Operator Theory Advances and Applications Vol.66 Editor I. Gohberg
Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel
"
Editorial Board: A. Atzmon (Tel Aviv) J.A. Ball (Blackburg) A. Ben-Artzi (Tel Aviv) A. Boettcher (Chemnitz) L. de Branges (West Lafayette) K. Clancey (Athens, USA) L.A. Coburn (Buffalo) K.R Davidson (Waterloo, Ontario) RG. Douglas (Stony Brook) H. Dym (Rehovot) A. Dynin (Columbus) P.A. Fillmore (Halifax) C. Foias (Bloomington) P.A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) B. Gramsch (Mainz) G. Heinig (Chemnitz) J.A. Helton (La Jolla) M.A. Kaashoek (Amsterdam)
T. Kailath (Stanford) H.G. Kaper (Argonne) S.T. Kuroda (Tokyo) P. Lancaster (Calgary) L.E. Lerer (Haifa) E. Meister (Darmstadt) B. Mityagin (Columbus) V.V. Peller (Manhattan, Kansas) J.D. Pincus (Stony Brook) M. Rosenblum (Charlottesville) J. Rovnyak (Charlottesville) D.E. Sarason (Berkeley) H. Upmeier (Lawrence) S.M. Verduyn-Lunel (Amsterdam) D. Voiculescu (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville) Honorary and Advisory Editorial Board: P.R Halmos (Santa Clara) T. Kato (Berkeley) P.D. Lax (New York) M.S. Livsic (Beer Sheva) R. Phillips (Stanford) B. Sz.-Nagy (Szeged)
Differentiable Operators and Nonlinear Equations
Victor Khatskevich David Shoiykhet
Translated from the Russian by Mircea Martin
Authors Victor Khatskevich Department of Mathematics University of Haifa Afula Research Institute Mount Carmel, Haifa 31905 Israel
David Shoiykhet Department of Mathematics International College of Technology Ort Braude, College Campus P.O.B.78 Karmiel20101 Israel
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Khatskevich, Victor: Differentiable operators and nonlinear equations / Victor Khatskevich ; David Shoiykhet. Trans!. from the Russ. by Mircea Martin. - Basel; Boston; Berlin: Birkhiiuser, 1994 (Operator theory; Vo!' 66) ISBN 3-7643-2929-7 (Basel ... ) ISBN 0-8176-2929-7 (Boston) NE: Soi~e!, Dllwid:; GT
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© 1994 Birkhiiuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Printed on acid-free paper produced from chlorine-free pulp Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN 3-7643-2929-7 ISBN 0-8176-2929-7
Table of Contents
Introduction Chapter 0: Preliminaries 1. Sets and relations ................................................. . 2. Topological spaces ................................................ . 3. Convergence. Directedness ........................................ . 4. Metric spaces ..................................................... . 5. Spaces of mappings ............................................... . 6. Linear topological spaces ........................................ '" 7. Normed spaces .................................................... . 8. Linear operators and functionals .................................. . 9. Conjugate space. Conjugate operator .............................. . 10. Weak topology and reflexivity ..................................... . 11. Hilbert spaces ..................................................... . Chapter I: Differential calculus in normed spaces 1. The derivate and the differential of a nonlinear operator ........... . 2. Lagrange formula and Lipschitz condition ......................... . 3. Examples of Frechet differentiable operators ....................... . 4. Lemmas about differentiable operators ............................ . 5. Partial derivatives ................................................. . 6. Multilinear operators. Duality. Homogeneous forms ............... . 7. Higher order derivatives ........................................... . 8. Complete continuity of operators and of their derivatives .......... .
