blocks Qkk (·), k = 1, 2, . . . , m. Each diagonal block in the Frobenius normal form is irreducible. Therefore we can apply to each of them [7, Theorem 4.10] and will obtain the assertions. In [9, Theorem 5] the main assumption is that S is substochastic. If r(S) = 1, this implies (see [9, Equations (55) and (56)]) that there exists a p ∈ {1, 2, . . . , m}

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such that M = {1, 2, . . . , p} and the Frobenius normal form of S(ρ) is ⎛ S11 (ρ) 0 ··· 0 0 ··· 0 ⎜ .. .. ⎜ . . 0 0 0 ··· 0 ⎜ ⎜ .. .. .. .. . . . . ⎜ . . . . . ··· . ⎜ ⎜ 0 0 0 0 ⎜ ⎜ 0 ··· 0 0 ··· 0 Spp (ρ) ⎜ ⎜ Sp+1,1 (ρ) · · · · · · Sp+1,p (ρ) Sp+1,p+1 (ρ) 0 · · · 0 ⎜ ⎜ .. . . . . . .. .. .. .. .. ⎜ . ⎜ ⎜ . . . . . . . . ⎝ . . . . 0 Sm1 (ρ) · · · · · · Smp (ρ) Sm,p+1 (ρ) · · · · · · Smm (ρ)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ . (3.11) ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Corollary 3.5. Let Q(λ) = λIn×n −(λ2 A+λB+C) with nonnegative n × n-matrices A, B and C. Assume that there exists a positive ρˆ such that r(S(ˆ ρ)) ≤ ρˆ. Then Q(·) has a nonnegative right root if one of the following assumptions is satisﬁed: ρ)) = ρˆ} is empty. 1. M := {k ∈ {1, 2, . . . , m} : r(Skk (ˆ 2. There exists a p ∈ {1, 2, . . . , m} such that ρ)) = ρˆ} = {1, 2, . . . , p}, M := {k ∈ {1, 2, . . . , m} : r(Skk (ˆ and (3.11) is the Frobenius normal form of S(ρ). Proof. Assumption 1. is equivalent to r(S(ˆ ρ)) < ρˆ. The assertion follows from [7, Theorem 3.4] or Theorem 3.2 with m = 1. Let Assumption 2. hold. Then r(S(ˆ ρ)) = ρˆ. For k = 1, 2, . . . , p each Qkk (·) has a nonnegative right root with spectral radius equal to ρˆ (see [7, Proposition 4.6]). Since the upper left corner of Q(·) (of the ﬁrst p × p blocks) is diagonal, it has a nonnegative right root with spectral radius equal to ρˆ (see Corollary 2.3). The right lower corner of S(·) has a spectral radius strictly less than ρ. ˆ Now an analogous argument as in the proof of Theorem 3.2 proves the assertion. As a completion to [9, Theorem 5] we obtain in the case of a quadratic matrix polynomial, the following: Corollary 3.6. Let Q(λ) = λIn×n −(λ2 A+λB+C) with nonnegative n × n-matrices A, B and C. If S = A + B + C is substochastic, then Q(·) has a nonnegative right root. The proof follows from the preceding corollary as the case ρˆ = 1. In [3] the authors consider the polynomial −Q(·) where Q(λ) = λIn×n − (λ2 In×n + C).

(3.12)

and C is a nonnegative n × n-matrix. In [3, Theorem 2.3] they give necessary and suﬃcient conditions for the existence of nonnegative roots of Q(·). Their proof uses analytical methods, for example the convergence of the binomial series of 14 I − C.

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We will give a diﬀerent proof of their results by methods in the spirit of this paper (see Theorem 3.10 below). We need some preparations. Let Q(·) be as in (3.12). Then r(S(ρ)) = ρ2 + r(C) for positive ρ. This implies that both equalities r(S(ˆ ρ)) = ρˆ and r (S(ˆ ρ)) = 1 hold if and only if both equalities r(C) = 1/4 and ρˆ = 1/2 hold. From [7, Theorem 3.4] it follows Proposition 3.7. Let Q(·) be as in (3.12) with a nonnegative square matrix C with r(C) < 1/4. Then Q(·) has a nonnegative right root. In addition, there exist a nonnegative matrix W and a nonsingular M -matrix V such that (3.13) Q(λ) = λIn×n − (λ2 In×n + C) = (V − λI)(λI − W ), −1 r(W ) < 1/2 and r(V ) < 2. From [7, Theorems 3.4 and 4.10] it follows Proposition 3.8. Let Q(·) be as in (3.12) with a nonnegative irreducible square matrix C. Then Q(·) has a nonnegative right root if and only if r(C) ≤ 1/4. In addition, there exist a nonnegative matrix W and a nonsingular M -matrix V such that (3.13) holds, and 1. r(W ) = 1/2 and r(V −1 ) = 2, if r(C) = 1/4, 2. r(W ) < 1/2 and r(V −1 ) < 2, if r(C) < 1/4. Note that for a nonnegative irreducible matrix its spectral radius is a simple root of its characteristic polynomial (see [16, Theorem 1.4.3]). This implies that it is in the terminology of [3] an eigenvalue value of degree 1, i.e., it is a semisimple eigenvalue. Therefore the ﬁrst part of the last proposition is exactly [3, Theorem 2.3] for the case of an irreducible matrix C. Lemma 3.9. 1. Let Q(·) be as in (3.12) with a nonnegative irreducible square matrix C and Q(λ) = λIn×n − (λ2 In×n + C) = (V − λI)(λI − W ), with nonnegative W and a nonsingular M -matrix V . Then W is irreducible and V −1 = (I − W )−1 is strictly positive, i.e., all entries of V −1 are positive. 2. Let W be nonnegative and irreducible, V have a strictly positive inverse, −1 r(W ) = r(V −1 ) and B ≥ 0n×n . Then the equation V X − XW = B has a nonnegative solution if and only if B = 0n×n ; here W, V, B, 0n×n and X are matrices of the size n × n. Proof. 1. The irreducibility of W is easily proved by contradiction using C = V W and V = I − W . The second assertion follows from [16, Theorem 2.3]. 2. The equation V X − XW = B is equivalent to X − 3(V −1 , W )X = X − V −1 XW = V −1 B. By the description of the spectrum of 3(V −1 , W ) following Corollary 2.5, we obtain r(3(V −1 , W )) = r(V −1 )r(W ) = 1. Now we will use some results from the local spectral theory of nonnegative operators in ﬁnite-dimensional spaces, see [5, 6]. In the following we write for short 3 instead of 3(V −1 , W ).

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Set ♦ = 0≤j≤n 3j . Then the linear maps ♦ and 3 (acting in the linear space of all complex n × n-matrices) commute. Since W is irreducible and V −1 is strictly positive, it follows that ♦V −1 B is strictly positive when B is nonnegative and nonzero. This implies that the local spectral radius r (♦V −1 B) of ♦V −1 B with respect to 3 is equal to the spectral radius of 3, see [6, Proposition 1]. Now 1 = r (♦V −1 B) ≤ r (V −1 B) ≤ r(3) = 1. We are in ﬁnite-dimensional spaces, therefore, by [5, Theorem 12] or [19, Theorem 3.1], the equation X − 3X = X − V −1 XW = V −1 B with nonnegative B has a nonnegative solution if and only if B = 0n×n . The following theorem was proved in [3, Theorem 2.3] by completely diﬀerent methods. Theorem 3.10. Let Q(λ) = λIn×n − (λ2 In×n + C).

(3.14)

where C is a nonnegative n × n-matrix. Then 1. r(C) < 1/4 implies that there is a nonnegative right root of Q(·). 2. r(C) > 1/4 implies that there is no nonnegative right root of Q(·). 3. If r(C) = 1/4, then Q(·) has a nonnegative right root if and only if 1/4 is a semisimple eigenvalue of C. Proof. For S(λ) = λ2 In×n + C we have r(S(ρ)) = ρ2 + r(C) for nonnegative ρ. Then r(C) < 1/4 implies that r(S(1/2)) < 1/2. From [7, Theorem 3.4] we obtain that Q(·) has a nonnegative right root, which proves 1. Let r(C) > 1/4. In the Frobenius normal form of C there exists an irreducible diagonal block Ckk with r(Ckk ) > 1/4. Then r(Skk (ρ)) = ρ2 + r(Ckk ) ≥ ρ + r(Ckk ) − 1/4 for all nonnegative ρ. By [7, Theorem 4.10(i)] Qkk (·) has no nonnegative right root, here Qkk (λ) = λ − (λ2 + Ckk ). By Theorem 3.1, Q(·) has no nonnegative right root, this proves 2. Now assume that r(C) = 1/4 and 1/4 is an eigenvalue of C of degree 1. By [17, Theorem 3.1] or [13, Proposition II.1.], C is (cogredient to) ⎛ ⎞ C11 0 0 ⎠, C˜ = ⎝ C21 C22 0 C31 C32 C33 where 1. C11 are C33 are possibly empty, but r(Ckk ) < 1/4 for k = 1, 3 if this is not the case, 2. C22 is block diagonal, C22 = Diag(C1 , C2 , . . . , Cp ) where p is the dimension of the eigenspace Ker(1/4I − C), Cl is irreducible and r(Cl ) = 1/4 for l = 1, 2, . . . , p.

On Reducible Nonmonic Matrix Polynomials

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From Proposition 3.8.1. we obtain Q22 (λ) = λI − (λ2 I + C22 ) = (V22 − λI)(λI − W22 ) with nonnegative W22 = Diag(W1 , W2 , . . . , Wp ), an invertible M -matrix V2 = −1 −1 Diag(V1 , V2 , . . . , Vp ) and r(W22 ) = r(Wl ) = 1/2 = r(V22 −1 ) = r(Vl −1 ) for l = 1, 2, . . . , p. From Proposition 3.8.2. we obtain for k = 1, 3 Qkk (λ) = λI − (λ2 I + Ckk ) = (Vkk − λI)(λI − Wkk ) with nonnegative Wkk , invertible M -matrices Vkk , r(Wkk ) < 1/2 and r(Vkk −1 ) < 2. This implies r(3(Vpp −1 , Wqq )) < 1 for 1 ≤ p < q ≤ 3. As in the proof of Theorem 3.2, it follows that for 1 ≤ p < q ≤ 3 the equations (3.5) have nonnegative solutions (note that in our case App = I, and Brs = Ars = 0 for r = s). Therefore Q(·) has a nonnegative right root, by Theorem 3.1. Now assume that r(C) = 1/4, and Q(·) has a nonnegative right root W . Then there exists a matrix V such that (3.13), and therefore V = In×n − W and C = V W = W V hold. By Theorem 3.1, we can and will assume that W and V have the same block structure as the Frobenius normal form of C. Further, assume that 1/4 is an eigenvalue of C of degree greater than 1. By [17, Theorem 3.1] or [13, (2.28)], there exists a principal submatrix C˜ of C such that ⎞ ⎛ 0 C11 0 ⎠, C˜ = ⎝ C21 C22 0 (3.15) C31 C33 C33 where 1. C11 and C33 are irreducible, 2. r(Ckk ) = 1/4 for k = 1, 3, 3. C22 may be empty (then C21 and C32 are also empty), but r(C22 ) < 1/4 if this is not the case, 4. (3.16) C31 + C32 C21 > 0, i.e., the matrix of the left-hand side is nonnegative and nonzero. ˜ ˜ = (V˜ − λI)(λI − W ˜ ), where W ˜ and V˜ are the Then Q(λ) = λI − (λ2 I + C) ˜ and V˜ corresponding principal submatrices of W and V , respectively. Further W have the same block structure as C˜ in (3.15). By Theorem 3.1, W31 is a nonnegative solution of V33 X − XW11 = C31 − V32 W21 . ˜ ≥ 0 and V˜ = I − W ˜. Now Vrs = −Wrs are nonpositive for 1 ≤ s < r ≤ 3, since W From Lemma 3.9.1 we obtain that W11 and W33 are nonnegative and irreducible, and V33 has a strictly positive inverse. By Proposition 3.8.1, we have r(W11 ) = 1/2 and r(V33 −1 ) = 2. Then, by Lemma 3.9.2, the last equation has a nonnegative solution if and only if its right-hand side is zero, which is equivalent to C31 = 0 and V32 W21 = 0; note that C and W are nonnegative, and V is an M -matrix.

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˜ =W ˜ V˜ , W ˜ ≥ 0 and V˜ = I − W ˜ we obtain 0 ≤ C21 = W21 V11 + From C˜ = V˜ W W22 V21 ≤ W21 V11 . Now the inverse of V11 exists and is nonnegative, therefore 0 ≤ C21 V11 −1 ≤ W21 ; in a similar way we obtain from 0 ≤ C32 = V32 W22 +V33 W32 that 0 ≤ V33 −1 C32 ≤ W32 = −V32 . But then 0 ≤ V33 −1 C32 C21 V11 −1 ≤ −V32 W21 = 0, which is equivalent to C32 C21 = 0. Together with C31 = 0 from above, we get a contradiction to (3.16). Therefore 1/4 is not an eigenvalue of C of degree greater than 1. Acknowledgment The authors wish to thank the referee for his/her careful reading and for several suggestions.

References [1] Berman, A., Plemmons, R.: Nonnegative Matrices in the Mathematical Sciences. SIAM: Philadelphia, 1994. [2] Bini, D.A., Latouche, G., Meini, B.: Numerical Methods for Structured Markov Chains, Oxford University Press, Oxford-New York, 2005. [3] Butler, G.J., Johnson, C.R., Wolkowicz, H.: Nonnegative Solutions of a Quadratic Matrix Equation Arising from Comparison Theorems in Ordinary Diﬀerential Equations. SIAM J. Alg. Disc. Meth. 6, 47–53 (1985). [4] Conway, J.B.: A Course in Functional Analysis, Springer: New York-Berlin-Heidelberg-Tokyo, 1985. [5] F¨ orster, K.-H., Nagy, B.: On the Local Spectral Theory of Positive Operators. Operator Theory, Advances and Applications 28, 71–81, Birkh¨ auser: Basel-Boston-Berlin, 1988. [6] F¨ orster, K.-H., Nagy, B.: On the Local Spectral Radius of a Nonnegative Element with Respect to an Irreducible Operator, Acta Sci. Math. 55, 155–166 (1991). [7] F¨ orster, K.-H., Nagy, B.: On Nonmonic Quadratic Matrix Polynomials with Nonnegative Coeﬃcients. Operator Theory, Advances and Applications 162, 145–163, Birkh¨ auser: Basel-Boston-Berlin, 2005. [8] F¨ orster, K.-H., Nagy, B.: Spectral Properties of Operator Polynomials with Nonnegative Coeﬃcients. Operator Theory, Advances and Applications 163, 147–162, Birkh¨ auser: Basel-Boston-Berlin, 2005. [9] Gail, H.R., Hantler, S.L., Taylor, B.A.: Spectral Analysis of M/G/1 and G/M/1 Type Markov Chains, Adv. Appl. Prob. 28, 114–165 (1996). [10] Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials, Academic Press: New York, 1982. [11] Horn, R.A., Johnson, C.R.: Matrix Analysis, Cambridge University Press: Cambridge, 1985. [12] Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis, Cambridge University Press: Cambridge, 1991. [13] Jang, R., Victory, Jr., H.D.: On the Ideal Structure of Positive, Eventually Compact Operators on Banach Lattices, Paciﬁc J. Math. 157, 57–85 (1993).

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[14] Latouche, G., Ramaswami, S.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Stochastics and Applied Probability. SIAM: Philadelphia, 1999. [15] Marcus, A.S.: Introduction to the Spectral Theory of Polynomial Operator Pencils. Translation of Mathematical Monographs, Vol. 71, Amer. Math. Soc., Providence, 1988. [16] Minc, H.: Nonnegative Matrices, Wiley: New York, 1988. [17] Rothblum, U.G.: Algebraic Eigenspaces of Nonnegative Matrices, Linear Algebra Appl. 12, 281–291 (1975). [18] Schneider, H.: The Inﬂuence of the Marked Reduced Graph of a Nonnegative Matrix on the Jordan Form and on Related Properties: A Survey. Linear Algebra Appl. 84, 169–189 (1986). [19] Tam, Bit-Shun, Schneider, H. Linear Equations over Cones and Collatz-Wielandt Numbers. Linear Algebra Appl. 363, 295–332 (2003). [20] Trampus, A.: A Spectral Mapping Theorem for Functions of Two Commuting Linear Operators, Proc. Amer. Math. Soc. 14, 893–895 (1963). K.-H. F¨ orster Technische Universit¨ at Berlin Institut f¨ ur Mathematik, MA 6-4 D-10623 Berlin, Germany e-mail: [email protected] B. Nagy University of Technology and Economics Department of Analysis Institute of Mathematics H-1521 Budapest, Hungary e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 175, 111–120 c 2007 Birkh¨ auser Verlag Basel/Switzerland

On Exceptional Extensions Close to the Generalized Friedrichs Extension of Symmetric Operators Seppo Hassi, Henk de Snoo and Henrik Winkler Abstract. If the Q-function Q corresponding to a closed symmetric operator S with defect numbers (1, 1) and one of its selfadjoint extensions belongs to the Kac class N1 then it is known that all except one of the Q-functions of S belong to N1 , too. In this note the situation that the given Q-function does not belong to the class N1 is considered. If Q ∈ Np , i.e., if the restriction of the spectral measure of Q on the positive or the negative axis corresponds to an N1 -function, then Q itself is the Q-function of the exceptional extension, and, hence, it is associated with the generalized Friedrichs extension of S. If Q or, equivalently, the spectral measure of Q is symmetric, or if the diﬀerence of Q and a symmetric Nevanlinna function belongs to the class N1 or Np , then Q is still exceptional in a wider sense. Similar results hold for the generalized Kre˘ın-von Neumann extension of the symmetric operator. Mathematics Subject Classiﬁcation (2000). Primary 47A06, 47B25; Secondary 47A57 . Keywords. Q-function, generalized Friedrichs extension, generalized Kre˘ın-von Neumann extension, Kac class.

1. Introduction Let S be a not necessarily densely deﬁned closed symmetric relation in a Hilbert space (H, (·, ·)) with defect numbers (1, 1) and let A be some canonical (i.e., dom A ⊂ H) selfadjoint extension of S with resolvent set ρ(A). Let χ(µ), µ ∈ C\R, be a nontrivial element of ker(S ∗ − µ), where S ∗ denotes the adjoint of S. Then the element χ(z) := (I + (z − µ)(A − z)−1 )χ(µ), z ∈ ρ(A), The research was partially supported by the Research Institute for Technology at the University of Vaasa. H. Winkler was supported by the “Fond zur F¨ orderung der wissenschaftlichen Forschung” (FWF, Austria), grant number P15540-N05.

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belongs to ker(S ∗ − z) and for each z ∈ ρ(A) the symmetric relation S can be recovered from A and χ(z) by S = { {f, g} ∈ A : (g − zf, χ(¯ z)) = 0 }. Let N be the set of Nevanlinna functions, i.e., the set of all functions which are z ), and map the open upper half-plane C+ into analytic on C\R, satisfy Q(z) = Q(¯ + C ∪ R. Recall that the Q-function associated with S and A is a solution (unique up to a real constant) of the equation Q(z) − Q(µ) = (χ(z), χ(µ)), z−µ ¯

z, µ ∈ ρ(A),

which implies that Q ∈ N. Each Nevanlinna function Q ∈ N which is not equal to a real constant is the Q-function of a closed symmetric relation S with defect numbers (1, 1) and a canonical selfadjoint extension A of S, see, e.g., [4]. Moreover, for a given Q ∈ N the relations S and A are uniquely determined up to isometric isomorphisms if S is completely nonselfadjoint, i.e., if there exists no nontrivial orthogonal decomposition of S such that one of the summands is selfadjoint. Note that a completely nonselfadjoint closed symmetric relation is automatically an operator. The canonical selfadjoint extensions of S can be uniquely parametrized by α ∈ (−π/2, π/2] : (A(α) − z)−1 = (A − z)−1 − χ(z)

1 (·, χ(¯ z )), Q(z) + tan α

(1.1)

where Q is the Q-function corresponding to S and A = A(π/2). If S is nonnegative, among all selfadjoint extensions of S there are two extremal nonnegative selfadjoint extensions, the so-called Friedrichs extension and the Kre˘ın-von Neumann extension. In [4] a generalization of the Friedrichs extension was introduced for a class of non-semibounded symmetric operators with defect numbers (1,1), which was further studied in, e.g., [1], cf. [2]. If one of the Q-functions of S belongs to the so-called Kac class N1 (see below), then all but one of the Q-functions belong to N1 . The Q-function which does not belong to N1 corresponds to the generalized Friedrichs extension. A corresponding characterization of the Kre˘ın-von Neumann extension is investigated in [3]. For an extension of such results to a class of Nevanlinna functions wider than N1 which is related to Sturm-Liouville equations, see [5]. A generalization of the above results in a diﬀerent direction can be found in [9] and will be considered in the present paper. In the above papers it is usually assumed that the Q-function corresponding 1 , respectively. In this note some to a given extension belongs to the class N1 or N results are presented for the cases that the given Q-function does not belong to the 1 (the precise deﬁnitions of various classes are given in Section 2). If class N1 or N p then it corresponds to the generalized Friedrichs Q belongs to the class Np or N or to the generalized Kre˘ın-von Neumann extension, respectively; see Theorems 4.2 and 5.2 below. A new situation occurs if Q is symmetric or if the diﬀerence between Q and a symmetric Nevanlinna function is a function from one of the

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above classes. Then there are at most two extensions which do not belong to the 1+ , see Theorems 4.3 and 5.3 below, cf. [9]. Nevertheless, it is classes N1+ or N also possible that there is no exceptional extension.

2. Preliminaries Let A be a canonical selfadjoint extension of S with a corresponding Q-function denoted by Q and let A(α) be the canonical selfadjoint extension of S which is determined by the identity (1.1). For α ∈ (−π/2, π/2) the Q-function Qα corresponding to A(α) and S satisﬁes Qα (z) =

1 + tan2 α Q(z) tan α − 1 = tan α − , Q(z) + tan α Q(z) + tan α

z ∈ C \ R,

(2.1)

(see [6], Proposition 4.4 and what follows). It follows from (2.1) that Im Qα (z) =

1 + tan2 α |Q(z) + tan α|2

Im Q(z).

Each function Q ∈ N has an integral representation of the form 1 λ Q(z) = bz + a + − dσ(λ), λ−z 1 + λ2 R

(2.2)

(2.3)

with a ∈ R, b ≥ 0, and a measure σ with the property that dσ(λ) < +∞. 2 R 1+λ The following identities are immediate from (2.3) dσ(λ) Im Q(iy) = by + y , y > 0, (2.4) 2 2 R λ +y and λdσ(λ) Re Q(iy) = a + (1 − y 2 ) , y > 0. (2.5) 2 + y 2 )(1 + λ2 ) (λ R Let Nγ , γ ∈ [0, 2), be the class of all functions Q ∈ N for which b = 0 in the representation (2.3) and for which the inequality dσ(λ) < +∞ (2.6) γ R 1 + |λ| is satisﬁed (see [7]). If Q ∈ N1 , the so-called Kac class, it has an integral representation of the form dσ(λ) λ Q(z) = a1 + with a1 = a − dσ(λ). (2.7) λ − z 1 + λ2 R R γ , γ ∈ [0, 2), be the class of all functions Q ∈ N with the property that Let N 1 dσ(λ) < +∞. (2.8) 2−γ |λ| −1

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1 , it has an integral representation of the form (see [3]) If Q ∈ N z 1 dσ(λ) with a ˆ1 = a + dσ(λ). (2.9) Q(z) = bz + a ˆ1 + 2 R λ(λ − z) R λ(1 + λ ) γ1 ⊂ N γ2 . Moreover, with Clearly, if 0 ≤ γ1 < γ2 < 2 then Nγ1 ⊂ Nγ2 and N 0 if and γ = 0 one has Q ∈ N0 if and only if limy→∞ yIm Q(iy) < ∞ and Q ∈ N only if limy→0+ y −1 Im Q(iy) < ∞. The remaining subclasses with γ ∈ (0, 2) can be characterized as follows (see [7], [8]): Proposition 2.1. Let γ ∈ (0, 2). A Nevanlinna function Q belongs to the class Nγ if and only if ∞ Im Q(iy) dy < +∞, (2.10) yγ 1 γ if and only if and it belongs to the class N 1 Im Q(iy) dy < +∞. (2.11) y 2−γ 0 In [9] a generalization of results from [4] and [3] led to the following two propositions: Proposition 2.2. Let Q ∈ Nγ for some γ ∈ (0, 1], and let Qα be given by the identity (2.1). Let α0 ∈ (−π/2, π/2) be the solution of tan α0 + a1 = 0, where a1 is the corresponding constant in the representation (2.7) of Q. Then Qα ∈ Nγ for each α ∈ (−π/2, π/2) \ {α0 }, whereas Qα0 , which corresponds to the generalized Friedrichs extension, does not belong to the class N2−γ : Qα0 ∈ N2−γ . If Q ∈ Nγ˜ for some γ˜ ∈ (0, γ) then Qα ∈ Nγ˜ for each α ∈ (−π/2, π/2) \ {α0 }. γ for some γ ∈ (0, 1], and let Qα be given by the Proposition 2.3. Let Q ∈ N identity (2.1). Let α0 ∈ (−π/2, π/2) be the solution of tan α0 + a ˆ1 = 0, where a ˆ1 is the corresponding constant in the representation (2.9) of Q. Then Qα ∈ Nγ for each α ∈ (−π/2, π/2) \ {α0 }, whereas Qα0 , which corresponds to the generalized 2−γ . Kre˘ın-von Neumann extension, does not belong to the class N2−γ : Qα0 ∈ N γ˜ for each α ∈ (−π/2, π/2) \ {α0 }. γ˜ for some γ˜ ∈ (0, γ), then Qα ∈ N If Q ∈ N

3. Some new subclasses of Nevanlinna functions For the sequel of the paper it is necessary to introduce some more subclasses of 1+ will be the class of all Nevanlinna functions N. First the classes N1+ and N deﬁned: 1+ := γ. N1+ := Nγ and N N 1<γ<2

Then Q ∈ N1+ if and only if

1

∞

1<γ<2

Im Q(iy) dy < +∞ y 1+δ

(3.1)

Exceptional extensions 1+ if and only if for each δ > 0, and Q ∈ N 1 Im Q(iy) dy < +∞ y 1−δ 0

115

(3.2)

for each δ > 0. In particular, N1 ⊂ N1+

1 ⊂ N 1+ , and N

and these inclusions are proper. p will be introduced. Let Q ∈ N have the Now the subclasses Np and N integral representation (2.3). Then there is a decomposition of Q in terms of the corresponding spectral function: deﬁne the functions Q+ and Q− by λ 1 − Q+ (z) := bz + a + dσ(λ) λ−z 1 + λ2 [0,∞) and Q− (z) := Q(z) − Q+ (z). Observe that Q ∈ N \ N1 if and only if either Q+ ∈ N \ N1 or Q− ∈ N \ N1 . The class of all Nevanlinna functions Q with the property that either Q+ ∈ N1 and Q− ∈ N \ N1 or Q− ∈ N1 and Q+ ∈ N \ N1 is denoted by Np . Note that 1 if and only if either Q+ ∈ N\ N 1 Np ⊂ N\ N1 . Similarly, observe that Q ∈ N\ N or Q− ∈ N \ N1 . The class of all Nevanlinna functions Q with the property that 1 and Q− ∈ N\ N 1 or Q− ∈ N 1 and Q+ ∈ N\ N 1 is denoted by N p. either Q+ ∈ N Finally, Ns stands for the class of all symmetric Nevanlinna functions Q, that is, functions Q ∈ N for which the identity Q(z) = −Q(−z) holds for all z ∈ C \ R. In particular, if Q ∈ Ns , then Re Q(iy) = 0, y > 0.

4. The case that Q ∈ N \ N1 In this section some functions in N \ N1 will be identiﬁed as exceptional functions for a family of Nevanlinna functions of the form (2.1). A ﬁrst observation concerns the limiting behavior of functions in N \ N1 . Lemma 4.1. Let Q ∈ N. If Q+ ∈ N \ N1 then limy→+∞ Re Q+ (iy) = −∞, and if Q− ∈ N \ N1 then limy→+∞ Re Q− (iy) = +∞. Proof. Assume that Q+ ∈ N \ N1 and deﬁne dσ1 (λ) :=

λ dσ(λ), 1 + λ2

λ > 0.

0∞ Then the identity 0 dσ1 (λ) = +∞ holds. Choose K > 0 and a corresponding 0l l > 0 such that 0 dσ1 (λ) ≥ K. Then for y ≥ l one has ∞ l y2 y2 K dσ (λ) ≥ dσ1 (λ) ≥ . 1 2 2 2 2 λ +y 2 0 0 l +y

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√ It follows from the identity (2.5) that for y ≥ max{ 2, l} the inequality 1 ∞ y2 K Re Q+ (iy) ≤ a − dσ1 (λ) ≤ a − 2 0 λ2 + y 2 4 holds. Hence limy→∞ Re Q+ (iy) = −∞, since K can be chosen arbitrary large. If Q− ∈ N \ N1 , the statement of the lemma can be shown in a similar way. Theorem 4.2. Let Q ∈ Np and let Qα be given by (2.1). Then Qα ∈ N1 for each α ∈ (−π/2, π/2). Proof. Assume that Q− ∈ N1 and Q ∈ N \ N1 . Let Q+α (z) := tan α − As in [4], for x < −1 one has

Q+ (x) − Q+ (−1) = [0,∞)

1 + tan2 α Q+ (z) + tan α

1 1 − λ−x λ+1

(4.1)

dσ(λ) < 0.

It follows that the Nevanlinna function Q1 (z) := −(Q+ (z)− Q+ (−1))−1 is positive on (−∞, −1). Since limy→+∞ y −1 Q1 (iy) = limy→+∞ (yQ+ (iy))−1 = 0 one can assume that Q1 has a representation of the form ∞ 1 λ dσ1 (λ). Q1 (z) = a + − λ − z 1 + λ2 −1 As in [7], for each λ ≥ −1 the function hλ (x) := (λ − x)−1 is increasing on (−∞, −1). By monotone convergence one concludes ∞ λ lim Q1 (x) = a − dσ1 (λ) ≥ 0, x→−∞ 1 + λ2 −1 hence Q1 ∈ N1 . Let tan α1 := −Q+ (−1). It follows that Qα1 + ∈ N1 , which implies by Proposition 2.2 that Q+α ∈ N1 for each α ∈ (−π/2, π/2) (see also [4]). Let c := limy→∞ Re Q− (iy) and let tan β := tan α + c + 1 for α ∈ (−π/2, π/2). Since limy→+∞ Re Q+ (iy) = −∞ by Lemma 4.1, it follows from (2.2) that for α ∈ (−π/2, π/2) and suﬃciently large y Im Q+ (iy) + Im Q− (iy) Im Qα (iy) = 2 (Re Q+ (iy) + Re Q− (iy) + tan α)2 + (Im Q− (iy) + Im Q+ (iy))2 1 + tan α Im Q+ (iy) ≤ + Im Q− (iy) (Re Q+ (iy) + c + 1 + tan α)2 + (Im Q+ (iy))2 Im Q+β (iy) + Im Q− (iy). < (4.2) 1 + tan2 β Hence Qα ∈ N1 , since Q+β ∈ N1 and Q− ∈ N1 . If Q+ ∈ N1 , the statement of the theorem can be shown in a similar way.

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Theorem 4.3. Assume that Q ∈ N \ N1 is of the form Q = Qs + Qp such that ˆ s (z) := Qs (z − xq ) Qp ∈ Np ∪ N1 and that for some xq ∈ R the shifted function Q belongs to Ns . Let Qα be given by the identity (2.1). Then: (i) if Qp ∈ N1 , there is at most one value α0 ∈ (−π/2, π/2) such that Qα ∈ N1+ for each α ∈ (−π/2, π/2) \ {α0 }; (ii) if Qp ∈ Np , then Qα ∈ N1+ for each α ∈ (−π/2, π/2). ˆ ˆ p (z) := Qp (z − xq ). Then also Q ˆ p ∈ Np ∪ N1 . Proof. Let Q(z) := Q(z − xq ) and Q ˆ ˆ ˆ s (iy) = ˆ Let Qα be given by (2.1) with Q instead of Q. Since Qs ∈ Ns , one has Re Q 0 and the identity (2.2) gives ˆ α (iy) = Im Q

ˆ (1 + tan2 α)Im Q(iy) 2 ˆ ˆ p (iy))2 (Im Q(iy)) + (tan α + Re Q

1 + tan2 α ≤ . ˆ p (iy)| 2| tan α + Re Q

(4.3)

ˆ p ∈ N1 . Let (i) Assume Qp ∈ N1 or equivalently that Q ˆ p (iy), a1 := lim Re Q y→+∞

and let α0 ∈ (−π/2, π/2) be the solution of tan α0 + a1 = 0. For α = α0 it ˆ α is uniformly bounded on the interval (i, i∞), hence follows from (4.3) that Im Q ˆ Qα ∈ N1+ . ˆ p ∈ Np . Then Lemma 4.1 shows that (ii) Assume that Qp ∈ Np or equivalently Q ˆ p (iy)| = +∞, lim |Re Q

y→+∞

ˆ α is uniformly bounded on the interval (i, i∞). and it follows from (4.3) that Im Q ˆ Hence Qα ∈ N1+ for each α ∈ (−π/2, π/2). ˆ α and therefore Qα ∈ Nγ if and only if Finally, observe that Qα (z − xq ) = Q ˆ α ∈ Nγ . This completes the proof. Q In particular, if Q ∈ N \ N1 is symmetric, i.e., if Q ∈ Ns then assertion (i) of Theorem 4.3 holds. However, it should be observed that there are functions in N \ N1 which do not serve as exceptional functions for a family of Nevanlinna functions in N1+ of the form (2.1). Consider, for example, Q = i. Then Q ∈ Ns and it is easy to see that Q ∈ N1+ \ N1 . Since Qα = i for each α ∈ (−π/2, π/2), there is no exceptional extension.

1 5. The case that Q ∈ N \ N 1 will be identiﬁed as exceptional functions In this section some functions in N \ N for a family of Nevanlinna functions of the form (2.1). Again, a ﬁrst observation 1. concerns the limiting behavior of functions in N \ N

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1 then limy→+0 Re Q+ (iy) = +∞, and if Lemma 5.1. Let Q ∈ N. If Q+ ∈ N \ N Q− ∈ N \ N1 then limy→+0 Re Q− (iy) = −∞. 1 . Choose K > 0 and a corresponding real l with Proof. Assume that Q+ ∈ N \ N √ 01 0 < l < 1/ 2 such that l dσ(λ)/λ ≥ K. If 0 ≤ y ≤ l then using 1 λ , ≥ λ2 + y 2 2λ one obtains the estimate ∞ 2 (1 − y ) 0

λ ≥ l,

λ dσ(λ) ≥ 2 2 (λ + y )(1 + λ2 )

l

1

K dσ(λ) ≥ . 8λ 8

This together with (2.5) gives K , 0 ≤ y ≤ l. 8 Hence limy→+0 Re Q+ (iy) = +∞, since K can be chosen arbitrary large. In case 1 , the statement of the lemma can be shown in a similar way. Q− ∈ N \ N Re Q+ (iy) ≥ a +

Of course, the result in Lemma 5.1 can be derived also from Lemma 4.1 by 1, ˜ means of the transform Q(z) = −Q(1/z), which connects the classes N1 and N see [3]. 1 for each p and let Qα be given by (2.1). Then Qα ∈ N Theorem 5.2. Let Q ∈ N α ∈ (−π/2, π/2). 1 and Q ∈ N \ N 1 . Let Q+α be given by (4.1). For Proof. Assume that Q− ∈ N −1 < x < 0 one has 1 1 − Q+ (x) − Q+ (−1) = dσ(λ) > 0. λ−x λ+1 [0,∞) Then the Nevanlinna function Q1 (z) := −(Q+ (z) − Q+ (−1))−1 is nonpositive on (−1, 0). Since the function Q1 is nonnegative on (−∞, −1), the support of the corresponding spectral measure σ1 on the interval (−∞, 0) must be concentrated at the point −1. Therefore Q1 has a representation of the form ∞ 1 λ σ1 ({−1}) + − Q1 (z) = bz + a − dσ1 (λ). 1+z λ−z 1 + λ2 0 For each λ ≥ 0 the function hλ (x) := (λ − x)−1 is increasing on (−1, 0). By monotone convergence one concludes ∞ dσ1 (λ) ≤ 0. lim Q1 (x) = a − σ1 ({−1}) + x→0− λ(1 + λ2 ) 0 1 . Let tan α1 := −Q+ (−1). It follows that Qα1 + ∈ N 1 , which Hence Q1 ∈ N 1 for each α ∈ (−π/2, π/2). Let c := implies by Proposition 2.3 that Qα+ ∈ N

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limy→+0 Re Q− (iy) and let tan β := tan α + c + 1 for α ∈ (−π/2, π/2). Since limy→+0 Re Q+ (iy) = +∞ by Lemma 5.1, it is seen as in (4.2) that Im Q+β (iy) Im Qα (iy) < + Im Q− (iy) 1 + tan2 α 1 + tan2 β 1 , since for α ∈ (−π/2, π/2) and suﬃciently small y. It follows that Qα ∈ N Q+β ∈ N1 and Q− ∈ N1 . Again if Q+ ∈ N1 , the statement of the theorem can be shown in a similar way. 1 is of the form Q = Qs + Qp such that Theorem 5.3. Assume that Q ∈ N \ N Qs ∈ Ns and Qp ∈ Np ∪ N1 . Let Qα be given by the identity (2.1). Then: 1 , then there is at most one value α0 ∈ (−π/2, π/2) such that (i) if Qp ∈ N 1+ for each α ∈ (−π/2, π/2) \ {α0 }; Qα ∈ N p , then Qα ∈ N 1+ for each α ∈ (−π/2, π/2). (ii) if Qp ∈ N Proof. Since Re Qs (iy) = 0, the estimate (4.3) still holds: ˆ α (iy) ≤ Im Q

1 + tan2 α . ˆ p (iy)| 2| tan α + Re Q

(5.1)

1, a Hence, if Qp ∈ N ˆ1 := limy→+0 Re Qp (iy), and α0 ∈ (−π/2, π/2) is the solution of tan α0 +ˆ a1 = 0, then with α = α0 (5.1) implies that Im Qα is uniformly bounded 1+ . On the other hand, if Qp ∈ N p then on the interval (0, i), so that Qα ∈ N ˆ Lemma 5.1 implies that limy→+0 |Re Qp (iy)| = +∞. Now Im Qα is uniformly 1+ for each α ∈ (−π/2, π/2) by (5.1). bounded on the interval (0, i) and Qα ∈ N Here again one observes that assertion (i) of Theorem 5.3 holds for symmetric 1 which do not serve 1 . Moreover, there are functions in N \ N functions in N \ N 1+ of the form as exceptional functions for a family of Nevanlinna functions in N (2.1).

References [1] 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. [2] S. Hassi, M. Kaltenb¨ ack, and H.S.V. de Snoo, “A characterization of semibounded selfadjoint operators”, Proc. Amer. Math. Soc., 125 (1997), 2681–2692. [3] S. Hassi, M. Kaltenb¨ ack, and H.S.V. de Snoo, “Generalized Kre˘ın-von Neumann extensions and associated operator models”, Acta Sci. Math. (Szeged), 64 (1998), 627– 655. [4] 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.

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[5] S. Hassi, M. M¨ oller, and H.S.V. de Snoo, “A class of Nevanlinna functions associated with Sturm-Liouville operators”, Proc. Amer. Math. Soc., 134 (2006), 2885–2893. [6] S. Hassi, H.S.V. de Snoo, and H. Winkler, “Boundary-value problems for twodimensional canonical systems”, Integral Equations Operator Theory, 36 (2000), 445– 479. [7] I.S. Kac and M.G. Kre˘ın, “R-functions-analytic functions mapping the upper halfplane into itself”, Amer. Math. Soc. Transl. (2), 103 (1974), 1–18. [8] H. Winkler, “Spectral estimations for canonical systems”, Math. Nachr., 220 (2000), 115–141. [9] H. Winkler, “On generalized Friedrichs and Kre˘ın-von Neumann extensions and canonical systems”, Math. Nachr., 236 (2002), 175–191. 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 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. 175, 121–158 c 2007 Birkh¨ auser Verlag Basel/Switzerland

On the Spectrum of the Self-adjoint Extensions of a Nonnegative Linear Relation of Defect One in a Krein Space P. Jonas and H. Langer Abstract. A nonnegative symmetric linear relation A0 with defect one in a Krein space H has self-adjoint extensions which are not nonnegative. If the resolvent set of such an extension A is not empty, A has a so-called exceptional eigenvalue α. For α = 0, ∞ this means that α is an eigenvalue in the open upper half-plane, or a positive eigenvalue with a nonpositive eigenvector, or a negative eigenvalue with a nonnegative eigenvector. In this paper we study these exceptional eigenvalues and their dependence on a parameter if the selfadjoint extensions of A0 are parametrized according to M. G. Krein’s resolvent formula. An essential tool is a family of generalized Nevanlinna functions of the class N1 and their zeros or generalized zeros of nonpositive type. Mathematics Subject Classiﬁcation (2000). Primary 47B50; Secondary 47A20, 47A55. Keywords. Linear relations in Krein spaces, nonnegative operators, operators with one negative square, selfadjoint extensions, generalized Nevanlinna functions.

1. Introduction It is well known that a nonnegative symmetric operator A0 in a Hilbert space H with defect one has self-adjoint extensions in H. They can be described, e.g., by Krein’s formula by means of a real parameter γ ∈ R (= R ∪ {∞}). If a suitable such parametrization is ﬁxed and the self-adjoint extensions of A0 are denoted by A(γ) , γ ∈ R, then there exists a nonempty open interval (γ− , γ+ ) ⊂ R, such that for γ ∈ (γ− , γ+ ) the symmetric form A(γ) ·, · on dom A(γ) has one negative square or, equivalently, the operator A(γ) has one negative eigenvalue, and for γ ∈ R\(γ− , γ+ ) The ﬁrst author was supported by the Hochschul- und Wissenschaftsprogramm des Bundes und der L¨ ander of Germany.

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the operator A(γ) is nonnegative. If for γ ∈ (γ− , γ+ ) the negative eigenvalue of A(γ) is denoted by α(γ), then α(γ) is a strictly monotonous function of γ on (γ− , γ+ ) and tends to zero or −∞ if γ tends to the boundary points γ− or γ+ . We shall illustrate this for the simple example of the nonnegative symmetd2 ric operator A0 in L2 (0, 1) generated by − 2 and the boundary conditions dx y(0) = y (0) = 0, y(1) = 0. Its self-adjoint extensions A(γ) are given by the same diﬀerential expression, and the boundary conditions y(1) = 0, y(0) − γy (0) = 0 with γ ∈ R; for γ = ∞ this boundary condition reads as y (0) = 0. Then the extension A(γ) has a negative eigenvalue α(γ) if and only if γ ∈ (−1, 0), and α(γ) is a strictly decreasing function on (−1, 0). Now let A be a self-adjoint relation in some Krein space H, [·, ·] with the property that the Hermitian sesquilinear form [·, ·]A on A, deﬁned by 2 3 f g g f := [f , g], , , ∈ A, (1.1) f g g f A has one negative square on A. Under some regularity assumptions this is equivalent to the fact that the self-adjoint relation A has exactly one so-called exceptional eigenvalue α; this is an eigenvalue of A in the open upper half-plane C+ or a negative eigenvalue with a nonnegative eigenelement or a positive eigenvalue with a nonpositive eigenelement, or α ∈ {0, ∞} and in this case to α there corresponds a neutral eigenelement with an ‘approximate associated vector’ with some sign property (see (3.3)). The main object of our studies is a nonnegative symmetric relation A0 of defect one in some Krein space H and its self-adjoint extensions in H. We make use of the parametrization A(γ) of the self-adjoint extensions of A0 by a parameter γ ∈ R considered in [JL3]. In this paper there was found an interval (γ− , γ+ ) with the property that γ belongs to this interval if and only if the Hermitian sesquilinear form [ · , · ]A(γ) has one negative square. Under an additional assumption all extensions have non-empty resolvent sets and then, according to what was said above, for γ ∈ (γ− , γ+ ) the self-adjoint relation A(γ) has one exceptional eigenvalue α(γ). The main results of this paper concern the dependence of this exceptional eigenvalue α(γ) on γ: we show that the function γ → α(γ) is continuous and that there exist two more numbers γ0 , γ∞ such that γ− ≤ γ0 ≤ γ∞ ≤ γ+ ,

(1.2)

and α(γ) = 0 for γ ∈ (γ− , γ0 ), α(γ) = ∞ for γ ∈ (γ∞ , γ+ ), and that on the interval (γ0 , γ∞ ) the function γ → |α(γ)| is nondecreasing and there is no subinterval of (γ0 , γ∞ ) on which the function γ → α(γ) is constant. We also describe the root subspace of A(γ) at a real exceptional eigenvalue = 0, ∞. Our study of the exceptional eigenvalues of the operators A(γ) is based on the study of the zeros of nonpositive type and their dependence on γ of the generalized Nevanlinna functions Fγ (z) = γz + G1 (z) + z 2 G2 (z),

γ ∈ R,

(1.3)

Self-adjoint Extensions of a Nonnegative Linear Relation

123

where G1 and G2 are Nevanlinna functions. These results may be of independent interest. For particular cases of families of generalized Nevanlinna functions the same questions were studied in [JL2] and [DaL2]. In the ﬁrst mentioned paper the results were applied to the description of the eigenvalues of nonpositive type of a family of self-adjoint operators in Pontryagin spaces with negative index one. In [DaL2] a periodic Sturm-Liouville problem with an indeﬁnite weight was studied. There a family of self-adjoint operators A(γ) in some Krein space as above arises from a self-adjoint boundary condition, which depends on some parameter, and the dependence of the exceptional eigenvalue on this parameter was described in the same way as in the more general abstract framework of the present paper. A brief synopsis is as follows. In the next section we consider the family of functions Fγ from (1.3). In Section 3 the exceptional eigenvalue of a self-adjoint relation A with the property that the Hermitian sesquilinear form [·, ·]A has one negative square is characterized in diﬀerent ways. Section 4 contains the main result about the existence of the four numbers γ− , γ0 , γ∞ , γ+ , the monotonicity of the modulus of the exceptional eigenvalue α(γ) on the interval (γ0 , γ∞ ), and the further properties of the curve γ → α(γ) as mentioned above. Finally, we describe the root subspace of A(γ) at a real exceptional eigenvalue, and we give some criteria for the coincidence of the numbers γ− and γ0 as well as of γ∞ and γ+ .

2. A family of holomorphic functions For κ ∈ N0 we denote by Nκ the set of all complex functions Q which are meromorphic in C \ R, such that Q(z) = Q(z), z ∈ hol Q (here hol Q denotes the domain of holomorphy of Q), and for which the kernel Q(z) − Q(ζ) , z−ζ

z, ζ ∈ hol Q, z = ζ,

has κ negative squares. Now let Q be a function of the class N1 . Recall that z0 ∈ R := R ∪ {∞} is called a generalized zero (generalized pole, respectively) of nonpositive type of Q if for each neighborhood U of z0 in C there exists a δU > 0 such that for 0 < ν < δU (δU < ν < ∞, respectively) the equation Q(z) = −iν has a solution z ∈ U ∩ C+ . If the function Q is holomorphic in a deleted neighborhood of its generalized zero (generalized pole, respectively) z0 of nonpositive type, then z0 is a zero (pole, respectively) of Q. Every function Q ∈ N1 has either a generalized zero of nonpositive type in R or a simple zero in the open upper half-plane C+ , see [KL]. This point, which is uniquely determined, is denoted by ζQ . Similarly, Q ∈ N1 has either a generalized pole of nonpositive type in R or a simple pole in the open upper half-plane C+ . This point, which is also uniquely determined, is denoted by ζQ ; clearly, ζQ = ζ−Q−1 . We mention that if ζQ ∈ R then it coincides with the uniquely determined point ζ ∈ R for which there exists a sequence (ζn ) ⊂ C+ converging to ζ in C such that lim inf n→∞ Im Q(ζn ) < 0. This

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is a direct consequence of the deﬁnition of ζQ mentioned above and well-known decomposition properties of functions of the class N1 . By N1∞ we denote the class of all functions Q ∈ N1 such that ζQ = ∞. This class will play a special role in this section. It was studied (with one replaced by any natural number) also in the papers [DLShZ], [DLSh], [DHdS]. The ﬁrst lemma is an easy consequence of results from [DaL1] and [L2], therefore the proof is omitted. Lemma 2.1. Let the function Q be holomorphic at zero. Then Q ∈ N1∞ if and only if it has a representation of the form 1 + tz Q(z) = z 2ρ dσ(t) + aν z ν , (2.1) t − z R ν=0 where ρ ∈ {0, 1}, σ is a bounded positive measure on R with 0 ∈ / supp σ and 2 t dσ(t) = ∞ if ρ = 1, ∈ {0, 1, 2, 3}, and aν ∈ R, ν = 0, . . . , , a = 0 if > 0, R

such that one of the following conditions is fulﬁlled: (a) ρ = 0, = 1, a1 < 0;

(b) ρ = 0, = 2;

(c) ρ ∈ {0, 1}, = 3, a3 > 0;

(d) ρ = 1, ∈ {0, 1, 2}.

The representation of a function Q ∈ N1∞ in the form (2.1) is unique. In the following, let G1 , G2 ∈ N0 , and for j = 1, 2, we write Gj in the form 1 + tz dσj (t), (2.2) Gj (z) = αj + βj z + R t−z where αj ∈ R, βj ≥ 0, and σj is a bounded positive measure on R (cf., e.g., [KK]). Theorem 2.2. The union N0 ∪ N1∞ is the set of all functions F of the form F (z) = γz + G1 (z) + z 2 G2 (z) with γ ∈ R, G1 , G2 ∈ N0 .

(2.3)

This function F belongs to the class N0 if and only if 2 β2 = 0, 1 + t dσ2 (t) < ∞, α2 = 1 + t2 dσ2 (t) − β1 . t dσ2 (t), γ ≥ R

R

R

. . Evidently, N0 ⊂ N Proof. Denote the set of functions F of the form (2.3) by N ∞ Every function F ∈ N1 can be written as a sum F = F0 +F1 where F0 ∈ N0 , F1 ∈ N1∞ , and F1 is holomorphic at zero ([L2, §2]). By Lemma 2.1, the function F1 , hence N0 ∪ N ∞ ⊂ N . belongs to N 1 , i.e., F (z) = γz + G1 (z) + z 2 G2 (z) where γ ∈ R and G1 , G2 ∈ Now let F ∈ N N0 with integral representations (2.2). Choose ∆0 := (−1, 1), ∆∞ := R \ ∆0 and deﬁne for j = 1, 2 1 + tz 1 + tz dσj (t), Gj,∞ (z) := βj z + dσj (t). Gj,0 (z) := αj + t − z ∆0 ∆∞ t − z

Self-adjoint Extensions of a Nonnegative Linear Relation Then

z G2,0 (z) = −

t dσ2 (t) − z

2

∆0

125

1 + t2 dσ2 (t)

∆0

t dσ2 (t) +

2 +z α2 − ∆0

∆0

1 + tz 2 t dσ2 (t) t−z

and F (z) = γz + G1,0 (z) + z 2 G2,0 (z) + G1,∞ (z) + z 2 G2,∞ (z) 2 2 1 + t dσ2 (t) + z α2 − t dσ2 (t) + z γ − t dσ2 (t) = − ∆0 ∆0 ∆0 1 + tz 2 + G1,0 (z) + t dσ2 (t) + G1,∞ (z) + z 2 G2,∞ (z). ∆0 t − z Hence F belongs to N0 or N1∞ if and only if the following function F∞ belongs to N0 or N1∞ , respectively: 2 2 F∞ (z) : = z γ − 1 + t dσ2 (t) + z α2 − t dσ2 (t) ∆0

=

If R

∆0

+G1,∞ (z) + z 2 G2,∞ (z) z γ + β1 − 1 + t2 dσ2 (t) + z 2 α2 − t dσ2 (t) ∆0 ∆0 1 + tz 1 + tz +z 3 β2 + dσ1 (t) + z 2 dσ2 (t). (2.4) ∆∞ t − z ∆∞ t − z

1 + t2 dσ2 (t) = ∞ then we replace the second last integral in (2.4): ∆∞

1 + tz dσ1 (t) = t−z

t−1 dσ1 (t) + z

∆∞

t−2 1 + t2 dσ1 (t)

∆∞

t−1 dσ1 (t) + z 2

+z 2

∆∞

∆∞

1 + tz −2 t dσ1 (t). t−z

It follows that F∞ is of the form (2.1) with ρ = 1 and a3 ≥ 0, therefore F∞ ∈ N1∞ . If 1 + t2 dσ2 (t) < ∞, then R

z2 ∆∞

1 + tz dσ2 (t) t−z

= −

1 + t2 dσ2 (t)

t dσ2 (t) − z ∆∞

∆∞

−z 2

t dσ2 (t) + ∆∞

∆∞

t2 (1 + tz) dσ2 (t). t−z

Inserting this expression for the last integral into (2.4) we see that F∞ has the form (2.1) with ρ = 0. Then, by Lemma 2.1, F∞ ∈ N1∞ if β2 = 0.

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It remains to consider the case β2 = 0. Then F∞ ∈ N1∞ if α2 = t dσ2 (t). R t dσ2 (t) then F∞ ∈ N1∞ if γ + β1 − If β2 = 0 and α2 = 1 + t2 dσ2 (t) < 0. If R R β2 = 0, α2 = 1 + t2 dσ2 (t) ≥ 0, then F∞ ∈ N0 . t dσ2 (t) and γ + β1 − R

R

For the rest of this section we ﬁx two functions G1 , G2 ∈ N0 and consider the family of functions Fγ , γ ∈ R, as on the right-hand side of (2.3): Fγ (z) = γz + G1 (z) + z 2 G2 (z). Deﬁne

⎧ 2 ⎪ ⎪ ⎪ (1 + t ) dσ2 (t) − β1 ⎨

γ∞ :=

if

R

⎪ ⎪ ⎪ ⎩

R

∞

(1 + t2 ) dσ2 (t) < ∞, β2 = 0, t dσ2 (t), and α2 = R

otherwise.

Then it follows that

Fγ ∈

N1

if γ ∈ (−∞, γ∞ ),

N0

if γ ∈ [γ∞ , ∞).

If γ ∈ (−∞, γ∞ ) we denote the zero of nonpositive type of Fγ in R ∪ C+ by ζ(γ) : ζ(γ) := ζFγ . Observe that γ∞ < ∞ holds if and only if 1 + t2 dσ2 (t), 1 + t2 dσ2 (t) < ∞ and G2 (z) = R R t−z which in the notation of [KK, 4] means that G2 belongs to the class (R0 ). In the following, if σ is a measure with σ({ω}) > 0 for some ω ∈ R then we set R

(t − ω)−2 dσ(t) = ∞.

Lemma 2.3. If γ ∈ (−∞, γ∞ ), then ζ(γ) = ∞ and the following statements hold: (i) ζ(γ) = 0 if and only if t−2 1 + t2 dσ1 (t) < ∞, γ ≤ σ2 ({0}) − β1 − t−2 1 + t2 dσ1 (t), R

α1 = −

R

(ii) ζ(γ) = ω ∈ R \ {0} if and only if |t − ω|−2 dσ1 (t) < ∞, R

R

t−1 dσ1 (t). R

|t − ω|−2 dσ2 (t) < ∞

(2.5)

(2.6)

Self-adjoint Extensions of a Nonnegative Linear Relation and

127

1 + tω dσ1 (t) R t − ω 1 + tω 2 + ω α2 + β2 ω + dσ2 (t) = 0, R t−ω 0 + 3ω 2 β2 + R |t − ω|−2 1 + t2 dσ1 (t) 2tω − 2tω 3 − ω 2 + 3t2 ω 2 dσ2 (t) ≤ 0. + (t − ω)2 R

γω + α1 + β1 ω +

γ + β1 + 2ωα2

(2.7)

(2.8)

The inequality (2.8) means that Fγ (ω) ≤ 0, where the derivative is to be understood in the Carath´eodory sense; the relation (2.7) means that Fγ (ω) = 0 if Fγ (ω) is the nontangential boundary value of Fγ at ω. Proof. By [L2, Corollary 2.1] the point ∞ cannot be a zero of nonpositive type of Fγ . It follows from [L2, Remark 3.2 and proof of Theorem 3.1] that ζ(γ) = ω ∈ R is a generalized zero of nonpositive type of Fγ if and only if lim η −1 Im Fγ (ω + iη) exists and is nonpositive

(2.9)

lim η −1 Re Fγ (ω + iη) = 0.

(2.10)

η↓0

and η↓0

For arbitrary z = ω + iη we ﬁnd η

−1

Im Fγ (z) =

γ + β1 + 2ωα2 + (3ω − η )β2 + |t − z|−2 1 + t2 dσ1 (t) R 2 2 2 3 + (2.11) 2ωt − ω + 3ω t − 2tω |t − z|−2 dσ2 (t) R − η2 1 + t2 − 2ωt |t − z|−2 dσ2 (t). 2

2

R

If ω = 0 and (2.9) holds then the last term on the right-hand side of (2.11) converges to −σ2 ({0}) if η → 0. Moreover, by Fatou’s lemma, −2 t dσ1 (t) < ∞, γ + β1 + t−2 1 + t2 dσ1 (t) − σ2 ({0}) ≤ 0. σ1 ({0}) = 0, R

R

Now it is easy to see that the relation (2.10) can be written as (2.5), that the inequalities in (i) imply (2.9) and that (2.5) implies (2.10). 2 If ζ(γ) = ω ∈ R \ {0}, then the last term in (2.11) converges for η → 0 to ω − 1 σ2 ({ω}), and we conclude from (2.9) and Fatou’s lemma that −2 |t − ω| dσ1 (t) < ∞, |t − ω|−2 dσ2 (t) < ∞, R

R

in particular, σ1 ({ω}) = σ2 ({ω}) = 0. Now it is easy to see that (2.9) implies (2.8) and that (2.10) implies (2.7), and also that, conversely, (2.6), (2.7) and (2.8) imply (2.9) and (2.10).

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We deﬁne γ0 as follows: ⎧ 1 + t2 ⎨ σ ({0}) − dσ1 (t) − β1 2 t2 γ0 := R ⎩ −∞

dσ1 (t) < ∞, α1 + t2 R otherwise.

if

R

dσ1 (t) = 0, t

The numbers γ0 , γ∞ satisfy the following inequalities: −∞ ≤ γ0 ≤ γ∞ ≤ ∞,

γ0 < ∞,

−∞ < γ∞ ,

and Lemma 2.3 implies that for γ ∈ (−∞, γ∞ ) the relation ζ(γ) = 0 holds if and only if γ ≤ γ0 . Evidently, if Fγ (z) = γz − cz with some c ∈ R then Fγ belongs to N1 if γ < c and to N0 if γ ≥ c, hence, for this family Fγ , γ ∈ R, we have γ0 = γ∞ = c. In the next theorem, where we study the dependence of the zero ζ(γ) of nonpositive type of Fγ on γ in the interval (−∞, γ∞ ), it will be shown that this is the only case where γ0 = γ∞ . Theorem 2.4. The equality γ0 = γ∞ holds if and only if Fγ (z) = γz − cz with some c ∈ R; in this case γ0 = γ∞ = c. If γ0 = γ∞ then the function ζ on (−∞, γ∞ ) has the following properties: (1) ζ is a continuous function, ζ(γ) = 0 if and only if γ ∈ (−∞, γ0 ], and ζ(γ1 ) = ζ(γ2 ) if γ1 = γ2 , γ1 , γ2 ∈ (γ0 , γ∞ ). (2) limγ↓γ0 ζ(γ) = 0, limγ↑γ∞ ζ(γ) = ∞. (3) ζ is real analytic at points γ ∈ (−∞, γ∞ ) with ζ(γ) ∈ / (supp σ1 ∪ supp σ2 ) with possible exception of branching points of order ≤ 3 for which ζ(γ) ∈ R. (4) σj {ζ(γ) : γ ∈ (γ0 , γ∞ )} ∩ R = 0, j = 1, 2. (5) The function |ζ|: |ζ|(γ) := |ζ(γ)| is nondecreasing; it is increasing on (γ0 , γ∞ ) if and only if Fγ is not of the form Fγ (z) = γz + α1 + β1 z + α2 z 2 with α1 α2 > 0.

1+t2 dσ2 (t) Proof. If γ0 = γ∞ , then the numbers γ0 and γ∞ are ﬁnite, β2 = 0, R t−2 1 + t2 dσ1 (t) are ﬁnite and α1 = − t−1 dσ1 (t), α2 = t dσ2 (t). and R

R

R

Moreover, 2 −2 2 1 + t dσ2 (t) = σ2 ({0}) − t 1 + t dσ1 (t) ≤ σ2 ({0}) ≤ 1 + t2 dσ2 (t), R

R

R

hence σ1 = 0, σ2 = c0 δ0 , where c0 ∈ R and δ0 is the unit mass at 0. It follows that Fγ (z) = γz + β1 z − c0 z. Conversely, evidently, Fγ (z) = γz − cz, c ∈ R, implies γ0 = γ∞ . For the rest of the proof we assume that γ0 = γ∞ . The continuous dependence of ζ(γ) on γ is an easy consequence of [J1, Corollary 3.2 and Proposition 3.5]. The

Self-adjoint Extensions of a Nonnegative Linear Relation

129

second assertion of (1) follows from Lemma 2.3: Suppose that for γ1 , γ2 ∈ (γ0 , γ∞ ) we have ζ(γ1 ) = ζ(γ2 ) =: ζ0 . Then ζ0 = 0 and the relations γj ζ0 + G1 (ζ0 ) + ζ02 G2 (ζ0 ) = 0 j = 1, 2, (understood as in (2.7)) imply γ1 = γ2 . In order to prove the ﬁrst relation of (2) we can suppose that γ0 = −∞ and consider only γ < 0. Then ζ(γ) is the generalized zero of nonpositive type of the 1 function Fγ , and the ﬁrst relation of (2) is again a consequence of [J1, Corollary |γ| 3.2 and Proposition 3.5]. In order to prove the second relation of (2) we introduce the function 2 (z) 1 (z) + z 2 G Fγ (z) := z 2 F−γ − z −1 = γz + G 1 (z) := G2 − z −1 , G 2 (z) := G1 − z −1 , and we write with G j (z) = α G j + βj z + (1 + tz)(t − z)−1 d σj (t), j = 1, 2, R

where α j , βj and σ j have the same properties as αj , βj and σj . Deﬁne γ 0 , γ ∞ for Fγ in the same way as γ0 and γ∞ were deﬁned for Fγ . Then [ADo, Section 2] yields the relations α 1 = α2 , β1 = σ2 ({0}), σ 1 ({0}) = β2 , d σ1 (t) = dσ2 (−t−1 ) on R \ {0}, and also the same relations with indices 1 and 2 interchanged. Further, an easy ∞ = −γ0 , and if γ ∈ (γ0 , γ∞ ) the generalized computation shows that γ0 = −γ∞ , γ = −ζ(−γ)−1 . Therefore, the relation zero ζ(γ) of nonpositive type of Fγ is ζ(γ) limγ↓γ0 ζ( γ ) = 0 implies limγ↑γ∞ ζ(γ) = +∞. Statement (3) follows from (2.7), (4) is a consequence of the relations (2.6) and [JL2, Lemma 2.1]. It remains to prove (5). We ﬁrst show that the function |ζ| is nondecreasing in (γ0 , γ∞ ). To this end we observe that there exist a sequence of real numbers bn and a sequence of compactly supported positive measures τn on R, such that the sequence of functions Hn : Hn (z) := bn z + (t − z)−1 dτn (t), n = 1, 2, . . . , R

converges for n → ∞ locally uniformly on C+ to G1 (z) + z 2 G2 (z). Indeed, it is easy to see that the functions n n 1 + tz 1 + tz 2 dσ1 (t) + z α2 + β2 z + dσ2 (t) , n = 1, 2, . . . , α1 + β1 z + −n t − z −n t − z converge for m → ∞ locally uniformly on C+ to G1 (z) + z 2 G2 (z) for n → ∞. Since for arbitrary α ∈ R and β ≥ 0 the functions −1 β 2 |α| m (sign α)m − z + m (m − z)−1 + (−m − z)−1 , m = 1, 2, . . . , 2

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converge locally uniformly to the function α + βz, there exist sequences of com (t), σ2,n (t), such that the functions pactly supported positive measures σ1,n (t − z)−1 dσ1,n (t) + z 2 (t − z)−1 dσ2,n (t) R

= R

R

−1

(t − z)

dσ1,n (t)

−z

R

dσ2,n (t)

−

R

t dσ2,n (t)

+ R

(t − z)−1 t2 dσ2,n (t)

converge locally uniformly to G1 (z) + z 2 G2 (z). If we approximate the constants as above by Nevanlinna functions we get a sequence of functions Hn with the desired properties. Consider γ ∈ (γ0 , γ∞ ). Since Fγ ∈ N1 , from the deﬁnition of the class N1 it follows that there exist a natural number n0 and an ε > 0 with ∆ := [γ −ε, γ +ε] ⊂ (γ0 , γ∞ ), such that for γ ∈ ∆ and n > n0 the functions Fγ,n : Fγ,n (z) := γz + Hn (z) belong to N1 . Denote the zero of nonpositive type of Fγ,n by ζn (γ). Then, according to [JL2, Theorem 3.1], for n > n0 the functions |ζn | are nondecreasing on ∆. By [J1, Corollary 3.2 and Proposition 3.5], limn→∞ ζn (γ) = ζ(γ), γ ∈ ∆, therefore also the function |ζ| is nondecreasing on ∆. For the proof of the second statement of (5) it is suﬃcient to consider the function |ζ| on an interval (γ1 , γ2 ) such that ζ(γ) ∈ C+ if γ ∈ [γ1 , γ2 ]. Suppose ﬁrst that σ1 = 0 or that (supp σ2 ) \ {0} = ∅. Then, by the above construction of Hn , we may assume that for each bounded interval ∆ there exists an n1 such that for n ≥ n1 the measure τn , restricted to ∆, is not zero and does not depend on n. Applying [JL2, (3.6)] and the reasoning which follows that relation, we see that there exists an a > 0 such that d|ζn (γ)| ≥ a if n ≥ n1 , inf dγ γ∈[γ1 ,γ2 ] which implies that the function |ζ| is increasing. Let now σ1 = 0 and supp σ2 ⊂ {0}. Then Fγ (z) = γz + α1 + β1 − σ2 ({0}) z + α2 z 2 + β2 z 3 . Set β := β1 − σ2 ({0}), suppose β2 > 0 and let (γ1 , γ2 ) be as above. Then a simple computation, starting from Fγ (ζ(γ)) = 0, yields d log |ζ(γ)| dγ

= =

d ζ(γ) Im ζ(γ)−1 dγ −2 β2 β2 Im ζ(γ) − i α2 + 3β2 Re ζ(γ) > 0

for all γ ∈ (γ1 , γ2 ). Therefore the function |ζ| is strictly increasing on (γ0 , γ∞ ). Finally, the case Fγ (z) = γ(z) + α1 + β z + α2 z 2 can be considered by straightforward calculations.

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For γ1 ∈ (γ0 , γ∞ ) and ζ1 = ζ(γ1 ) ∈ R \ (supp σ1 ∪ supp σ2 ∪ {0}), in a neighborhood of ζ1 the graph of the function ζ can be described similarly to [JL2, Proposition 3.2]. To this end we denote by a± the directions of the (one-sided) tangents of the curve γ → ζ(γ) at γ1 : a± := ±

lim

γ→γ1 ±0

ζ(γ) − ζ(γ1 ) . |ζ(γ) − ζ(γ1 )|

(2.12)

Then one of the following four cases prevails for some ε > 0: (rr) (cr) (rc) (cc)

ζ(γ) ∈ R if |γ − γ1 | < ε. ζ(γ) ∈ C+ if γ ∈ (γ1 − ε, γ1 ), ζ(γ) ∈ R if γ ∈ (γ1 , γ1 + ε), and a− = −i. ζ(γ) ∈ R if γ ∈ (γ1 − ε, γ1 ), ζ(γ) ∈ C+ if γ ∈ (γ1 , γ1 + ε), and a+ = i. ζ(γ) ∈ C+ if 0 < |γ − γ1 | < ε and a− = (sign γ0 ) exp (−(sign γ0 )iπ/3) , a+ = (sign γ0 ) exp ((sign γ0 )iπ/3) .

If we denote the inverse function of ζ by γ, that is, γ(ζ(γ)) = γ for γ ∈ (γ0 , γ∞ ), then γ(ζ) = −ζ −1 G1 (ζ) + ζ 2 G2 (ζ) . The case (rr) holds if and only if γ (ζ1 ) = 0, in the other cases we have γ (ζ1 ) = 0 and, additionally, γ (ζ1 ) > 0, < 0, = 0, respectively, in the case (cr), (rc) and (cc), respectively. The graph of the function ζ ‘visits’ an open interval I with 0 ∈ / I and such that G1 and G2 are holomorphic on I at most once. This is a consequence of the following theorem. Theorem 2.5. If I is a component of the open set R \ (supp σ1 ∪ supp σ2 ∪ {0}), then the set I ∩ ζ (γ0 , γ∞ ) is empty or connected. Proof. If Fγ (z) = γz + α1 + β z + α2 z 2 , β := β1 − σ2 ({0}),

(2.13)

the theorem can be proved by a straightforward computation. Suppose that Fγ is not of the form (2.13). Then the third derivative of the function F (ζ) := −ζγ(ζ) = G1 (ζ) + ζ 2 G2 (ζ) is positive in real points of holomorphy of F . We consider the case I ⊂ (0, ∞), , γ∞ ) is not the case I ⊂ (−∞, 0) can be treated similarly. Assume that I ∩ ζ (γ 0 connected. Then there exist points ζ1 , ζ2 ∈ I ∩ ζ (γ0 , γ∞ ) , ζ1 < ζ2 , such that γ (ζ1 ) = γ (ζ2 ) = 0,

γ (ζ1 ) ≤ 0,

γ (ζ2 ) ≥ 0.

therefore F (ζ1 ) ≥ 0, F (ζ2 ) ≤ 0. But this is impossible since F (ζ) is positive on (ζ1 , ζ2 ).

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3. The exceptional eigenvalue of a self-adjoint relation with one negative square

Let H, [ · , · ] be a Krein space, and let · be a Hilbert norm on H such that [·, ·] is · -continuous. A closed linear relation A in H is a closed linear subspace of H2 . In the sequel a closed linear A in H is identiﬁed with the closed operator f linear relation given by its graph : f ∈ D(A) . For the deﬁnitions of the Af domain D(A), the kernel ker A, the range ran A, the adjoint A+ , the resolvent set ρ(A), the spectrum σ(A) and the point spectrum σp (A) we refer to [DdS1], [DdS2], [JL3]. The extended spectrum σ (A) is deﬁned as follows: σ(A) if 0∈ / σ A−1 , σ (A) := σ(A) ∪ {∞} if 0 ∈ σ A−1 . 0 If A is not an operator, that is A(0) = g : ∈ A = {0}, then ∞ is called an g eigenvalue of A with the nonzero elements of A(0) as corresponding eigenvectors. The set if A(0) = {0}, σp (A) σ p (A) := σp (A) ∪ {∞} if A(0) = {0} is called the extended point spectrum of A. The linear relation A in the Krein space H is called symmetric if the sesquilinear form [·, ·]A (see (1.1)) is Hermitian, that is f g , ∈ A, [f , g] = [f, g ], f g and A is called self-adjoint if for elements g, g ∈ H the relation f [f , g] = [f, g ] for all ∈A f g yields ∈ A, see, e.g., [DdS1], [DdS2]. The symmetric linear relation A in H g is called nonnegative if 2 3 f f f ≡ [f , f ] ≥ 0, , ∈ A, f f f A and it is said to have one negative square if the Hermitian sesquilinear form 2 3 f f g g ≡ [f , g], , , ∈ A, (3.1) f f g A g has this property. Let A be a self-adjoint relation with ρ(A) = ∅, let µ ∈ ρ(A) and denote by { · , ·} the following Hermitian sesquilinear form on H: {f, g} := I + µ(A − µ)−1 f, (A − µ)−1 g , f, g ∈ H. (3.2)

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133

For κ ∈ N0 and a Krein space H, by Nκ (H) we denote the set of all functions Q with values in L(H) (the set of bounded linear operators in H), which are meromorphic in C \ R and such that Q(z) = Q(z)+ for all z ∈ hol (Q) and that the kernel Q(z) − Q(ζ)+ , z, ζ ∈ hol (Q), z−ζ has κ negative squares; here + denotes the Krein space adjoint. Then the following statements are equivalent: (a) A is nonnegative (has one negative square, respectively). (b) For some (and hence for all) µ ∈ ρ(A) the Hermitian sesquilinear form {·, ·} on H is nonnegative (has one negative square, respectively). (c) The Krein space operator function z → z(A−z)−1 belongs to N0 (H) (N1 (H), respectively). In particular, the fact that for a nonnegative self-adjoint relation A with ρ(A) = ∅ the operator function in (c) is a Nevanlinna function implies that in this case C \ R ⊂ ρ(A). The main result of this section is the following theorem. Theorem 3.1. Let A be a self-adjoint relation with one negative square in the Krein space H such that ρ(A) = ∅. Then exactly one of the following two statements holds true: (i) The non-real spectrum of A consists of one pair of complex conjugate points α ∈ C+ and α which are simple normal eigenvalues and such that the twodimensional linear span of the corresponding eigenvectors is a non-degenerate indeﬁnite subspace. In the orthogonal complement of this span the relation A induces a nonnegative self-adjoint linear relation with non-empty resolvent set. (ii) The spectrum of A is real and there exists a unique point α ∈ R with the following properties: If α = 0, ∞ then it is an eigenvalue of A with an eigenelement e0 such that (sgn α) [e0 , e0 ] ≤ 0; if α = 0 then α is an eigenvalue of A with a neutral eigenelement e0 and en there exists a sequence ⊂ A such that en − e0 → 0 and en [en − em , en − em ] → 0,

lim [en , en ] ≤ 0 if m, n → ∞; (3.3) en if α = ∞ there exists a sequence ⊂ A such that en − e0 → 0 if en n → ∞ for some nonzero neutral element e0 ∈ A(0) and (3.3) holds. n→∞

In the proof of this theorem we need the notion of a deﬁnitizable self-adjoint relation in a Krein space and its spectral function. The self-adjoint relation A in the Krein space H is called deﬁnitizable if ρ(A) = ∅ and if there exists a real

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polynomial p such that the self-adjoint relation p(A) is nonnegative (see [DdS2]). Obviously, a nonnegative self-adjoint relation A with ρ(A) = ∅ is deﬁnitizable. The self-adjoint relation A with µ ∈ ρ(A), µ = µ, is deﬁnitizable if and only if the unitary operator U in the Krein space H deﬁned by U := −I + (µ − µ)(A − µ)−1 ,

(3.4)

i.e., the Cayley transform of A, is deﬁnitizable in the sense of [L1]. The spectrum of a deﬁnitizable unitary operator U lies on the unit circle T with possible exception of a ﬁnite set of eigenvalues, which is symmetric with respect to T. On T the operator U has a spectral function with possibly a ﬁnite number of critical points, see [L1]; the set of these critical points is denoted by c(U ). The Cayley transform U of the deﬁnitizable self-adjoint relation A with µ ∈ ρ(A) can be written as U = ψ(A) with ψ(z) := −(z − µ)(z − µ)−1 .

(3.5)

According to [DdS2, Proposition 3.2] we have ψ( σ (A)) = σ(U ) and hence σ (A) lies on R with possible exception of a ﬁnite set of complex eigenvalues, which is symmetric with respect to R. If EU denotes the spectral function of U then the spectral function EA of A is deﬁned by the relation EA (∆) = EU (ψ(∆)) , where ∆ ∈ B(A). Here B(A) denotes the Boolean algebra generated by the connected subsets of R (R is considered to be homeomorphic to the unit circle) for which the endpoints are not in c(A) := ψ −1 (c(U )); this set c(A) is called the set of critical points of A. The spectral function EA deﬁnes a homomorphism from B(A) into the set of self-adjoint projections in H such that for all ∆ ∈ B(A) the following holds: 1. EA R = 1 − E0 , where E0 is the Riesz–Dunford projection corresponding to the set of nonreal eigenvalues of A. 2 2 2. A is the direct sum of the three subspaces A ∩ (E0 H) , A ∩ EA (∆)H , and 2 A∩ EA R \ ∆ H of H2 . 2 3. σ A ∩ EA (∆)H ⊂ ∆. The point λ ∈ σ(A) is said to be a spectral point of positive (negative, respectively) type of the deﬁnitizable self-adjoint relation A if for some open set ∆ ∈ B(A) such that λ ∈ ∆ the set EA (∆)H \ {0} consists of positive (negative, respectively) elements or, equivalently, the space (EA (∆)H, [ · , · ]) is a Hilbert space (anti-Hilbert space, respectively). The spectral function EA has also the following properties: 4. If λ ∈ σ(A) and for a deﬁnitizing polynomial p of A we have p(λ) > 0 (< 0, respectively) then λ is a spectral point of positive (negative, respectively) type of A. 5. λ ∈ c(A) if and only if for all ∆ ∈ B(A) with λ ∈ ∆ the range EA (∆)H contains positive as well as negative elements.

Self-adjoint Extensions of a Nonnegative Linear Relation In the following lemma we set ϕ(U ) := (2 Im µ)−2 µ(1 + U ) + µ(1 + U −1 ) = (A − µ)−1 A(A − µ)−1 .

135

(3.6)

Then the Hermitian sesquilinear form in (3.2) becomes {f, g} = [ϕ(U )f, g],

f, g ∈ H.

Lemma 3.2. Let A be a self-adjoint relation with one negative square in the Krein space H, suppose that ρ(A) = ∅, and choose µ ∈ ρ(A) ∩ C+ . Then A and also its Cayley transform U = ψ(A) (see (3.4)) are deﬁnitizable. There exists a unique point β ∈ C, 0 < |β| ≤ 1, such that with the operator ϕ(U ) from (3.6) the relation (3.7) ϕ(U )(U − β)(U −1 − β)x, x ≥ 0 for all x ∈ H holds. The point β is an eigenvalue of U with a [ϕ(U ) · , · ]-nonpositive eigenelement e, and α = ψ −1 (β) is an eigenvalue of the self-adjoint relation A with the same eigenelement e. −1 If |β| < 1, that is α ∈ C+ , then σ(U ) \ T = β, β and σ(A) \ R = {α, α}, β and α are simple eigenvalues of U and A, respectively, the corresponding eigenvectors of U and A coincide and the two-dimensional subspace −1 = ker (A − α) + ker (A − α) ker (U − β) + ker U − β is non-degenerated and indeﬁnite. If |β| = 1, that is if α ∈ R then σ(U ) ⊂ T and σ(A) ⊂ R. For ∆ ∈ B(A) it holds α ∈ ∆ (α ∈ / ∆, respectively) if and only if the Hermitian sesquilinear form [·, ·]A has one negative square (is positive semi-deﬁnite, respectively) on A ∩ 2 EA (∆)H . := H/ ker ϕ(U ). Since ker ϕ(U ) is the Proof. We consider the factor space H {·, ·} isotropic subspace of the Hermitian sesquilinear form (3.2), the space H, is a non-degenerated inner space with negative index one. We complete product it to a Pontryagin space H, { · , · } with one negative square. The operator U in H, which maps H onto itself and leaves the induces a bounded operator U is a unitary operator. inner product { · , · } invariant. The closure U of U in H ), |β| ≤ 1, with According to Pontryagin’s Theorem there exists a point β ∈ σp (U −1 a nonpositive eigenelement e. The relation ran U − β , e = {0} implies −1 −β U −β x U , x ≥ 0 for all x ∈ H, consequently, the relation (3.7) holds and U is a deﬁnitizable operator. The point β is an eigenvalue of U since otherwise the range of U − β would be dense in H and hence [ϕ(U )x, x] ≥ 0 for all x ∈ H, which is impossible since this form has one negative square. On ker (U − β) we have [ϕ(U ) · , · ] = ϕ(β)[ · , · ]. If this form [⊥] would be positive deﬁnite there, then ker (U − β)+ (ker (U − β)) would be dense

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P. Jonas and H. Langer [⊥]

in H. Since, by (3.7), [ϕ(U ) · , · ] is positive semi-deﬁnite on (ker (U − β)) there exists an x ∈ ker (U − β) with [ϕ(U )x, x] < 0, a contradiction. Therefore there exists a [ϕ(U ) · , · ]-nonpositive nonzero element e ∈ ker (U − β). The point β, |β| ≤ 1, with the property (3.7) is uniquely determined. Indeed, if there would be a point β = β with the same properties as β, then also ker(U −β ) would contain a nonpositive element and this element is orthogonal to e, which is impossible since the inner product [ϕ(U ) · , · ] has only one negative square. This argument also proves that β with |β| < 1 is a simple eigenvalue of U . According to the spectral mapping theorem α = ψ −1 (β) is an eigenvalue of A with the same root subspace as that of U at β. For β ∈ T or α ∈ R it follows as above that σ(U ) ⊂ T and σ(A) ⊂ R. If ∆ ∈ / ψ(∆), EU (ψ(∆))H = (U − β)EU (ψ(∆))H B(A) such that α ∈ / ∆, then β = ψ(α) ∈ and we get from (3.7) (A − µ)−1 A(A − µ)−1 EA (∆)x, EA (∆)x = ϕ(U )EU (ψ(∆))x, EU (ψ(∆))x ≥ 0. From this relation the last claim of the lemma follows easily.