VI Chapter II: Integration in normed spaces 1. Riemann - Stieltjes integrals of vector-functions ....................... 61 2. Pettis integral and the connection with Riemann - Stiltjes integral ..... 65 3. Antiderivatives of vector-functions. Integral representations ............ 66 4. Integrals of operators in Banach spaces ................................ 71 Chapter III: Holomorphic (analytic) operators and vector-functions on complex Banach spaces 1. Differentiability in complex and real sense. Cauchy - Riemann conditions ......................................... 76 2. The p-topology and holomorphy ....................................... 80 3. Cauchy integral theorems and their consequences ...................... 85 4. Uniqueness theorems and maximum principles 5. Schwartz Lemma and its generalizations ............................... 92 6. Uniformly bounded families of p-holomorphic (holomorphic) operators. Montel property ...................................................... 98 Capter IV: Linear operators 1. The spectrum and the resolvent of a linear operator .................. 103 2. Spectral radius ...................................................... 108 3. Resolvent and spectrum of the adjoint operator ...................... 111 4. The spectrum of a completely continuous operator .................... 114 5. Normally solvable operators .......................................... 117 6. Noether and Fredholm operators ..................................... 119 7. Projections. Split able operators ...................................... 122 8. Invariant subs paces .................................................. 127 Chapter V: Nonlinear equations with differentiable operators 1. Fixed points. Banach principle ....................................... 133 2. Non-expansive operators ............................................. 137 3. Fixed points for differentiable operators .............................. 143 4. Some applications of fixed point principle ............................ 147 5. Implicite and inverse operators. Connection with fixed points ......... 160 Chapter VI: Nonlinear equations with holomorphic operators 1. s-fixed points for holomorphic operators. A converse of Banach principle ....................................... 171 2. Criterions for the existence of an s-fixed point and its extension with respect to a parameter ............................. 177 3. Regular fixed points. Geometric criterions ............................ 182
VII 4. Apriori estimates and the extension of an s-solution to the boundary of the domain ....................................... 189 5. Local inversion of holomorphic operators and a posteriori error estimates ........................................... 195 6. Single-valued small solutions in some degenerate cases ................ 200 Chapter VII: Banach manifolds 1. Basic definitions ..................................................... 211 2. Smooth mappings .................................................... 213 3. Submanifolds ........................................................ 214 4. Complex manifolds and Stein manifolds .............................. 218 Chapter VIII: Non-regular solutions of nonlinear equations 1. Ramification of solutions. Statement of the problem .................. 223 2. Equations of ramification ............................................ 225 3. Equations of ramification for an analytic operator. The problem of the coefficients ....................................... 231 4. The description of the set of fixed points for an analytic operator .................................................... 232 Chapter IX: Operators on spaces with indefinite metric 1. Spaces with indefinite metric ......................................... 239 2. Angle operators ...................................................... 242 3. Plus-operators ....................................................... 244 4. Symmetric properties of a plus-operator and its adjoint ............... 249 5. The problem of invariant semi-definite subspaces ..................... 258 6. An application of fixed point principles for holomorphic operators to the invariant semi-definite subspace problem .................................................... 262 References ..................................................................... 267 List of Symbols ................................................................ 277 Subject Index .................................................................. 279
Introd uction We have considered writing the present book for a long time, since the lack of a sufficiently complete textbook about complex analysis in infinite dimensional spaces was apparent. There are, however, some separate topics on this subject covered in the mathematical literature. For instance, the elementary theory of holomorphic vectorfunctions. and mappings on Banach spaces is presented in the monographs of E. Hille and R. Phillips [1] and L. Schwartz [1], whereas some results on Banach algebras of holomorphic functions and holomorphic operator-functions are discussed in the books of W. Rudin [1] and T. Kato [1]. Apparently, the need to study holomorphic mappings in infinite dimensional spaces arose for the first time in connection with the development of nonlinear analysis. A systematic study of integral equations with an analytic nonlinear part was started at the end of the 19th and the beginning of the 20th centuries by A. Liapunov, E. Schmidt, A. Nekrasov and others. Their research work was directed towards the theory of nonlinear waves and used mainly the undetermined coefficients and the majorant power series methods. The most complete presentation of these methods comes from N. Nazarov.