Proof of Theorem 3.1. First we shall show that the point α of Lemma 3.2 has the properties stated in the theorem. Evidently, if α ∈ C+ then the case (i) prevails. If α ∈ R\{0}, it follows from Lemma 3.2 that the corresponding point β = ψ(α) is an eigenvalue of U and α is an eigenvalue of A with the same [ϕ(U ) · , · ]-nonpositive eigenelement e, and hence 0 ≥ [ψ(U )e, e] = (A − µ)−1 A(A − µ)e, e = |α − µ|2 α [e, e]. It follows that [e, e] is nonpositive if α > 0 and nonnegative if α < 0. It remains to consider the cases α = 0 and α = ∞. We restrict ourselves to the ﬁrst case, for α = ∞ a similar reasoning applies. If α = 0 we choose a bounded open interval ∆ ∈ B(A) with 0 ∈ ∆ and consider in the Krein space H∆ := E(∆)H the restriction A∆ := A|H∆ , which is a bounded self-adjoint operator. Repeating in fact considerations in the proof of ∆ := H∆ / ker A∆ with the inner product Lemma 3.2, we equip the factor space H f, g ∆ := A∆ f, g , f ∈ f , g ∈ g, which has one negative square. The completion ∆ . of H∆ , which is a Pontryagin space with negative index one, is denoted by H The operator A∆ induces bounded self-adjoint operators A∆ and A∆ in H∆ and ∆ , respectively, and since in H ∆ all the nonzero spectral points of A ∆ are of H ∆ with { x, x }∆ ≤ 0. For positive type, there exists a nonzero element x ∈ ker A ∆ this means that there exists a Cauchy sequence ( ∆ which the space H xn ) ⊂ H n → 0 in H∆ and lim { xn , x n } ≤ 0 does not converge to zero and is such that A∆ x if n → ∞. That ( xn ) is a Cauchy sequence means that m , x n − x m }∆ −→ 0, { xn − x

{ xn − x m , y}∆ −→ 0

∆ , or, equivalently, for m, n → ∞ and all y ∈ H A∆ (xn − xm ), xn − xm −→ 0, A∆ (xn − xm ), y −→ 0

(3.8)

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137

n . By the second relation in (3.8), the for m, n → ∞ and all y ∈ H∆ , xn ∈ x elements A∆ xn converge weakly to some e0 ∈ H∆ , and limn→∞ [A∆ xn , y0 ] = 0 for some y0 ∈ H∆ implies e0 = 0, and limn→∞ [A2∆ xn , y] = 0 for all y ∈ H∆ implies A∆ e0 = 0. Finally, limn→∞ [A∆ xn , xn ] ≤ 0. Now we replace the sequence (xn ) by the sequence (en ) of arithmetic means of a subsequence of (xn ) such that A∆ en converges strongly to e0 . Then the sequence (en ) has all the properties of the sequence (xn ) mentioned above, and therefore en −e0 → 0 if n → ∞ and the relations in (3.3) hold. Since e0 ∈ ker A∆ ∩ ran A∆ , the element e0 is neutral. It remains to be shown that any α ∈ R, which is diﬀerent from α in Lemma 3.2, does not satisfy the condition (ii) of Theorem 3.1 with α replaced by α . By Lemma 3.2 there exists a connected open set ∆ ∈ B(A) with α ∈ ∆ such that the form [·, ·]A is positive semi-deﬁnite on A ∩ (EA (∆)H)2 . Let α = 0, ∞ and assume, in addition, that 0, ∞ ∈ / ∆. Then for any f0 ∈ ker (A − α ), f0 = 0, we have α [f0 , f0 ] > 0 and hence (ii) with α replaced by α does not hold. Now let α = 0 and assume in addition that ∆ is bounded. Suppose that there fn exist an f0 ∈ ker A, f0 = 0, and a sequence ⊂ A such that fn − f0 → 0 fn for n → ∞ and the following limit exists and is nonpositive: lim fn , fn ≤ 0. (3.9) n→∞

We claim that lim

n→∞

(I − EA (∆)) fn , fn = 0.

Indeed, consider y ∈ H. Then y := (I − EA (∆)) y ∈ ran A, and if relation

(3.10) u ∈ A the y

y, (I − EA (∆)) fn = y, fn = u, fn , n = 1, 2, . . . , and the boundedness of the sequence fn imply that sup I − EA (∆) fn : n = 1, 2, . . . < ∞.

From this inequality and 4 4 4 4 lim 4 I − EA (∆) fn 4 = 4 I − EA (∆) f0 4 = 0 n→∞

the relation (3.10) follows. Now (3.9) and (3.10) imply lim [EA (∆)fn , EA (∆)fn ] = lim [EA (∆)fn , fn ] ≤ 0,

n→∞

n→∞

2 and, since [·, ·]A is positive semi-deﬁnite on A ∩ EA (∆)H , we ﬁnd lim [EA (∆)fn , EA (∆)fn ] = 0.

n→∞

(3.11)

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If J∆ is a fundamental symmetry of the Krein space (EA (∆)H, [·, ·]) then for n = 1, 2, . . . EA (∆)fn , J∆ EA (∆)fn 1/2 1/2 ≤ EA (∆)fn , EA (∆)fn AJ∆ EA (∆)fn , J∆ EA (∆)fn . This relation and (3.11) imply that EA (∆)fn → 0 and hence f0 = 0, a contradiction. For α = ∞ a similar reasoning applies.. Remark 3.3. The proof of Theorem 3.1 shows that this theorem remains true if (3.3) is replaced by limn→∞ en , en ≤ 0. In order to explain Theorem 3.1 we ﬁrst mention that for a nonnegative selfadjoint relation A with nonempty resolvent set the positive spectral points are of positive type and the negative spectral points are of negative type. In particular, all the eigenvectors corresponding to positive eigenvalues are positive and all eigenvectors corresponding to negative eigenvalues are negative, and there is no non-real spectrum. If A has one negative square, it has either a pair α, α of simple non-real complex conjugate eigenvalues, or on the real axis there is a positive eigenvalue α with a nonpositive eigenvector, or a negative eigenvalue α with a nonnegative eigenvector, or at 0 or ∞ there is a neutral eigenvector to which there corresponds an ‘approximate associated vector’ with a certain sign property, see (3.3). The following deﬁnition of the exceptional eigenvalue will play an important role in this paper. Deﬁnition 3.4. Let A be a self-adjoint relation with ρ(A) = ∅ and one negative square in the Krein space H. The point α ∈ C+ ∪ R from Theorem 3.1 is called the exceptional eigenvalue of A and it is denoted in the following by α(A). In the following theorem we give some equivalent descriptions of the exceptional eigenvalue α(A). For the deﬁnition of a subset of R of positive type with respect to an operator function in the class N1 (H), which will be used in the formulation and the proof of this theorem, we refer the reader to [J2, § 3.1]. Theorem 3.5. Consider the self-adjoint relation A with one negative square and ρ(A) = ∅ in the Krein space H, and let µ ∈ ρ(A) \ R, α ∈ C+ ∪ R. Then each of the following conditions (i–iv) is equivalent to the fact that α is the exceptional eigenvalue of A. (i) If A has an eigenvalue in C+ then this eigenvalue is α. If σ(A) is real then α is the unique point in R such that for all open sets ∆ ∈ B(A) with α ∈ ∆ the 2 self-adjoint linear relation A ∩ EA (∆)H is not nonnegative in EA (∆)H. (ii) For the rational function r, r(t) := ψ(t) − ψ(α) ψ(t)−1 − ψ(α) , we have (A − µ)−1 A(A − µ)−1 r(A)x, x ≥ 0, x ∈ H.

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(iii) α is a pole in C+ of the operator function z → z(A − z)−1 , or α ∈ R and there exists an x ∈ H and a sequence (zn) ⊂ C+ converging to α (in C) such that lim inf n→∞ Im zn [(A − zn )−1 x, x] < 0. (iv) α is a pole in C+ of the operator function z → z(A − z)−1 , or α ∈ R and α does not belong to an open subset of R of positive type with respect to the operator function z → z(A − z)−1 . Proof. It was proved in Lemma 3.2 that (i) and (ii) are equivalent to the fact that α is the exceptional eigenvalue of A. For the proof that (i), (iii), and (iv) are equivalent it is suﬃcient to consider only the case α ∈ R. Therefore in the rest of this proof we assume that σ(A) ⊂ R. Suppose that (i) holds. Then there exists an x ∈ H such that the function Gx : z → z (A − z)−1 x, x belongs to the class N1 . If α ∈ R is the generalized pole of nonpositive type of Gx then there exists a sequence (zn ) ⊂ C+ which converges to α (in C) and for which (3.12) lim inf Im Gx (zn ) < 0. n→∞

Suppose that α = α . Let ∆ be an open subset of R with ∆ ∈ B(A) such that α ∈ ∆ and α ∈ / ∆. Then (3.13) Gx (z) = z EA (∆)(A − z)−1 x, x + z EA (R \ ∆)(A − z)−1 x, x , and by (i) the function z → EA (R \ ∆)(A − z)−1 x, x belongs to the class N0 . This is in contradiction with (3.12) since the ﬁrst term on the right-hand side of (3.13) is holomorphic at α . Therefore α = α and (iii) holds. Assume now that (iii) holds. Then for every open ∆ ∈ B(A) containing α and for the element x ∈ H from (iii) the function Gx belongs to N1 . Therefore 2 A ∩ EA (∆)H is not nonnegative. If the point α in (i) would not be unique then A would have more than one negative square, which is not the case. This shows that (i) holds. By assumption of Theorem 3.5 the function G : z → z(A − z)−1 belongs to N1 (H). Then on account of the additional assumption σ(A) ⊂ R there exists a unique point α0 ∈ R such that R \ {α0 } is of positive type with respect to G and every open subset ∆0 of R with α0 ∈ ∆0 is not of positive type with respect to G. With this point α0 the statement (iv) is equivalent to α = α0 . Now [J3, Lemma 3.8, (a)⇔(c)] implies that (iii) and (iv) are equivalent. Remark 3.6. For α ∈ R the conditions (i)–(iv) are equivalent to the fact that α is the generalized pole of nonpositive type of the function G : z → z(A − z)−1 in the following sense: For each neighborhood U of α in C there exists a δU > 0 such that for every µ with δU < µ < ∞ there exists a z ∈ U ∩ C+ such that G(z) + iµJ is not injective; here J is an arbitrary ﬁxed fundamental symmetry of H. That this condition is equivalent to (iv) follows from the similar statement [J1,

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Proposition 3.5] for generalized Carath´eodory functions by means of a fractional linear transformation of the independent variable. Finally we prove a continuity property of the exceptional eigenvalue. Theorem 3.7. Let A and An , n = 1, 2, . . . , be self-adjoint relations with one negative square in the Krein space H. Suppose that there exists a point µ ∈ C \ R and an open neighborhood U of µ such that U ⊂ ρ(An ) for n = 1, 2, . . . , U ⊂ ρ(A), and that lim (An − µ)−1 = (A − µ)−1 ,

n→∞

lim (An − µ)−1 = (A − µ)−1

n→∞

(3.14)

in the strong operator topology. Then for the exceptional eigenvalues it holds lim α(An ) = α(A).

n→∞

(3.15)

Proof. We consider the sequence of exceptional eigenvalues α(An ). Let α Anν be a subsequence which converges in C+ ∪ R with respect to the topology of the closed complex plane; denote the limit by α0 . From (3.7) we obtain 5 6 ϕ ψ(Anν ) ψ(Anν ) − ψ(α(Anν )) ψ(Anν )−1 − ψ(α(Anν ) f, f ≥ 0, f ∈ H, where ψ is as in (3.5). If we pass to the limit ν → ∞ it follows easily from the assumptions (3.14) and α(Anν ) → α0 that 5 6 ϕ ψ(A) ψ(A) − ψ(α0 ) ψ(A)−1 − ψ(α0 ) f, f ≥ 0 for all f ∈ H. Since, on the other hand, the point β = ψ(α) in (3.7) is uniquely determined we must have α0 = α(A). Therefore each convergent subsequence of the sequence α(An ) has the same limit and (3.15) follows. Remark 3.8. It is easy to see that Theorem 3.7 remains true if the assumption (3.14) is replaced by lim (An − µ)−k = (A − µ)−k ,

n→∞

lim (An − µ)−k = (A − µ)−k for k = 1, 2,

n→∞

with the limits existing only in the weak operator topology.

4. The exceptional eigenvalues of the self-adjoint extensions of a nonnegative closed linear relation of regular defect one 4.1. The family of self-adjoint extensions We recall some notions and results from [JL3]. Let A0 be a closed nonnegative linear relation in the Krein space (H, [·, ·]). By r(A0 ) we denote the set of all points of regular type of A0 , i.e., the set of all z ∈ C for which there exists a cz > 0 such that f f − zf ≥ cz f , ∈ A0 . f

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A0 is said to be of regular defect one if there exist non-real points z0 , z¯0 ∈ r(A0 ) [⊥] [⊥] with dim ran(A0 − z0 ) = dim ran (A0 − z 0 ) = 1. By [JL3, Theorem 3.2] the closed nonnegative relations of regular defect one are precisely those closed nonnegative relations of defect one which have at least one self-adjoint extension with nonempty resolvent set; here and in the sequel we consider only self-adjoint extensions without exit, that is within the given Krein space H. The self-adjoint extensions of a closed nonnegative relation of regular defect one are either nonnegative or have one negative square, and their resolvent sets are nonempty with the possible exception of one extension. In the rest of this paper, A0 is a closed nonnegative relation in (H, [·, ·]) of regular defect one. It is our aim to study the exceptional eigenvalue of the selfadjoint extensions of A0 which have one negative square. First we consider the special case where no nonnegative self-adjoint extension A of A0 with ρ(A) = ∅ exists. Then A0 has exactly one nonnegative self-adjoint ˆ this extension has the property ρ(A) ˆ = ∅, and all self-adjoint exextension A, ˆ tensions of A0 except A have nonempty resolvent set and one negative square. The relations A0 with this property were characterized in [JL3, Theorem 8.1]. All self-adjoint extensions A of A0 diﬀerent from Aˆ have the same spectrum σ , the same spectral function E and the same exceptional eigenvalue α , which is also 2 an eigenvalue of A0 . If α ∈ C \ R then A ∩ E (R)H ⊂ A0 . If α ∈ R then σ ⊂ R and for every connected closed subset ∆ ⊂ R \ {α } such that E (∆) is deﬁned we 2 have A ∩ E (∆)H ⊂ A0 , see [JL3, Section 4 and Theorem 7.4]. In the rest of this section we exclude this special case, that is we assume that there exists a nonnegative self-adjoint extension A of A0 with ρ(A) = ∅. We set R(z) := (A − z)−1 , z ∈ ρ(A), and denote by EA the spectral function of A in the [⊥] Krein space H. We ﬁx a defect element g ∈ ran (A0 + i) and deﬁne (4.1) g(z) := I + (z − i)R(z) g, Q(z) := z[g, g] + (z 2 + 1)[R(z)g, g]. If Q is constant on C+ then, according to [JL3, Lemma 7.1] Q(z) = 0 for z ∈ C+ . According to [JL3, Theorem 7.2] the self-adjoint extensions of A0 coincide with the linear relations A(γ) , γ ∈ R, deﬁned as follows: A(∞) = A, −1 [ · , g(¯ z )] g(z) , = R(z) − A(γ) − z γ + Q(z)

γ ∈ R \ {0},

(4.2)

for some (and then for all) z ∈ ρ(A) with γ + Q(z) = 0, and A(0) :=

lim

γ→0,γ=0

A(γ) ,

where the limit is to be understood in the sense of the gap metric in H2 . Here A(0) is the only self-adjoint extension of A0 which can have an empty resolvent set, see [JL3, Theorem 7.2]. If Q is not identically equal to zero then the relation

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(4.2) holds also for γ = 0. For every γ ∈ R we have ρ(A(γ) ) ∩ ρ(A) = {z ∈ ρ(A) : γ + Q(z) = 0}.

(4.3)

According to [JL3, Theorem 8.2] there exists a nonempty open interval (γ− , γ+ ), −∞ ≤ γ− ≤ 0 ≤ γ+ ≤ ∞, such that A(γ) has one negative square ⇐⇒ A(γ) is nonnegative ⇐⇒

γ ∈ (γ− , γ+ ), γ ∈ R \ (γ− , γ+ ).

Here γ+ and γ− can be expressed in terms of A and g (see [JL3, Section 8]): If J is a fundamental symmetry of H, xJ := [Jx, x]1/2 , x ∈ H, and if we denote by Top the operator part of the self-adjoint relation T in the Hilbert space (H, [J · , · ]), then 1/2 1/2 if g ∈ D((JA)op ), (JA)op g2J γ+ = +∞ otherwise, (4.4) 1/2 1/2 −(JA−1 )op g2J if g ∈ D((JA−1 )op ), γ− = −∞ otherwise. If γ ∈ (γ− , γ+ ) and ρ(A(γ) ) = ∅, the exceptional eigenvalue of A(γ) is denoted by α(γ): α(γ) := α(A(γ) ), see Deﬁnition 3.4. Recall that these values α(γ), γ ∈ (γ− , γ+ ), belong to R ∪ C+ . In the sequel we shall study the function (γ− , γ+ ) γ → α(γ). 4.2. The family of functions z Q(z) + γ , γ ∈ R In this subsection we set for γ ∈ R Fγ (z) := z(Q(z) + γ) = (1 + z 2 )z[R(z)g, g] + z 2 [g, g] + γz.

(4.5)

A simple calculation shows that the function z → z[R(z)g, g] belongs to N0 since A is nonnegative. Therefore, the functions Fγ form a family as considered in Section 2: Fγ (z) = γz +G1 (z)+z 2 G2 (z),

G1 (z) = z [R(z)g, g], G2 (z) = [g, g]+z [R(z)g, g].

We express the numbers αj , βj , the measures σj , j = 1, 2, and also γ0 and γ∞ , which were introduced in Section 2, in terms of A and g. Evidently, α1 = Re G1 (i) = −Im [R(i)g, g] = −[R(i)g, R(i)g], α2 = Re G2 (i) = −[R(i)g, R(i)g] + [g, g], β : = β1 = β2 = lim η −1 Im iη[R(iη)g, g] η↑∞

1 (R(iη) + R(−iη))g, g . 2 Further, σ1 = σ2 =: σ, and an application of the Stieltjes-Livˇsic inversion formula to the relation 1 + zt 2 −1 2 −1 (1 + z ) z[R(z)g, g] = (1 + z ) dσ(t) α1 + βz + R t−z = lim Re [R(iη)g, g] = lim η↑∞

η↑∞

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143

yields, for every interval (a, b) with a, b = 0, the formula b−0 dσ(t) = R(−i)AR(i)EA ((a, b))g, g .

(4.6)

a+0

On the other hand,

[R(−i)AR(i)g, g] = Im i [R(i)g, g] 1 + it = Im α + β i + dσ(t) = β + dσ(t), R t−i R

and hence, with ∆n := (−n, n), n = 1, 2, . . . , β = lim [R(−i)AR(i)(1 − EA (∆n ))g, g]. (4.7) n→∞ Lemma 4.1. Assume that t2 dσ(t) < ∞ or, equivalently, R sup AEA (∆n )g, g : n = 1, 2, . . . < ∞. (4.8) 1/2 Then the sequence EA (∆n ) g converges for n → ∞ in D (JA)op with respect to := g − g(∞) , the graph norm of this domain. If g(∞) := limn→∞ EA (∆n )g and g∞ −1 then g∞ belongs to the root subspace of A corresponding to 0 and we have −1 , g∞ dσ(t) = [g∞ , g∞ ]. β = A−1 g∞ , α2 − t 1 + t2 R

In particular, under the assumption (4.8), γ∞ < ∞ if and only if −1 = g∞ , g∞ = 0, A g∞ , g∞ and in this case

AEA (∆n )g, g . (4.9) Proof. On span EA (∆n \ ∆1 )H : n = 2, . . . the norm [A · , · ]1/2 is equivalent to 1/2 the graph norm of (JA)op . If n1 , n2 ∈ N, n2 ≥ n1 , then [A(EA (∆n2 ) − EA (∆n1 ))g, (EA (∆n2 ) − EA (∆n1 ))g] = 1 + t2 dσ(t), γ∞ = lim

n→∞

∆n2 \∆n1

which implies the ﬁrst assertion. For every bounded interval ∆ we have EA (∆)g∞ = EA (∆)g − lim EA (∆)EA (∆n )g = 0. n→∞

g∞

A−1 g∞

and belong to the root space of A−1 at zero, and [A−1 g∞ , g(∞) ]= Hence [g∞ , g(∞) ] = 0. In the subspace (I − EA (∆1 ))H the following identities hold: = I − R(−i)R(i) A−2 , (4.10) R(−i)R(i) (I−EA (∆1 ))H

R(−i)AR(i)

(I−EA (∆1 ))H

(I−EA (∆1 ))H

= (A−1 −A−1 R(−i)R(i))

(I−EA (∆1 ))H

.

(4.11)

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Since the operator A−1 (I−EA (∆1 ))H is nonnegative, its root subspace at zero co incides with the kernel of its square. Therefore A−2 g∞ = 0 and, by (4.10) and (4.11), R(−i)R(i)g∞ = 0, R(−i)AR(i)g∞ = A−1 g∞ . (4.12) In view of (4.7), β = lim R(−i)AR(i)(I − EA (∆n ))g, g n→∞ −1 −1 = R(−i)AR(i)g∞ = A g∞ , g(∞) = A g∞ , g∞ . , g(∞) + g∞ + g∞ Moreover, α2 − t dσ(t)

=

R

= = =

− R(−i)R(i)g, g + [g, g] − lim t dσ(t) n→∞ ∆ n (I − R(−i)R(i))g, g − lim (I − R(−i)R(i))EA (∆n )g, g n→∞ lim (I − R(−i)R(i))(I − EA (∆n ))g, g n→∞ = g∞ , g∞ . , g∞ (I − R(−i)R(i))g∞

Here the last equality is a consequence of (4.12). Finally, the last assertion of the lemma follows from the deﬁnition of γ∞ . Remark 4.2. In the terminology of [KK, Section 4], the inequality γ∞ < ∞ means that the function z −→ z[R(z)g, g] belongs to the class (R0 ). In this case (4.9) and [KK, Theorem S 1.4.2] give also other expressions for γ∞ , e.g., γ∞ = sup η 2 Re [R(iη)g, g] . η>0

Set Γn := R \ [−n−1 , n−1 ], n = 1, 2, . . . . The following lemma can be proved similarly to Lemma 4.1. 0 Lemma 4.3. Assume that R\{0} t−2 dσ(t) < ∞ or, equivalently, (4.13) sup A−1 EA (Γn )g, g : n = 1, 2, . . . < ∞. 1/2 Then the sequence EA (Γn )g converges for n → ∞ in D (JA−1 )op with respect to the graph norm of this domain. If g(0) := limn→∞ EA (Γn )g and g0 := g − g(0) , then g0 belongs to the root subspace of A at zero, Ag0 , g(0) = g0 , g(0) = 0, and we have

σ({0}) = Ag0 , g0 ,

α1 + lim

n→∞

Γn

t−1 dσ(t) = − g0 , g0 .

In particular, under the assumption (4.13), γ0 > −∞ if and only if [Ag0 , g0 ] = [g0 , g0 ] = 0, and in this case γ0 = − lim

n→∞

A−1 EA (Γn )g, g .

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145

and g0 we obtain the From the above lemmas, with the deﬁnitions of g∞ following relations: ⎧ if supn AEA (∆n )g, g < ∞, ⎪ ⎨ limn→∞ AEA (∆n )g, g −1 = g∞ = 0, , g∞ A g∞ , g∞ γ∞ = ⎪ ⎩ ∞ otherwise, (4.14) −1 −1 ⎧ − lim A A E (Γ )g, g if sup E (Γ )g, g < ∞, n→∞ A n A n ⎪ ⎨ n Ag0 , g0 = g0 , g0 = 0, γ0 = ⎪ ⎩ −∞ otherwise.

Trivially, γ0 ≤ 0 ≤ γ∞ , hence γ0 = γ∞ ⇐⇒ γ0 = γ∞ = 0.

(4.15)

4.3. The case ρ(A(0) ) = ∅ We shall show that in this case the exceptional eigenvalue α(γ), γ ∈ (γ− , γ+ ), is 0 or ∞. To prove this, we rely on the results of [JL3]. Theorem 4.4. Assume that there exists a nonnegative self-adjoint extension A of A0 with ρ(A) = ∅. Then the following conditions are equivalent. (i) ρ(A(0) ) = ∅. (ii) The element g admits the unique representation g = g0 + g∞ , g0 ∈ ker A0 , g∞ ∈ A0 (0)1 , and [g0 , g0 ] = [g∞ , g∞ ] = 0. (iii) γ0 = γ∞ . If these conditions are fulﬁlled, then γ+ = 0 ⇐⇒ g∞ = 0,

and α(γ) =

0 ∞

γ− = 0 ⇐⇒ g0 = 0,

(4.16)

γ ∈ (γ− , 0), γ ∈ (0, γ+ ).

[⊥] Proof. If (i) holds then A(0) = A. Then, by [JL3, Corollary 6.2], ran A0 + i is spanned by an element of the form g0 + g∞ where g0 ∈ ker A0 , g∞ ∈ A0 (0), = g , g , g and g 0 0 ∞ ∞ = 0, that is, (ii) holds. Assume now that (ii) holds. Since g0 , g∞ = 0 we have Q(z) := z g0 + g∞ , g0 + g∞ + z 2 + 1 R(z)(g0 + g∞ ), g0 + g∞ = 0 for every z ∈ ρ(A). Then, by [JL3, Lemma 7.1] there exists a self-adjoint exten = ∅. By [JL3, Theorem 7.2], A coincides with A(0) and, of A0 with ρ(A) sion A therefore, (i) holds. Moreover, (ii) implies sup AEA (∆n )g, g : n = 1, 2, . . . = sup A−1 EA (Γn )g, g : n = 1, 2, . . . = 0, 1 By

[JL3, (3.1)] we have ker A0 ∩ A0 (0) = {0}.

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and g0 in Lemma 4.1 and Lemma 4.3 we have g0 = g0 and by the deﬁnition of g∞ and g∞ = g∞ . Then, by (4.14), γ0 = γ∞ = 0, that is, (iii) is true. of (4.14), this implies If (iii) holds, then, by (4.15), γ0 = γ∞ = 0. On account for every bounded real interval ∆ with 0 ∈ / ∆ that AEA (∆)g, g = 0 and hence EA (∆)g = 0. By the deﬁnition of g∞ and g0 we obtain = I − EA (−1, 1) g. g0 = EA (−1, 1) g, g∞ Therefore, g = g0 + g∞ and, in view of Ag0 , g0 = 0 (see (4.14)), for every x ∈ EA (−1, 1) H, 1/2 1/2 Ag , x ≤ Ag , g Ax, x = 0, 0 0 0 which implies g0 ∈ ker A. Similarly, g∞ ∈ A(0). Again by (4.14) we have g0 , g0 = g∞ , g∞ = 0. Hence (ii) holds with A0 replaced by A. This yields [g(z), g(z)] = 0 for all z ∈ C \ R, and, by [JL3, Theorem 6.1], (i) is true. Assume now that the conditions (i–iii) are satisﬁed. By (4.4) the relation γ+ = 0 is equivalent to g ∈ ker A. In view of (ii) this is equivalent to g∞ ∈ (ker A) ∩ A(0). The latter holds, by the assumption on A, if and only if g∞ = 0 (see [JL3, Theorem 1.3]). Similarly it follows that γ− = 0 is equivalent to g0 = 0. It remains to prove the last assertion about the exceptional eigenvalues. Let γ ∈ (γ− , 0) ∪ (0, γ+ ). Then, by (ii) and [JL3, Theorem 7.2], −1 = R(z) − γ −1 ·, iz −1 g0 + g∞ iz −1 g0 + g∞ , A(γ) − z

and the following function has one negative square: −1 z → z A(γ) − z = zR(z) − γ −1 z −1 ·, g0 g0 (4.17) −1 −1 − γ z ·, g∞ g∞ − γ − i ·, g0 g∞ + i ·, g∞ g0 . If γ > 0 then γ+ > 0 and, by (4.16), g∞ = 0. In this case, the terms on the righthand side of (4.17) with the exception of the third term are Nevanlinna functions. We choose some x ∈ H such that the function Gx : Gx (z) := z (A(γ) − z)−1 x, x belongs to N1 . Then we have [x, g∞ ] = 0 and the generalized pole of the function z → −γ −1 z|[x, g∞ ]|2 , which is the point ∞, coincides with the generalized pole of Gx . Therefore there exists a sequence (zn ) ⊂ C+ converging to ∞ in C such that lim inf n→∞ Im Gx (zn ) < 0. Then, on account of Theorem 3.5, α(γ) = ∞. If γ < 0 then γ− < 0 and, by (4.16), g0 = 0. Now all the terms on the right-hand side of (4.17) with the exception of the second term are Nevanlinna functions and, as above, using Theorem 3.5 we ﬁnd α(γ) = 0. 4.4. The generic case Now we describe the exceptional eigenvalue α(γ), γ ∈ (γ− , γ+ ), if ρ(A(γ) ) = ∅ for all γ ∈ R. To this end we apply the results of Section 2 to the functions Fγ from (4.5). Recall that the function Fγ belongs to the class N1 if and only if γ ∈ (−∞, γ∞ ), and as in Section 2, for these γ we denote the zero of nonpositive type of Fγ by ζ(γ).

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147

Lemma 4.5. Suppose that ρ A(γ) = ∅ for all γ ∈ R. Then −∞ ≤ γ− ≤ γ0 ≤ 0 ≤ γ∞ ≤ γ+ ≤ +∞,

(4.18)

and α(γ) = ζ(γ) for γ ∈ (γ0 , γ∞ ). For these γ, if α(γ) ∈ C+ then g(α(γ)) (see (4.1)) is an eigenelement of A(γ) at the eigenvalue α(γ), if α(γ) = α(γ) = 0 then the limit g(α(γ)) := limε→0 g(α(γ) + iε) exists and is an eigenelement of A(γ) at α(γ) and (sign α(γ)) [g(α(γ)), g(α(γ))] ≤ 0. (4.19) Proof. By Theorem 4.4 we have γ0 = γ∞ . The relations (4.14) show that γ0 ≤ 0 ≤ γ∞ holds. Let γ ∈ (γ0 , γ∞ ). Then, according to Lemma 2.3 and Theorem 2.4, ζ(γ) = 0, ∞. We prove Lemma 4.5 in the two cases ζ(γ) ∈ / R and ζ(γ) ∈ R \ {0} separately. If ζ(γ) ∈ / R then the second term on the right-hand side of (4.2) has a pole at ζ(γ) and ζ(γ) is an eigenvalue of A(γ) in C+ . It follows that A(γ) is not nonnegative, i.e., γ ∈ (γ− , γ+ ) and α(γ) = ζ(γ). Since g(α(γ)) is a defect element of A0 at the point α(γ) it is an eigenelement of the extension A(γ) of A0 to the eigenvalue α(γ). |t − ω|−2 dσ(t) < ∞. Let Let ω := ζ(γ) ∈ R \ {0}. Then, by Lemma 2.3, R

¯ We denote by ι the ∆ be a bounded open interval, such that ω ∈ ∆, 0 ∈ / ∆. linear mapping φ → EA (∆)φ(A)g deﬁned for all rational functions φ with only non-real poles. Then ι is an isometry from the linear manifold of these elements φ of L2 (∆; σ) into EA (∆)H, [R(−i)AR(i)·, ·] . We extend ι to an isometry from 2 L (∆; σ) onto a closed subspace of EA (∆)H, [R(−i)AR(i)·, ·] . For any fundamental symmetry J the scalar products [J·, ·] and [R(−i)AR(i)·, ·] are equivalent on EA (∆)H. Since the function ( · − ω)−1 belongs to L2 (∆; σ) and the functions ( · − ω − i)−1 converge in L2 (∆; σ) to ( · − ω)−1 for → 0, also lim R(ω + i)g =: gω

→0

(4.20)

exists. For arbitrary z ∈ ρ(A) we have R(z)gω = (z − ω)−1 (R(z)g − gω ).

(4.21)

One veriﬁes without diﬃculty that for the function Fγ considered in the present section, ω ∈ R \ {0} and under the assumption (4.20) the relation (2.9) holds if and only if ω(ω 2 + 1)[gω , gω ] + (3ω 2 + 1)[gω , g] + 2ω[g, g] + γ ≤ 0,

(4.22)

and (2.10) holds if and only if γω + ω 2 [g, g] + ω(1 + ω 2 )[gω , g] = 0.

(4.23)

Therefore the relations (4.22) and (4.23) hold. Inserting (4.23) into (4.22) we obtain ω(ω 2 + 1)[gω , gω ] + 2ω 2 [gω , g] + ω[g, g] ≤ 0.

(4.24)

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Using the relations (4.21) and (4.23) it can be veriﬁed by an easy calculation that the element (4.25) g(ω) = g + (ω − i)gω is not the zero element and an eigenelement of A(γ) to the eigenvalue ω. Further, [g(ω), g(ω)] = [g, g] + (ω − i)[gω , g] + (ω + i)[g, gω ] + (ω 2 + 1)[gω , gω ]

(4.26)

and, by (4.24), (sign ω)[g(ω), g(ω)] ≤ 0. (4.27) Hence A(γ) is not nonnegative, i.e., γ ∈ (γ− , γ+ ), and, by Theorem 3.1, ω = ζ(γ) coincides with α(γ). Lemma 4.6. If γ ∈ (γ− , γ+ ) and α(γ) = 0, ∞, then γ ∈ (γ0 , γ∞ ), α(γ) = ζ(γ), and we have the direct orthogonal decomposition ˙ ker(A − α(γ)). (4.28) ker A(γ) − α(γ) = span {g(α(γ))}+ Proof. If α(γ) ∈ C+ then, by (4.2), Fγ has a pole at α(γ) and therefore ζ(γ) = α(γ). Then Theorem 2.4,(1) implies γ ∈ (γ0 , γ∞ ). In this case (4.28) holds with ker (A − α(γ)) = {0}. Assume that ω := α(γ) ∈ R \ {0}. Then there exists an x ∈ H, x = 0, such x that ∈ A(γ) and ωx (sign ω)[x, x] ≤ 0. (4.29) By [JL3, Corollary 7.3], there exists an x0 ∈ H, x0 = 0, such that g x R(−i)x0 . (4.30) − [x0 , g] = (γ + i[g, g]) ig x0 − iR(−i)x0 ωx We have γ + i[g, g] = 0 since otherwise (4.30) would imply ω = i. We also have x [x0 , g] = 0. Indeed, [x0 , g] = 0 would imply that ∈ A and that x is an ωx eigenvector of the bounded nonnegative operator R(−i)AR(i), corresponding to the eigenvalue ω(ω 2 + 1)−1 , and hence that (sign ω)[x, x] > 0. This contradicts (4.29). From (4.30) we get 1 − (ω + i)R(−i) x0 = Cg with C := (i − ω)[x0 , g](γ + i[g, g])−1 , (4.31) which implies (4.32) EA ({ω})g = 0, where EA denotes again the spectral function of A. If we set x0 := EA ({ω})x0 , x ˜0 := x0 − x0 , then [x0 , g] = [˜ x0 , g], and we shall show that for x0 , g]g (4.33) x˜ := (γ + i[g, g])R(−i)˜ x0 − [˜ x ˜ it holds ∈ A(γ) and ωx ˜ (sign ω)[˜ x, x ˜] ≤ 0. (4.34)

Self-adjoint Extensions of a Nonnegative Linear Relation Indeed,

x˜ = x − γ + i[g, g] R(−i)x0 = x − γ + i[g, g] (ω + i)−1 x0

149

(4.35)

and

x ˜ x (ω + i)−1 x0 = − (γ + i[g, g]) ω(ω + i)−1 x0 ωx ˜ ωx R(−i)x0 R(−i)x0 − (4.36) = (γ + i[g, g]) x0−iR(−i)x0 x0 − iR(−i)x0 g R(−i)˜ x0 g = (γ + i[g, g]) ∈ A(γ) . − [x0 , g] − [˜ x0 , g] ig ig x ˜0 − iR(−i)˜ x0 To verify (4.34) observe that because of [ x0 , x0 ] = [g, x0 ] = 0 we have [ x, x0 ] = 0.

(4.37)

As A is nonnegative,

(signω) [x0 , x0 ] ≥ 0. Using (4.29), (4.35), (4.37), and (4.38) we ﬁnd 0 ≥ = ≥

(4.38)

(sign ω) [x, x] + (γ + i[g, g])(ω + i)−1 x0 (sign ω) x + (γ + i[g, g])(ω + i)−1 x0 , x (sign ω) [ x, x ].

By (4.31) we have (I − (ω + i)R(−i))˜ x0 = Cg.

(4.39) ¯ Let ∆ be a bounded open interval such that ω ∈ ∆ and 0 ∈ / ∆. Then, because of (4.6) and (4.39), for every ε > 0, |C|2 ((t − ω)2 + 2 )−1 dσ(t) ∆ = |C|2 R(ω + iε)R(ω − iε)R(−i)AR(i)EA (∆)g, EA (∆)g x0 , x 0 = |C|2 R(ω + iε)(A−ω)R(ω −iε)(A−ω)R(−i)2AR(i)2 EA (∆) = |C|2 (t − ω)2 ((t − ω)2 + 2 )−1 (1 + t2 )−2 t d[EA (t) x0 , x 0 ], ∆

and therefore,

(t − ω)−2 dσ(t) < ∞,

(4.40)

∆ −1

i.e., the function t → (t − ω) belongs to L2 (∆; σ). Now we see as in the proof of Lemma 4.5 that limε↓0 R(ω + iε)g =: gω exists. Applying the operator I + (ω + i + iε)R(ω + iε) to (4.39) we ﬁnd I + iεR(ω + iε) x 0 = C g + (ω + i + iε)R(ω + iε)g , x0 = 0, and for ε ↓ 0, on account of EA ({ω}) x 0 = C g + (ω + i)gω .

(4.41)

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Taking the inner product with g = I − EA ({ω}) g on both sides of this relation and inserting C from (4.31) we obtain −1 g + (ω + i)gω , [x0 , g] = (i − ω)[x0 , g] γ + i[g, g] i.e., γ + ω[g, g] + (ω 2 + 1)[gω , g] = 0. The relations (4.33) and (4.41) yield x = (i − ω) x 0 , g R(−i) g + (ω + i)gω − x 0 , g g = − x 0 , g g − (i − ω) R(−i)g + (ω − i)R(−i)gω .

(4.42)

This implies in view of (ω + i)R(−i)gω = gω − R(−i)g (see (4.21)) the relation

x0 , g] g(ω). x =− x 0 , g g − (i − ω)gω = −[

(4.43)

From (4.34) we obtain 0

≥ ω g − (i − ω)gω , g − (i − ω)gω = ω(ω 2 + 1) gω , gω + 2ω 2 gω , g + ω[g, g].

Adding (4.42) it follows that 0 ≥ ω(ω 2 + 1) gω , gω + (3ω 2 + 1) gω , g + 2ω[g, g] + γ.

(4.44)

Now (4.44) and (4.42) imply ω = ζ(γ), hence γ ∈ (γ0 , γ∞ ). The ﬁrst equality in (4.36) and (4.43) give (4.28). Now we prove the main result about the curve γ → α(γ) of the exceptional see Lemma 4.1 and Lemma eigenvalue. For the deﬁnition of the elements g0 , g∞ 4.3, the measure σ and the number β were deﬁned in Subsection 4.2. Theorem 4.7. Suppose that all self-adjoint extensions of A0 have nonempty resolvent set. Then the following statements hold. (i) The function α : (γ− , γ+ ) → C+ ∪ R is continuous and lim α(γ) = 0, γ↓γ−

lim α(γ) = ∞.

γ↑γ+

(ii) We have −∞ ≤ γ− ≤ γ0 ≤ 0 ≤ γ∞ ≤ γ+ ≤ ∞, γ0 < γ∞ , and 0 if γ ∈ (γ− , γ0 ), α(γ) = ∞ if γ ∈ (γ∞ , γ+ ). (iii) The function |α| : |α|(γ) := |α(γ)|, γ ∈ (γ0 , γ∞ ), is nondecreasing and γ1 , γ2 ∈ (γ0 , γ∞ ), γ1 = γ2 implies α(γ1 ) = α(γ2 ). If σ = 0 or β = 0, then this function is strictly increasing, if σ = 0 and β = 0, then it is strictly , g∞ ] ≥ 0. increasing if and only if [g0 , g0 ] [g∞ (iv) The function α is real analytic in γ ∈ (γ0 , γ∞ ) if α(γ) ∈ / supp σ with possible exception of branch points of order ≤ 3 for which α(γ) ∈ R.

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(v) σ α(γ) : γ ∈ (γ0 , γ∞ ) ∩ R = 0. (vi) For every component I of the open set R \ supp σ ∪ {0} the set I ∩ α(γ) : γ ∈ (γ0 , γ∞ ) is empty or connected. The statement (v) means that α(γ) cannot move within the support of σ, (vi) means that every bounded component I as in the statement is ‘visited’ by the exceptional eigenvalue at most once. Proof of Theorem 4.7. The ﬁrst assertion of (i) follows from (4.2) and Theorem 3.7. By Theorem 4.4 we have γ0 < γ∞ . Then, according to Lemma 4.5, α(γ) = ζ(γ) for γ ∈ (γ0 , γ∞ ), and Theorem 2.4 implies lim α(γ) = 0, γ↓γ0

lim α(γ) = ∞.