In the forties and fifties the interest in Liapunov's and Schmidt's analytic methods diminished temporarily due to the appearence of variational calculus methods (M. Golomb, A. Hammerstein and others) and also to the rapid development of the mapping degree theory (J. Leray, J. Schauder, G. Birkhoff, O. Kellog and others). These new methods were particularly attractive since they enabled the study of many classes of nonlinear equations, and therefore they were highly developed. (Important results were obtained by M. Krasnoselski, P. Zabreiko, V. Odinetz, Yu. Borisovich and B. Sadovski.) However, these new techniques retarded the development of spe-
x
INTRODUCTION
cific methods for solving equations with an analytic nonlinear part. That is why in the sixties some mathematicians (P. Rybin, V. Pokornyi, M. Vainberg, V. Trenogyn and others) interested in the theory of integral equations and their applications returned to the Liapunov-Schmidt and Nekrasov-Nazarov analytic methods. At the same time the theory of functions of one or several complex variables was enriched with more significant and subtle results. Parallel with these achievements, the first results on holomorphic mappings on infinite dimensional spaces appeared in the works of A. Cartan, R. Phillips, L. Nachbin, L.Harris, T. Suffridge, W.Rudin, M. Herve, E. Vesentini, J.-P. Vigue, P. Mazet, K. Goebel, and of many others. We consider that it is now about the right time "to set a bridge" between nonlinear analysis and the theory of holomorphic mappings on infinite dimensional spaces. Of course, to this end it is necessary to put together results and techniques from the homology theory, sheaf theory, vector fields theory and from a lot of other modern theories in analysis - a task difficult to achieve within the limits of but one book. That is why we decided to start this vast project, by presenting only the theory of differentiable and holomorphic mappings on Banach spaces, as well as some prerequisites from functional analysis and topology. In all chapters with the exception of Chapter 0 which has the character of a dictionary, we tried to give a complete account of definitions and proofs, and to make this book accesible not only to specialists, but also to students and to those engineers who are currently using the solutions of some specific integral and differentiable equations. We conclude the work by mentioning the interesting relationship between the theory of holomorphic mappings and the theory of linear operators on spaces with indefinite metrics. More precisely, our last chapter is a brief exposition of the theory of spaces with indefinite metrics and of some relevant applications of the holomorphic mappings theory in this setting. In closing, we draw our readers to a few technical points. Throughout the book we strove to use a uniform notation for objects of the same type. The most used notations are presented in Chapter O. At the end of the book we give a list of some standard symbols, and also a subject index. We used the symbols "<1" and "." for the beginning and the end of a proof, respectively. The references in the text contain the name(s) of the author(s) followed by a number in brackets, which corresponds to the one in the reference list. THE AUTHORS
Chapter 0 Preliminaries This is an auxiliary chapter: we collect here the basic definitions and theorems which will be used in what follows. Accordingly, some results are mentioned here merely for a relative completeness and orderliness of the presentation, and not only for pure pragmatic reasons.
§1. Sets and relations 1. We will work with sets and with their elements. For us the words "set", "class",
"collection", "family" will have the same meaning. Generally, we denote the sets by Gothic capitals, and their elements or -
as one also says -
their points, by small
Latin characters. The fact that an element x belongs to a set X is denoted by x We write X
=
SV when x
E X
if and only if x E
SV. The notation {x : x
E X}
E
X.
means
that we consider all elements of the set X. Any (not necessarily proper) part of a set is called a subset. The fact that a set l.2t is a subset of the set X is denoted by l.2t ~ X. In this respect, the empty set 0 is considered a subset of any set. As usually, the union and the intersection of the sets X and SV are denoted by X U SV and X n SV, respectively. The difference of the sets X and SV (or the complement of SV in X) is denoted by X\SV. Union and intersection are dual operations by de Morgan formulas: X\(l.2t U~)
=
(X\l.2t) n (X\~),
where X is a set, and l.2t and
~
X\(l.2t n~)
are subsets of X.