(4.45)

γ↑γ∞

If γ ∈ (γ− , γ0 ) ∪ (γ∞ , γ+ ) then by Lemma 4.6 either α(γ) = 0 or α(γ) = ∞. Hence, on account of the continuity of γ → α(γ) and (4.45), α(γ) = 0 for γ ∈ (γ− , γ0 ) and α(γ) = ∞ for γ ∈ (γ∞ , γ+ ). This proves the assertions (i) and (ii). Lemma 4.6 and Theorem 2.4 imply the ﬁrst assertion of (iii) and that the function γ → |α(γ)| is nondecreasing. The criterion on strict monotonicity follows from Theorem 2.4 in view of the fact that we have α2 = − g0 , g0 + [g, g] = g∞ . , g∞ α1 = − g0 , g0 , This proves assertion (iii). The assertions (iv), (v), and (vi) are consequences of Theorem 2.4 and Theorem 2.5. The following two examples show that intervals of constancy of the functions γ → α(γ) and γ → |α(γ)| may occur, even in ﬁnite-dimensional situations. 0 1 ·, · we consider the nonnegative Example 4.8. In the Krein space C2 , 1 0 0 0 , a > 0. Deﬁne H0 := H∞ := C2 , H := H0 × H∞ and the operator Ta := a 0 nonnegative relation A = Ta0 × Ta−1 for some a0 , a∞ > 0. Let g = (b1 , b2 , c1 , c2 )T . ∞ With the help of Lemmas 4.1 and 4.3 we can easily compute the quantities γ+ , γ∞ , γ− , γ0 : ∞ if c1 = 0 ∞ if c1 = 0 γ+ = = , γ , ∞ 2 −1 2 2 a0 |b1 | + a∞ |c2 | if c1 = 0 a0 |b1 | if c1 = 0 −∞ if b1 = 0 −∞ if b1 = 0 = γ− = , γ . 0 −1 2 2 2 −a∞ |c1 | if b1 = 0 −a∞ |c1 | − a0 |b2 | if b1 = 0 Example 4.9. Let (H, [·, ·]) be the orthogonal direct sum of the Krein subspaces (H , [·, ·]) and (H , [·, ·]). Then A := yx : x ∈ H , y ∈ H is a nonnegative self-adjoint relation. We choose some g = g + g with g ∈ H , g ∈ H and [g , g ][g , g ] < 0. Then (γ0 , γ∞ ) γ → |α(γ)| is not strictly increasing.

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In the following theorem we give some criteria for the coincidence of the numbers γ∞ and γ+ as well as of γo and γ− . Recall that ∆n = (−n, n), Γn = R \ [−n−1 , n−1 ]. Theorem 4.10. (1) Each of the following conditions implies γ∞ = γ+ : (i) limn→∞ [AE(∆n )g, g] = ∞. (ii) A(0) is trivial or a deﬁnite subspace. (2) Each of the following conditions implies γ0 = γ− : (i) limn→∞ [A−1 E(Γn )g, g] = ∞. (ii) ker A is trivial or a deﬁnite subspace. Proof. We prove assertion (1), assertion (2) can be proved by a similar reasoning. Condition (i) implies γ∞ = ∞ (see (4.14)), and therefore γ∞ = γ+ = ∞. Assume that (ii) holds. If limn→∞ [AE(∆n )g, g] < ∞, then, by Lemma 4.1, (E(∆n )g) 1/2 with respect to the graph norm, and converges in D((JA)op ) to an element g(∞) g∞ := g − g(∞) belongs to the root space of A−1 corresponding to 0. Then, by (ii), we have either [g∞ , g∞ ] = 0 or g∞ = 0. In the ﬁrst case Lemma 4.1 implies 1/2 ∈ D (JA)op , and Lemma 4.1 and γ∞ = ∞. In the second case we have g = g(∞) (4.4) give 42 4 4 γ∞ = lim AE(∆n )g, g = lim 4(JA)1/2 op E(∆n )g J = γ+ . n→∞

n→∞

4.5. The root subspace at a real nonzero exceptional eigenvalue We suppose as in Subsection 4.4 that all self-adjoint extensions of A0 have nonempty resolvent set. Let γ ∈ (γ0 , γ∞ ) be such that ω := α(γ) ∈ R \ {0}. We show that modulo ker(A − ω) the elements of the Jordan chain of A(γ) at ω can be expressed as limits of the defect elements g(ω + iε), ε ↓ 0. First we recall that by the Lemmas 4.5 and 4.6 with g(ω) = limε↓0 g(ω + iε) we have ˙ ker (A − ω) ker (A(γ) − ω) = span {g(ω)}+ and (sign ω)[g(ω), g(ω)] ≤ 0. Moreover, g(ω) = g + (ω − i)gω , where gω := limε↓0 R(ω + iε)g (see (4.20)). If E(γ) denotes the spectral function of A(γ) and ∆ is a bounded open interval such that ω ∈ ∆, 0 ∈ / ∆, then E(γ) (∆)H, [A(γ) · , · ] is a Pontryagin space with negative index 1, and A(γ) is self-adjoint in this space. According to [JL2], the length of any Jordan chain of A(γ) at ω is ≤ 3, 7 dim ker(A(γ) − ω)k+1 ) ker(A(γ) − ω)k ≤ 1, k = 1, 2, and exactly one of the following cases holds: (e) ker(A(γ) −ω) = ker(A(γ) −ω)2 and ω is not a singular critical point of A(γ) . (p1s ) ker(A(γ) − ω) = ker(A(γ) − ω)2 and ω is a singular critical point of A(γ) . (p2r ) ker (A(γ) − ω) = ker (A(γ) − ω)2 = ker (A(γ) − ω)3 and ω is a regular critical point of A(γ) .

Self-adjoint Extensions of a Nonnegative Linear Relation (p2s ) (p3 )

153

ker (A(γ) − ω) = ker (A(γ) − ω)2 = ker (A(γ) − ω)3 and ω is a singular critical point of A(γ) . ker (A(γ) − ω) = ker (A(γ) − ω)2 = ker (A(γ) − ω)3 .

In the case (p3 ) ω is a regular critical point of A(γ) . Theorem 4.11. Assume that all self-adjoint extensions of A0 have nonempty resolvent set and let γ ∈ (γ0 , γ∞ ) be such that ω := α(γ) ∈ R \ {0}. Then A(γ) − ω g(ω) = 0, and the following equivalences hold: (e) ⇐⇒ (sign ω)[g(ω), g(ω)] < 0. (p1s ) ⇐⇒ [g(ω), g(ω)] = 0, lim R(ω + iε)g(ω + iε) does not exist. ε↓0

(p2r ) ⇐⇒ [g(ω), g(ω)] = 0, f1 := lim R(ω + iε)g(ω + iε) exists, [g(ω), f1 ] = 0. ε↓0

(p2s ) ⇐⇒ [g(ω), g(ω)] = 0, f1 := lim R(ω + iε)g(ω + iε) exists, [g(ω), f1 ] = 0, ε↓0

lim R(ω + iε)2 g(ω + iε) does not exist. ε↓0

(p3 ) ⇐⇒ [g(ω), g(ω)] = 0, f1 := lim R(ω + iε)g(ω + iε) exists, [g(ω), f1 ] = 0, ε↓0

f2 := lim R(ω + iε)2 g(ω + iε) exists. ε↓0

Moreover, in the cases (p2r ) and (p2s ), the elements g(ω), f1 form a Jordan chain of A(γ) at ω and ˙ ker(A − ω); ker (A(γ) − ω)2 = span{g(ω), f1 } [+] in the case (p3 ) the elements g(ω), f1 , f2 form a Jordan chain of A(γ) at ω and ˙ ker(A − ω). ker (A(γ) − ω)3 = span{g(ω), f1 , f2 } [+] Proof. First we mention that a real eigenvalue of a self-adjoint relation B with non-empty resolvent set in a Pontryagin space is a regular critical point of B if and only if the corresponding root subspace is non-degenerated, see [KL1]. The statement of the theorem is now a consequence of the following two claims and of Lemma 4.6. Claim 1: If [g(ω), g(ω)] = 0 then there exists a Jordan chain of A(γ) at ω of length ≥ 2 if and only if f1 := lim R(ω + iε)g(ω + iε) ε↓0

(4.46)

exists; in this case the elements g(ω), f1 form a Jordan chain of A(γ) at ω, and this Jordan chain is orthogonal to ker(A − ω).

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To prove this, assume ﬁrst that f1 = limε↓0 R(ω + iε)g(ω + iε) exists. If z, ζ ∈ C \ R, z = ζ, we ﬁnd from (4.2) by a straightforward calculation that (A(γ) − ζ)−1 − (z − ζ)−1 R(z)g(z) = −(z − ζ)−2 g(z) −

[g(z), g(z)] − (γ + Q(z))(z − ζ)−1 ) g(ζ). (z − ζ)(γ + Q(ζ))

If we set z = ω + iε, then in view of lim [g(ω + iε), g(ω − iε)] = [g(ω), g(ω)] = 0

ε→0

and

lim γ + Q(ω + iε) = γ + ω[g, g] + ω 2 + 1 [gω , g] = 0

ε→0

(see (4.23)), we ﬁnd (A(γ) − ζ)−1 − (ω − ζ)−1 f1 = −(ω − ζ)−2 g(ω). This relation is equivalent to

f1 ∈ A(γ) − ω. g(ω)

(4.47)

(4.48)

By (4.32), for z = z we have

E({ω})R(z)g(z) = R(z) I + (z − i)R(z) E({ω})g = 0, therefore f1 [⊥] ker (A − ω) and span g(ω), f1 [⊥] ker (A − ω). Conversely, assume that there exist elements x ∈ H, y ∈ ker A(γ) − ω , y = x 0, such that ∈ A(γ) − ω. This implies [y, y] = 0, and by Lemma 4.6 and y [g(ω), g(ω)] = 0 we ﬁnd y = αg(ω) for some number α = 0. Hence x (4.49) ∈ A(γ) − ω for x := α−1 x . g(ω) If ζ = ζ then

or

x ∈ (A(γ) − ζ)−1 − (ω − ζ)−1 , −(ω − ζ)−2 g(ω)

R(ζ) − (ω − ζ)−1 x − (γ + Q(ζ))−1 [x, g(ζ)]g(ζ) = −(ω − ζ)−2 g(ω) = −(ω − ζ)−2 g − (ω − ζ)−2 (ω − i)gω

(4.50)

(see (4.20), (4.25)). Let ∆ be a real bounded open interval such that ω ∈ ∆, 0 ∈ / ∆, and denote the spectral function of A again by EA . Evidently, R(ζ) − (ω− ζ)−1 EA (∆)x (4.51) =−(ω− ζ)−1 EA (∆)(A− ω)R(ζ)EA (∆)x ∈ ran (A− ω)EA (∆)H .

Self-adjoint Extensions of a Nonnegative Linear Relation

155

For ζ = ζ we have

EA (∆)g(ζ) = EA (∆) I + (ζ − i)R(ζ) g = EA (∆) I+ (ζ− i)R(ζ) (A− ω)EA (∆)gω∈ ran (A− ω)EA (∆)H .

(4.52)

We apply EA (∆) to the left- and the right-hand side of (4.50). Then (4.51) and (4.52) yield EA (∆)gω ∈ ran (A− ω)EA (∆)H , and that the set R(ω + iε)EA (∆)gω : ε ∈ R \ {0} is bounded in EA (∆)H. Making use of the isometry ι deﬁned in the proof of Lemma 4.5 we conclude that the integrals −1 |t − ω|−2 (t − ω)2 + ε2 dσ(t), ε > 0, ∆

are uniformly bounded. Therefore the function t → (t − ω)−4 is σ-integrable. This implies that the functions ∆ t → (t − i)(t − ω − iε)−2 converge in L2 (∆; σ) for ε ↓ 0, which is equivalent to the fact that the limit (4.46) exists. Then, by (4.48) and (4.49), x − f1 ∈ ker A(γ) − ω , which on account of Lemma 4.6 gives 2 = span {f1 } + ker A(γ) − ω ker A(γ) − ω ˙ ker (A − ω). = span {g(ω), f1 }[+] Claim 2: If [g(ω), g(ω)] = 0, f1 := limε↓0 R(ω + iε)g(ω + iε) exists and [f0 , f1 ] = 0 then there exists a Jordan chain of A(γ) at ω of length 3 if and only if f2 := limε↓0 R(ω + iε)2 g(ω + iε) exists; in this case the elements g(ω), f1 , f2 form a Jordan chain of A(γ) at ω, and this chain is orthogonal to ker(A − ω). The proof of this claim is analogous to the proof of the Claim 1; e.g., the relation (4.47) is to be replaced by (A(γ) − ζ)−1 − (ω − ζ)−1 f2 = −(ω − ζ)−2 f1 + (ω − ζ)−3 g(ω). Remark 4.12. If the assumptions of Theorem 4.11 are fulﬁlled, it follows from the proof of Theorem 4.11 that limε↓0 R(ω+iε)g(ω+iε) exists if and only if the function t → (t − ω)−4 is σ-integrable. Analogously, the limits limε↓0 R(ω + iε)g(ω + iε) and limε↓0 R(ω + iε)2 g(ω + iε) exist if and only if the function t → (t − ω)−6 is σ-integrable. If, in addition to the assumptions of Theorem 4.11, ω ∈ / supp σ, then only the cases (e), (p2r ) and (p3 ) can occur. With the reasoning at the end of Section 2, the behavior of the curve γ → α(γ) in a neighborhood of ω can be characterized by the structure of the root subspace of A(γ) at ω. Proposition 4.13. Let the assumptions of Theorem 4.11 be fulﬁlled with γ replaced by some γ1 ∈ (γ0 , γ∞ ) such that ω := α(γ1 ) ∈ / supp σ. If by (rr), (cr), (rc), and (cc) we denote the properties of a curve in the neighborhood of some real point

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as introduced at the end of Section 2 (with ζ1 replaced by ω), then the following equivalences hold: (e) (p2r ) and [f1 , g(ω)] < 0 (p2r ) and [f1 , g(ω)] > 0 (p3 )

⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

ω ω ω ω

is is is is

of of of of

type type type type

(rr), (cr), (rc), (cc).

4.6. Rank one perturbations of a nonnegative self-adjoint relation In the preceding considerations the starting point was a nonnegative relation A0 of regular defect one. After excluding some special cases a nonnegative self-adjoint extension A of A0 with ρ(A) = ∅ was ﬁxed, and this self-adjoint extension played an essential role. This viewpoint may be changed by starting from a nonnegative self-adjoint relation A with ρ(A) = ∅ and a rank one perturbation B of A, either in resolvent sense or in the sense of generalized sums. In the ﬁrst case, let A be a nonnegative self-adjoint relation A with ρ(A) = ∅ and let B be a self-adjoint relation with ρ(B) = ∅, B = A, such that for some (and hence for all) λ ∈ ρ(A) ∩ ρ(B) the operator (A − λ)−1 − (B − λ)−1 is of rank one, say (A − λ)−1 − (B − λ)−1 = [ · , h] h with some h, h ∈ H. Setting g := I + (i − λ)(A − i)−1 h, then A and B are self-adjoint extensions of the relation x x x x : ∈ A, [y − λx, h] = 0 = : ∈ A, [y − ix, g] = 0 . A0 := y y y y This relation A0 is nonnegative and of regular defect one. Hence the relation B belongs to the family of operators A(γ) as constructed above with the help of A0 and γ, and the results of this paper can be used in order to study the spectral properties of the relation B. To deﬁne rank one perturbations B of a nonnegative self-adjoint operator A with ρ(A) = ∅ in the sense of generalized sums we recall some deﬁnitions from [JL1]. Denote by J a fundamental symmetry of the Krein space H and by H 12 (A) the linear space D(JA)1/2 provided with the graph norm 8 x 21 = x2 + (JA)1/2 x2 , x ∈ D(JA)1/2 . Deﬁne the negative norm

x− 12 := sup |[x, y]| : y ∈ H 12 (A), y 21 ≤ 1 ,

and let H− 12 (A) be the completion of H with respect to this norm. Then [ · , · ] can be extended by continuity to H 12 (A) × H− 12 (A). The operator A can be extended from H 1 (A) into H 1 (A), and (A − z)−1 , z ∈ ρ(A), to a bounded operator A 2

−2

can be extended by continuity to an isomorphism R(z) of H− 12 (A) onto H 12 (A). If f ∈ H− 12 (A) and α ∈ R we deﬁne the operator A α [ · , f]f as the restriction + + α [ · , f]f ∈ L H 1 (A), H− 1 (A) to those x ∈ H 1 (A) for which Ax of A 2 2 2

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α [x, f]f ∈ H. The operator A α [ · , f]f is self-adjoint and its resolvent set is not empty. Now we consider for some f ∈ H− 12 (A) the family of self-adjoint operators A[α] := A α [ · , f] f,

α ∈ R.

f, then [x, f] = [(A + i)x, g], x ∈ In particular, f may belong to H. If g := R(i) D(A), and we set A0 := A|{x∈D(A): [(A+i)x,g]=0} . Starting from A and the defect element g, we deﬁne the self-adjoint extensions A(γ) R(i) f, f] < ∞, and Theorem 4.10 implies that of A0 as above. Then γ+ = [R(−i) A in this case γ∞ = γ+ . It was shown in [JL1, 2.2] that A[α] = A(γ) , where γ = α−1 + γ+ , α ∈ R. −1 in terms of f see [JL1, Proposition 2.1]. For an expression of γ− − γ+

References [ADo]

Aronszajn, N., Donoghue, W.F.: On exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. d’Analyse Math. 5 (1956-57), 321–388.

[DaL1]

Daho, K., Langer, H.: Matrix functions of the class Nκ , Math. Nachr. 120 (1985), 275–294.

[DaL2]

Daho, K., Langer, H.: Sturm–Liouville operators with an indeﬁnite weight function: the periodic case, Radovi Matematiˇcki 2 (1986), 165–188.

[DHdS] Derkach, V., Hassi, S., de Snoo, H.S.V.: Rank one perturbations in a Pontryagin space with one negative square, J. Funct. Anal. 188 (2002), 317–349. [DdS1]

Dijksma, A., de Snoo, H.S.V.: Symmetric and selfadjoint relations in Krein spaces I, Operator Theory: Advances and Applications, vol. 24 (1987), Birkh¨ auser Verlag Basel, 145–166.

[DdS2]

Dijksma, A., de Snoo, H.S.V.: Symmetric and selfadjoint relations in Krein spaces II, Ann. Acad. Sci. Fenn., Ser. A. I. Mathematica 12 (1987), 199–216.

[DLShZ] Dijksma, A., Langer, H., Shondin, Yu., Zeinstra, C.: Selfadjoint diﬀerential operators with inner singularities and Pontryagin spaces, Operator Theory: Adv. Appl., vol. 118, Birkh¨ auser Verlag, Basel, 2000, 105–175. [DLSh]

Dijksma, A., Luger, A., Shondin, Yu.: Minimal models for Nκ∞ -functions, Operator Theory: Adv. Appl., vol. 163, Birkh¨ auser Verlag, Basel, 2006, 97–134.

[J1]

Jonas, P.: A class of operator-valued meromorphic functions on the unit disc, Ann. Acad. Sci. Fenn. Ser. A. I: Mathematica 17 (1992), 257–284.

[J2]

Jonas, P.: Operator representations of deﬁnitizable functions, Ann. Acad. Sci. Fenn. Ser. A. I: Mathematica 25 (2000), 41–72.

[J3]

Jonas, P.: On locally deﬁnite operators in Krein spaces, in: Spectral Theory and Its Applications, Bucharest, Theta 2003, 95–127.

158 [JL1] [JL2] [JL3]

[KK]

[KL1]

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[L2]

P. Jonas and H. Langer Jonas, P., Langer, H.: Some questions in the perturbation theory of Jnonnegative operators in Krein spaces, Math. Nachr. 114 (1983), 205–226. Jonas, P., Langer, H.: A model for π-self-adjoint operators and a special linear pencil, Integral Equations Operator Theory 8 (1985), 13–35. Jonas, P., Langer, H.: Selfadjoint extensions of a closed linear relation of defect one in a Krein space, Operator Theory: Advances and Applications, vol. 80 (1995), Birkh¨ auser Verlag Basel, 176–205. Kac, I.S., Krein, M.G.: R-functions - analytic functions mapping the upper half-plane into itself, Supplement I of the Russian translation of the book by F.V. Atkinson, Discrete and continuous boundary problems, Moscow 1968. English translation: Amer. Math. Soc. Transl. (2) vol. 103 (1974), 1–18. Krein, M.G., Langer, H.: On the spectral function of a selfadjoint operator in a space with an indeﬁnite metric, Dokl. Akad. Nauk SSSR 152 (1963), 39–42 (Russian) Krein, M.G., Langer, H.: Some propositions on analytic matrix functions related to the theory of operators in the space Πκ , Acta Sci. Math. 43 (1981), 181–205. Langer, H.: Spektraltheorie linearer Operatoren in J-R¨ aumen und einige Anwendungen auf die Schar L(λ) = λ2 I + λB + C, Habilitationsschrift Technische Universit¨ at Dresden, 1965. Langer, H.: Characterization of generalized zeros of negative type of functions of the class Nκ , Operator Theory: Advances and Applications, vol. 17 (1986), Birkh¨ auser Verlag Basel, 202–212.

P. Jonas Institut f¨ ur Mathematik Technische Universit¨ at Berlin Straße des 17. Juni 135 D-10623 Berlin, Germany e-mail: [email protected] H. Langer Institut f¨ ur Analysis und Computational Mathematics Technische Universit¨ at Wien Wiedner Hauptstrasse 8–10 A-1040 Vienna, Austria e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 175, 159–168 c 2007 Birkh¨ auser Verlag Basel/Switzerland

Canonical Diﬀerential Equations of Hilbert-Schmidt Type Michael Kaltenb¨ack and Harald Woracek Abstract. A canonical system of diﬀerential equations, or Hamiltonian system, is a system of order two of the form Jy (x) = −zH(x)y(x), x ∈ R+ . We characterize the property that the selfadjoint operators associated to a canonical system have resolvents of Hilbert-Schmidt type in terms of the Hamiltonian H as well as in terms of the associated Titchmarsh-Weyl coeﬃcient. Mathematics Subject Classiﬁcation (2000). Primary 47B25, 47E05, 34B20; Secondary 34A55, 34L05, 47A57. Keywords. Canonical diﬀerential equation, Hilbert-Schmidt.

1. Introduction A canonical system of diﬀerential equations is an equation of the form Jy (x) = −zH(x)y(x), x ∈ R+ ,

(1.1)

where y(x) is a C -valued function on R , 0 −1 J= , 1 0 2

+

and H(x) is a R2×2 -valued function on R+ such that H(x) ≥ 0 and H|(0,x] ∈ L1 for all x ∈ R+ . The function H(x) is called the Hamiltonian corresponding to the canonical diﬀerential equation (1.1). We always assume that moreover tr(H(x)) = 1, x ∈ R+ . Canonical diﬀerential equations are usually studied with operator theoretic methods. A very good and detailed account on the operator model associated with the equation (1.1) can be found in [5]. Let us recall the basic notions: Denote by L2 (H, R+ ) the Hilbert space of all measurable C2 -valued functions f (x) on R+ such that +∞

f 2 = 0

f (x)∗ H(x)f (x)dx < +∞.

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One considers the closed subspace L2s (H, R+ ) of L2 (H, R+ ) consisting of all f ∈ L2 (H, R+ ) such that H(x)f (x) is constant on H-indivisible intervals. Thereby an interval I ⊆ R+ is called H-indivisible if for some ϕ ∈ R cos ϕ H(x) = ξϕ ξϕT , x ∈ I a.e., ξϕ = . sin ϕ Moreover one considers the linear relation Tmax,s = {(f ; g) ∈ L2s (H, R+ )2 : f is loc. abs. cont., Jf = −Hg},

(1.2)

and its restriction Tmin,s = {(f ; g) ∈ Tmax,s : f (0+) = 0},

(1.3)

which turns out to be a symmetric operator such that ∗ Tmin,s = Tmax,s .

All the selfadjoint extensions of Tmin,s are given by A(ν) = {(f ; g) ∈ Tmax,s : sin νf1 (0+) = cos νf2 (0+)}, ν ∈ R ,

(1.4)

and thereby we have A(ν1 ) = A(ν2 ) if and only if ν1 − ν2 ∈ πZ. For a detailed discussion of linear relations in Hilbert spaces see [2]. A fundamental notion in the theory of canonical systems is the TitchmarshWeyl coeﬃcient associated with the equation (1.1). Let W (x,z) = (wij (x,z))i,j=1,2 , x ∈ R+ , z ∈ C, be the 2 × 2-matrix-valued solution of the initial value problem d W (x, z) J = zW (x, z)H(x), x > 0, W (0+, z) = I, dx and deﬁne w11 (x, z)τ + w12 (x, z) . (1.5) w21 (x, z)τ + w22 (x, z) This limit exists for z ∈ C \ R and does not depend on τ ∈ R. The function qH is called the Titchmarsh-Weyl coeﬃcient of (1.1). It belongs to the Nevanlinna class z ) = qH (z), z ∈ C \ R, and has the property N0 , i.e., is holomorphic, satisﬁes qH (¯ that the kernel qH (z) − qH (w) LqH (w, z) = z−w ¯ is positive semideﬁnite. The inverse spectral theorem, a deep result due to L. de Branges (see [1]), shows that (1.5) sets up a bijective correspondence between the set of all trace normed Hamiltonians and the Nevanlinna class N0 . In a generalization to the indeﬁnite setting one allows the Titchmarsh-Weyl coeﬃcient q to belong to the generalized Nevanlinna class N<∞ , i.e., allows the kernel Lq to have a ﬁnite number of negative squares, and asks for an equation of similar type as (1.1), or for an operator model similar to L2s (H, R+ ), Tmax,s , such that again there is a bijective correspondence between N<∞ and the respective indeﬁnite Hamiltonians. qH (z) = lim

x→+∞

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It turns out that an indeﬁnite Hamiltonian is composed out of a ﬁnite number H1 , . . . , Hn of positive Hamiltonians and a ﬁnite number of real parameters. Thereby the occurring Hamiltonians and parameters are subject to certain conditions. One among them is that selfadjoint extensions of the symmetry Tmin,s in L2s (Hi , R+ ) have resolvents of Hilbert-Schmidt type. It is the aim of this note to characterize this property explicitly in terms of the Hamiltonian as well as in terms of the spectral measure of the associated Titchmarsh-Weyl coeﬃcient. In §2 we prove our main result Theorem 2.4 which gives a necessary and suﬃcient integrability condition on H in order that selfadjoint extensions have resolvents of Hilbert-Schmidt type. To this end we reduce the problem to a speciﬁc boundary condition and apply classical criteria for integral operators which can be found, e.g., in [3]. It is the subject of §3 to reformulate these conditions in terms of the spectral measure of the Titchmarsh-Weyl coeﬃcient qH , see Theorem 3.1. Thereby we use the theory of integral representations and operator models for Nevanlinna functions as developed, e.g., in [4]. In general our exposition relies on the material presented in [5] and the classical theory of integral operators, otherwise is fairly elementary.

2. Integral operators of Hilbert-Schmidt type In the subsequent lemma we are going to use a well-known criterion on whether an integral operator is of Hilbert-Schmidt type. Concerning this lemma note that the square root of H(t) exists because H(t) is assumed to be positive semideﬁnite. Lemma 2.1. Let H(t) = (hij (t))i,j=1,2 be a trace normed Hamiltonian on R+ such that +∞ h11 (t)dt < +∞. (2.1) 0

Consider the space L (R ) of all C2 -valued functions on R+ which are square integrable with respect to the Lebesgue measure, and consider the kernel 1 1 0 −χ{y<x} K(x, y) = H(x) 2 (2.2) H(y) 2 . −χ{y>x} 0 2

+ 2

On L2 (R+ )2 deﬁne the integral operator +∞ (Cg)(x) = K(x, y)g(y)dy

(2.3)

0

with g ∈ dom(C) if the integral exists for almost every x and Cg ∈ L2 (R+ )2 . Moreover, we set x M (x) = (mij (x))i,j=1,2 = H(t)dt. 0

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Then C is a continuous, everywhere deﬁned operator which belongs to the HilbertSchmidt class if and only if +∞ 2 m22 (t)h11 (t)dt < +∞. (2.4) 0

In this case the Hilbert-Schmidt norm of C coincides with the square root of this number. Proof. It is well known (see for example [3]) that C is a continuous, everywhere deﬁned operator which belongs to the Hilbert-Schmidt class if and only if +∞ +∞ tr(K(x, y)∗ K(x, y))dxdy < +∞. 0

0

In this case C22 coincides with the value of integral. We calculate K(x, y)∗ K(x, y) 1 1 0 −χ{y<x} 0 −χ{y>x} H(x) H(y) 2 = H(y) 2 −χ{y<x} 0 −χ{y>x} 0 1 1 0 1 0 0 0 0 0 1 = H(y) 2 χ{y>x} H(x) + χ{y<x} H(x) H(y) 2 0 0 1 0 1 0 0 0 1 1 1 0 0 0 = H(y) 2 χ{y>x} h22 (x) + χ{y<x} h11 (x) H(y) 2 . (2.5) 0 0 0 1 1

Set (kij (y))i,j=1,2 = H(y) 2 , and note that k12 = k21 . We then rewrite (2.5) as k11 (y)k11 (y) k11 (y)k12 (y) χ{y>x} h22 (x) k11 (y)k12 (y) k12 (y)k12 (y) k (y)k12 (y) k12 (y)k22 (y) +χ{y<x} h11 (x) 12 . k12 (y)k22 (y) k22 (y)k22 (y) 2 2 2 2 As k11 (y) + k12 (y) = h11 (y) and k12 (y) + k22 (y) = h22 (y) we obtain

tr(K(x, y)∗ K(x, y)) = χ{y>x} h22 (x)h11 (y) + χ{y<x} h11 (x)h22 (y), and by Fubini’s theorem +∞ +∞ tr(K(x, y)∗ K(x, y))dxdy 0

0 +∞

=

0

0

+∞

0

h22 (y) 0

y

h22 (x)dxdy = 2 0

+∞

h22 (x)dxdy +

h11 (y)

=2

y

h11 (y)

+∞

h11 (x)dxdy y

+∞

m22 (y)h11 (y)dy.

0

Lemma 2.2. Let H(t) = (hij (t))i,j=1,2 be a trace normed Hamiltonian on R+ such that (2.1) holds. Moreover, let B be the operator deﬁned in L2s (H, R+ ) by +∞ x 0 0 (Bf )(x) = JH(t)f (t)dt − JH(t)f (t)dt, 0 1 0 0

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with f ∈ dom(B) if f ∈ L2s (H, R+ ) such that (Bf )(x) exists for almost every x ∈ R+ and Bf ∈ L2s (H, R+ ). Then B is a continuous, everywhere deﬁned operator which belongs to the Hilbert-Schmidt class if and only if (2.4) holds true. Proof. The integral operator B can be written as +∞ (Bf )(x) = k(x, y)f (y)dy, 0

where k(x, y) is the kernel χ{y<x} k(x, y) = 0

0

−χ{y>x}

JH(y).

Now we consider the mapping φ from L2s (H, R+ ) into L2 (R+ )2 given by 1

φ(f )(t) = H(t) 2 f (t). It is straightforward to show that φ is an isometry. Let K(x, y) be as in (2.2) and the operator C deﬁned as in (2.3). By Lemma 2.1 the operator C is a continuous, everywhere deﬁned operator which belongs to the Hilbert-Schmidt class if and only if (2.4) holds true. We have Cφ = φB. In fact, φB ⊆ Cφ is obvious from the deﬁnitions of the respective kernels. For the other inclusion let f ∈ dom(Cφ). From (2.2) we see 1 that C(φ(f ))(x) is of the form H(x) 2 g(x) for g ∈ L2 (H, R+ ) with +∞ g(x) = k(x, y)f (y)dy 0 +∞ x 0 0 JH(t)f (t)dt − JH(t)f (t)dt. = 0 1 0 0 It remains to show that g ∈ L2s (H, R+ ). So let (a, b) be an indivisible interval of type ϕ ∈ R and calculate for x ∈ (a, b) d T ξ g(x) = ξϕT JH(x)f (x) = ξϕT Jξϕ ξϕT f (x) = 0. dx ϕ Hence g ∈ L2s (H, R+ ), and we proved that Cφ = φB. If g ∈ L2 (R+ )2 φ(L2s (H, R+ )), then ' & +∞ T x 1 1 1 0 1 JH(y) 2 g(y)dy − H(y) 2 g(y)dy . (Cg)(x) = H(x) 2 1 0 0 0 The second integral vanishes because 1 1 2 H(y) ∈ φ(L2s (H, R+ )). 0

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If x is contained in an indivisible interval (a, b) of type ϕ ∈ R, we assume that (a, b) is maximal, i.e., (a, b) is not contained in a larger indivisible interval. For a < y < x we have 1 1 H(x) 2 JH(y) 2 = ξϕ ξϕT Jξϕ ξϕT = 0. Therefore, 1

(Cg)(x) = H(x) 2

a

1

JH(y) 2 g(y)dy. 0 1

Since the columns of the matrix function χ{y≤a} H(y) 2 J T belong to φ(L2s (H, R+ )), we see that Cg = 0. We conclude that dom(C) = φ(dom(B))⊕φ(L2s (H, R+ ))⊥ and C = φBφ−1 P , where P is the orthogonal projection onto φ(L2s (H, R+ )). Thus C is a HilbertSchmidt operator if and only if B is. The previous lemma is of importance if we consider the operator theoretical background of canonical diﬀerential equations. Using the notation from the introduction we obtain the subsequent result. Corollary 2.3. Let H(t) = (hij (t))i,j=1,2 be a trace normed Hamiltonian on R+ such that (2.1) holds true. Then the selfadjoint extensions A(ν), ν ∈ (− π2 , π2 ] of Tmin,s have a resolvent (A(ν) − z)−1 , z ∈ ρ(A(ν)) which is of Hilbert-Schmidt type if and only if (2.4) holds. Proof. Because of the resolvent identity and since the product of a bounded operator and a Hilbert-Schmidt operator is of Hilbert-Schmidt type, the selfadjoint extension A(ν) of Tmin,s has a resolvent which is of Hilbert-Schmidt type if and only if (A(ν) − z)−1 is of Hilbert-Schmidt type for one z ∈ ρ(A(ν)). Moreover, by Krein’s formula (see Proposition 4.4 in [5]) the resolvents of diﬀerent selfadjoint extensions are one-dimensional perturbations of each other. Thus, the resolvents of A(ν), ν ∈ (− π2 , π2 ] being of Hilbert-Schmidt type is equivalent to the fact that π −1 A −z (2.6) 2 is a Hilbert-Schmidt operator for some z ∈ ρ(A( π2 )). Note that by our assumption (2.1) 1 span = ker Tmax,s , (2.7) 0 and hence ker A( π2 ) = {0}. This means that A( π2 )−1 is a densely deﬁned selfadjoint operator on L2s (H, R+ ). Moreover, as the elements f ∈ dom A( π2 ) vanish at 0 in the upper entry we see that B ⊆ A( π2 )−1 , where B is deﬁned as in Lemma 2.2. If (2.4) is satisﬁed, then B is an everywhere deﬁned, bounded operator belonging to the Hilbert-Schmidt class. We conclude B = A( π2 )−1 , and 0 ∈ ρ(A( π2 )).

Canonical Diﬀerential Equations of Hilbert-Schmidt Type

165

For the converse direction assume that (2.6) is of Hilbert-Schmidt type. We are going to show that A( π2 )−1 is an everywhere deﬁned, bounded operator belonging to the Hilbert-Schmidt class. If z = 0, we are done. Otherwise, besides zero the spectrum of (2.6) consists of eigenvalues. Since ker A( π2 ) = {0}, the number −z −1 cannot be an eigenvalue of (2.6). Because of −1 1 π −1 −1 π −1 1 π A −z + A −z A = , (2.8) 2 z 2 z 2 A( π2 )−1 is an everywhere deﬁned, bounded operator belonging to the HilbertSchmidt class. If g ∈ L2s (H, R+ ) and f = A( π2 )−1 g, then f = JHg. It follows from Lemma 7.8, [5] and from (2.7) that f (x) tends to zero in the lower component for x → +∞. Moreover, f (0+) is zero in the upper component. Thus Bg exists and coincides with f = A( π2 )−1 g, and we proved that B = A( π2 )−1 . By Lemma 2.2 condition (2.4) holds true. The above assertions yield a criterion for a canonical diﬀerential equation to have a compact resolvent of Hilbert-Schmidt type. Put cos ϕ ξϕ := . sin ϕ Theorem 2.4. Let H(t) = (hij (t))i,j=1,2 be a trace normed Hamiltonian on R+ . Then the selfadjoint extensions A(ν), ν ∈ (− π2 , π2 ] of Tmin,s have a resolvent (A(ν) − z)−1 , z ∈ ρ(A(ν)), which is of Hilbert-Schmidt type if and only if there exists a real ϕ such that +∞ ξϕT H(t)ξϕ dt < +∞, (2.9) 0

and

+∞

0

T T π M (t)ξϕ+ π ξ H(t)ξϕ dt < +∞. ξϕ+ ϕ 2 2

Here M (x) is deﬁned by

M (x) = (mij (x))i,j=1,2 =

(2.10)

x

H(t)dt. 0

Proof. First we consider a transformation of the Hamiltonian H(t). For µ ∈ R set sin µ − cos µ , Nµ = cos µ sin µ and Hµ (t) = Nµ∗ H(t)Nµ (see [5]). It is straightforward to verify that ψ : f → Nµ∗ f is an isomorphism from L2s (H, R+ ) onto L2s (Hµ , R+ ).

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µ µ (Tmax,s ) the respective diﬀerential operators on the If we denote by Tmin,s µ 2 + µ space Ls (Hµ , R ) and by A (ν) the corresponding selfadjoint extensions of Tmin,s , then µ µ ψTmin,s = Tmin,s ψ, ψTmax,s = Tmax,s ψ,

and ψA(ν) = Aµ (ν + µ − π2 )ψ. Thus, the A(ν)’s have resolvents of Hilbert-Schmidt type if and only if the Aµ (ν)’s do so. Condition (2.9) for the given H(t) is equivalent to condition (2.1) when h11 (t) now denotes the left upper entry of Hµ (t) with µ = π2 − ϕ. Similarly condition (2.10) for H(t) is equivalent to condition (2.4) when h11 (t), h22 (t) (m11 (t), m22 (t)) now denote the diagonal entries of Hµ (t) (the primitive of Hµ (t)) with µ = π2 − ϕ. If there exists a real ϕ such that (2.9) and (2.10) hold, then by Corollary 2.3 the resolvents of the Aµ (ν)’s are Hilbert-Schmidt type for µ = π2 − ϕ. As we mentioned above this is equivalent to the A(ν)’s having resolvents of HilbertSchmidt type. Conversely, assume that A(ν), ν ∈ (− π2 , π2 ], have resolvents of HilbertSchmidt type. If there exists a ϕ ∈ R such that ξϕ ∈ L2s (H, R+ ), then we set µ = π2 − ϕ and see that (2.9) holds. We just mentioned that this is the same as saying that (2.1) holds true when h11 (t) denotes the left upper entry of Hµ (t). Since also the Aµ (ν)’s have resolvents of Hilbert-Schmidt type, Corollary 2.3 shows that (2.4) holds for Hµ (t) which in turn yields (2.10). It remains to exclude the case that none of the constant functions ξϕ belong to L2s (H, R+ ). In fact, if we were in this situation, then all of the selfadjoint relations A(ν) would have a trivial kernel because the scalar multiples of ξϕ are the only possible candidates for elements of L2s (H, R+ ) to belong to ker A(ϕ). Thus all A(ν)−1 are operators, and by (2.8) with A( π2 ) replaced by A(ν) obtain that the operators A(ν)−1 are everywhere deﬁned, bounded and of Hilbert-Schmidt type. Hence the closed one-dimensional restriction (Tmin,s )−1 is a bounded, symmetric operator with a closed domain of co-dimension one. It is easy to check that D = (Tmin,s )−1 ⊕ ({0} × dom((Tmin,s )−1 )⊥ ) is a selfadjoint extension of (Tmin,s )−1 . Its inverse would then be a selfadjoint extension of Tmin,s , and would, therefore, coincide with A(ν) for some ν. But this contradicts the fact that D−1 has a non-trivial kernel.