=
(X\l.2t) U (X\~),
(1.1)
PRELIMINARIES
2
De Morgan formulas are valid not only for two, but for any family of subsets of the set X. The direct (or Cartezian) product X x ~ of two sets X and ~ is the set of all ordered pairs (x, y), where x E X and y E ~.
2. Any subset of the direct product X x ~.
~
is called a relation between the sets X and
If R is a relation, then the notations xRy and (x, y) E R are equivalent; in this
case we are saying that x is in the relation R with y. The domain of a relation R is the set D(R) of all first coordinates of the pairs in R, and the range of R is the set
R(R) of their second coordinates. The inverse of the relation R is denoted by R -1 and is defined by R -1 =
= {(y, x) : (x, y)
E
R}; it follows that xRy if and only if yR- 1 x.
The composition, or the product, of the relations Rand S is the relation RoS (denoted also by RS) consisting of all pairs (x, z) such that there exists an y with
(x, y) E Sand (y, z) E R. Composition is not commutative, since R are, in general, different.
0
Sand So R
The identity relation on a set X (or the identity on X) is the set of pairs of the form (x,x) with x E X. It is denoted by
Ix
or Idx and it has the property that
R 0 Ix = Ix 0 R = R for any relation with the domain and the range included in X. (The subscript X could by omitted when no confusions arise.) It is easy to see that R- 1 0 R ;;2 IDCR), R 0 R- 1 ;;2 IR(R) for any relation R. The next result is useful when working with relations:
Let R, S and T be relations. Then: (a) (R- 1 )-1 = Rand (R 0 S)-l = S-l 0 R- 1 ;
THEOREM 1.1.
(b) R
0
(S 0 T) = (R 0 S)
0
T.
Let us consider now some special types of relations, which are very important in all mathematics. A relation R is reflexive if xRx; in other words I ~ R, that is, R contains the diagonal. A relation R is called symmetric if xRy always implies yRx, that is, R = R- 1 • On the other hand, R is called antisymmetric if xRy and yRx do not hold simultaneously, that is, R n R- 1 = 0. A relation R is called transitive if xRy and yRz imply xRz. In terms of composition, R is transitive if and only if R 0 R ~ R. In this case R- 1 0 R- 1 = (R 0 R)-l ~ R- 1 , so the inverse of a transitive relation is also transitive. If R is transitive and reflexive, then R 0 R = R.
3
Sets and relations
A transitive relation on a set X is called an ordering (partial ordering, order relation) on X; it is often denoted by ;;::. When x ;;:: y one usually says that x follows
y (y precedes x), or that x is greater than y (y is smaller than x). Let
~
be a subset of a set X ordered by the relation ;;::. An element x E
is called maximal (in
~)
if for all y
E ~,
~
either x follows y, or x does not precede y.
In the latter case, x is not connected with y by the relation R, or, as one says, x is not comparable with y. An element x E X is called an upper bound of the set for any y E
~,
~
if
either x;;:: y or x = y.
The set
~
is, for any x, y E
is called linearly ordered if all its elements are comparable, that
~
either x;;:: y or y:::;; x, and, in addition, from x;;:: y and y;;:: x it
follows that x = y. LEMMA 1.1 (Zorn). If any linearly ordered subset of a partially ordered set X has an
upper bound, then X has a maximal element.
A direction in X is a reflexive order relation which fulfils the condition: (a) for any x, y E X there exists z E X such that z;;:: x and z;;:: y. A nonempty subset
1)
with a given direction ;;:: on it is called a directed set
and is denoted by CD, ;;:: ). An equivalence relation (or simply an equivalence) on X is a reflexive, symmetric and transitive relation. 3. We consider now a special class of relations called mappings.