3. Titchmarsh-Weyl coeﬃcients and the Hilbert-Schmidt property Recall that a Nevanlinna function has a unique integral representation 1 t a + bz + − 2 dσ(t), (3.1) t−z t +1 R where a, b ∈ R, b ≥ 0 and σ is a non-negative Borel measure on R such that 1 dσ(t) < +∞. 2 R t +1

Canonical Diﬀerential Equations of Hilbert-Schmidt Type

167

Theorem 3.1. Let H(t) = (hij (t))i,j=1,2 be a trace normed Hamiltonian on R+ , and let qH (z) be the Titchmarsh-Weyl coeﬃcient of the canonical diﬀerential equation (1.1). Assume that (3.1) is the integral representation of qH (z). Then H(x) satisﬁes (2.9) and (2.10) if and only of σ is a discrete measure σ= αn δtn , (3.2) n

such that

- 1 < +∞. t2n n

(3.3)

Proof. Corollary 4.2, [5] shows that Tmin,s is a completely non-selfadjoint symmetric relation with defect index (1, 1), and by Theorem 4.3, [5] the Titchmarsh-Weyl coeﬃcient qH (z) is the Q-function of A( π2 ) and Tmin,s . Thus according to Theorem 2.5, [4] the triplet (L2s (H, R+ ), Tmin,s , A( π2 )) is unitarily equivalent to (H, S, A), where H = L2 (σ), A = {(ϕ(t); tϕ(t)) : ϕ(t), tϕ(t) ∈ L2 (σ)}, S = {(ϕ(t); tϕ(t)) : ϕ(t), tϕ(t) ∈ L2 (σ),

ϕ(t)dσ(t) = 0}, R

in the case that b = 0 in the representation (3.1) of qH (z) and H = L2 (σ) ⊕ C, ϕ(t) tϕ(t) A= ; : ϕ(t), tϕ(t) ∈ L2 (σ), c ∈ C , 0 c ϕ(t) tϕ(t) S= ; : ϕ(t), tϕ(t) ∈ L2 (σ), bc + ϕ(t)dσ(t) = 0 , 0 c R otherwise. We conclude that A( π2 − z)−1 is of Hilbert-Schmidt type if and only if the operator ϕ(t) , ϕ(t) → t−z on L2 (σ) is of Hilbert-Schmidt type. This in turn is equivalent to the fact that σ is a discrete measure (3.2) such that (3.3) holds.

References [1] L. de Branges, Hilbert spaces of entire functions Prentice-Hall, London 1968. [2] A. Dijksma, H. de Snoo, Self-adjoint extensions of symmetric subspaces Paciﬁc J. Math. 54 (1974), 71–100. [3] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of linear operators. Vol. I Operator Theory: Advances and Applications, 49, Birkh¨ auser Verlag, Basel, 1990.

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[4] S. Hassi, H. Langer, H. de Snoo, Selfadjoint extensions for a class of symmetric operators with defect numbers (1, 1) Topics in operator theory, operator algebras and applications (Timi¸soara, 1994), 115–145, Rom. Acad., Bucharest, 1995. [5] S. Hassi, H. de Snoo, H. Winkler, Boundary-value problems for two-dimensional canonical systems Integral Equations Operator Theory 36 (2000), no. 4, 445–479. Michael Kaltenb¨ ack and Harald Woracek Institut f¨ ur Analysis und Scientiﬁc Computing Technische Universit¨ at Wien Wiedner Hauptstraße 8-10 A 1040 Wien, Austria e-mail: [email protected] e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 175, 169–191 c 2007 Birkh¨ auser Verlag Basel/Switzerland

Spectral Analysis of Diﬀerential Operators with Indeﬁnite Weights and a Local Point Interaction Ilia Karabash and Aleksey Kostenko Abstract. We consider quasi-self-adjoint extensions of the symmetric operator 2 d2 A = −(sgn x) dx 2 , dom(A) = {f ∈ W2 (R) : f (0) = f (0) = 0}, in the Hilbert 2 space L (R). The main result is a criterion of similarity to a normal operator for operators of this class. The spectra and resolvents of these extensions are described. As an application we describe the spectral properties of the main d2 d2 . and (sgn x) − dx operators (sgn x) − dx 2 + cδ 2 + cδ Mathematics Subject Classiﬁcation (2000). Primary 47A45; Secondary 47B50. Keywords. Symmetric operator, quasi-self-adjoint extensions, similarity problem, boundary triplets, Weyl functions.

Introduction Consider the symmetric operator A in the Hilbert space L2 (R) deﬁned by dom(A) = {f ∈ W22 (R) : f (0) = f (0) = 0}, (Af )(x) = −(sgn x)f (x)

for

f ∈ dom A.

(0.1)

The object of investigation is the similarity of quasi-self-adjoint extensions of A (see [1]) to a normal operator. Let us recall that two operators T1 and T2 in a Hilbert space H are called similar if there exists a bounded operator C with bounded inverse C −1 such that T1 = C −1 T2 C. Spectral problems (Ly)(x) = λr(x)y(x),

(0.2)

where L is an elliptic operator and the function r(x) change sign, occur in certain physical models (see [4] and references therein). The question whether the system of eigenfunctions of the problem (0.2) forms a Riesz basis was studied in [3], [4],

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[19], [32] [33], [34] (see also references in [34]). If the operator 1r L has a nonempty continuous spectrum, then the corresponding problem is the similarity of 1r L to a self-adjoint (normal) operator. In [9], [6], [15], [7], [8], [16] the Krein–Langer spectral theory of deﬁnitizable operators (see [28]) was applied to similarity problems for quasi J-nonnegative ´ and B. Najman [7] operators (see [15]) of the form 1r L. In particular, B. Curgus showed that the operator d2 ˜ = W 2 (R), A˜ = −(sgn x) 2 , dom(A) (0.3) 2 dx is similar to a self-adjoint one. This result was proved by another method in [21]; the method is based on the Naboko–Malamud criterion of similarity to a self-adjoint operator [31], [29] (see also [5]). One more proof is presented in [20]. In the recent papers [22], [13], [14], [24] the Naboko-Malamud criterion was applied to diﬀerent J-self-adjoint diﬀerential operators. Diﬀerential operators with an indeﬁnite weight are of interest from one more point of view. The characteristic function W (·) of the operator 1r L as well as the corresponding J-form J − W ∗ JW is unbounded in C+ . Therefore known suﬃcient conditions of similarity to a self-adjoint operator cannot be applied here (see [30], [20] and bibliography therein). In the present paper we describe quasi-self-adjoint extensions AB of the symmetric operator A in terms of boundary triplets (see [18], [11]). In Sections 3–4 we formulate a criterion of similarity of AB to a normal (self-adjoint) operator. In order to illustrate these results in Section 5 we obtain simple similarity criteria for operators with local point interactions at zero d2 d2 ˜ ˜ A1 := sgn x − 2 + c1 δ , c1 ∈ C, A2 := sgn x − 2 + c2 δ , c2 ∈ C. dx dx (See deﬁnitions of the operators A˜1 , A˜2 in [2] and also in Section 5 of the present paper.) The results of the paper were announced in [23]. Notation: By H, H we denote separable Hilbert spaces. The set of all bounded linear operators from H to H is denoted by [H, H] or [H] if H = H. C(H) stands for the set of closed densely deﬁned operators in H. Let T be a linear operator in a Hilbert space H. In what follows dom(T ), ker(T ), ran(T ) are the domain, kernel, range of T , respectively. We denote by σ(T ), σr (T ), σc (T ) the point, residual and continuous spectra of T . By σp (T ) the set of eigenvalues of T is indicated. −1 We denote the resolvent set by ρ(T ); RT (λ) := (T − λI) , λ ∈ ρ(T ), is the resolvent of T . Recall that σr (T ) = {λ ∈ σ(T ) \ σp (T ) : ran(T − λI) = H}, σc (T ) = σ(T ) \ (σp (T ) σr (T )). We set C± := {λ ∈ C : ± Im λ > 0}, R+ := (0, +∞), R− := (−∞, 0). By χI (t) we denote the characteristic function of the interval I, i.e., χI (t) = 1 for t ∈ I, χI (t) = 0 for t ∈ I. Finally, we set χ± (t) := χR± (t).

Diﬀerential Operators with Indeﬁnite Weights and a Point Interaction 171

1. Preliminaries 1.1. A similarity criterion Our approach is based on the concept of boundary triplets (see [18], [11]) and the resolvent similarity criterion obtained by S.N. Naboko [31] and M.M. Malamud [29] (in [5] this criterion was obtained under an additional assumption). Theorem 1.1 ([29, 31]). A closed operator T in a Hilbert space H is similar to a self-adjoint one if and only if σ(A) ⊂ R and for all f ∈ H the inequalities +∞ 2 2 ε RT (µ + iε) f dµ ≤ C f , sup ε>0 −∞

+∞ 2 2 sup ε RT ∗ (µ + iε) f dµ ≤ C ∗ f ,

(1.1)

ε>0 −∞

are valid with constants C and C ∗ independent of f . 1.2. Linear relations Deﬁnition 1.1. (i) A closed linear relation Θ in H is a closed subspace Θ of H ⊕ H. (ii) The closed linear relation Θ is symmetric if for all {f1 , g1 }, {f2, g2 } ∈ Θ the condition (g1 , f2 ) − (f1 , g2 ) = 0,

(1.2)

is satisﬁed. (iii) The closed linear relation Θ is self-adjoint if it is maximal symmetric, i.e., ˜ such Θ is symmetric and there does not exist a closed symmetric relation Θ ˜ that Θ is properly contained in Θ. Let us illustrate closed linear relations by simple examples. Example 1.1. (i) Let B be a closed operator in H, not necessarily bounded. Then the graph G(B) of B is a closed relation in H. Moreover, if B = B ∗ is a self-adjoint operator, then G(B) is a self-adjoint relation in H. (ii) The subspaces Θ0 := {0} × H, Θ1 := H × {0} of H × H are self-adjoint relations in H. Obviously, Θ0 is not the graph of any operator. 1.3. Boundary triplets Let A ∈ C(H) be a closed symmetric operator with equal deﬁciency indices n+ (A) = n− (A) (n± (T ) := dim N±i and by Nλ := ker(T ∗ − λ) the deﬁciency subspaces of A are indicated). Without loss of generality we can assume that A is simple. This means that A has no self-adjoint parts.

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Deﬁnition 1.2 ([1]). (i) A closed extension A˜ of A is called a proper extension if A ⊂ A˜ ⊂ A∗ . The set of all proper extensions is denoted by ExtA . (ii) A proper extension A˜ is called a quasi-self-adjoint if ˜ dom(A)) = n± (A). (1.3) dim(dom(A)/ We recall the deﬁnition of a boundary triplet which may be considered as an abstract version of the second Green formula. Deﬁnition 1.3 ([18]). A triplet Π = {H, Γ0 , Γ1 } consisting of an auxiliary Hilbert space H and linear mappings Γj : dom(A∗ ) −→ H,

j ∈ {0, 1},

(1.4)

is called a boundary triplet for the adjoint operator A∗ of A if the following two conditions are satisﬁed: (i) The second Green formula (A∗ f, g) − (f, A∗ g) = (Γ1 f, Γ0 g)H − (Γ0 f, Γ1 g)H ,

f, g ∈ dom(A∗ ),

(1.5)

takes place and (ii) the mapping Γ : dom(A∗ ) −→ H ⊕ H,

Γf := {Γ0 f, Γ1 f },

(1.6)

is surjective. The above deﬁnition allows one to describe the set ExtA in the following way (see [10, 11]). Proposition 1.1 ([10, 11]). Let Π = {H, Γ0 , Γ1 } be a boundary triplet for A∗ . Then ˜ between the mapping Γ establishes a bijective correspondence A˜ → Θ := Γ(dom(A)) the set ExtA and the set of closed linear relations in H. By Proposition 1.1 the following deﬁnition is natural. Deﬁnition 1.4. Let Π = {H, Γ0 , Γ1 } be a boundary triplet for the operator A∗ . ˜ that is (i) Denote AΘ = A˜ if Θ = Γ(dom(A)), AΘ := A∗ |DΘ ,

where

DΘ := {f ∈ dom(A∗ ) : {Γ0 f, Γ1 f } ∈ Θ}.

(1.7)

(ii) If Θ = G(B) is the graph of B ∈ C(H), then dom(AΘ ) is determined by the equation dom(AB ) = DB := DΘ = ker(Γ1 − BΓ0 ). We set AB := AΘ . Let us make the following remarks. Remark 1.1. 1) The deﬁciency indices n± (A) are equal to the dimension of H, i.e., dim(H) = n± (A). 2) There exist two self-adjoint extensions Aj := A∗ | ker(Γj ) which are naturally associated to a boundary triplet. According to Deﬁnition 1.4 Aj = AΘj , j ∈ {0, 1}, where Θ0 = {0}×H, Θ1 = H×{0}. Conversely, if A0 is a self-adjoint

Diﬀerential Operators with Indeﬁnite Weights and a Point Interaction 173 extension of A, then there exists a boundary triplet Π = {H, Γ0 , Γ1 } such that A0 = A∗ | ker(Γ0 ). 3) Θ is the graph of an operator B ∈ C(H) iﬀ A˜ and A0 are disjoint, i.e., ˜ ∩ dom(A0 ) = dom(A). dom(A) 4) Θ = G(B) with B ∈ [H] iﬀ A˜ and A0 are transversal, i.e., A˜ and A0 are ˜ + dom(A0 ) = dom(A∗ ). disjoint and dom(A) Deﬁnition 1.5 ([12]). The proper extension A˜ ∈ ExtA is called almost solvable if there exists a boundary triplet Π = {H, Γ0 , Γ1 } and an operator B ∈ [H] such that ˜ = dom(AB ) := ker(Γ1 − BΓ0 ). dom(A)

(1.8)

The set of almost solvable extensions is denoted by AsA . Note that the class AsA is suﬃciently wide. Proper extensions having two regular points λ1 , λ2 ∈ C such that Im λ1 · Im λ2 < 0 belong to AsA . All quasi-self-adjoint extensions are in AsA if n± (A) < ∞. 1.4. Weyl functions It is well known that Weyl functions are an important tool in the direct and inverse spectral theory of singular Sturm–Liouville operators. In [10, 11] the concept of Weyl function was generalized to an arbitrary symmetric operator A with inﬁnite deﬁciency indices n+ (A) = n− (A). In this subsection we recall basic facts about Weyl functions. Deﬁnition 1.6 ([10, 11]). Let Π = {H, Γ0 , Γ1 } be a boundary triplet for the operator A∗ . The Weyl function of A corresponding to the boundary triplet {H, Γ0 , Γ1 } is a unique mapping M (·) : ρ(A0 ) −→ [H]

(1.9)

satisfying Γ1 fλ = M (λ)Γ0 fλ

for all

fλ ∈ Nλ ,

λ ∈ ρ(A0 ),

(1.10)

where Nλ := ker(A∗ − λI). It is well known (see [10, 11]) that the above implicit deﬁnition of the Weyl function is correct and M (·) is an R-function obeying 0 ∈ ρ(Im(M (i))). The Weyl function immediately provides some information about the “spectral properties” of proper extensions. We conﬁne ourselves to the case of almost solvable extensions of the symmetric operator A. Proposition 1.2 ([11, 12]). Suppose that Π = {H, Γ0 , Γ1 } is a boundary triplet for A∗ , M (·) is the corresponding Weyl function, λ ∈ ρ(A0 ) and B ∈ [H]. Then: 1) λ ∈ ρ(AB ) if and only if 0 ∈ ρ(B − M (λ)); 2) λ ∈ σi (AB ) if and only if 0 ∈ σi (B − M (λ)), i ∈ {p, r, c}.

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1.5. γ-ﬁelds With each boundary triplet we can associate a so-called γ-ﬁeld. Deﬁnition 1.7 ([11]). Let Π = {H, Γ0 , Γ1 } be a boundary triplet for A∗ . The γ-ﬁeld γ(·) corresponding to Π is deﬁned by γ(λ) := (Γ0 |Nλ )−1 : H −→ Nλ ,

λ ∈ ρ(A0 ).

(1.11)

λ, λ0 ∈ ρ(A0 ),

(1.12)

One can easily check that γ(λ) = (A0 − λ0 )(A0 − λ)−1 γ(λ0 ),

and consequently γ(·) is a γ-ﬁeld in the sense of [26]. It is shown in [11] that the γ-ﬁeld γ(·) and the Weyl function M (·) are related by ¯0 )γ(λ0 )∗ γ(λ), λ, λ0 ∈ ρ(A0 ). (1.13) M (λ) − M (λ0 )∗ = (λ − λ The relation (1.13) means the M (·) is a Q-function in the sense of [26]. The following version of the Krein-Naimark formula for canonical resolvents (see for instance [26]) is based on the notion of boundary triplets. Theorem 1.2 ([10, 11]). Let A˜ be an almost solvable extension of A (A˜ ∈ AsA ), i.e., A˜ = AB with B ∈ [H] for some boundary triplet Π = {H, Γ0 , Γ1 }. Then (AB − λ)−1 = (A0 − λ)−1 + γ(λ)(B − M (λ))−1 γ ∗ (λ),

λ ∈ ρ(AB ).

(1.14)

Here M (·) and γ(·) are the Weyl function and γ-ﬁeld corresponding to the triplet Π.

2. Extensions of the minimal operator 2.1. Boundary conditions Consider the operator A of the form (0.1). It is obvious that A is a closed simple symmetric operator √ with deﬁciency indices n± (A) = 2. We denote by z the branch of the multifunction √ on the complex plane C with the cut along R− , singled out by the condition −1 + i0 = i. Theorem 2.1. (i) The adjoint operator A∗ has the form d2 , dom(A∗ ) = W22 (R \ {0}) := W22 (R− ) ⊕ W22 (R+ ). dx2 (ii) Let mappings Γj : W22 (R \ {0}) → C2 , j = {0, 1}, be given by f (+0) f (+0) Γ0 f = . , Γ1 f = f (−0) −f (−0) A∗ = −(sgn x)

Then Π = {C2 , Γ0 , Γ1 } is a boundary triplet for A∗ . (iii) The corresponding Weyl function M (·) is √ − −λ 0√ M (λ) := , λ ∈ C \ R. 0 −1/ λ

(2.1)

(2.2)

(2.3)

Diﬀerential Operators with Indeﬁnite Weights and a Point Interaction 175 (iv) The corresponding γ-ﬁeld γ(λ) : C2 → Nλ is √ √ c− c+ γ(λ) := c+ · e− −λx χ+ (x) + √ · e λx χ− (x), c− λ

c± ∈ C.

(2.4)

Proof. The ﬁrst statement is obvious. Moreover, we have (A∗ f, g) − (f, A∗ g) = f (+0)g(+0) + f (−0)g(−0)− f, g ∈ dom(A∗ ). (2.5)

− f (+0)g (+0) − f (−0)g (−0), Hence (ii) follows from Deﬁnition 1.3. Note that Nλ = {fλ (x) := c+ · e−

√ −λx

χ+ (x) + c− · e

Combining (2.2) and (2.6), one gets c√ + Γ0 f λ = , c− λ

√

λx

Γ1 f λ =

χ− (x) : c± ∈ C}.

(2.6)

√ −c+ −λ . −c−

(2.7)

By Deﬁnitions 1.6 and 1.7, we easily obtain (2.3) and (2.4).

Let us introduce the following boundary conditions at zero a11 f (−0) + a12 f (−0) + a13 f (+0) + a14 f (+0) = 0 a21 f (−0) + a22 f (−0) + a23 f (+0) + a24 f (+0) = 0

aij ∈ C.

,

(2.8)

By Deﬁnition 1.2, a quasi-self-adjoint extension A˜ of the operator A has the form A˜ = A(aij ) = A∗ | dom(A(aij ) ), dom(A(aij ) ) = {f ∈ W22 (R \ {0}) : f satisﬁes conditions (2.8)}, (2.9) with the matrix

(aij ) :=

a11 a21

a12 a22

a13 a23

a14 a24

such that rank(aij ) = n± (A) = 2.

(2.10)

Consider three cases. 1) Let

a11 a21

a12 a22

=

0 0

0 0

or

a13 a23

a14 a24

=

0 0 0 0

.

Then A˜ = A− ⊕ A+ , where A± is an operator in L2 (R± ). By condition (2.10), we see that one of the operators A− , A+ is a symmetric with deﬁciency indices (1,1) and another one is an adjoint to a symmetric operator with deﬁciency indices ˜ and A˜ is not similar to a normal operator. (1,1). Hence C \ R ⊂ σp (A)

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a13 a14 a11 a12 and have a zero column. Since 2) Suppose that a21 a22 a23 a24 rank(aij ) = 2, it follows that A˜ = A− ⊕ A+ , where A+ and A− are self-adjoint operators in L2 (R+ ) and L2 (R− ), respectively. Thus A˜ = A˜∗ .

3) Suppose that there are three determinants a ∆1 = 11 a21 a12 ∆3 = a22

nonzero columns in (aij ). In this case one of the a a14 , ∆2 = 12 a24 a22 a14 a , ∆4 = 11 a24 a21

a13 , a23 a13 a23

(2.11)

does not vanish. Evidently, only the case (3) is of interest to us. 2.2. The case ∆1 = 0 Let ∆1 = 0. (The cases ∆2 = 0, ∆3 = 0, and ∆4 = 0 will be considered in Section 4.) Then conditions (2.8) take the form f (+0) = b11 f (+0) + b12 f (−0) (2.12) −f (−0) = b21 f (+0) + b22 f (−0). Hence, by Deﬁnition 1.5, A(aij ) = AB = A∗ | ker(Γ1 − BΓ0 ). Here b11 b12 B= ∈ C2×2 b21 b22 and the boundary triplet Π = {C2 , Γ0 , Γ1 } is of the form (2.2). In what follows AB stands for the operator AB := − sgn x

d2 , dx2

dom(AB ) = {f ∈ W22 (R \ {0}) : f satisﬁes (2.12)}. (2.13)

For B ∈ C2×2 and the Weyl function M (·) of the form (2.3) we deﬁne the function ϕB (·) : C \ R → C by ϕB (λ) := det(B − M (λ)),

λ ∈ C \ R.

(2.14)

Lemma 2.1. Suppose AB is the operator of the form (2.13) and |b12 | + |b21 | = 0; then: (i) σc (AB ) = R; (ii) σr (AB ) = ∅; (iii) σp (AB ) = {λ ∈ C \ R : ϕB (λ) = 0} = {λ ∈ C+ : ϕB (λ) = 0} ∪ {λ ∈ C− : ϕB ∗ (λ) = 0}. Proof. Simple calculations show that there are no eigenvalues on the real axis and σc (AB ) = R. The second and the third statements evidently follow from Proposition 1.2.

Diﬀerential Operators with Indeﬁnite Weights and a Point Interaction 177 Lemma 2.2. Let Π = {H, Γ0 , Γ1 } be a boundary triplet of the form (2.2). Then ⎛ +∞ ⎞ √ 0 − −λt f (t)e dt ⎜ 0 ⎟ ⎟, (2.15) γ ∗ (λ)f = ⎜ f ∈ L2 (R). √ ⎝ 1 00 ⎠ λt √ f (t)e dt λ −∞

Proof. By Theorem 2.1.(iv), γ ∗ (·) is a map from L2 (R) to C2 . To ﬁnd γ ∗ (·) we use the equation c1 ∗ (γ(λ)c, f )L2 (R) = (c, γ (λ)f )C2 , c= (2.16) ∈ C2 , f ∈ L2 (R). c2 Combining (2.4) and (2.16), one gets +∞ 0 √ √ c2 − −λt c1 · f (t)e dt + √ · f (t)e λt dt = c1 · (γ ∗ (λ)f )1 + c2 · (γ ∗ (λ)f )2 . (2.17) λ −∞

0

Hence (2.15) immediately follows from (2.17). Let us denote +∞ √ f (t)e− −λt dt, y+ (f, λ) := 0

0 y− (f, λ) :=

f (t)e

√ λt

dt,

λ ∈ C \ R.

−∞

(2.18) Lemma 2.3. Let the operator AB be of the form (2.13) and A0 = A∗ | ker Γ0 . Then (AB − λ)−1 f − (A0 − λ)−1 f (x) √ e− −λx · χ+ (x) 1 b12 = y+ (f, λ) − √ · y− (f, λ) b22 + √ ϕB (λ) λ λ √ √ λx e · χ− (x) b11 + −λ √ + √ · y− (f, λ) − b21 · y+ (f, λ) , λ · ϕB (λ) λ f ∈ L2 (R),

λ ∈ ρ(AB ).

(2.19)

Here ϕB (·) and y± (f, ·) are given by (2.14) and (2.18), respectively. Proof. By (2.3), for λ ∈ ρ(AB ) (see Lemma 2.1) one obtains √ −1 b11 + −λ b12 √ −1 (B − M (λ)) = b21 b22 + 1/ λ √ 1 −b√ b22 + 1/ λ 12 = . −b21 b11 + −λ ϕB (λ) Combining (2.4), (2.15), (2.20) with formula (1.14), we get (2.19).

(2.20)

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3. Similarity to a normal operator 3.1. The main result For each B ∈ C2×2 let us deﬁne the function ϕ+ B : C+ → C in the following way. We set ϕ+ for λ ∈ C+ , (3.1) B (λ) := ϕB (λ) = det(B − M (λ)) and for x ∈ R by ϕ+ B (x) we denote the boundary values of ϕB (λ) in C+ , lim det(B − M (z)), ϕ+ B (x) := z→x

x ∈ R ∪ {∞}.

(3.2)

z∈C+

Note that the function ϕ+ B is analytic on C+ and continuous on C+ \ {0}. The following similarity criterion is the main result of the paper. Theorem 3.1 (Main Theorem). Assume that ∆1 = 0 and the operator AB is deﬁned + by (2.13). Let ϕ+ B and ϕB ∗ be the functions deﬁned in (3.1)–(3.2) and |b12 |+|b21 | = 0. Then AB is similar to a normal operator if and only if the following conditions hold: + (i) ϕ+ B and ϕB ∗ have no zeroes in the set R ∪ {∞}; + (ii) ϕB and ϕ+ B ∗ have no zeroes of the second order in C+ . Remark 3.1. Suppose that |b12 | + |b21 | = 0. Then the operator AB has the form AB = A− ⊕ A+ , where the operators A± : L (R± ) → L2 (R± ) are given by 2

d2 , dom(A± ) = {f ∈ W22 (R± ) : f (±0) + b± · f (±0) = 0}. (3.3) dx2 Here b+ := −1/b11 , b− := b22 . Operators A± are well studied. A± := ∓

Remark 3.2. The function ϕ+ B has a simple form. Indeed, by (2.14) and (2.3), we have √ b11 ϕ+ λ ∈ C+ \ {0}. (3.4) B (λ) = −ib22 λ + (b11 b22 − b12 b21 ) − i + √ , λ So the conditions of Theorem 3.1 can be easily checked (see Section 5). Let us remark that the function ϕ+ B has at most two zeroes (a zero of multiplicity k is counted as k zeroes). A criterion of similarity to a self-adjoint operator immediately follows from Theorem 3.1. Theorem 3.2. Let |b12 |+|b21 | = 0. Then the operator AB is similar to a self-adjoint + one iﬀ the functions ϕ+ B and ϕB ∗ do not vanish in C+ . + Proof. By Lemma 2.1.(iii), σ(AB ) = R iﬀ the functions ϕ+ B and ϕB ∗ have no zeroes in C+ . Combining this fact with Theorem 3.1, we get Theorem 3.2.

To prove the main theorem we recall the following Lemma.

Diﬀerential Operators with Indeﬁnite Weights and a Point Interaction 179 Lemma 3.1. If an operator T is similar to a normal one, then the inequality (T − λI)−1 ≤

C dist(λ, σ(T ))

(3.5)

holds with some constant C > 0. 3.2. Some estimates We start with the following lemma. Lemma 3.2. Let |b12 | + |b21 | = 0. Suppose there exists λ0 ∈ R ∪ {∞} such that + ϕ+ B (λ0 ) = 0 or ϕB ∗ (λ0 ) = 0. Then the operator AB of the form (2.13) is not similar to a normal operator. Proof. Without loss of generality suppose that b21 = 0 and ϕ+ B (λ0 ) = 0 for some λ0 ∈ R. It is obvious that for λ ∈ C \ R 2 √ 2 − −λt sup |y+ (f, λ)| = sup f (t)e dt f ≤1 f ≤1 R+ (3.6) √ 1 1 − −λx 2 √ . √ = e χ+ (x)L2 = = |2 Re −λ| |2 Im λ| Further, we set f+ (·) := f (·)χ+ (·), f ∈ L2 (R). By (2.19), we have 4 4 4(AB − λI)−1 f+ − (A0 − λI)−1 f+ 42 2 L 4 √ 42 4 e− −λx · χ (x) 4 1 4 4 + √ b =4 + (f , λ) y 4 22 + + 4 4 ϕ+ (λ) λ B L2 4 √ 42 4 e λx · χ (x) 4 4 4 − + |b21 | · 4 √ · y+ (f+ , λ)4 , λ ∈ ρ(AB ) ∩ C+ . 4 λ · ϕ+ 4 (λ) B

(3.7)

L2

Combining (3.6) and (3.7), one obtains 4 4 4(AB − λI)−1 − (A0 − λI)−1 42 2 (3.8) L 2 2 1 1 1 b21 √ + √ √ √ . ≥ b22 + √ · + + 4 λ · ϕB (λ) | Re λ · Im λ| λ 2ϕB (λ) · Im λ Now if we recall Lemma 2.1, we obtain that dist(λ, σ(AB )) = | Im λ| in some neighborhood of λ0 . Therefore, for suﬃciently small ε and λ = λ0 + iε 42 4 dist(λ, σ(AB ))2 · 4(AB − λI)−1 − (A0 − λI)−1 4L2 b · Im λ 2 1 1 21 √ √ ≥ C1 · . (3.9) ≥ √ · + 4 λ · ϕB (λ) | Re λ · Im λ| | Im λ|

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Then the left part of inequality (3.9) is unbounded in the neighborhood of λ0 . Note that A0 is a self-adjoint operator. Hence inequality (3.5) is valid for A0 . Therefore, the function 4 4 dist(λ, σ(AB )) · 4(AB − λI)−1 4L2 (3.10) is unbounded in the neighborhood of λ0 . By Lemma 3.1, the operator AB is not similar to a normal operator. √ + If ϕ+ B (∞) = 0, then formula (3.4) implies ϕB (λ) = b11 / λ for λ ∈ C+ . Hence for ε large enough 4 42 dist(iε, σ(AB ))2 · 4(AB − iεI)−1 − (A0 − iεI)−1 4L2 ≥

ε2 |b21 | √ √ = C2 ε . · 4|b11 | | Re iε · Im iε|

(3.11)

Since the right part of (3.11) is unbounded in C+ , we see that AB is not similar to a normal operator. Lemma 3.3. Let |b12 | + |b21 | = 0. Suppose that the function ϕ+ B has a zero of algebraic multiplicity 2 in C+ . Then AB is not similar to a normal operator. Proof. Let b21 = 0 (the case b12 = 0 can be considered in the same way). Suppose λ0 ∈ C+ is a zero of multiplicity 2 of ϕ+ B (·). By (3.8), we have for λ ∈ ρ(AB ) ∩ C+ 4 4 1 4(AB − λI)−1 − (A0 − λI)−1 4 2 ≥ √ b21 √ √ . (3.12) · + L 4 λ · ϕB (λ) | Re λ · Im λ|1/2 Since λ0 ∈ ρ(A0 ), we see that (3.12) implies 4 4 4(AB − λI)−1 4 2 ≥ L

Cλ0 , |ϕ+ B (λ)|

Cλ0 = const > 0,

(3.13)

in some neighborhood of λ0 . Therefore λ0 is a pole of multiplicity 2 of the resolvent (AB − λI)−1 . Consequently, the operator AB is not similar to a normal one. We also need the following estimates. Lemma 3.4. Let λ = µ + iε, (ε > 0). Let y± (f, λ) be of the form (2.18). Then the following inequalities +∞4 42 4 y± (f, λ) −√−λx 4 4 √ 4 dµ ≤ 2π · C1 f 2 2 , ε · e χ (x) + L 4 4 2 λ L

(3.14)

−∞

+∞4 42 4 y± (f, λ) √λx 4 4 √ ε ·e χ− (x)4 dµ ≤ 2π · C2 f 2L2 4 4 λ L2 −∞

are valid for all f ∈ L2 (R) with constants C1 , C2 independent of ε and f .

(3.15)

Diﬀerential Operators with Indeﬁnite Weights and a Point Interaction 181 Proof. Let us prove the inequality (3.14) for y− (f, λ). Put f− (·) := f (·)χ− (·), f ∈ L2 (R). Denote by F (z) the Fourier transform of f− , 1 F (z) := √ 2π

+∞ f− (t)e−izt dt,

z ∈ C+ .

(3.16)

−∞

Note that F (·) ∈ H 2 (C+ ) and F H 2 = f− L2 ≤ f L2 . Further, we obtain +∞4 42 4 y− (f, λ) −√−λx 4 4 √ χ+ (x)4 ·e 4 4 2 dµ λ L

−∞

+∞

= 2π −∞ +∞

= 2π −∞

√ √ 1 |F (i λ)|2 e− −λx 2L2 dµ |λ|

√ 2 dµ 1 √ √ F (i λ) ) 2| λ Im λ| 2 |λ|

iε+∞

≤ 2π

√ 2 √ 1 √ √ F (i λ) |d λ| 2| Im λ Re λ|

iε−∞ iε+∞

= 2π

√ 2 √ 1 F (i λ) |d λ| Im λ

iε−∞

2π = ε

iε+∞

√ 2 √ F (i λ) |d λ|.

(3.17)

iε−∞

Hence we ﬁnd the estimate +∞4 iε+∞ 42 √ 2 √ 4 y− (f, λ) −√−λx 4 4 √ 4 · e ε χ (x) dµ ≤ 2π F (i λ) |d λ|. + 4 4 λ L2 −∞

(3.18)

iε−∞

√ Finally, let us remark that |d λ| is the Carleson measure for all ε > 0 (see [17]). It is easy to see that the Carleson norms of these measures are uniformly bounded. Then, by the Carleson embedding theorem (see [17]), there exists C1 > 0 such that for all ε > 0 the inequality iε+∞

√ 2 √ 2 F (i λ) |d λ| ≤ C1 F H 2 = C1 f− L2 ≤ C1 f L2

(3.19)

iε−∞

holds. Combining (3.18) and (3.19), one gets (3.14). Other inequalities can be obtained analogously.

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I.M. Karabash and A.S. Kostenko

3.3. Proof of Theorem 3.1 3.3.1. Necessity. Follows immediately from Lemmas 3.2 and 3.3. 3.3.2. Suﬃciency. a) Suppose that conditions (i) and (ii) of Theorem 3.1 hold. + Suppose also that λ1 ∈ C+ is a unique zero of ϕ+ B (·) and ϕB ∗ (·) does not vanish in C+ . Then, by Lemma 2.1, σ(AB ) = R ∪ {λ1 }. Denote by B(λ1 ) a closed neighborhood of λ1 such that B(λ1 ) ⊂ C+ . Let us consider the Riesz projection 1 P1 := (AB − λ)−1 dλ , (3.20) 2πi ∂B(λ1 ) where ∂B(λ1 ) is the boundary of B(λ1 ). Then (see [25]) P1 ∈ [L2 (R)] and AB P1 = P1 AB . Since A0 is a self-adjoint operator, it follows that 1 1 −1 −1 (AB − λ) − (A0 − λ) dλ = (AB − λ)−1 dλ. (3.21) 2πi ∂B(λ1 ) 2πi ∂B(λ1 ) It is not hard to show that P1 is a one-dimensional operator in L2 (R). Actually, 1 we set mλ1 := lim λ−λ . Using (3.21) and (2.19), one gets λ→λ1 ϕ(λ) √ (P1 f )(x) 1 b12 − −λ1 x =e · χ+ (x) · b22 + √ y+ (f,λ1 ) − √ · y− (f,λ1 ) mλ1 λ1 λ1 √ √ λ1 x · χ− (x) b11 + −λ1 e √ √ + · y− (f,λ1 ) − b21 · y+ (f,λ1 ) , f ∈ L2 (R). (3.22) λ1 λ1 Let us write (3.22) in the following form 9 : √ √ e λ1 x · χ− (x) (P1 f )(x) 1 − −λ1 x √ = y+ (f,λ1 ) · e χ+ (x) · b22 + √ − b21 mλ1 λ1 λ1 9 : √ √ ) y− (f,λ1 ) e λ1 x · χ− (x) − −λ1 x √ + √ · e χ+ (x) · (−b12 ) + · b11 + −λ1 . (3.23) λ1 λ1 Note that (b22 + √1λ ) −b12 1 √ = det(B − M (λ1 )) = ϕB (λ1 ) = 0. det −b21 (b11 − i λ1 )

(3.24)

Hence P1 is a one-dimensional operator. b) By step (a), the space H = L2 (R) can be decomposed as (see [25]) H = H0 H1 ,

Hj := Pj H,

j ∈ {0, 1},

P0 := I − P1 .

(3.25)

Moreover, AB Pj = Pj AB , j ∈ {0, 1}, and the operator AB admits the following decomposition AB = A0B A1B , We also have

σ(A1B )

= {λ1 } and

AjB := Pj AB Pj , σ(A0B )

= R.

j ∈ {0, 1}.

(3.26)

Diﬀerential Operators with Indeﬁnite Weights and a Point Interaction 183 Let us show that the inequality +∞ 4 42 4 4 sup ε 4RA0B (µ + iε) f 4 dµ ≤ C f 2 ,

f ∈ H0 ,

ε>0 −∞

(3.27)

holds with some constant C > 0. Since ϕ+ B (·) does not vanish in R ∪ {∞}, we see that Lemma 3.4 and (2.19) imply sup RAB (µ + iε)f 2 dµ ≤ C1 f 2 , f ∈ L2 (R) . (3.28) ε>0

µ∈R µ+iε∈B(λ1 )

Therefore

sup ε>0

RA0B (µ + iε)f 2 dµ ≤ C1 f 2,

f ∈ H0 .

(3.29)

µ∈R µ+iε∈B(λ1 )

Further, let us recall that λ1 ∈ ρ(A0B ). It means that the operator function RA0B (λ) is bounded on B(λ1 ). If we combine this with (3.29), we get (3.27). Since σ(A0B ) = R and ϕ+ B ∗ (·) have no zeroes in C+ , we can obtain in the same way the estimate +∞ R(A0B )∗ (µ + iε)f 2 dµ ≤ C ∗ f 2 , f ∈ H0 , C ∗ = const > 0. sup ε>0

−∞

(3.30) Hence, by Theorem 1.2, the operator A0B is similar to a self-adjoint operator. Moreover, A1B is a one-dimensional operator. Thus the operator AB is similar to a normal one. + c) General case. Suppose that conditions (i) and (ii) hold, i.e., ϕ+ B (·) and ϕB ∗ (·) do not vanish on R ∪ {∞} and have only simple zeros in C+ . Let us denote by + + + n(ϕ+ B ) the number of zeroes of ϕB (·) in C+ . Then n(AB ) := n(ϕB ) + n(ϕB ∗ ) is the number of eigenvalues of AB , σp (AB ) = {λi : i = 1, . . . , n(AB )}, (σp (AB ) = ∅ if n(AB ) = 0). By (3.4), the function ϕ+ B (·) has at most two zeroes in C+ . Hence n(AB ) ≤ 4. It can be shown in the same way as in step (b) that exists a decomposition

L2 (R) = H0 H1 · · · Hn(AB ) , n(AB )

AB = A0B A1B · · · AB

.

(3.31)

Here A0B is similar to a self-adjoint operator and AiB are one-dimensional operators. Thus, AB is similar to a normal operator. This completes the proof.

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4. Other boundary conditions 4.1. The case ∆2 = 0 The following theorem is an obvious corollary of Theorem 2.1. Theorem 4.1. (i) The triplet Π2 = {C2 , Γ0 , Γ1 }, where Γj : W22 (R \ {0}) → C2 , j ∈ {0, 1}, −f (+0) f (+0) , Γ1 f = , (4.1) Γ0 f = f (−0) f (−0) is a boundary triplet for A∗ . (ii) The corresponding Weyl function M (·) is √ 1/ −λ √0 M (λ) = M2 (λ) := , 0 λ

λ ∈ C \ R.

(iii) The corresponding γ-ﬁeld γ(λ) : C2 → Nλ is √ √ 1 c+ γ(λ) := c+ √ · e− −λx χ+ (x) + c− · e λx χ− (x), c− −λ

c± ∈ C.

If ∆2 = 0 then the boundary conditions (2.8) take the form f (+0) = −b11 f (+0) + b12 f (−0) . f (−0) = −b21 f (+0) + b22 f (−0) Further, by Deﬁnition 1.5, we get A(aij ) = AB = A∗ | ker(Γ1 − BΓ0 ),

B=

b11 b21

b12 b22

(4.2)

(4.3)

(4.4)

∈ C2×2 .

(4.5)

As before, denote by ϕB (·) the function as in (2.14) with M (·) as in (4.2). Combining Proposition 1.2 and Theorem 1.2, one obtains Lemma 4.1. If AB is the operator (4.5) and |b12 | + |b21 | = 0 then: (i) σc (AB ) = R, σr (AB ) = ∅; (ii) σp (AB ) = {λ ∈ C \ R : ϕB (λ) = 0} = {λ ∈ C+ : ϕB (λ) = 0} ∪ {λ ∈ C− : ϕB ∗ (λ) = 0}. (iii) The Krein formula has the form (AB − λ)−1 f − (A0 − λ)−1 f (x) & ' √ √ e− −λx · χ+ (x) b22 − λ √ √ · y+ (f, λ) − b12 · y− (f, λ) = ϕB (λ) · −λ −λ √ e λx · χ− (x) 1 b21 √ √ + · y+ (f, λ) + b11 − · y− (f, λ) , − ϕB (λ) −λ −λ (4.6) f ∈ L2 (R), where A0 = A∗ | ker Γ0 , λ ∈ ρ(AB )∩ρ(A0 ), and y± (f, λ) are deﬁned by (2.18).