A mapping is a single-valued relation. That is, a relation R between the sets
X and ~ is a mapping if xRy and xRz imply y = z, where x E X, y, z E~. We make no distinction between a mapping and its graph. For us the terms correspondence, transformation, mapping, operator have the same meaning. A mapping is more often denoted by the letter f. If f is mapping and x E D(f), then f(x) (or fx, or (x,!)) is the only element y E R(f) with xfy, that is, the element corresponding to x by the relation f. The point f(x) is called the value of f at the point x, or the image of x by f, while x is a preimage of y; one also says that f maps x into y. A mapping f is defined on X if D(f)
= X;
when D(f) is only a part of X, it is said that f is defined ~ ~, then one says, that f acts from
in X. If the mapping f is defined in X and R(f) X into ~; this is written f: X -+ ~.
We will use the term function for a mapping defined in the set lR of real numbers or in the set C of complex numbers, and the term functional for a mapping
PRELIMINARIES
4
which ranges in lR. or C. A mapping f: X
----+
!:V
is called injective (or an injection) if it is one-to-one,
that is f(x) = f(y) implies x = y for x, y E D(f). A mapping f: X
surjective (or a surjection, or onto) if R(f)
=
!:V.
!:V
!:V
is called
A mapping defined on X which is
injective and surjective is called bijective, or a bijection. In this case, an isomorphism between X and
----+
and we say that X and
!:V
f is also called
are isomorphic. A set
X is called countable if either X is finite or is isomorphic with the set N of natural numbers. Notice that any injection is an isomorphism between D(f) and R(f), and that the inverse relation f- 1 is in this case a mapping from R(f) onto D(f). An ~.
isomorphism is often denoted by
A mapping is a set, so two mappings f and g coincide if and only if D(f) =
D(g) and f(x) = g(x) for all x
E
D(f). Consequently, a mapping is defined when
its domain and its value at each point of this domain are specified. If f: X mapping and
~ ~
X, then f n (~ x
f to (the set) of the mapping f if f mapping
!:V)
----+
!:V
is a
is also a mapping, called the restriction of the
f I ~. A mapping g is called an extension
~
and denoted by
~
g. In other words, the mapping g is an extension of the
mapping f if and only if f is the restriction of g to a subset of D(g). Let f be a mapping and
~ ~
D(f). We define
f(~) =
{y I y
=
f(x) for some
the set f(~) is called the image of ~ under f. Analogously, if ~ ~ R(f), we put f-l(93) = {x I x E D(f), f(x) E ~}. (Note that f- 1 is not necessarily a
x
E ~};
mapping.) The set f-l(~) is called the preimage of ~ by
93 we have f(~ U~) = f(~) u f(~) and preimagine is concerned, we have: THEOREM 1.2. Let
f: X
(a) f-l(~\~)
=
----+
!:V
f.
f(~ n~) ~ f(~)
be a mapping
and~, ~ ~
For any two sets ~ and
n
f(~).
As far as the
R(f). Then:
f-l(~)\f-l(~);
(b)
f-l(~ U~) = f-l(~)
(c)
f-l(~ n~)
U f-l(~);
= f-l(~) n f-l(~).
Rules (b) and (c) can be extended for the union and the intersection of an arbitrary family of sets. The composition of two mappings is still a mapping. For any mapping f the composition f- 1 0 f is an equivalence relation, and f 0 f- 1 is the identity of R(f). 4. We close this paragraph by pointing out how the notion of mapping can be used to define the direct product of an arbitrary family of factors.
Topological spaces
5
Let 2t be an arbitrary set, called in what follows the indexing set. Its elements are called indexes, and for any a E 2t let Xa be a given set. The direct product of the sets X a , denoted by
II
X a , is defined as the set of all mappings x defined on 2t, aE2t such that x(a) E Xa for each a E 2t. In this context, the argument is usually written as index; accordingly, Xa = {x I x is a mapping defined on 2t such that Xa E Xa
II
aEQ(
2t}. The set Xa is called the a-th coordinate set and the point Xa is called the a-th coordinate of the point x in the product. The mapping Pa , which maps each point x into its a-th coordinate Xa is called the projector, or the projection operator, or the coordinate projector on the a-th coordinate set. for all a
E
Suppose that all sets Xa are nonempty. Is their direct product nonempty? Zorn lemma implies a positive answer to this question. AXIOM OF CHOICE.