Diﬀerential Operators with Indeﬁnite Weights and a Point Interaction 185 4.2. The case ∆3 = 0 Let ∆3 = 0. In this case we write the boundary conditions (2.8) in the following form f (+0) = b11 f (+0) + b12 f (−0) (4.7) f (−0) = b21 f (+0) + b22 f (−0). b11 b12 and Let B = b21 b22 AB := A∗ | dom(AB ),

dom(AB ) = {f ∈ W22 (R \ {0}) : f satisﬁes (4.7)}. (4.8)

Theorem 4.2. (i) The triplet Π3 = {C2 , Γ0 , Γ1 }, where Γj : W22 (R \ {0}) → C2 , j ∈ {0, 1}, f (+0) f (+0) , Γ1 f = , (4.9) Γ0 f = f (−0) f (−0) is a boundary triplet for A∗ . (ii) The corresponding Weyl function M (·) is √ − −λ √0 M (λ) = M3 (λ) := , λ 0

λ ∈ C \ R.

(iii) The corresponding γ-ﬁeld γ(λ) : C2 → Nλ is √ √ c+ γ(λ) := c+ · e− −λx χ+ (x) + c− · e λx χ− (x), c−

c± ∈ C.

(4.10)

(4.11)

Therefore, AB of the form (4.8) is an almost solvable extension of A and AB = A∗ | ker(Γ1 − BΓ0 ), where Γi , i ∈ {0, 1}, are deﬁned by (4.9). Lemma 4.2. Let the function ϕB (·) be of the form (2.14) with M (·) deﬁned by (4.10); let the operator AB is given by (4.8) and |b12 | + |b21 | = 0. Then: (i) σc (AB ) = R, σr (AB ) = ∅; (ii) σp (AB ) = {λ ∈ C \ R : ϕB (λ) = 0} = {λ ∈ C+ : ϕB (λ) = 0} ∪ {λ ∈ C− : ϕB ∗ (λ) = 0}. (iii) The Krein formula has the form (AB − λ)−1 f − (A0 − λ)−1 f (x) √ √ e− −λx · χ+ (x) = (b22 − λ) · y+ (f, λ) − b12 · y− (f, λ) ϕB (λ) √ √ e λx · χ− (x) −b21 · y+ (f, λ) + (b11 + −λ) · y− (f, λ) , + ϕB (λ) f ∈ L2 (R),

(4.12)

where A0 = A∗ | ker Γ0 , λ ∈ ρ(AB ) ∩ ρ(A0 ), and y± (f, λ) are given by (2.18).

186

I.M. Karabash and A.S. Kostenko

4.3. The case ∆4 = 0 If ∆4 = 0, then boundary conditions (2.8) take the form −f (+0) = b11 f (+0) + b12 f (−0) For B =

b11 b21

b12 b22

(4.13)

−f (−0) = b21 f (+0) + b22 f (−0). we set

AB := A∗ | dom(AB ), dom(AB ) = {f ∈ W22 (R \ {0}) : f satisﬁes (4.13)}. (4.14) Theorem 4.3. (i) The triplet Π4 = {C2 , Γ0 , Γ1 }, where Γj : W22 (R \ {0}) → C2 , j ∈ {0, 1}, f (+0) −f (+0) Γ0 f = , (4.15) , Γ1 f = −f (−0) f (−0) is a boundary triplet for A∗ . (ii) The corresponding Weyl function M (·) is √ 1/ −λ 0√ M (λ) = M4 (λ) := , 0 −1/ λ

λ ∈ C \ R.

(iii) The corresponding γ-ﬁeld γ(λ) : C2 → Nλ is √ √ c+ c− c+ · e− −λx χ+ (x) + √ · e λx χ− (x), := − √ γ(λ) c− −λ λ

(4.16)

c± ∈ C. (4.17)

Therefore, AB as in (4.14) is an almost solvable extension of A and AB = A∗ | ker(Γ1 − BΓ0 ), where Γj , j ∈ {0, 1} are given by (4.15). Lemma 4.3. Suppose the function ϕB (·) is given by (2.14) with M (·) as in (4.16). If the operator AB is given by (4.14) and |b12 | + |b21 | = 0 then: (i) σc (AB ) = R, σr (AB ) = ∅; (ii) σp (AB ) = {λ ∈ C \ R : ϕB (λ) = 0} = {λ ∈ C+ : ϕB (λ) = 0} ∪ {λ ∈ C− : ϕB ∗ (λ) = 0}. (iii) The Krein formula has the form (AB − λ)−1 f − (A0 − λ)−1 f (x) √ √ y+ (f, λ) y− (f, λ) e− −λx · χ+ (x) √ + b12 · √ = (b22 + 1/ λ) · √ ϕB (λ) · −λ −λ λ √ √ y+ (f, λ) y− (f, λ) e λx · χ− (x) √ + (b11 − 1/ −λ) · √ + b21 · √ , −λ ϕB (λ) · λ λ f ∈ L2 (R),

(4.18)

where A0 = A∗ | ker Γ0 , λ ∈ ρ(AB ) ∩ ρ(A0 ), and y± (f, λ) are given by (2.18).

Diﬀerential Operators with Indeﬁnite Weights and a Point Interaction 187 4.4. The similarity criterion Arguing as above, we see that for the cases considered in Subsections 4.1–4.3 analogues of Theorems 3.1 and 3.2 hold. Theorem 4.4. Theorems 3.1 and 3.2 are valid for the extensions AB with boundary conditions (4.4), (4.7) or (4.13) if the function M (·) is replaced by (4.2), (4.10) or (4.16), respectively.

2 2 d d 5. On similarity of (sgn x) − dx + cδ and (sgn x) − + cδ 2 dx2 to normal operators 5.1. Let us illustrate the previous results by several examples. We start with the d2 ˜ operator A˜ = −(sgn x) dx 2 , see (0.3). It is obvious that A = AB , where AB is an 0 1 almost solvable extension of the form (2.13) with B = . It follows from −1 0 (3.4) that in this case + ϕ+ B (λ) = ϕB ∗ (λ) ≡ 1 − i,

λ ∈ C+ .

Hence, by Theorem 3.2, one obtains the result of [7]. d2 Theorem 5.1 ([7]). The operator A˜ = −(sgn x) dx 2 is similar to a self-adjoint operator.

5.2. Let δ be the Dirac delta. Let us introduce the following diﬀerential expression d2 + cδ, c ∈ C \ {0}. (5.1) dx2 Here δ formally represents a contact interaction at zero if c ∈ R. In L2 (R), expression (5.1) generates the diﬀerential operator Lcδ deﬁned by −

dom(Lcδ ) = {f ∈ W22 (R \ {0}) : f (+0) = f (−0), f (+0) − f (−0) = cf (−0)}, d2 (5.2) dx2 (see for example [2, 35]). We put Acδ := JLcδ , where (Jf )(x) = (sgn x)f (x), i.e., the operator Acδ is deﬁned in L2 (R) by the diﬀerential expression d2 (sgn x) − 2 + cδ , c ∈ C \ {0}. (5.3) dx Lcδ := −

It is clear that the operator Acδ is an extension of the operator A of the form (0.1). Moreover, Acδ = AB where AB is given by (2.13) and c 1 . (5.4) B = Bc := −1 0

188

I.M. Karabash and A.S. Kostenko

Theorem 5.2. Let c = 0. (i) The operator Acδ is similar to a normal one if and only if Re c = −| Im c|. (ii) Acδ is similar to a self-adjoint operator if and only if Re c > −| Im c|. Proof. Let B be given by (5.4). By (3.4), one gets c ϕ+ B ∗ (λ) = 1 − i + √ , λ

c ϕ+ B (λ) = 1 − i + √ , λ

λ ∈ C+ \ {0}.

(5.5)

+ Note that ϕ+ B (·) or ϕB ∗ (·) have a real zero iﬀ Re(c) = −| Im(c)|. Furthermore, + and ϕB ∗ (·) do not vanish in C+ iﬀ Re(c) > −| Im c|. Hence the statements of Theorem 5.2 obviously follow from Theorems 3.1 and 3.2.

ϕ+ B (·)

5.3. Let us consider the extension AB of the form (2.13) with 0 1 =B c = B , c = 0. −1 c This is a so-called “operator with δ -interaction” (see [2]). The formal diﬀerential expression corresponding to AB is d2 (sgn x) − 2 + cδ , c ∈ C \ {0}. (5.6) dx Therefore we will denote the operator AB by Acδ . Theorem 5.3. Let c = 0. (i) The operator Acδ is similar to a normal one if and only if Re c = −| Im c|. (ii) Acδ is similar to a self-adjoint operator if and only if Re c > −| Im c|. Proof. So, by (3.4), we have √ ϕ+ cδ (λ) = 1 − i − ic λ,

√ ϕ+ cδ (λ) = 1 − i − ic λ,

λ ∈ C+ .

(5.7)

These functions have a real zero iﬀ Re c = −| Im c|, have no zeros in C+ iﬀ Re c > −| Im c|. Hence the statements of Theorem 5.3 follow from Theorems 3.1 and 3.2. be a self-adjoint extension of the symmetric operator Remark 5.1. Let L (Lf )(x) := −f (x),

dom(L) = {f ∈ W22 (R) : f (0) = f (0) = 0}.

(5.8)

are Assume that the boundary conditions at 0 associated with the extension L does not admit the following decomposition L + ⊕ nonseparate, i.e., the operator L 2 L− , where L± := L |(dom(L) ∩ L (R± )). Then L can be considered as an operator with a singular interaction (see [27] for the details). Using the arguments of this section, one can describe the main spectral properties of the corresponding J-self := J L. adjoint operator A

Diﬀerential Operators with Indeﬁnite Weights and a Point Interaction 189 Acknowledgment The authors are deeply grateful to professor M.M. Malamud for statement of the problem and his constant attention to this work. The authors thanks a referee for a careful reading of the manuscript and a number of remarks, which helped to improve it. Also, it is a pleasure for the ﬁrst author to acknowledge Technische Universit¨ at Berlin for support of his participation in 4th Workshops “Operator Theory in Krein Spaces”.

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Diﬀerential Operators with Indeﬁnite Weights and a Point Interaction 191 [33] S.G. Pyatkov, Some properties of eigenfunctions of linear pencils. Sibirskii Mat. Zh. 30, no. 4 (1989), 111–124 [in Russian]; English translation: Siberian Math. J. 30 (1989), 587–597. [34] S.G. Pyatkov, Indeﬁnite elliptic spectral problems. Sibirskii Mat. Zh. 39, no. 2 (1998), 409–426 [in Russian]; English translation: Siberian Math. J. 30 (1989), 358–372. [35] M. Reed, B. Simon, Methods of modern mathematical physics. II. Academic Press, New York-San Francisco-London, 1975. Ilia Karabash and Aleksey Kostenko Department of Mathematics Donetsk National University Universitetskaja 24 83055 Donetsk, Ukraine e-mail: [email protected]; [email protected] e-mail: [email protected]; [email protected]

Operator Theory: Advances and Applications, Vol. 175, 193–209 c 2007 Birkh¨ auser Verlag Basel/Switzerland

Normal Matrices in Degenerate Indeﬁnite Inner Product Spaces Christian Mehl and Carsten Trunk Abstract. Complex matrices that are structured with respect to a possibly degenerate indeﬁnite inner product are studied. Based on the theory of linear relations, the notion of an adjoint is introduced: the adjoint of a matrix is deﬁned as a linear relation which is a matrix if and only if the inner product is nondegenerate. This notion is then used to give alternative deﬁnitions of selfadjoint and unitary matrices in degenerate inner product spaces and it is shown that those coincide with the deﬁnitions that have been used in the literature before. Finally, a new deﬁnition for normal matrices is given which allows the generalization of an extension result for positive invariant subspaces from the case of nondegenerate inner products to the case of degenerate inner products. Mathematics Subject Classiﬁcation (2000). Primary 15A57; Secondary 15A63. Keywords. Degenerate inner product space, adjoint, linear relations, H-selfadjoint, H-unitary, H-normal.

1. Introduction We consider the space Cn equipped with an indeﬁnite inner product induced by a Hermitian matrix H ∈ Cn×n via [x, y] := [x, y]H := Hx, y = y ∗ Hx,

(1.1)

where · , · denotes the standard Euclidean scalar product on C . We will suppress the subscript H when it is clear that H induces the indeﬁnite inner product. If H is invertible, i.e., if the indeﬁnite inner product is nondegenerate, then for a matrix M ∈ Cn×n there exists a unique matrix M [∗]H satisfying n

[x, M y] = [M [∗]H x, y]

for all x, y ∈ Cn .

(1.2)

(Again, we will suppress the subscript H if it is clear from the context which inner product is under consideration.) This matrix M [∗] is called the H-adjoint of M .

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From (1.2), one easily obtains the matrix identity M [∗] = H −1 M ∗ H.

(1.3)

A matrix M ∈ Cn×n is called H-selfadjoint, H-skewadjoint, or H-unitary, respectively, if M [∗] = M , M [∗] = −M , or M [∗] = M −1 , respectively. Using (1.3), we obtain the matrix identities M ∗ H = HM,

M ∗ H + HM = 0,

or M ∗ HM = H

(1.4)

for H-selfadjoint, H-skewadjoint, or H-unitary matrices, respectively. These three types of matrices have been widely discussed in the literature, both in terms of theory and numerical analysis. Extensive lists of references can be found in [1, 9, 13, 21]. H-selfadjoint, H-skewadjoint, and H-unitary matrices are special cases of Hnormal matrices. A matrix M ∈ Cn×n is called H-normal if M commutes with its H-adjoint, i.e., if M M [∗] = M [∗] M , or, in other words, if and only if M H −1 M ∗ H = H −1 M ∗ HM,

or,

HM H −1 M ∗ H = M ∗ HM.

(1.5)

In recent years, there has been great interest in H-normal matrices, see [8, 9, 10, 17, 18] and the references therein. Spaces with degenerate inner products, that is, H is singular, are less familiar, although this case does appear in applications [14]. Some works here, primarily concerning inﬁnite-dimensional degenerate Pontryagin spaces, include [23], [11] (and references there), [2], [12], and parts of the book [3]. The main problem in the context of degenerate inner products is that there is no straightforward deﬁnition of an H-adjoint. Indeed, if H is singular, then for a matrix M ∈ Cn×n an H-adjoint, i.e., a matrix N ∈ Cn×n satisfying [x, M y] = [N x, y] for all x, y ∈ Cn need not exist, and if it exists, it need not be unique. Example 1.1. Let a, b ∈ C and 1 0 1 0 H= , M1 = , 0 0 0 1

M2 =

1 0

1 1

,

N (a, b) =

1 a

0 b

.

Then for all possible choices a, b ∈ C, N (a, b) satisﬁes [x, M1 y] = [N (a, b)x, y] for all x, y ∈ Cn . On the other hand, there is no matrix N ∈ C2×2 such that [x, M2 y] = [N x, y] for all x, y ∈ Cn . Despite the lack of the notion of an H-adjoint, one can deﬁne H-selfadjoint, H-skewadjoint and H-unitary for the case of singular H by the matrix identities (1.4) and this deﬁnition has been used in many sources, see, e.g., [20] and the references therein. The corresponding matrix identities (1.5) for H-normal matrices, however, require an inverse of H. A standard approach to circumvent this diﬃculty is the use of some generalized inverse, in particular the Moore-Penrose generalized inverse H † , instead. However, it seems that the use of the Moore-Penrose inverse leads to some inconsistencies in the theory of degenerate inner products. We illustrate this by an example.

Normal Matrices in Inner Product Spaces Example 1.2. Let H=

1 0

0 0

,

A=

1 1

0 1

,

N=

0 1 1 0

195 .

Then we have H † = H. Observe that A∗ H = HA, but A = H † A∗ H. So, should A be considered as an H-selfadjoint matrix or not? On the other hand, note that N H † N ∗ H = H † N ∗ HN , but HN H † N ∗ H = N ∗ HN . So, should N be considered as an H-normal matrix or not? In [16] H-normal matrices have been deﬁned as matrices M satisfying HM H † M ∗ H = M ∗ HM

(1.6)

and this deﬁnition has later been taken up in [4, 17]. (This way minimizes the number of times the Moore-Penrose generalized inverse of H appears in the deﬁning equation of H-normal matrices.) In this paper, we will call matrices M satisfying (1.6) Moore-Penrose H-normal matrices in order to highlight the occurrence of the Moore-Penrose generalized inverse in the deﬁnition. Although it is possible to prove some interesting results for Moore-Penrose Hnormal matrices (e.g., concerning existence of invariant maximal semideﬁnite subspaces), there is an unpleasant mismatch between H-selfadjoint, H-skewadjoint, and H-unitary matrices on the one hand and Moore-Penrose H-normal matrices on the other hand. It is easy to check (see also [20]) that the kernel of H is always an invariant subspace for H-selfadjoint, H-skewadjoint, and H-unitary matrices (deﬁned as in (1.4)). However, it has been shown in [17, Example 6.1] that there exist Moore-Penrose H-normal matrices A such that ker H is not A-invariant. It is exactly the fact that ker H need not be invariant which makes the investigation of Moore-Penrose H-normal matrices challenging. For example, let us consider the problem of existence of semideﬁnite invariant subspaces. Recall that a subspace M ⊆ Cn is called H-nonnegative if [x, x] ≥ 0 for every x ∈ M, Hpositive if [x, x] > 0 for every nonzero x ∈ M, and H-neutral if [x, x] = 0 for every x ∈ M. An H-nonnegative subspace is said to be maximal H-nonnegative if it is not properly contained in any larger H-nonnegative subspace. It is easy to see that an H-nonnegative subspace is maximal if and only if its dimension is equal to ν+ (H) + ν0 (H), where ν+ (H) and ν0 (H) denote the number (counted with multiplicities) of positive and zero eigenvalues of H, respectively. It has been shown in [17, Theorem 6.6] that any Moore-Penrose H-normal matrix has a maximal H-nonnegative invariant subspace, but the problem which H-nonnegative, H-positive, or H-neutral invariant subspaces of Moore-Penrose H-normal matrices can be extended to a maximal H-nonnegative invariant subspaces has only been solved for the case of invertible H so far [18, 19]. It is the aim of this paper to propose a diﬀerent deﬁnition of H-normal matrices in degenerate inner product spaces that is based on a generalization of the H-adjoint A[∗]H of a matrix A for singular H. This generalization is obtained by dropping the assumption that the H-adjoint of a matrix is a matrix itself. Instead, the H-adjoint A[∗]H is deﬁned to be a linear relation in Cn , i.e., a linear

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subspace of C2n . For basic facts on linear relations and further references see, e.g., [5, 6, 7, 22]. Throughout this paper we identify a matrix A ∈ Cn×n with its graph x : x ∈ Cn ⊆ C2n Ax which is a linear relation, also denoted by A. If H ∈ Cn×n is invertible, then, by (1.2), we obtain that A[∗]H coincides with the linear relation y ∈ C2n : [y, Ax]H = [z, x]H for all x ∈ Cn . A[∗]H = z This representation allows a direct generalization of the concept of adjoint to the case of degenerate inner products and even to the case of starting with a linear relation rather than a matrix A, see [22]. Deﬁnition 1.3. Let H ∈ Cn×n be Hermitian and let A be a linear relation in Cn . Then the linear relation y x A[∗]H = ∈ C2n : [y, w]H = [z, x]H for all ∈A z w is called the H-adjoint of A. Again, if there is no risk of ambiguity, we will suppress the subscript H in the notation. In this setting, we obtain (see Proposition 2.6 below) that A[∗] = H −1 A∗ H, where H −1 is the inverse of H in the sense of linear relations (see Section 2). Observe that this coincides with (1.3) if H is invertible. Hence, the H-adjoint in degenerate inner product spaces is a natural generalization of the H-adjoint in nondegenerate inner product spaces. The remainder of the paper is organized as follows. In Section 2, we discuss basic properties of the H-adjoint. In Section 3 we use the H-adjoint to deﬁne H-symmetric and H-isometric linear relations and we show that in the case of matrices these deﬁnitions coincide with the deﬁnitions of H-selfadjoint and Hunitary matrices via the identities (1.4). In Section 4, we present a new deﬁnition for H-normal matrices. We show that the set of H-normal matrices is a proper subset of the set of Moore-Penrose H-normal matrices and that H-normal matrices share the property with H-selfadjoint and H-unitary matrices that the kernel of H is always an invariant subspace. The latter fact allows us to obtain suﬃcient conditions for an H-positive invariant subspace of an H-normal matrix to be contained in a maximal H-nonnegative invariant subspace. This generalizes a result obtained in [18].

2. The adjoint in degenerate inner product spaces We study linear relations in Cn , i.e., linear subspaces of C2n . For the deﬁnitions of linear operations with relations and the inverse of relations we refer to [6]. We

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only mention the following. For linear relations A, B ⊆ C2n we deﬁne x dom A = x : ∈A , the domain of A, y x ran A = y : ∈A , the range of A, y 0 mul A = y : ∈A , the multivalued part of A, y y x : ∈ A , the inverse of A A−1 = x y and the product of A and B, y x x n ∈ A, ∈B . AB = : there exists a y ∈ C with z y z In all cases, x, y, z are understood to be from Cn . For a subset M ⊆ Cn we deﬁne M [⊥] = {x : [x, y] = 0 for all y ∈ M } . The following lemma is needed for the proof of Proposition 2.2 below. It is contained in [15, proof of Lemma 2.2], but for the sake of completeness we give a separate proof. Lemma 2.1. Let M ⊆ Cn be a subspace. Then (M [⊥] )[⊥] = M + ker H. / M + ker H then, as the Proof. Obviously, we have M + ker H ⊆ (M [⊥] )[⊥] . If u ∈ quotient space (Cn /ker H, [., .]∼ ) with [x+ ker H, y + ker H]∼ := [x, y], x, y ∈ Cn , is nondegenerate, there exists an v ∈ Cn such that [v, M ] = [v+ker H, M +ker H]∼ = / (M [⊥] )[⊥] , which {0} and [v, u] = [v + ker H, u + ker H]∼ = 0. Therefore u ∈ completes the proof. In the next proposition we collect some properties of the H-adjoint. Proposition 2.2. Let A, B ⊆ C2n be linear relations. Then we have (i) A ⊆ B =⇒ B [∗] ⊆ A[∗] . (ii) mul A[∗] = (dom A)[⊥] . If A is a matrix, we have mul A[∗] = ker H. (iii) ker A[∗] = (ran A)[⊥] . (iv) (A[∗] )[∗] = A + (ker H × ker H). Proof. Assertions (i), (ii), (iii) are easy consequences of Deﬁnition 1.3 and can be found, e.g., in [22]. In order to prove assertion (iv) we equip the space Cn × Cn with the inner product 22 33 ; < y x 0 −iH y x , := i [y, w] − [z, x] = , z w iH 0 z w

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where x, y, w, z ∈ Cn . Then A[∗] = A[[⊥]]

and (A[∗] )[∗] = (A[[⊥]] )[[⊥]] .

(2.1)

Lemma 2.1 remains true if we replace Cn by C2n , [⊥] by [[⊥]] and ker H by (ker H×ker H). This and (2.1) shows assertion (iv) of Proposition 2.2. If A is a matrix, then, by Proposition 2.2 (iv), we have that y ∈ mul (A[∗] )[∗] if and only if there exists f, g ∈ ker H such that 0 f −f 0 = + = . y Af g Af + g This proves the following lemma. Lemma 2.3. Let A ∈ Cn×n be a matrix. Then ker H ⊆ mul (A[∗] )[∗] . If , in addition, ker H is A-invariant, then we have mul (A[∗] )[∗] = ker H. The following formula for the H-adjoint of matrices will be used frequently. Lemma 2.4. Let A ∈ Cn×n be a matrix. Then y A[∗] = ∈ C2n : A∗ Hy = Hz . z

(2.2)

In particular, A[∗] is a matrix if and only if H is invertible. Proof. Clearly, since A is a matrix, we have y ∈ A[∗] ⇐⇒ x∗ A∗ Hy = x∗ Hz for all x ∈ Cn . z This implies (2.2). The remaining assertion of Lemma 2.4 follows from Proposition 2.2 (ii). By Lemma 2.4 we have that dom A[∗] = Cn if and only if ran (A∗ H) ⊆ ran H. Example 2.5. Using the notations of Example 1.1 we have [∗]

[∗]

M1 = graph I + ({0} × {0} × {0} × C) and M2 = {0} × C × {0} × C, where + denotes the sum of linear subspaces. We then obtain [∗]

dom M1 = C2 , [∗] dom M2 = {0} × C,

[∗]

mul M1 = {0} × C, [∗] mul M2 = {0} × C.

Let A ⊆ C2n be a linear relation. Introducing a change of basis x → P x on C , where P ∈ Cn×n is a nonsingular matrix, yields −1 P x x −1 : ∈A . (2.3) P AP = P −1 w w n

Normal Matrices in Inner Product Spaces Therefore, for

y z

∈ (P

−1

[∗]P ∗ HP

AP )

and arbitrary

x w

199

∈A

we have 0 = [y, P −1 w]P ∗ HP − [z, P −1 x]P ∗ HP = [P y, w]H − [P z, x]H , i.e., Py y ∈ A[∗]H or, equivalently, ∈ P −1 A[∗]H P. Pz z This gives (P −1 AP )[∗]P ∗ HP = P −1 A[∗]H P.

(2.4)

Moreover, with the help of (2.3), it is easily deduced that (P −1 AP )−1 = P −1 A−1 P,

(2.5)

where the inverses are understood in the sense of linear relations. Thus, if A is a matrix, then changing the basis of Cn accordingly, we may always assume that H and A have the forms

H=

m n−m

2

m

n−m

H1 0

0 0

3 and

A=

m n−m

2

m

n−m

A1 A3

A2 A4

3 ∈ Cn×n ,

(2.6)

where H1 ∈ Cm×m is nonsingular. When using these forms and identifying A with the linear relation A ⊆ C2n , then for the ease of simple notation we will usually omit the indication of dimensions of vectors if those are clear from the context. Thus, for example, we write ⎧⎛ ⎫ ⎧⎛ ⎞ ⎞⎫ x1 x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎬ ⎜ ⎜ ⎟ ⎟⎬ x x 2 2 m n−m ⎜ ⎟ : x1 ∈ C , x2 ∈ C ⎟ . = A= ⎜ ⎝ A1 x1 + A2 x2 ⎠ ⎪⎝ A1 x1 + A2 x2 ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎪ ⎭ A3 x1 + A4 x2 A3 x1 + A4 x2 Proposition 2.6. Let A ∈ Cn×n be a matrix. Then A[∗] = H −1 A∗ H,

(2.7)

where H −1 is the inverse in the sense of linear relations. In particular, if H and A have the forms as in (2.6) then ⎧⎛ ⎫ ⎞ y1 ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎬ ⎟ y2 [∗] ∗ ⎜ ⎟ . (2.8) H y = 0 A = ⎝ [∗]H1 : A 1 1 2 ⎪ ⎪ A1 y1 ⎠ ⎪ ⎪ ⎩ ⎭ z2 Moreover, we have dom A[∗] = Cn if and only if A2 = 0.

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Proof. We have H

−1

=

and, using Lemma 2.4, H

−1

∗

−1

x z

:

y A∗ Hy

z x

∈H

−1

=

A H=H =H y ∗ = : Hz = A Hy = A[∗] z

and (2.7) is proved. Let H and A be in the ⎧⎛ y1 ⎪ ⎪ ⎨⎜ ⎜ y2 A[∗] = ⎝ z1 ⎪ ⎪ ⎩ z2 ⎧⎛ y1 ⎪ ⎪ ⎨⎜ y ⎜ 2 = ⎝ z1 ⎪ ⎪ ⎩ z2

y Hz

Hz z

∗

: Hz = A Hy

forms (2.6). By Lemma 2.4, we have that ⎫ ⎞ ⎪ ⎪ ⎬ ⎟ z1 ⎟ : A∗ H y1 = H ⎠ y2 z2 ⎪ ⎪ ⎭ ⎫ ⎞ ⎪ ⎪ ⎬ ⎟ ⎟ : A∗1 H1 y1 = H1 z1 and A∗2 H1 y1 = 0 . ⎠ ⎪ ⎪ ⎭

Since H1−1 A∗1 H1 y1 = A1 H1 y1 , we obtain (2.8) and since H1 is invertible, we have dom A[∗] = Cn if and only if A2 = 0. [∗]

3. H-symmetric and H-isometric matrices In the nondegenerate case, H-selfadjoint matrices are deﬁned as matrices A satisfying A = A[∗] . When generalizing this concept to the degenerate case, however, we have to take into account that for any A ∈ Cn×n , the relation A[∗] is never a matrix when H is singular. Thus, matrices satisfying A = A[∗] do not exist. Instead, it is natural to consider matrices that are H-symmetric relations in the following sense. Deﬁnition 3.1. A linear relation A in Cn is called H-symmetric if A ⊆ A[∗] . We mention that H-symmetric relations in degenerate inner product spaces have been introduced in [22]. Example 3.2. In Example 1.1 we have that M1 is H-symmetric. Clearly, A is H-symmetric if and only if P −1 AP is P ∗ HP -symmetric for any invertible P ∈ Cn×n , cf. (2.3). At ﬁrst sight, Deﬁnition 3.1 in the case of a matrix A ∈ Cn×n may look a little bit weird, but the following proposition shows that this deﬁnition does make sense, because we will show that H-symmetry is equivalent to the condition A∗ H = HA which has been used as the deﬁnition for H-selfadjoint matrices in degenerate inner products in various sources.

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Proposition 3.3. Let A ∈ Cn×n be a matrix. Then the following statements are equivalent. i) A is H-symmetric, i.e., A ⊆ A[∗] . ii) A∗ H = HA. If one of the conditions is satisﬁed, then ker H is A-invariant. In particular, if H and A have the forms as in (2.6) then A is H-symmetric if and only if A1 is H1 -selfadjoint and A2 = 0. y Proof. Let ∈ A. Then, by Lemma 2.4, we have Ay y ∈ A[∗] ⇐⇒ A∗ Hy = HAy. Ay This shows the equivalence of the two statements in Proposition 3.3. (For the implication i) ⇒ ii) observe that A ⊆ A[∗] implies that the domain of A[∗] is Cn .) For the remainder of the proof, let H and A be in the forms as in (2.6) and assume that A is H-symmetric. Then using that the domain of A[∗] is Cn , we [∗] obtain from (2.8) that A1 y1 + A2 y2 = A1 H1 y1 and A∗2 H1 y1 = 0 for all y1 ∈ Cm . [∗] Since H1 is invertible, this implies A2 = 0 and we also obtain A1 = A1 H1 , i.e., A1 is H1 -selfadjoint. Thus, a matrix is H-symmetric (in the sense of linear relations) if and only if it is H-selfadjoint (in the sense of matrices). At ﬁrst sight, it may look a little bit disappointing that only one inclusion of the usual concept of “selfadjointness” (in the sense of linear relations) is fulﬁlled. However, this changes if we adjoin the inclusion A ⊆ A[∗] once more, as the following proposition shows. Proposition 3.4. Let A ∈ Cn×n be a matrix. Then A is H-symmetric if and only if A[∗] = (A[∗] )[∗] . Proof. If A[∗] = (A[∗] )[∗] then, by Proposition 2.2 (iv), A is H-symmetric. For the converse assume A ⊆ A[∗] . Then Proposition 2.2 (i) implies (A[∗] )[∗] ⊆ A[∗] . For the other inclusion observe that A ⊆ A[∗] and Proposition 2.2 (iv) give dom A[∗] = dom (A[∗] )[∗] = Cn . Moreover, Proposition 2.2 (ii) and Lemma 2.3 together with Proposition 3.3 imply mul A[∗] = mul (A[∗] )[∗] = ker H.

Hence, A[∗] = (A[∗] )[∗] . Proposition 2.2 (iv) together with Proposition 3.4 imply the following

Corollary 3.5. Let A ∈ Cn×n be a matrix. Then A is H-symmetric if and only if A[∗] = A + (ker H × ker H).

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Similar to H-symmetric matrices, H-isometric matrices can be deﬁned by passing to the concept of linear relations. (We mention that H-isometric relations in degenerate inner product spaces have been introduced in [22].) Deﬁnition 3.6. A linear relation U in Cn is called H-isometric if U −1 ⊆ U [∗] . We note that in the deﬁnition above U −1 is the inverse in the sense of linear relations. E.g., if H = 0, then every matrix is H-isometric. It follows from (2.4) and (2.5) that U is H-isometric if and only if P −1 U P is P ∗ HP -isometric for any invertible P ∈ Cn×n . Proposition 3.7. Let U ∈ Cn×n be a matrix. Then the following statements are equivalent. i) U is H-isometric, i.e., U −1 ⊆ U [∗] . ii) U ∗ HU = H. If one of the conditions is satisﬁed, then ker H is U -invariant. In particular, if H and U have the forms as in (2.6), i.e., H1 0 U1 U2 , H= U= , (3.1) U3 U4 0 0 where H1 is invertible, then U is H-isometric if and only if U1 is H1 -unitary and U2 = 0. Moreover, we have (ran U )[⊥] = ker H.

(3.2)

Proof. By Lemma 2.4, we have Uy U −1 ⊆ U [∗] ⇐⇒ ∈ U [∗] for all y ∈ Cn ⇐⇒ U ∗ HU y = Hy for all y ∈ Cn . y This shows the equivalence of the two statements in Proposition 3.7. For the remainder of the proof, let H and U be in the forms as in (3.1) and assume that U is H-isometric. Then we obtain from the identity U ∗ HU = H that ∗ U1 H1 U1 U1∗ H1 U2 H1 0 = 0 0 U2∗ H1 U1 U2∗ H1 U2 This immediately implies U1∗ H1 U1 = H1 , i.e., U1 is H1 -unitary. In particular, with H1 also U1 must be invertible. This ﬁnally yields U2 = 0 and (3.2). If we adjoin the inclusion U −1 ⊆ U [∗] once more we get, similar to Proposition 3.4, the following characterization of H-isometric matrices. Proposition 3.8. Let U ∈ Cn×n be a matrix. Then U is H-isometric if and only if (U −1 )[∗] = (U [∗] )[∗] Proof. Let U be H-isometric, i.e., U −1 ⊆ U [∗] . Then, by Proposition 2.2 (i), we have (U [∗] )[∗] ⊆ (U −1 )[∗] . For the other inclusion, observe that, by Proposition 2.2 (iv), we have dom (U [∗] )[∗] = Cn .

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Thus, using (U [∗] )[∗] ⊆ (U −1 )[∗] , we obtain that dom (U [∗] )[∗] = dom (U −1 )[∗] = Cn . Proposition 2.2 (ii) and (3.2) imply mul (U −1 )[∗] = (dom U −1 )[⊥] = (ran U )[⊥] = ker H and, with Lemma 2.3 and the U -invariance of ker H (Proposition 3.7), we conclude mul (U [∗] )[∗] = ker H = mul (U −1 )[∗] . Hence (U −1 )[∗] = (U [∗] )[∗] . The contrary follows from Proposition 2.2 and U −1 ⊂ ((U −1 )[∗] )[∗] = ((U [∗] )[∗] )[∗] ⊂ U [∗] .

4. H-normal matrices Recall that in the case of invertible H, a matrix A is called H-normal if and only if AA[∗] = A[∗] A. For the case that H is singular and that H and A are given in the forms as in (2.6), a straightforward computation reveals ⎧⎛ ⎫ ⎞ y1 ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎬ ⎟ y 2 [∗] ∗ ⎜ ⎟ (4.1) AA = ⎝ : A H y = 0 [∗] 1 1 2 A1 A1 y1 + A2 z2 ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ [∗] A3 A1 y1 + A4 z2 and

⎧⎛ y1 ⎪ ⎪ ⎨⎜ y2 A[∗] A = ⎜ [∗] [∗] ⎝ ⎪ A1 A1 y1 + A1 A2 y2 ⎪ ⎩ z2

⎞

⎫ ⎪ ⎪ ⎬

⎟ ⎟ : A∗2 H1 A1 y1 + A∗2 H1 A2 y2 = 0 . ⎠ ⎪ ⎪ ⎭

(4.2)

However, even in the case that A is H-symmetric (i.e., in the identities (4.1) [∗] [∗] and (4.2) we have A1 A1 = A1 A1 = A21 and A2 = 0), we only obtain the inclusion AA[∗] ⊆ A[∗] A, while the other inclusion A[∗] A ⊆ AA[∗] is only satisﬁed if A4 is invertible. This motivates the following deﬁnition. Deﬁnition 4.1. A relation A in Cn is called H-normal if AA[∗] ⊆ A[∗] A. As for the case of H-isometric and H-symmetric matrices, we obtain that the kernel of H is always an invariant subspace for H-normal matrices. Proposition 4.2. Let A ∈ Cn×n be an H-normal matrix. Then ker H is A-invariant. In particular, if A and H are in the forms as in (2.6), then A is H-normal if and only if A1 is H1 -normal and A2 = 0. Proof. Without loss of generality, we may assume that A and H are in the forms as in (2.6). Clearly, if A1 is H1 -normal and A2 = 0 then it follows directly from (4.1) and (4.2) that AA[∗] ⊆ A[∗] A, i.e., A is H-normal. For the converse, assume that

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A is H-normal. Then AA[∗] ⊆ A[∗] A implies in particular that AA[∗] ∩ ({0} × Cn ) ⊆ A[∗] A ∩ ({0} × Cn ) Comparing the third block components of (4.1) and (4.2) this reduces to A2 z = 0 for all z2 ∈ Cn−m and this is only possible if A2 = 0. But then AA[∗] ⊆ A[∗] A [∗] [∗] implies A1 A1 y1 = A1 A1 y1 for all y1 ∈ Cm and we obtain that A1 is H1 -normal. Clearly, ker H is A-invariant, because of A2 = 0. This concludes the proof. With Propositions 3.3 and 3.7 we immediately obtain the following corollary. Corollary 4.3. H-symmetric and H-isometric matrices are H-normal. The question arises, if we obtain a diﬀerent characterization of H-normality in the style of Propositions 3.4 and 3.8 by the identity A[∗] (A[∗] )[∗] = (A[∗] )[∗] A[∗] .

(4.3)

Let us investigate this question in detail. Without loss of generality assume that the matrix A ∈ Cn×n and H are given in the forms (2.6). Then using Proposition 2.2 (iv), we obtain ⎧⎛ ⎞ ⎛ ⎞ ⎫ ⎧⎛ ⎞⎫ x1 x1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎬ ⎨⎜ ⎬ ⎟ ⎜ ⎟ ⎟ x w w 2 2 2 [∗] [∗] ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ = . (A ) = ⎝ + A1 x1 + A2 x2 ⎠ ⎝ 0 ⎠⎪ ⎪⎝ A1 x1 + A2 x2 ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎪ ⎭ A3 x1 + A4 x2 z2 z2 Together with (2.8) this implies ⎧⎛ x1 ⎪ ⎪ ⎨⎜ w 2 A[∗] (A[∗] )[∗] = ⎜ ⎝ A[∗] A1 x1 + A[∗] A2 x2 ⎪ ⎪ 1 1 ⎩ z2 and (A[∗] )[∗] A[∗]

⎫ ⎪ ⎪ ⎬

⎞

⎟ ⎟ : A∗2 H1 (A1 x1 + A2 x2 ) = 0 (4.4) ⎠ ⎪ ⎪ ⎭

⎧⎛ y1 ⎪ ⎪ ⎨⎜ y2 = ⎜ ⎝ A1 A[∗] y1 + A2 x2 ⎪ ⎪ 1 ⎩ z2

⎫ ⎪ ⎪ ⎬

⎞

⎟ ⎟ : A∗2 H1 y1 = 0 . ⎠ ⎪ ⎪ ⎭

(4.5)

With the help of these formulas, we obtain that (4.3) may be satisﬁed even if the matrix A ∈ Cn×n is not H-normal, see Example 4.4 below. Example 4.4. Let A and H be ⎛ 1 A1 A2 A= := ⎝ 0 A3 A4 0

given as ⎞ 0 1 1 0 ⎠, 0 0

H=

H1 0

0 0

Then A is not H-normal, because ker H is not A-invariant.

⎛

0 := ⎝ 1 0

⎞ 1 0 0 0 ⎠. 0 0

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However, from (4.4) and (4.5), we immediately obtain ⎧⎛ ⎫ ⎞ x1 ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎬ ⎟ w2 [∗] [∗] [∗] ∗ ⎜ ⎟ = (A[∗] )[∗] A[∗] . H x = 0 A (A ) = ⎝ : A 1 1 2 x1 + A2 x2 ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ z2 The following result shows that the set of matrices satisfying (4.3) contains the set of H-normal matrices. Proposition 4.5. Let A ∈ Cn×n be a matrix. If A is H-normal, then A[∗] (A[∗] )[∗] = (A[∗] )[∗] A[∗] . Proof. Without loss of generality let A and H be in the forms (2.6). Then we obtain by Proposition 4.2 that A2 = 0. The identities (4.4) and (4.5) imply ⎧⎛ ⎧⎛ ⎞⎫ ⎞⎫ y1 x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎨ ⎬ ⎜ ⎟ ⎟⎬ y2 x2 [∗] [∗] [∗] [∗] [∗] [∗] ⎜ ⎜ ⎟ ⎟ and A (A ) A = ⎝ (A ) = [∗] ⎝ A[∗] A1 x1 ⎠⎪ . ⎪ ⎪ A1 A1 y1 ⎠⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎩ ⎩ ⎭ ⎭ w2 z2 Indeed, these two sets are equal because of the H1 -normality of A1 which is guaranteed by Proposition 4.2. Next, let us compare H-normal matrices with Moore-Penrose H-normal matrices. We obtain the following result. Proposition 4.6. Let A ∈ Cn×n be a matrix. Then the following statements are equivalent. i) A is H-normal, i.e., AA[∗] ⊆ A[∗] A. ii) A is Moore-Penrose H-normal and A[∗] (A[∗] )[∗] = (A[∗] )[∗] A[∗] . Proof. Without loss of generality, let A and H have the forms (2.6). Then the Moore-Penrose generalized inverse of H is given by −1 0 H1 † H = 0 0 and the matrix A is Moore-Penrose H-normal if and only if ∗ A1 H1 A1 A∗1 H1 A2 H1 A1 H1−1 A∗1 H1 = ∗ ∗ A2 H1 A1 A2 H1 A2 0

0 0

.

(4.6)

“i) ⇒ ii)”: If A is H-normal, then by Proposition 4.2 we have that A1 is H1 normal and that A2 = 0. Then (4.6) is satisﬁed and the remainder follows from Proposition 4.5. “ii) ⇒ i)”: Let A be Moore-Penrose H-normal and let A[∗] (A[∗] )[∗] = (A[∗] )[∗] A[∗] . Then by (4.6) we have A∗2 H1 A1 = 0 and A∗2 H1 A2 = 0. Comparing this with (4.4), we ﬁnd that dom A[∗] (A[∗] )[∗] = Cn . But then, we must have dom (A[∗] )[∗] A[∗] = Cn as well which, using (4.5), implies A∗2 H1 y1 = 0 for all y1 ∈ Cm . From this, we obtain [∗] [∗] A2 = 0. But then A[∗] (A[∗] )[∗] = (A[∗] )[∗] A[∗] reduces to A1 A1 x1 = A1 A1 x1 for

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all x1 ∈ Cm which implies H1 -normality of A1 . From this and Proposition 4.2, we ﬁnally obtain that A is H-normal. As a consequence, we obtain that the set of H-normal matrices is a strict subset of the set of Moore-Penrose H-normal matrices, because it has been shown in [17, Example 6.1] that there exist Moore-Penrose H-normal matrices A such that ker H is not A-invariant. The fact that the kernel of H is invariant for H-normal matrices allows the generalization of extension results for H-semideﬁnite invariant subspaces of normal matrices for invertible H to the case of singular H. For example, if A is H-normal and H is invertible, then any H-nonnegative subspace that is invariant for both A and A[∗] can be extended to an A-invariant maximal H-nonnegative subspace, see [15]. This result now easily generalizes to the case of singular H. Here, an invariant subspace U ⊆ Cn of a linear relation A in Cn is deﬁned by the implication x x ∈ U and ∈ A =⇒ y ∈ U. y Theorem 4.7. Let A ∈ Cn×n be H-normal, and let M0 be an H-nonnegative A-invariant subspace that is also invariant for A[∗] . Then there exists an A-invariant maximal H-nonnegative subspace M containing M0 that is also invariant for A[∗] . Proof. Without loss of generality assume that A and H are in the forms (2.6) and ˙ 2 , where M2 ⊆ ker H and that M0 can be written as a direct sum M0 = M1 +M x1 (1 : x1 ∈ M M1 = 0 (1 ⊆ Cm . It is easy to verify that the A- and A[∗] -invariance for some subspace M (1 is A1 -invariant as well as A[∗] -invariant. Thus, by [15] of M0 imply that M 1 there exists an A1 -invariant maximal H-nonnegative subspace Mmax that is also [∗] ˙ ker H, we obtain that M contains M0 invariant for A1 . Setting M = Mmax + and is A-invariant and maximal H-nonnegative. It is easy to check that M is also A[∗] -invariant. If we drop the assumption that M0 is A[∗] -invariant, then extension results are not as immediate. Indeed, it has been shown in [18] that there exist H-normal matrices (in the case of invertible H) that have an invariant H-nonnegative subspace that cannot be extended to an invariant maximal H-nonnegative subspace. (There still exist such counterexamples if one restricts the subspace to be Hpositive rather than H-nonnegative.) Thus, stronger conditions have to be imposed on an H-normal matrix such that extension of semideﬁnite invariant subspaces can be guaranteed, see [18, 19]. We conclude the paper by generalizing a result concerning the extension of H-positive invariant subspaces of H-normal matrices obtained in [18] to the case of singular H. The fact that ker H is always an invariant subspace for H-normal matrices plays a key role in this proof.

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Theorem 4.8. Let A ∈ Cn×n be H-normal, and let M0 be an H-positive A-in[⊥] variant subspace. Let Mcom be a direct complement of ker H in M0 , that is, [⊥] ˙ ker H. Deﬁne M0 = Mcom + A22 := P A|Mcom : Mcom → Mcom , ˙ ker H. Then Mcom is nondewhere P is the projection onto Mcom along M0 + generate. Equip Mcom with the indeﬁnite inner product induced by H. Assume that [∗] [∗] (4.7) σ(A22 + A22 ) ⊆ R or σ(A22 − A22 ) ⊆ iR. Then there exists an A-invariant maximal H-nonnegative subspace M that contains M0 and that is also A[∗] -invariant. The condition (4.7) is independent of the particular choice of a direct com[⊥] plement Mcom of ker H in M0 . [⊥]

Proof. Since M0 is H-positive, we have that M0 is a direct complement of M0 . Moreover, it is clear that a complement of ker H is nondegenerate. (By default, i.e., nonexistence of H-neutral vectors, {0} is a nondegenerate subspace.) Thus, without loss of generality, we may assume that M0 = span(e1 , . . . , ek ) and [⊥] Mcom = span(ek+1 , . . . , ek+l ), l ≥ 0, and M0 = span(ek+1 , . . . , en ), where ej denotes the jth unit vector. Then A and H have the corresponding block forms ⎛ ⎞ ⎛ ⎞ A11 A12 A13 0 0 Ik A=⎝ 0 A22 A23 ⎠ , H = ⎝ 0 H22 0 ⎠ , 0 A32 A33 0 0 0 where H22 is invertible. In this representation of A, we have A11 A12 Ik 0 A1 = , and H1 = , 0 A22 0 H22 where A1 and H1 are deﬁned in analogy to the decomposition (2.6). By Proposition 4.2 we have that A1 is H1 -normal and that ker H is A-invariant which implies A13 = 0 and A23 = 0. Setting Ik ( M0 = ran ⊆ Ck+l 0 which is an A1 -invariant H1 -positive subspace, we obtain, given the condition on A22 , that by [18, Theorem 4.4] there exists an A1 -invariant maximal H1 (0 . We .0 of dimension ν+ (H) = ν+ (H1 ) containing M nonnegative subspace M . choose appropriate matrices M11 , M12 and write M0 in the following way .0 = ran Ik M11 . M 0 M12 Let

⎛

Ik M := ran ⎝ 0 0

M11 M12 0

0 0 In−k−l

⎞ ⎠.

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Then it is straightforward to check that M is an A-invariant, H-nonnegative subspace of dimension ν+ (H) + ν0 (H) containing M0 . Clearly, M is also invariant for A[∗] . It remains to show that the condition (4.7) is independent of the particular choice of the direct complement Mcom . But choosing a diﬀerent direct complement [⊥] Mnew of ker H in M0 = span(ek+1 , . . . , en ) amounts to a change of basis given by a matrix of the form ⎞ ⎛ 0 0 Ik ⎠ S = ⎝ 0 S22 0 0 S32 In−k−l ˙ new + ˙ ker H with S22 invertible. With respect to the decomposition Cn = M0 +M and the new basis, A and H take the forms ⎞ ⎛ ⎛ ⎞ ∗ 0 A11 0 0 Ik −1 ∗ ˜ = S ∗ HS =⎝ 0 S22 H22 S22 0 ⎠ . A˜ = S −1 AS =⎝ 0 S22 A22 S22 0 ⎠, H 0 0 0 0 ∗ A33 ˜ to Mnew are S −1 A22 S22 and S ∗ H22 S22 . Clearly, The compressions of A˜ and H 22 22 condition (4.7) is satisﬁed for these compressions if and only if (4.7) is satisﬁed for A22 and H22 . Acknowledgment We thank Heinz Langer for asking a question that initiated this research and Peter Jonas for fruitful comments on an earlier version of this paper.

References [1] G. Ammar, C. Mehl, and V. Mehrmann, Schur-like Forms for Matrix Lie Groups, Lie Algebras and Jordan Algebras. Linear Algebra Appl. 287 (1999), 11–39. [2] T.Ya. Azizov, Completely Continuous Operators that are Selfadjoint with Respect to a Degenerate Indeﬁnite Metric (Russian). Mat. Issled. 7 (1972), 237–240, 259. [3] T.Ya. Azizov and I.S. Iohvidov, Linear Operators in Spaces with an Indeﬁnite Metric. John Wiley and Sons, Ltd., Chichester, 1989. (Translated from Russian.) [4] V. Bolotnikov, C.K. Li, P. Meade, C. Mehl, and L. Rodman, Shells of Matrices in Indeﬁnite Inner Product Spaces. Electron. J. Linear Algebra 9 (2002), 67–92. [5] R. Cross, Multivalued Linear Operators. Marcel Dekker Inc., 1998. [6] A. Dijksma and H.S.V. de Snoo, Symmetric and Selfadjoint Relations in Krein Spaces I. Oper. Theory Adv. Appl. 24 (1987), 145–166. [7] A. Dijksma and H.S.V. de Snoo, Symmetric and Selfadjoint Relations in Krein Spaces II. Ann. Acad. Sci. Fenn. Math. 12 (1987), 199–216. [8] I. Gohberg, P. Lancaster, and L. Rodman, Matrices and Indeﬁnite Scalar Products. Birkh¨ auser, 1983. [9] I. Gohberg, P. Lancaster, and L. Rodman, Indeﬁnite Linear Algebra. Birkh¨ auser, 2005.

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[10] I. Gohberg and B. Reichstein, On Classiﬁcation of Normal Matrices in an Indeﬁnite Scalar Product. Integral Equations Operator Theory 13 (1990), 364–394. [11] M. Kaltenb¨ ack and H. Woracek, Selfadjoint Extensions of Symmetric operators in Degenerated Inner Product Spaces, Integral Equations Operator Theory 28 (1997), 289–320. [12] P. Lancaster, A.S. Markus, and P. Zizler, The Order of Neutrality for Linear Operators on Inner Product Spaces. Linear Algebra Appl. 259 (1997), 25–29. [13] P. Lancaster and L. Rodman, Algebraic Riccati Equations. Clarendon Press, 1995. [14] H. Langer, R. Mennicken, and C. Tretter, A Self-Adjoint Linear Pencil Q − λP of Ordinary Diﬀerential Operators. Methods Funct. Anal. Topology 2 (1996), 38–54. [15] H. Langer, Invariante Teilr¨ aume deﬁnisierbarer J-selbstadjungierter Operatoren. Ann. Acad. Sci. Fenn. Ser. A.I. Math. 475 (1971), 1–23 [16] C.K. Li, N.K. Tsing, and F. Uhlig, Numerical Ranges of an Operator on an Indeﬁnite Inner Product Space. Electron. J. Linear Algebra 1 (1996), 1–17. [17] C. Mehl, A. Ran, and L. Rodman, Semideﬁnite Invariant Subspaces: Degenerate Inner Products. Oper. Theory Adv. Appl. 149 (2004), 467–486. [18] C. Mehl, A. Ran, and L. Rodman, Hyponormal Matrices and Semideﬁnite Invariant Subspaces in Indeﬁnite Inner Products. Electron. J. Linear Algebra 11 (2004), 192– 204. [19] C. Mehl, A. Ran, and L. Rodman, Extension to Maximal Semideﬁnite Invariant Subspaces for Hyponormal Matrices in Indeﬁnite Inner Products. Linear Algebra Appl. 421 (2007), 110–116. [20] C. Mehl and L. Rodman, Symmetric Matrices with Respect to Sesquilinear Forms. Linear Algebra Appl. 349 (2002), 55–75. [21] V. Mehrmann, Existence, Uniqueness, and Stability of Solutions to Singular Linear Quadratic Optimal Control Problems. Linear Algebra Appl. 121 (1989), 291–331. [22] B. C. Ritsner, The Theory of Linear Relations. (Russian) Voronezh, Dep. VINITI, No. 846-82, 1982. [23] H. Woracek, Resolvent Matrices in Degenerated Inner Product Spaces. Math. Nachr. 213 (2000), 155–175. Christian Mehl Institut f¨ ur Mathematik, MA 4-5 Technische Universit¨ at Berlin Str. des 17. Juni 136 D-10623 Berlin, Germany e-mail: [email protected] Carsten Trunk Institut f¨ ur Mathematik, MA 6-3 Technische Universit¨ at Berlin Str. des 17. Juni 136 D-10623 Berlin, Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 175, 211–224 c 2007 Birkh¨ auser Verlag Basel/Switzerland

Symmetric Hermite-Biehler Polynomials with Defect Vyacheslav Pivovarchik Abstract. The polynomial ω = P (λ) + iQ(λ) with real P (λ) and Q(λ) which belongs to Hermite-Biehler class (all its zeros lie in the open upper half-plane) and is symmetric (ω(−λ) = ω(λ)) is modiﬁed as follows ˜ ˆ 2 + c), c > 0. ωc (λ) = P˜ (λ2 + c) + iλQ(λ ˜ 2 ˆ Here P˜ (λ2 ) = P (λ), Q(λ ) = λ−1 Q(λ) and ω(λ) + ω(−λ) ω(λ) − ω(−λ) , Q(λ) = . 2 2i The conditions are obtained necessary and suﬃcient for a set of complex numbers to be the zeros of a polynomial of the form ωc (λ). P (λ) =

Mathematics Subject Classiﬁcation (2000). Primary 12D10; Secondary 26C10, 30C15. Keywords. Hermite-Biehler polynomial, Hurwitz polynomials, zeros in the lower half-plane.

1. Introduction In [1] the following so-called generalized Regge boundary problem was considered: y − q(x)y = λ2 y, x ∈ (0, a),

(1.1)

y(0) = y (a) + (iλα + β)y(a) = 0, α > 0, β ∈ R. It was shown that the spectrum of this problem coincides with the set of zeros of the characteristic function ϕ(λ) = s (λ, a) + (iαλ + β)s(λ, a),

(1.2)

where s(λ, x) is the solution of (1.1) which satisﬁes the conditions s(λ, 0) = s (λ, 0) − 1 = 0. The function ϕ(λ) is symmetric with respect to the imaginary axis, i.e., ϕ(−λ) = ϕ(λ), λ ∈ C.

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This function is of exponential type. Moreover, if the corresponding operator A acting in L2 (0, a) deﬁned by D(A) = {f : f ∈ W22 (0, a), f (0) = f (a) + βf (a) = 0}, Af = −f + q(x)f, is strictly positive, then the function ϕ(λ) belongs to the Hermite-Biehler class. The function ω(λ) of exponential type is said to belong to SHB (symmetric Hermite-Biehler) class if 1)

ω(−λ) = ω(λ), λ ∈ C

(1.3)

2) all its zeros are located in the open upper half-plane and ω(λ) 3) ω(λ) < 1 for Imλ > 0. But in general the operator A is not strictly positive but bounded below only and ϕ(λ) in (1.2) does not belong to SHB. However, it belongs to a more general class deﬁned as follows. Due to the symmetry of the problem ϕ(λ) + ϕ(−λ) , (1.4) 2 ϕ(λ) − ϕ(−λ) (1.5) Q(λ) = 2i ˆ and the functions P (λ) and Q(λ) = λ−1 Q(λ) are even and, therefore ˜ˆ 2 ϕ(λ) = P˜ (λ2 ) + iλQ(λ ), (1.6) def ˜ ˆ 2 ) def ˆ where P˜ (λ2 ) = P (λ) and Q(λ = Q(λ) are entire functions. The functions of the class which occurred in [1] can be obtained from the symmetric Hermite-Biehler functions (1.6) by shifting: ˜ˆ 2 + c), (1.7) ϕ(λ, c) = P˜ (λ2 + c) + iλQ(λ P (λ) =

where c is a real constant. Of course, if c < 0 the function ω(λ, c) is still symmetric Hermite-Biehler function. But if c > 0 is large suﬃciently, then ϕ(λ, c) does not belong to Hermite-Biehler class. Also it was discovered in [1] that ϕ(λ, c) can have only a ﬁnite number of zeros in the closed lower half-plane. These zeros are purely imaginary and simple. To prove this the authors used the theory of operator polynomials (polynomial operator pencils). A certain order was found also in location of the purely imaginary zeros lying in the upper half-plane. It was shown also that these properties of purely imaginary zeros together with the corresponding asymptotics are suﬃcient for a sequence to be the set of zeros of an entire function of the form (1.2), where ϕ(λ) is symmetric Hermite-Biehler function. The mentioned order in location of purely imaginary zeros is a common feature of many entire functions occurring in applications (see [2], [3]). A more complicated order is present in location of pure imaginary eigenvalues of boundary problems generated by the fourth order diﬀerential equation (see [4]). Here we show that this

Symmetric Hermite-Biehler Polynomials with Defect

213

order is valid for pure imaginary zeros of polynomials of the corresponding class. Moreover, we show that the symmetry with respect to the imaginary axis and the mentioned order in location of the zeros (see below) are the characteristic properties of the zeros of a polynomial of the form (1.7). Thus, we obtain a generalization of the Hermite-Biehler theorem (see Theorem 2.3 below) to the case of polynomials possessing certain symmetry with respect to the imaginary axis and having pure imaginary zeros in the lower half-plane. It is well known that problem (1.1) is connected with the problem of small transversal vibrations of a nonhomogeneous smooth string with one end ﬁxed and the other end free to move with damping in the direction orthogonal to the equilibrium position of the string. If we assume that the string consists of weightless intervals (threads) joining a ﬁnite number of point masses, then the characteristic function will be a polynomial whose degree depends on the number of point masses.

2. Direct problem We are going to obtain a generalization of the well-known Hermite-Biehler theorem [5]. Let us recall it. Deﬁnition 2.1. A polynomial is said to be real if it takes on real values on the real axis. Deﬁnition 2.2. A polynomial is said to be Hermite-Biehler (HB) if all its zeros lie in the open upper half-plane. It should be mentioned that the transformation λ → iλ transforms a HB polynomial into a so-called Hurwitz polynomial [6]. Theorem 2.3. (Hermite-Biehler theorem, see [5]). In order that the polynomial ω (λ) = P (λ) + iQ(λ) where P (λ) and Q(λ) are real polynomials, not have any zeros in the closed lower half-plane Im λ ≤ 0, i.e., belong to HB, it is necessary and suﬃcient that the following conditions be satisﬁed: (1) the polynomials P (λ) and Q(λ) have only simple real zeros, while these zeros separate one another, i.e., between two successive zeros of one of these polynomials there lies exactly one zero of the other; (2) at some point λ0 of the real axis Q (λ0 )P (λ0 ) − Q(λ0 )P (λ0 ) > 0.

(2.1)

The fact that the two polynomials satisfy condition (1) will be expressed by saying that “the zeros of the polynomials P (λ) and Q(λ) are interlaced”. Deﬁnition 2.4. The polynomial ω(λ) is said to be symmetric if (1.3) is true. The polynomial ω(λ) is said to belong to the class SHB if it is Hermite-Biehler and symmetric.

214

V. Pivovarchik For a symmetric polynomial ω(λ) the following is valid ˜ˆ 2 ˆ ω(λ) = P (λ) + iQ(λ) = P (λ) + iλQ(λ) = P˜ (λ2 ) + iλQ(λ ),

(2.2)

ˆ where P (λ) and Q(λ) are real even function. Here P˜ (λ2 ) = P (λ),

(2.3)

˜ ˆ 2 ) = Q(λ). ˆ Q(λ

(2.4)

Lemma 2.5. Let the polynomial ω(λ) of the form (2.2) belong to SHB. Then the polynomial ˜ ˆ P˜ (λ) + iQ(λ), ˜ ˆ are given by (2.3) and (2.4) belongs to HB. where P˜ and Q Proof. Due to Theorem 2.3

n = P (λ) = (λ2 − a2k )

and

n+n

Q(λ) = (−1)

n = λ (λ2 − b2k ),

1

1 def

def

where n = n or n = n − 1, a−k = −ak , b−k = −bk , and the zeros ak and bk interlace as follows: · · · < b−1 < a−1 < 0 = b0 < a1 < b1 < a2 < b2 < · · · . We assume in above formulae

0 / (λ2 − b2k ) ≡ 1 by deﬁnition. 1

Then P˜ (λ) =

n = (λ − a2k )

n

and

= ˜ ˆ Q(λ) = (−1)n+n (λ − b2k ).

1

1

Here 0 < a21 < b21 < a22 < b22 < · · · ,

(2.5)

˜ ˆ i.e., the zeros of P˜ (λ) and Q(λ) interlace and therefore, condition (1) of Theorem 2.3 is satisﬁed. Let us prove that condition (2) is satisﬁed too. It is clear that P˜ (0) = (−1)n

n =

n

= ˜ ˆ Q(0) = (−1)n+2n b2k ,

a2k ,

1

P˜ (0) = (−1)n−1

1 n -

n =

k =1

k=1,k=k

a2k ,

n ˜ n+2n −1 ˆ Q (0) = (−1) k =1

n = k=1,k=k

b2k .

Symmetric Hermite-Biehler Polynomials with Defect Using the above expressions we obtain ⎛ n n = ˜ ˜ ˜ ˆ ˜ ˆ ˆ Q (0)P (0) − Q(0)P (0) = − ⎝

b2k

k =1 k=1,k=k

=

n =

b2k

k=1

⎛

n =

a2k ⎝

k=1

n -

n =

n -

a2k −

k=1

a2k

k =1 k=1,k=k

k=1

a−2 k −

n =

215

n -

⎞

⎠. b−2 k

n =

⎞ b2k ⎠

k=1

(2.6)

k=1

Taking into account (2.5) and the inequality n ≤ n we have n k=1

a−2 k

−

n -

b−2 k > 0.

(2.7)

k=1

Now (2.6) and (2.7) imply ˜ ˜ ˆ (0)P˜ (0) − Q(0) ˆ P˜ ˆ (0) > 0. Q Using again Theorem 2.3 we ﬁnish the proof.

Corollary 2.6. Let the polynomial ω(λ) of the form (2.2) belong to SHB. Then the polynomial ˜ ˆ + c) P˜ (λ + c) + iQ(λ belongs to HB for every real c. Deﬁnition 2.7. Two real polynomials P (λ) and Q(λ) are said to compose a real pair, if they have no common zeros and if any linear combination µP (λ) + νQ(λ) of them with real coeﬃcients µ and ν has no complex zeros. (We refer to not lying on the real axis as complex zeros). By Theorem 4 ([7], Chap. 7, Sec. 2, p. 315) we have the following statement. ˜ ˆ Corollary 2.8. The polynomials P˜ (λ) and Q(λ) form a real pair and ∂ ˜ ˜ ˆ + c) − Q(λ ˆ + c) ∂ P˜ (λ + c) > 0, forλ ∈ R, c ∈ R. P˜ (λ + c) Q(λ ∂λ ∂λ

(2.8)

To describe the sets of zeros of polynomials of the form (2.2) let us make use of the following deﬁnitions introduced in [1]. Deﬁnition 2.9. A ﬁnite or inﬁnite sequence of complex numbers {ζk }k∈Z or {ζk }0=k∈Z is said to be properly enumerated if 1) Re ζk ≥ Re ζp for all k > p. 2) ζ−k = −ζk for all ζk not purely imaginary. 3) A certain complex number appears in the sequence at most ﬁnitely many times.

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Deﬁnition 2.10. Let κ be a nonnegative integer. Then the properly enumerated sequence {ζk }nk=−n ({ζk }nk=−n, k=0 ) is said to have the SHBκ+ (resp., SHBκ− ) property if 1) All but κ terms of the sequence lie in the open upper half-plane. 2) All terms in the closed lower half-plane are purely imaginary and occur only once. If κ ≥ 1, we denote them as ζ−j = −i|ζ−j | (j = 1, . . . , κ). We assume that |ζ−j | < |ζ−(j+1) | (j = 1, . . . , κ − 1). 3) If κ ≥ 1, the numbers i|ζ−j | (j = 1, . . . , κ) (with the exception of ζ−1 if it equals zero ) are not terms of the sequence. 4) If κ ≥ 2, then the number of terms in the intervals (i|ζ−j |, i|ζ−(j+1) |) (j = 1, . . . , κ − 1) is odd. 5) If |ζ−1 | > 0, then the interval (0, i|ζ−1 |) contains no terms at all or an even number of terms. 6) If κ ≥ 1, then the interval (i|ζ−κ |, i∞) contains a nonzero even (in the case of SHBκ+ ) or odd (in the case of SHBκ− ) number of terms. 7) If κ = 0, then the sequence has an odd (in the case of SHBκ+ ) or even or zero (in the case of SHBκ− ) number of positive imaginary terms. Now we introduce the class of polynomials to which this paper is devoted. Deﬁnition 2.11. Let the polynomial ω(λ) ∈ SHB. Then the polynomial ˜ˆ 2 ω(λ, c) = P˜ (λ2 + c) + iλQ(λ + c), ˜ ˆ are given by (2.3)–(2.4) is said to belong to the class of shifted symwhere P˜ and Q metric Hermite-Biehler polynomials or in other words to the symmetric HermiteBiehler class with a defect (SHBc ). Lemma 2.12. If the polynomial ω(λ) ∈ SHBc , then it has no real zeros except of possible zero at λ = 0. Proof. If ˜ ˆ 2 + c) = 0 P˜ (λ2 + c) + iλQ(λ for real λ = 0, then

˜ ˆ 2 + c) = 0, P˜ (λ2 + c) = Q(λ what contradicts Corollary 2.6.

Lemma 2.13. If −iτ (τ ≥ 0) is a zero of ω(λ, c) ∈ SHBc at c > 0 ﬁxed, then this zero is simple. Proof. If −iτ (τ > 0) is a multiple zero of ω(λ, c) , then −2iτ P˜ (λ)

λ=−τ 2 +c

˜ˆ 2 P˜ (−τ 2 + c) + τ Q(−τ + c) = 0, ˜ˆ ˜ 2 ˆ + iQ(−τ + c) − 2iτ 2 Q (λ) = 0. 2 λ=−τ +c

(2.9) (2.10)

Symmetric Hermite-Biehler Polynomials with Defect Combining (2.9) with (2.10) we obtain ˜ ˜ˆ 2 ˆ 2P˜ (−τ 2 + c) P˜ (λ) Q(−τ + c) − P˜ (−τ 2 + c) Q (λ) λ=−τ 2 +c

˜ ˆ 3 (−τ 2 + c) = 0 +Q ˜ 2 ˆ Q(−τ + c) −2τ P˜ (λ)

217

λ=−τ 2 +c

(2.11) .

˜ ˜ˆ 2 ˆ Q(−τ + c) − P˜ (−τ 2 + c) Q (λ) λ=−τ 2 +c λ=−τ 2 +c ˜ ˆ 2 (−τ 2 + c) = 0 (2.12) +Q

˜ ˜ˆ 2 ˆ As the zeros of P˜ (λ) and Q(λ) interlace we obtain from (2.9) that Q(−τ + c) = 0 for τ > 0. Then (2.12) implies ˜ ˜ˆ ˜ˆ 2 2 ˆ Q(−τ −2τ P˜ (λ) + c) − P˜ (−τ 2 + c) Q (λ) (−τ 2 + c) = 0, +Q λ=−τ 2 +c

λ=−τ 2 +c

what is impossible for τ > 0 due to Corollary 2.8. Let now τ = 0. Then from (2.9) we obtain P˜ (c) = 0.

(2.13)

Suppose λ = 0 is a multiple zero. Then (2.11) implies ˜ ˆ Q(c) =0 what together with (2.13) contradicts the interlacing conditions.

Now we are going to prove that the set of zeros of any polynomial from SHBc belongs to SHBκ+ or SHBκ− . First we consider the zeros in the lower half-plane. Lemma 2.14. Let ω(λ, c) ∈ SHBc . Then all the zeros of ω(λ, c) in the closed lower half-plane are purely imaginary. Proof. Let λ0 = −iτ , (τ > 0) be a zero of ω(λ, c0 ), where c0 > 0. The polynomial ω(λ, c) is analytic in λ for c ﬁxed and analytic in c for λ ﬁxed. Therefore (see for example [8]), the zeros λj (c) of ω are continuous and piece-wise analytic functions of c, i.e., in some neighborhood of c0 the following is true ∞ k 1/r λj (c) = λ0 + βk (c − c0 )j , (j = 1, r), (2.14) k=1 1/r (c−c0 )j

(j = 1, r) means the set of all branches of the root. Here due to the where symmetry of the function each of the βj is either real or purely imaginary. When c changes from 0 to c1 > 0 the zeros of ω(λ, c) can come into the lower half-plane from inﬁnity or crossing the real axis. The order of the polynomial ω(λ, c) does not depend on c. Thus, the zeros cannot come to the lower half-plane from inﬁnity. Due to Lemma 2.12 they can cross the real axis only at the origin. Moving along the negative imaginary half-axis the zeros do not collide because at the collision a

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multiple zero should appear what would contradict Lemma 2.13. The zeros remain purely imaginary due to the symmetry of the problem. Now we are ready to state the main theorem of this section. Theorem 2.15. Let the polynomial ω(λ, c) belong to SHBc . Then the set of its zeros belongs to SHBκ+ (if the order of ω(λ, c) is odd) or to SHBκ− (if the √ order of ω(λ, c) is even) with κ equal to the number of zeros of P (λ) lying on (0, c] if c > 0 and κ = 0 if c ≤ 0. Proof. Let us consider the following polynomial of two variables λ and α (c is ﬁxed): ˜ˆ 2 Ω(λ, α) = P˜ (λ2 + c) + iαλQ(λ + c). It is clear that Ω(λ, 0) = P˜ (λ2 + c) and Ω(λ, 1) = ω(λ, c). To continue the proof of Theorem 2.15 we need the following four lemmas. Lemma 2.16. If Ω(λ, 1) ∈ SHBc , then Ω(λ, α) ∈ SHBc for all α > 0. Proof. It is clear that if P (λ) and Q(λ) satisfy conditions (1) and (2) of Theorem 2.3, then P (λ) and αQ(λ) also satisfy (1) and (2) for all α > 0. The assertion of Lemma 2.16 follows. Lemma 2.17. 1) If Ω(0, 0) = 0 then Ω(0, α) = 0 for all α ∈ R. 2) If Ω(0, 0) = 0 then λ = 0 is a double zero of Ω(λ, 0) and a simple zero of Ω(λ, α) for all α > 0. Proof. The ﬁrst statement follows from the identity Ω(0, α) = P˜ (c). If Ω(0, 0) = 0 then P˜ (c) = 0. This zero is double because the function P˜ (λ2 + c) is even. The simplicity of λ = 0 as the zero of Ω(λ, α) for all α > 0 follows from Lemma 2.13. Lemma 2.18. The number of zeros of Ω(λ, α) in the closed lower half-plane (all the zeros are simple and purely imaginary) does not depend on α for α ∈ (0, ∞). Proof. Let λk (α) be a purely imaginary zero of Ω(λ, α). Then diﬀerentiating ˜ ˆ 2k (α) + c) = 0 P˜ (λ2k (α) + c) + iαλk (α)Q(λ (2.15) with respect to α we obtain ˜ ˆ 2 (α) + c) −iλk (α)Q(λ k . ˜ ˜ˆ 2 2 2 ˜ ˆ 2λk (α)P (λk (α) + c) + iαQ(λk (α) + c) + 2iαλ2k (α)Q (λk (α) + c) (2.16) From (2.15) we express P˜ (λ2k (α) + c) λk (α) = i (2.17) ˜ ˆ 2 (α) + c) αQ(λ k λk (α) =

Symmetric Hermite-Biehler Polynomials with Defect

219

and substitute it into (2.16) and obtain ˜ ˆ 2 (λ2 (α) + c)) λk (α) = (−iλk (α)Q k ˜ ˜ˆ 2 ˆ 2k (α) + c)P˜ (λ2k (α) + c) − P˜ (λ2k (α) + c)Q 2λk (α)(Q(λ (λk (α) + c)) −1 ˜ˆ 2 2 +iαQ (λk (α) + c) . (2.18) Due to Corollary 2.8 ˜ ˜ˆ 2 ˆ 2 (α) + c)P˜ (λ2 (α) + c) − P˜ (λ2 (α) + c)Q Q(λ (λk (α) + c) < 0.. k k k

(2.19)

Using (2.19) we obtain from (2.18) that if Reλk (α) = 0, Imλk (α) < 0 (we keep α > 0), then Re λk (α) = 0 and Im λk (α) > 0. Combining this result with Lemmas 2.13, 2.16 and 2.17 we ﬁnish the proof of Lemma 2.18. Let us continue the proof of Theorem 2.15. If λk (0) = 0 then from (2.18) we obtain ˜ ˆ 2 (λ2 (0) + c) −iQ k λk (0) = . (2.20) ˜ ˜ˆ 2 ˆ 2 (0) + c)P˜ (λ2 (0) + c) − P˜ (λ2 (0) + c)Q 2(Q(λ (λk (0) + c) k k k Due to (2.19) equation (2.20) implies Re λk (0) = 0 and Im λk (0) > 0. Remark 2.19. One can prove Lemmas 2.13, 2.14 and 2.18 using the results of [9] on quadratic operator pencils. Remark 2.20. If α > 0 then the denominator in (2.18) is equal to zero if and only if the corresponding purely imaginary zero λk (α) is multiple (see the proof of Lemma 2.13). All the other (simple) purely imaginary zeros can be classiﬁed according to the sign of the mentioned denominator. An analogue of such a classiﬁcation for eigenvalues of quadratic operator pencils can be found in [10]. Deﬁnition 2.21. The purely imaginary zero λk of a symmetric polynomial is said to be of the ﬁrst (second) kind if ˜ ˜ ˜ˆ 2 2 ˆ 2 + c)P˜ (λ2 + c) − P˜ (λ2 + c)Q ˆ (λ2 + c)) + Q −2iλk (Q(λ (λk + c) > 0 (< 0). k k k k Remark 2.22. The purely imaginary zeros in the open lower half-plane are all of the ﬁrst kind and according to (2.18) and (2.19) they all move upwards when α > 0 grows. The purely imaginary zeros of the ﬁrst (second) kind in the open upper half-plane move downwards (upwards). Lemma 2.23. If −iτ (τ > 0) is a zero of ω(λ, c) ∈ SHBc then iτ is not a zero of it. Proof. Suppose the both −iτ and iτ to be zeros of ω(λ, c). Then ˜ ˜ˆ 2 2 ˆ P˜ (−τ 2 + c) + τ Q(−τ + c) = 0 and P˜ (−τ 2 + c) − τ Q(−τ + c) = 0 ˜ 2 ˆ and, consequently, P˜ (−τ 2 + c) = Q(−τ + c) = 0, what is impossible.

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Let us continue proving Theorem 2.15. Taking into account the symmetry of the problem on reﬂection with respect to the imaginary axis, we have λ−k (α) = −λk (α) for all not purely imaginary λ−k (α) with α ≥ 0, and hence new zeros can appear on the imaginary axis in pairs, which implies the statements of Theorem 2.15.

3. Inverse problem Let us consider the inverse theorem, i.e., we are going to prove that the condition of Theorem 2.15 is not only suﬃcient but also necessary. Theorem 3.1. Let the set {λk }nk=−n, with κ ≥ 0 Then the polynomial n =

ϕ(λ) =

k=0

({λk }nk=−n ) belong to SHBκ− (SHBκ+ ) &

(λ − λk ) ϕ(λ) =

k=−n, k=0

'

n =

(λ − λk )

(3.1)

k=−n

belongs to SHBc , with some c > 0. Proof. Let {λk }nk=−n,

k=0

∈ SHBκ− . Consider the auxiliary polynomial

ϕ0 (λ) =

n =

(0)

(0)

(λ − λk )(λ − λ−k ),

k=1 (0) λk

= k + i, > 0. It is clear that ϕ0 ∈ SHB. where Put ϕ0 (λ) + ϕ0 (−λ) P0 (λ) = , (3.2) 2 ˆ 0 (λ) = ϕ0 (λ) − ϕ0 (−λ) . Q (3.3) 2iλ ˆ 0 (λ) are both even polynomials. Let us introduce the following Then P0 (λ) and Q real polynomials √ P˜0 (λ) = P0 ( λ), (3.4) √ ˜ ˆ 0 (λ) = Q ˆ 0 ( λ). Q (3.5) (0)

(0)

k=0

ˆ 0 (λ) of Q

· · · < b−2 < a−2 < b−1 < a−1 < 0 < a1 < b1 < a2 < b2 < · · · .

(3.6)

Since the zeros {ak }nk=−n, interlace, we have (0)

(0)

(0)

k=0

of P0 (λ) and the zeros {bk }nk=−n,

(0)

(0)

(0)

(0)

(0)

Now we need the following proposition. Proposition 3.2. There exists a set of continuous and piecewise analytic functions (0) {λk (t)}nk=−n, k=0 such that λk (0) = λk and λk (1) = λk , (k = ±1, ±2, . . . , ±n) and − {λk (t)}nk=−n, k=0 ∈ SHBκ(t) for any ﬁxed t ∈ [0, 1] and κ(t) ≥ 0 depending on t stepwise.

Symmetric Hermite-Biehler Polynomials with Defect

221

Proof. If there are not purely imaginary λk then there exists ∈ N ∪ {0} such that Re λs > 0 for s ≥ + 1 and Re λs < 0 for s ≤ −( + 1). For such λs we deﬁne (0) λs (t) = λ(0) s + (λs − λs )t.

So it remains to construct the piecewise analytic functions involving the purely imaginary eigenvalues λs (0 < |s| ≤ ). Now let λ−j be a (purely imaginary) term in the closed lower half-plane. Consider the interval (i|λ−j |, i|λ−(j+1) |). Then the number of terms on this interval is odd. Let us choose one of them and denote it by λ+j , thus resulting in the positive imaginary, so-called principal, eigenvalues λ+1 , . . . , λ+κ . The remaining positive imaginary eigenvalues, which are even in number when multiplicities are taken into account, are grouped into pairs {λ+s , λ−s } (s = κ + 1, . . . , ) such that each of the intervals (i|λ−j |, i|λ−(j+1) |) contains an integer number of such pairs. (0)

Let us move the points λs (0 < |s| ≤ ) in various steps. First, we deﬁne for 0 ≤ t ≤ (1/(κ + 2)) ⎧ (0) ⎪ ⎪ ⎨ λj , 0 < |j| < |κ|, (0) (0) λ−κ +λ+κ λ±κ + (κ + 2)t − λ±κ , j = ±κ, λj (t) = 2 ⎪ ⎪ ⎩ λ(0) , |κ| < |j| ≤ . j

1+s 2+s , κ+2 ] we deﬁne Next, for s = 0, . . . , κ − 2 and t ∈ [ κ+2 ⎧ λj +λ−j λ −λ + ((κ + 2)t − s −1) j 2 −j , ⎪ ⎪ ⎪ (0)2 (0) λj +λ−j ⎨ , − λj λj + ((κ + 2)t − s − 1) 2 λj (t) = (0) ⎪ ⎪ λj , ⎪ ⎩ λj ,

j = ±(κ−s), j = ±(κ−s−1), 0 < |j| < κ−s−1, κ − s < |j| ≤ .

κ Next, for t ∈ [ κ+2 , κ+1 κ+2 ] we deﬁne ⎧ λ +λ 1 −1 ∓1 ⎪ + ((κ + 2)t − κ) λ±1 −λ , j = ±1, ⎨ 2 2 λ , 1 < |j| ≤ κ, j λj (t) = ⎪ ⎩ λ(0) + ((κ + 2)t − κ) λj +λ−j − λ(0) , κ < |j| ≤ . j j 2

Finally, for t ∈ [ κ+1 κ+2 , 1] we deﬁne λj , 0 < |j| ≤ κ, λj (t) = λj +λ−j λj −λ−j + ((κ + 2)t − κ − 1) , κ < |j| ≤ , 2 2

which completes the proof. Continuing the proof of Theorem 3.1, let us construct the function ϕ(λ, t) =

n = k=−n, k=0

(λ − λk (t)).

(3.7)

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Then ϕ(λ, 0) =

ϕ(λ) := ϕ(λ, 1) =

n = k=−n, k=0 n =

(0)

(λ − λk ), (3.8) (λ − λk ).

k=−n, k=0

Put ϕ(λ, t) + ϕ(−λ, t) , 2 ˆ t) = ϕ(λ, t) − ϕ(−λ, t) , Q(λ, 2iλ ˆ ˆ 1). Next, put and then deﬁne P (λ) = P (λ, 1) and Q(λ) = Q(λ, √ P˜ (λ, t) = P ( λ, t), √ ˜ ˆ t) = Q( ˆ λ, t), Q(λ, P (λ, t) =

(3.9) (3.10)

˜ ˜ ˆ 1). ˆ = Q(λ, and then deﬁne P˜ (λ) = P˜ (λ, 1) and Q(λ) n Denote by {ak (t)}k=−n, k=0 the set of zeros of P (λ, t) and by {bk (t)}nk=−n, k=0 ˆ t). Then {ak (t)2 }n are the zeros of P˜ (λ, t) and {bk (t)2 }n the set of zeros of Q(λ, k=1 k=1 ˜ ˆ t). are the zeros of Q(λ, Proposition 3.3. For any ﬁxed t ∈ [0, 1], the sets of zeros {ak (t)2 }nk=1 and {bk (t)2 }nk=1 interlace, i.e., −∞ < a1 (t)2 < b1 (t)2 < a2 (t)2 < b2 (t)2 < · · · < bn (t)2 .

(3.11)

Proof of Proposition 3.3. Due to (3.6), this statement is true for t = 0. The func˜ ˆ t) are entire functions of λ for every t ∈ [0, 1] and continuous tions P˜ (λ, t) and Q(λ, functions of t ∈ [0, 1] for every λ ∈ C. This means that their zeros are continuous functions of t. Therefore, inequalities (3.11) can be violated only if for some t1 ∈ [0, 1] and some k we have bk (t1 )2 = ak (t1 )2 or bk (t1 )2 = ak+1 (t1 )2 . But each of the two identities implies ˜ ˆ k (t1 )2 , t1 ) = 0, P˜ (bk (t1 )2 , t1 ) = Q(b and, consequently, ϕ(bk (t1 ), t1 ) = ϕ(−bk (t1 ), t1 ) = 0, where bk (t1 ) is real or purely imaginary. Since Lemma 2.12 implies that ϕ(λ, t) can have a real zero only at the origin for any t ∈ [0, 1], we conclude that bk (t1 ) ˆ t) are even functions of λ, we is purely imaginary or zero. Since P (λ, t) and Q(λ, have P (bk (t1 ), t1 ) = P (−bk (t1 ), t1 ) = 0 and ˆ k (t1 ), t1 ) = Q(−b ˆ Q(b k (t1 ), t1 ) = 0,

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223

and consequently ϕ(bk (t1 ), t1 ) = ϕ(−bk (t1 ), t1 ) = 0, which contradicts Proposition 3.2 if bk (t1 ) = 0. On the other hand, if bk (t1 ) = 0, then it is a double zero, because ˜ˆ t1 ) and consequently a zero of it is a zero of the even functions P˜ (λ, t1 ) and Q(λ, ω(λ, t1 ), which contradicts Proposition 3.2. Let us continue proving Theorem 3.1. Let us choose c such that a1 (1)2 > c.

(3.12)

Let us consider the polynomial ˜ˆ 2 ω(λ, c) = P˜ (λ2 + c, 1) + iλQ(λ + c, 1). ) ˜ ˆ 2 + c, 1) are { a2 (1) − c}n The zeros of P˜ (λ2 + c, 1) and of Q(λ k=−n, k=0 and k ) n 2 { ak (1) − c}k=−n,k=0 , respectively. They interlace as follows 8 8 8 8 − b2−n (1) − c < − a2−n (1) − c < − b2−n+1 (1) − c < · · · < − a2−1 (1) − c 8 ) < 0 < a21 (1) − c < · · · < b2n (1) − c. (3.13) Thus, ω(λ, c) fulﬁls condition (1) of Theorem 2.3. To show the validity of condition (2) for ω(λ, c) we notice that ∂ ˜ ˜ˆ 2 ˆ λQ(λ + c, 1) = Q(c, 1), ∂λ λ=0 ˜ ˆ 2 + c, 1) = 0. λQ(λ λ=0

Using these equations we obtain ∂ ˜ ˜ ˆ 2 + c, 1) − λQ(λ ˆ 2 + c, 1) ∂ P˜ (λ2 + c, 1) P˜ (λ2 + c, 1) λQ(λ ∂λ ∂λ λ=0 ˜ˆ ˜ = Q(c, 1)P (c, 1). Using (3.12) and (3.13) we obtain ˜ ˆ 1)P˜ (c, 1) > 0 Q(c, and therefore ∂ ˜ ˜ ˆ 2 + c, 1) − λQ(λ ˆ 2 + c, 1) ∂ P˜ (λ2 + c, 1) P˜ (λ2 + c, 1) λQ(λ > 0. ∂λ ∂λ λ=0 That means condition (2) of Theorem 2.3 is satisﬁed too. The ﬁrst statement of Theorem 3.1 follows. In the same way one can prove the second statement, i.e., that if {λk }n−n ∈ SHBκ+ , then ϕ(λ) =

n = k=−n, k=0

(λ − λk ) ∈ SHBc+ .

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Acknowledgment This work is partially supported by Grants UM1-2567-OD-03 and UKM2-2811OD-06 of Civil Research and Development Foundation.

References [1] V. Pivovarchik, C. van der Mee, The inverse generalized Regge problem. Inverse Problems, 17 (2000), 1831–1845. [2] C. van der Mee, V. Pivovarchik, A Sturm-Liouville Spectral Problem with Boundary Conditions Depending on the Spectral Parameter. Functional Analysis and Its Applications, 36, No. 4 (2002), 315–317. [3] C. van der Mee, V. Pivovarchik, The Sturm-Liouville inverse spectral problem with boundary conditions depending on the spectral parameter. To appear in Opuscula Mathematica. [4] M. M¨ oller, V. Pivovarchik, Spectral properties of a forth order diﬀerential equation. To appear in Zeitschrift f¨ ur Analysis und ihre Anwendungen. [5] C. Hermite, Extract d’une lettre de M. Ch. Hermite de Paris ` a Mr. Borchardt de Berlin sur le nombre des racines d’une ´equation alg´ebrique comprises entre des limites donn´ees. J. Reine Angew. Math. 52 (1856), 39–51; reprinted in his Oeuvres, Vol. 1, Gauthier-Villars, Paris, 1905, 397–414. [6] F.R. Gantmaher, Theory of Matrices (in Russian). Nauka, Moscow, 1988. [7] B.Ja. Levin, Distribution of Zeros of Entire Functions, Trans. Math. Monographs. 5, Amer. Math. Soc., Providence, RI, 1980. [8] A.I. Markushevich, Theory of Analytic Functions, (in Russian), 1, Nauka, Moscow, 1968. [9] V. Pivovarchik. On Positive Spectra of One Class of Polynomial Operator Pencils. Integral Equations and Operator Theory, 19 (1994), 314–326. [10] A.G. Kostyuchenko, M.B. Orazov, The problem of oscillations of an elastic halfcylinder and related self-adjoint quadratic pencil (in Russian). Trudy Sem. Petrovsk., N6 (1981), 97–147. Vyacheslav Pivovarchik Preobrazhenskaya str. 59/61, a.17 65045 Odessa, Ukraine e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 175, 225–229 c 2007 Birkh¨ auser Verlag Basel/Switzerland

A Note on Indeﬁnite Douglas’ Lemma Leiba Rodman Abstract. The Douglas lemma on majorization and factorization of Hilbert space operators is extended to the setting of Krein space operators. Mathematics Subject Classiﬁcation (2000). 47A68, 47A63, 46C20. Keywords. Majorization, factorization, Hilbert space operators, Krein spaces.

1. Introduction and main results We let E, F , G be (complex) Hilbert spaces, with the inner products (x, y)E , (x, y)F , and (x, y)G , respectively, and let L(E, F ) denote the Banach space of all linear bounded operators acting from E into F . The celebrated Douglas lemma, or more precisely one of its several versions, states that if H ∈ L(E, F ) and G ∈ L(E, G) are such that H ∗ H ≥ G∗ G, in other words, H ∗ H − G∗ G is positive semideﬁnite, then there exists an operator F ∈ L(F , G) satisfying Gx = F Hx for every x ∈ F. Although generally speaking the operator F is not unique, it can be always chosen so that F ≤ 1. The original reference for the Douglas lemma is [2]. It has been extensively used in operator theory, in particular in studies of division and quotients of operators (see [8, 4]), operator range inclusions [3], and operator inequalities [5]; see, e.g., [6] for connections with lifting and Leech’s theorem. A survey of results around the Douglas lemma in the context of Banach spaces is given in a recent paper [1]. In this note we extend the Douglas lemma to the setting of Krein space ˇ operators. Yu. L. Smul’jan [7, 9] was the ﬁrst who obtained some interesting results in this direction; in particular, he pointed out that generally the Douglas lemma does not hold verbatim in Krein spaces, even the ﬁnite-dimensional ones. In what follows, for a bounded selfadjoint operator X, we denote by i− (X) the dimension (ﬁnite or inﬁnite) of the spectral X-invariant subspace corresponding to the negative part of σ(X). Let JE ∈ L(E, E), JF ∈ L(F , F ), JG ∈ L(G, G) be operators that are simultaneously selfadjoint and unitary, and such that i− (JG ) < ∞. The operators JE , JF ,

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and JG introduce the Krein space structure in E, F , and G, using the indeﬁnite inner products [x, y]JE = (JE x, y)E ,

x, y ∈ E;

[x, y]JF = (JF x, y)F ,

x, y ∈ F;

[x, y]JG = (JG x, y)G , x, y ∈ G; respectively. An operator Q ∈ L(E, E) is called JE -nonnegative if [Qx, x]JE ≥ 0 for all x ∈ E. The (JF , JE )-adjoint of an operator H ∈ L(E, F ) is by deﬁnition the unique operator H ∗JF ,JE ∈ L(F , E) with the property that [Hx, y]JF = [x, H ∗JF ,JE y]JE ,

∀ x ∈ E, y ∈ F.

We now state the extension of the Douglas lemma to the Krein space operators. Theorem 1.1. Let H ∈ L(E, F ), G ∈ L(E, G), and V ∈ L(E, E) be such that the operator H ∗JF ,JE H − G∗JG ,JE G + V (1.1) is JE -nonnegative and V has ﬁnite rank. Then there exists an operator F ∈ L(F , G) such that F ≤ 1 and Gx = F Hx

for every x ∈ N ,

(1.2)

where N is a subspace of E of codimension at most i− (JG ) + rank V. It is of interest to ﬁnd out when (1.2) holds on the whole of E, not only on a certain subspace N . A criterion for this to happen is given in the next theorem. Theorem 1.2. Assume the hypotheses of Theorem 1.1. Then there exists an operator F ∈ L(F , G) such that Gx = F Hx for every x ∈ E if and only if the following condition is satisﬁed: If {xm }∞ m=1 is a sequence of vectors in E such that Hxm → Hf and Gxm → q for some vectors f and q, then necessarily q = Gf . In the next section we develop preliminary results for the proofs of Theorems 1.1 and 1.2.

2. Preliminary results Lemma 2.1. Let there be given operators H ∈ L(E, F ) and G ∈ L(E, G), such that κ := i− (H ∗ H − G∗ G) < ∞.

(2.3)

Then there exists an operator F ∈ L(F , G) such that Gx = F Hx

for every x ∈ N ,

(2.4)

where N is a (closed) subspace of E of codimension at most κ. Moreover, F can be chosen with the additional property that F ≤ 1. Conversely, if (2.4) holds for some F ∈ L(F , G), F ≤ 1, and some subspace N ⊆ E of codimension κ < ∞, then i− (H ∗ H − G∗ G) ≤ κ.

(2.5)

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Proof. We prove ﬁrst the converse statement. If Gx = F Hx for every x ∈ N , where F ≤ 1 and codim N = κ < ∞, then H ∗ H − G∗ G = H ∗ (I − F ∗ F )H ≥ 0

(2.6)

on N . If (2.5) were false, then we would ﬁnd a (κ + 1)-dimensional subspace N0 such that ((H ∗ H − G∗ G)y, y)E > 0 for every y ∈ N0 \ {0}. Since N0 ∩ N = {0}, we obtain a contradiction with (2.6). The direct statement follows upon applying the Douglas lemma to the operators G and H restricted to the kernel of the (ﬁnite rank) orthogonal projection on the spectral subspace of H ∗ H − G∗ G corresponding to the negative part of σ (H ∗ H − G∗ G). The Douglas lemma shows that without additional hypotheses, one cannot expect that the operator F of Lemma 2.1 be deﬁned on the whole of E. Indeed, such a H exists if and only if λH ∗ H − G∗ G ≥ 0 for some positive λ. Other type of conditions are speciﬁed in the next result. Lemma 2.2. Under the hypotheses of Lemma 2.1, there exists an operator F ∈ L(F , G) such that Gx = F Hx for every x ∈ E if and only if the following property (A) holds true: (A): If {xm }∞ m=1 is a sequence of vectors in E such that Hxm → Hf and Gxm → q for some vectors f and q, then necessarily q = Gf . Proof. It is easy to see that the hypothesis (A) is necessary for existence of an operator F ∈ L(F , G) such that Gx = F Hx for every x ∈ E. Assume now that (A) holds true. By Lemma 2.1, let N be the subspace of codimension at most κ such that Gx = F0 Hx for every x ∈ N , F0 ∈ L(F , G). If for some f ∈ N ⊥ we have Hf ∈ closure of {Hx | x ∈ N }, in other words, Hf = lim (Hxm ), m→∞

xm ∈ N ,

then F0 Hf = F0 ( lim (Hxm )) = lim (F0 Hxm ) = lim Gxm = Gf, m→∞

m→∞

m→∞

where the latter equality follows in view of (A). Thus, Gx = F0 Hx holds also for x ∈ N0 := {x ∈ E | Hx ∈ closure of {Hy | y ∈ N }}. Clearly, N0 is closed. If N0 = E, we are done. Otherwise, let ξ1 , . . . , ξp be a basis for E modulo N0 , and for every linear combination pj=1 αj ξj , αj ∈ C, let p p F (H( αj ξj )) = G( αj ξj ). j=1

j=1

(2.7)

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Now deﬁne F = F0 on the closure of {Hy | y ∈ N }, and deﬁne F as in (2.7) on {Hy | y ∈ Span {ξ1 , . . . , ξp }}. The deﬁnition is correct, because, as one checks easily, {Hy | y ∈ N } ∩ {Hy | y ∈ Span {ξ1 , . . . , ξp }} = {0}. It follows that we have G = F H on E.

3. Proofs of Theorems 1.1 and 1.2. We start with Theorem 1.1. Using the easily veriﬁable equalities H ∗JF ,JE = JE−1 H ∗ JF ,

G∗JG ,JE = JE−1 G∗ JG ,

the condition that the operator Q given by the left-hand side of (1.1) is JE nonnegative takes the form H ∗ JF H − G∗ JG G + JE V ≥ 0. Rewrite this inequality: H ∗ H + G∗ G ≥ −JE V + H ∗ (I − JF )H − G∗ (I − JG )G. Note that H ∗ (I − JF )H ≥ 0 and i− (−JE V − G∗ (I − JG )G) ≤ rank V + i− (G∗ (JG − I)G) ≤ rank V + i− (JG ). Thus,

i− (H ∗ H + G∗ G) ≤ rank V + i− (JG ), and it remains to apply Lemma 2.1.

The proof of Theorem 1.2 follows the same line of argument, using Lemma 2.2. Acknowledgments The author is grateful to V. Bolotnikov for constructive discussions concerning the subject matter of this note, and to T. Azizov for useful suggestions and for pointing out the references [7, 9]. The research is partially supported by NSF Grant DMS-0456625.

References [1] A.B. Barnes. Majorization, range inclusion, and factorization for bounded linear operators. Proc. Amer. Math. Soc. 133 (2005), 155–162. [2] R.G. Douglas. On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Amer. Math. Soc. 17 (1966), 413–415. [3] P.A. Fillmore and J.P. Williams. On operator ranges. Advances in Math. 7 (1971), 254–281. [4] S. Izumino. Quotients of bounded operators. Proc. Amer. Math. Soc. 106 (1989), 427–435. [5] C.-S. Lin. On Douglas’s majorization and factorization theorem with applications. Int. J. Math. Sci. 3 (2004), 1–11.

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[6] M. Rosenblum and J. Rovnyak. Hardy classes and operator theory. Oxford University Press, New York, 1985. ˇ [7] Ju.L. Smul’jan. Non-expanding operators in a ﬁnite-dimensional space with an indeﬁnite metric. Uspehi Mat. Nauk, 18 (1963), 225–230. (Russian.) ˇ [8] Ju.L. Smul’jan. Two-sided division in the ring of operators. Mat. Zametki, 1 (1967), 605–610. (Russian). ˇ [9] Ju.L. Smul’jan. Division in the class of J-expansive operators. Matem. Sbornik (N.S.), 74 (116) (1967), 516–525. (Russian.) Leiba Rodman Department of Mathematics The College of William and Mary Williamsburg VA 23187-8795, USA e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 175, 231–240 c 2007 Birkh¨ auser Verlag Basel/Switzerland

Some Basic Properties of Polynomials in a Linear Relation in Linear Spaces Adrian Sandovici Abstract. The behavior of the domain, the range, the kernel and the multivalued part of a polynomial in a linear relation is analyzed, respectively. Mathematics Subject Classiﬁcation (2000). Primary 47A05; Secondary 47A06. Keywords. Polynomial, linear space, linear relation, domain, range, kernel, multivalued part.

1. Introduction Let S be a linear relation in a linear space H over the ﬁeld K of real or complex numbers, let n and mi , 1 ≤ i ≤ n be some positive integers, and let λi ∈ K, 1 ≤ i ≤ n be some distinct constants. Then, the polynomial p in S given by p(S) =

n =

(S − λi )mi

(1.1)

i=1

is a linear relation in H too. Certain properties of polynomials in a linear relation have been obtained long time ago (see for instance [1, 4, 5]). However, it seems that the behavior of the domain, the range, the kernel and the multivalued part of p(S) has not yet been described, cf. [3]. It is the goal of this technical note to cover this gap. For instance, if α and β are two constants and p and q are two positive integers, then the results in this note show that dom (S − α)p (S − β)q = dom S p+q ,

(1.2)

mul (S − α)p (S − β)q = mul S p+q .

(1.3)

and If additionally, α = β then also ran (S − α)p (S − β)q = ran (S − α)p ∩ ran (S − β)q ,

(1.4)

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and ker (S − α)p (S − β)q = ker (S − α)p + ker (S − β)q . (1.5) The results in this note generalize and complete some algebraic ones in [2]. The notion of standard symmetric linear relation has been introduced in [2], and in connection with this notion, a von Neumann like formula has been proved for the case of standard symmetric linear operators. The main ingredients of the proof of that formula are some descriptions of the range and the kernel of certain polynomial with simple squares in a linear operator. The results proved in this note will be used in [6] in order to develop an extension theory of standard symmetric linear relations in Pontryagin spaces. A generalization of the von Neumann like formula in [2] will be also stated. Next the contents of this note are shortly presented. For the convenience of the reader Section 2 contains some basic notions concerning linear relations in linear spaces. Also, certain elementary results about the commutation of linear relations are presented. The main results are proved in Section 3.

2. Linear relations in linear spaces A linear relation A in a linear space H is a linear subspace of the space H × H, the Cartesian product of H and itself. All linear spaces in this paper are assumed to be over the ﬁled K of real or complex numbers. 2.1. Basic deﬁnitions and properties of linear relations The notations dom A and ran A denote the domain and the range of A, deﬁned by dom A = { x : {x, y} ∈ A },

ran A = { y : {x, y} ∈ A }.

Furthermore, ker A and mul A denote the kernel and the multivalued part of A, deﬁned by ker A = { x : {x, 0} ∈ A },

mul A = { y : {0, y} ∈ A }.

A linear relation A is the graph of an operator if and only if mul A = {0} and the inverse A−1 is given by { {y, x} : {x, y} ∈ A }. The following identities express the duality between A and its inverse A−1 : dom A−1 = ran A, ran A−1 = dom A, ker A−1 = mul A, mul A−1 = ker A. For linear relations A and B in a linear space H the operator-like sum A + B is the linear relation in H deﬁned by A + B = { {x, y + z} : {x, y} ∈ A, {x, z} ∈ B }. For λ ∈ K the linear relation λA in H is deﬁned by λA = { {x, λy} : {x, y} ∈ A }, while A − λ stands for A − λI, where I is the identity operator on H. From A − λ = {{x, y − λx} : {x, y} ∈ A} it follows that ker (A − λ) = { x : {x, λx} ∈ A }.

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233

Furthermore, the following equality holds (A − λ) + λI = A

for all λ ∈ K.

(2.1)

For linear relations A and B in a linear space H the product AB is deﬁned as the relation AB = { {x, y} : {x, z} ∈ B, {z, y} ∈ A for some z ∈ H }. Clearly, the following inclusions hold dom AB ⊂ dom B,

ran AB ⊂ ran A,

(2.2)

mul A ⊂ mul AB.

(2.3)

and ker B ⊂ ker AB,

For λ ∈ K the notation λA agrees in this sense with (λI)A. The product of relations is clearly associative. Hence An , n ∈ Z, is deﬁned as usual with A0 = I and A1 = A. Then for all n ∈ N ∪ {0} dom An+1 ⊂ dom An , ker An+1 ⊃ ker An ,

ran An+1 ⊂ ran An ,

(2.4)

mul An+1 ⊃ mul An ,

(2.5)

mul Ap ⊂ ran Ak .

(2.6)

and for all p, k ∈ N ∪ {0} ker Ap ⊂ dom Ak ,

2.2. Some commutation properties of linear relations Let A and B be two linear relations in a linear space H. Assume that A and B commute, i.e., AB = BA in the sense of the product of linear relations. It is easy to see that (A − α)B = B(A − α) (2.7) holds true for all α ∈ K, and that the following equality An B m = B m An

(2.8)

holds for all n, m ∈ N ∪ {0}. Applying (2.7) with B − β instead of B, it follows that (A − α)(B − β) = (B − β)(A − α). Using (2.8) with A − α and B − β instead of A and B it follows that (A − α)n (B − β)m = (B − β)m (A − α)n

(2.9)

for all α, β ∈ K, and for all n, m ∈ N ∪ {0}. Furthermore, the next result is straightforward; it is a generalization of [3, Proposition VI.5.1]. Corollary 2.1. Let S be a linear relation in a linear space H, let n be a positive number, let λi ∈ C, 1 ≤ i ≤ n, and let mi ∈ N, 1 ≤ i ≤ n. Then the identity n = i=1

(S − λi )mi =

n = (S − λσ(i) )mσ(i) , i=1

holds for every permutation σ of the set {1, . . . , n}.

(2.10)

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3. Main results This section contains the main results of this note. They represent the generalization of (1.2)–(1.5) to the case of a polynomial in a linear relation S in a linear space H. Lemma 3.1. Let S be a linear relation in a linear space H and let λ ∈ K. Then dom (S − λ)n = dom S n

n ∈ N.

for all

(3.1)

Proof. If λ = 0 then (3.1) is obvious. Assume that λ = 0 and proceed by induction. The case n = 1 is clear. Assume now that n > 1 and let x0 ∈ dom (S − λ)n , so that {xi , xi+1 } ∈ S− λ for some xi ∈ H, 0 ≤ i ≤ n − 1. Deﬁne the vectors yp , p 0 ≤ p ≤ n by yp = j=1 pj λj xp−j , 0 ≤ p ≤ n, so that p p p−j {yp , yp+1 } = λ {xj , xj+1 + λxj } ∈ S, 0 ≤ p ≤ n, j j=0 which implies that {y0 , yn } ∈ S n . Hence, x0 = y0 ∈ dom S n , which shows that dom (S − λ)n ⊂ dom S n . The converse inclusion follows using similar arguments. Theorem 3.2. Let S be a linear relation in a linear space H, let n ∈ N, λj ∈ K, mj ∈ N, 1 ≤ j ≤ n. Assume that λj , 1 ≤ j ≤ n are distinct. Then dom p(S) = dom S

n

j=1

mj

.

(3.2)

Proof. Proceed by induction. The case n = 1 follows from Lemma 3.1. Assume that (3.2) holds true for some n = k ≥ 1 and denote Mk = kj=1 mj . Let x0 ∈ /k+1 /k dom j=1 (S − λj )mj , so that {x0 , y0 } ∈ (S − λk+1 )mk+1 and {y0 , z0 } ∈ j=1 (S − λj )mj for some y0 , z0 ∈ H. By induction hypothesis it follows that y0 ∈ dom S Mk , so that {yj , yj+1 } ∈ S, 0 ≤ j ≤ Mk − 1 for some yi ∈ H, 1 ≤ j ≤ Mk . It follows from {x0 , y0 } ∈ (S − λk+1 )mk+1 that {xp , xp+1 } ∈ S − λk+1 ,

0 ≤ p ≤ mk+1 − 1

for some xp ∈ H, 1 ≤ p ≤ mk+1 , with xmk+1 = y0 . This implies that {xp , xp+1 + λk+1 xp } ∈ S,

0 ≤ p ≤ mk+1 − 1.

Furthermore, deﬁne the vectors wp ∈ H, 0 ≤ p ≤ mk+1 by p p j wp = xp−j , 0 ≤ p ≤ mk+1 , λ j k+1 j=0 so that

p p p−j {wp , wp+1 } = {xj , xj+1 + λk+1 xj } ∈ S, λ j k+1 j=0

0 ≤ p ≤ mk+1 − 1.

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Deﬁne also the vectors wmk+1 +q ∈ H, 1 ≤ q ≤ Mk by mk+1 −1 q mk+1 + q mk+1 +q−j mk+1 − 1 + q − i q−i wmk+1 +q = xj + λk+1 λk+1 yi , j mk+1 − 1 j=0 i=0 so that

mk+1 −1

{wmk+1 +q , wmk+1 +q+1 } =

mk+1 + q mk+1 +q−j {xj , xj+1 + λk+1 xj } λk+1 j j=0 q mk+1 − 1 + q − i q−i + λk+1 {yi , yi+1 } ∈ S mk+1 − 1 i=0 -

for all 0 ≤ q ≤ Mk − 1. Hence, {w0 , wmk+1 +Mk } ∈ S mk+1 +Mk , which leads to x0 = w0 ∈ dom S

k+1 j=1

mk

. This shows that

dom

k+1 =

(S − λj )mj ⊂ dom S

k+1 j=1

mj

.

j=1

Thus, the induction is complete. The converse inclusion will be next proved. Denote n M = j=1 mj and let µj , 1 ≤ j ≤ M be the λi ’s counted with their multiplicities, /n /M q so that j=1 (S − µj ) = j=1 (S − λj )mj . Deﬁne the constants ηpp , 0 ≤ p ≤ M , 0 0 ≤ qp ≤ p by η0 = 1 for p = 0, and µi1 · · · µiqp , 1 ≤ qp ≤ p ηpp = 1, ηpp−qp = (−1)qp 1≤i1 <···
for p ≥ 1. Let now x0 ∈ dom S , so that {xp , xp+1 } ∈ S, 0 ≤ p ≤ M − 1 for some vectors xp ∈ H, 1 ≤ p ≤ M . Deﬁne the vectors yp ∈ H, 0 ≤ p ≤ M by p j yp = η j=0 p xj , so that y0 = x0 . Let p be arbitrarily chosen between 0 and M − 1. Now it is easily seen that {yp , yp+1 } ∈ S − µp+1 , 0 ≤ p ≤ M − 1, which /M /M implies that {y0 , yM } ∈ j=1 (S − µj ). Therefore x0 = y0 ∈ dom j=1 (S − µj ) = /n dom i=1 (S − λi )mi , which completes the proof. M

Theorem 3.3. Let S be a linear relation in a linear space H, let n ∈ N, λj ∈ K, mj ∈ K, 1 ≤ j ≤ n. Assume that λj , 1 ≤ j ≤ n are distinct. Then ran p(S) =

n

ran (S − λi )mi .

(3.3)

i=1

Proof. In order to prove that n n = ran (S − λi )mi ⊂ ran (S − λi )mi , i=i

i=1

(3.4)

/n let y ∈ ran i=i (S − λi )mi , and let j be arbitrarily chosen /n between 1 and n. Applying Corollary 2.1 it follows that y ∈ ran (S − λj )mj i=1, i=j (S − λi )mi . By (2.2) it follows that y ∈ ran (S − λj )mj for all 1 ≤ j ≤ n. Thus, (3.4) holds true.

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The converse inclusion in (3.3) will be next proved by induction on n. The case n = 1 is obvious. Assume that the inclusion n n = ran (S − λi )mi ⊂ ran (S − λi )mi (3.5) i=1

i=i

holds for some n = k and it will be proved for n = k + 1. For simplicity denote >k+1 λ = λk+1 and m = mk+1 . Assume that y ∈ i=1 ran (S − λi )mi , so that y∈

k

ran (S − λi )

mi

ran (S − λ)

m

k = ⊂ ran (S − λi )mi ran (S − λ)m .

i=1

i=i

Since y ∈ ran (S − λ)m it follows that {zi+1 , zi } ∈ S − λ,

0≤i≤m−1

/k for some vectors zi ∈ H, 0 ≤ i ≤ m with z0 = y. Also, y ∈ ran i=1 (S − λi )mi implies that (3.6) {xi , xi+1 } ∈ (S − λi )mi , 1 ≤ i ≤ k, for some vectors xi ∈ H, 1 ≤ i ≤ k + 1, with xk+1 = y. Furthermore, it follows from (3.6) that {xp,qp , xp,qp −1 } ∈ S − λp ,

1 ≤ p ≤ k,

1 ≤ qp ≤ mp

(3.7)

for some vectors xp,qp ∈ H, with xk,0 = y, xp,mp = xp , 1 ≤ p ≤ k, and xp−1,0 = p,q xp,mp , 2 ≤ p ≤ k. Deﬁne the constants ξt p , 1 ≤ p ≤ k, 1 ≤ qp ≤ mp , 1 ≤ t ≤ m by m1 − q1 + t − 1 ξt1,q1 = (−1)t (λ − λ1 )−(m1 −q1 +t) , t−1 for p = 1, and p−1 t−1 = mp − qp + l p p,qp t −mj −(mp +1−qp ) ξt = (−1) (λ − λj ) (λ − λp ) ψt,l (λ − λp )−l l j=1 l=0

for p > 1, where p ψt,l =

p−1

h=1

p−1 =

rh =t−1−l j=1

mj − 1 + rj (λ − λj )−rj , rj

0 ≤ l ≤ t − 1.

Deﬁne also the constants ηts , 1 ≤ s ≤ t ≤ m by ηts = (−1)t+s

k =

(λ − λj )−mj

j=1

k

h=1

rh =t−s

k = mj − 1 + rj (λ − λj )−rj , r j j=1

and deﬁne the vectors wt ∈ H, 0 ≤ t ≤ m by w0 = x1,m1 , and wt =

mp k p=1 qp =1

p,q ξt p xp,qp

+

t s=1

ηts zs ,

1 ≤ t ≤ m.

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237

By a straightforward computation it follows that {wt , wt−1 } =

mp k -

p,qp

ξt

{xp,qp , xp,qp −1 −(λ−λp )xp,qp }+

p=1 qp =1

t -

ηts {zs , zs−1 } ∈ S −λ

s=1

for all t between 1 and m. Therefore, {wm , w0 } ∈ (S − λ) , so that {wm , x1,m1 } ∈ /k (S − λ)m . Since {x1,m1 , y} ∈ i=1 (S − λi )mi it follows that ' & k = mi {wm , y} ∈ (S − λ)m . (S − λi ) m

i=1

Thus, y ∈ ran

/k+1 i=1

(S − λi )mi and the proof is now complete.

Theorem 3.4. Let S be a linear relation in a linear space H, let n ∈ N, λj ∈ K, mj ∈ K, 1 ≤ j ≤ n. Assume that λj , 1 ≤ j ≤ n are distinct. Then ker p(S) =

n -

ker (S − λi )mi .

(3.8)

i=i

n mi Proof. , so that i=i ker (S − λi ) nTo prove the inclusion “⊃” in (3.8) let x ∈ mi x = i=1 xi , 1 ≤ i ≤ n for some xi ∈ ker (S − λi ) . It follows using (2.3) that xi ∈ ker (S − λi )mi n

/n

n =

(S − λj )mj = ker

n =

(S − λj )mj .

j=1

j=1, j=i

Hence x = i=1 xi ∈ ker j=1 (S − λj )mj . The inclusion “⊂” in (3.8) will be next proved by induction on n. The case n = 1 is obvious. /k+1 Assume that the inclusion holds true for some n = k ≥ 1 and let x0 ∈ ker i=1 (S − λi )mi . Without lose of generality assume that mi ≤ mk+1 for all 1 ≤ i ≤ k. For simplicity denote m = mk+1 and λ = λk+1 . Since & k ' k+1 = = mi mi {x0 , 0} ∈ (S − λi ) = (S − λi ) (S − λ)m , i=1

i=1

/k it follows that {x0 , y0 } ∈ (S − λ)m , and {y0 , 0} ∈ i=1 (S − λi )mi for some vector y0 ∈ H, which further implies that {xi , xi+1 } ∈ S − λ, 0 ≤ i ≤ m − 1 for some /k vectors xi ∈ H, 1 ≤ i ≤ m, with xm = y0 . Since y0 ∈ ker i=1 (S − λi )mi it follows k that y0 = i=1 yi0 , 1 ≤ i ≤ k for some vectors yi0 ∈ ker (S − λi )mi , 1 ≤ i ≤ k. Since mi ≤ m, it follows that ker (S − λi )mi ⊂ ker (S − λi )m , which implies that {yi,ji , yi,ji +1 } ∈ S − λi ,

1 ≤ i ≤ k,

0 ≤ ji ≤ m − 1

for some vectors yi,ji ∈ H with yi,m = 0. Thus {yi,ji , yi,ji +1 − (λ − λi )yi,ji } ∈ S − λ,

1 ≤ i ≤ k,

0 ≤ ji ≤ m − 1.

238

A. Sandovici

Deﬁne now the vectors wp ∈ H, 0 ≤ p ≤ m − 1 by wp =

m−1 j=p

k m−1 - m−p−1 q j+g xj + (−1) (λ − λi )−(j+q+1) yij , q i=1 j=0 q=0

and deﬁne wm = 0. By a straightforward computation it follows that {wp , wp+1 } =

m−1 -

{xj , xj+1 }

j=p

+

k m−1 - m−p−1 j+g (−1)q (λ − λi )−(j+q+1) {yij , yi,ji +1 − (λ − λi )yi,ji }, q i=1 j=0 q=0

so that {wp , wp+1 } ∈ S − λ for all 0 ≤ p ≤ m − 1. Since wm = 0 it follows that wp ∈ ker (S − λ)m−p ⊂ ker (S − λ)m for all 0 ≤ p ≤ m − 1, which implies k inductively that xp ∈ ker (S − λ)m−p + i=1 ker (S − λi )mi . Therefore, x0 ∈ ker (S − λ) + m

k -

ker (S − λi )

mi

i=1

=

k+1 -

ker (S − λi )mi ,

i=1

which shows that the induction’s step holds true. This completes the proof. Lemma 3.5. Let S be a linear relation in a linear space H, and let λ ∈ K. Then mul (S − λ)n = mul S n ,

n ∈ N.

(3.9)

Proof. The case λ = 0 is trivial so assume that λ = 0. The proof is done by induction. If n = 1 let x ∈ mul (S − λ), so that {0, x} ∈ S − λ, which implies that {0, x} ∈ S and then mul (S − λ) ⊂ mul S. The converse inclusion, i.e., mul S ⊂ mul (S − λ) follows using a similar argument, so that the case n = 1 is clear. Assume next that (3.9) holds for some n = k ≥ 1 and let x ∈ mul (S − λ)k+1 , so that {0, x} ∈ (S − λ)k+1 . Then {0, y} ∈ (S − λ)k and {y, x} ∈ S − λ for some y ∈ H. By induction hypothesis it follows that y ∈ mul S k . Furthermore, it follows from {y, x} ∈ S − λ that {y, x + λy} ∈ S, which together with the fact that y ∈ mul S k implies that x + λy ∈ mul S k+1 . Hence, x = (x + λy) − λy ∈ mul S k+1 + mul S k = mul S k+1 . Therefore, mul (S − λ)k+1 ⊂ mul S k+1 . Similar arguments show that the inclusion mul S k+1 ⊂ mul (S − λ)k+1 holds also true. This completes the proof. Theorem 3.6. Let S be a linear relation in a linear space H, let n ∈ N, λj ∈ K, mj ∈ N, 1 ≤ j ≤ n. Assume that λj , 1 ≤ j ≤ n are distinct. Then mul p(S) = mul S

n

j=1

mj

.

(3.10)

Polynomials in a Linear Relation

239

Proof. The proof is done by induction. The case n = 1 follows from Lemma 3.5. /k+1 Assume now that (3.10) holds for some n = k ≥ 1 and let x0 ∈ mul j=1 (S − λj )mj , so that {0, y0 } ∈

k =

(S − λj )mj ,

{y0 , x0 } ∈ (S − λk+1 )mk+1

(3.11)

j=1

for some y0 ∈ H. By induction hypothesis it follows that y0 ∈ mul S Mk where k Mk = j=1 mj . Furthermore, the second relation in (3.11) implies that there exist the vectors yj ∈ H, 1 ≤ j ≤ mk+1 , with ymk+1 = x0 , such that {yj , yj+1 } ∈ S − λk+1 ,

0 ≤ j ≤ mk+1 − 1,

which implies that {yj , yj+1 + λk+1 yj } ∈ S,

0 ≤ j ≤ mk+1 − 1.

(3.12)

0 ≤ p ≤ mk+1 .

(3.13)

Deﬁne the vectors zp ∈ H, 0 ≤ p ≤ mk+1 by p p j yp−j , zp = λ j k+1 j=0

It follows from (3.12) that z0 = y0 and also that p p p−j {zp , zp+1 } = {yj , yj+1 + λk+1 yj } ∈ S, λ j k+1 j=0

0 ≤ p ≤ mk+1 − 1,

so that zp ∈ mul S Mk +p for all 0 ≤ p ≤ mk+1 , which implies that yp ∈ mul S Mk +p k+1 for all 0 ≤ p ≤ mk+1 . In particular, x0 = ymk+1 ∈ mul S j=i mj . Therefore, mul

k+1 =

(S − λj )mj ⊂ mul S

k+1 j=1

mj

.

j=1

Conversely, let x0 ∈ mul

/k+1 j=1

{0, u0 } ∈

S mj , so that k =

S mj ,

{u0 , x0 } ∈ S mk+1 ,

(3.14)

j=1

/ for some u0 ∈ H. By induction hypothesis it follows that u0 ∈ mul kj=1 (S −λj )mj . Furthermore, the second relation in (3.14) implies that there exist the vectors uj ∈ H, 1 ≤ j ≤ mk+1 , with umk+1 = x0 , such that {uj , uj+1 } ∈ S,

0 ≤ j ≤ mk+1 − 1,

which implies that {uj , uj+1 − λk+1 yj } ∈ S − λk+1 ,

0 ≤ j ≤ mk+1 − 1.

(3.15)

240

A. Sandovici

Deﬁne the vectors wp ∈ H, 0 ≤ p ≤ mk+1 by p p (−λk+1 )j yp−j , wp = j j=0

0 ≤ p ≤ mk+1 .

(3.16)

It follows from (3.15) that w0 = u0 and also that p p {wp , wp+1 } = (−λk+1 )p−j {uj , uj+1 − λk+1 uj } ∈ S − λk+1 j j=0 / for all 0 ≤ p ≤ mk+1 − 1, so that wp ∈ mul (S − λk+1 )p kj=1 (S − λj ), for all /k 0 ≤ p ≤ mk+1 , which implies by (2.3) that up ∈ mul (S − λk+1 )p j=1 (S − λj ), /k+1 for all 0 ≤ p ≤ mk+1 . In particular, x0 = umk+1 ∈ mul j=1 (S − λj )mj , so that mul S

k+1 j=1

mj

⊂ mul

k+1 =

(S − λj )mj .

j=1

This completes the proof.

Remark 3.7. Let S be a linear relation in a linear space H, let λ ∈ K and let p, k ∈ N ∪ {0}. A combination of (2.6), (3.1) and (3.9) leads to ker (A − λ)p ⊂ dom Ak ,

mul Ap ⊂ ran (A − λ)k .

References [1] R. Arens, Operational calculus of linear relations. Paciﬁc J. Math. 11 (1961), 9–23. [2] T. Azizov, B. Curgus and A. Dijksma, Standard symmetric operators in Pontryagin spaces: a generalized von Neumann formula and minimality of boundary coeﬃcients. J. Funct. Anal. 198 (2003), 361–412. [3] R. Cross, Multivalued linear operators. Marcel Dekker, New York, 1998. [4] M.J. Kascic, Polynomials in linear relations. Paciﬁc J. Math. 24 (1968), 291–295. [5] M.J. Kascic, Polynomials in linear relations. II. Paciﬁc J. Math. 29 (1969), 593–607. [6] A. Sandovici, Standard symmetric linear relations in Pontryagin spaces (in preparation). Adrian Sandovici Department of Mathematics University of Groningen P.O. Box 800 9700 AV Groningen, The Netherland e-mail: [email protected]

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