Let Xa be a nonempty set for each element a in an indexing
set 2t. Then there exists a mapping c on 2t such that c( a) The case of equal factors is of special interest. coordinate set Xa coincides with a fixed set ~. Then
E
Xa for any a
2t.
Let us assume that each
II Xa = II ~ =
aEQ(
E
{x I x : 2t ~ ~
aEQ(
is a mapping}. In other words, in this case the direct product is exactly the set of all mappings from 2t into ~; this is often denoted by ~Q(. An example is the n-dimensional Euclidean space.
§2. Topological spaces 1. Let X be a set. A topology on X is a non empty family T of subsets of the set X,
satisfying three conditions: a) the intersection of any two elements of T is an element of b) the union of the elements of any subfamily of the family
T;
T
belongs to
c) the set X is contained in the union of the elements of the family
T,
T;
i.e., X
~
~U{lllllET}.
The last condition is obviously equivalent to the equality X and therefore (using (b)) X E T.
=
U{ll
III E T}
The set X is called the space of the topology T and the pair (X, T) is called a topological space. When no confusions arise, we simply write "X is a topological space". The elements of the topology T are called T-open (or simply open) subsets.
6
PRELIMINARIES
The set lR (C) of real (complex) numbers is an example of a topological space, with the usual topology consisting of unions and finite intersections of open intervals (open disks). The space of a topology is always open. The empty set is also open (as the union of the empty subfamily of the family 7 -
see (b)). The topology consisting
of X and 0 only, is called antidiscrete or trivial; in this case (X,7) is an antidiscrete (topological) space. Obviously, the antidiscrete topology is the smallest of all possible topologies on X. The largest topology on X is the one containing all subsets of X. This is called the discrete topology and a space endowed with the discrete topology is said to be discrete. In a discrete space every set is open. Any topology on X contains the antidiscrete topology and is contained in the discrete topology. Let 71 and 72 be two topologies on X; 71 is weaker (smaller, rougher) than 72, or 72 is stronger (greater, finer) than 72, if 71 ~ 72. It is possible that, for two given topologies 71 and 72 on X, neither 71 is greater that 72, nor 72 is greater that 71; in this case 71 and 72 are said to be not comparable.
2. A neighborhood (7-neighborhood) of a point x in a topological space (X,7) is any subset of this space which contains an element II of the topology 7 with the property x E ll. For example, in the case of the space C of complex numbers with the usual topology, a neighborhood of a point is any subset of C containing an open disk which contains the considered point. THEOREM 2.1. A set 2t ~ X is open if and only iffor every x E 2t, 2t is a neighborhood
ofx. The family of all neighborhoods of a given point is called the neighborhood
system of the point. A subfamily {3 of a topology 7 is a base for 7 if any element in 7 is the union of some elements of {3. A family {3; of neighborhoods of a point x is called a base of the neighborhood system of x, or a base of the topology in x, if any neighborhood of x contains a neighborhood from (3x. We are saying that the topological space (X,7) satisfies the nrst axiom of countability if the neighborhood system of any point has a countable base. If the topology 7 has a countable base, then it is said that X satisfies the second axiom of countability. Spaces satisfying the first, or better the second, axiom of count ability have many useful properties; some of these properties will be mentioned below.
Topological spaces 3. Let X be a topological space and let
2(
7
be a subset of it. A point x E X is adherent
to 2( if every neighborhood of x contains points of 2(. A point x E X is an accumulation
point (limit point) of 2( if in every neighborhood of x there exist points of 2( different from x. A set
2( ~
X is closed if it contains all its adherent points. In an antidiscrete
space X only X and 0 are closed sets, but in a discrete space X every set {x} (having only one element x E X) is closed. Open and closed sets are connected by: THEOREM 2.2.
A subset
of a topological space (X, T) is open if and only if its
2(
complement X\2( is closed. The closure of a set
2(
in the topological space (X, T) is, by definition, the
union of 2( with the set of its limit points, and is denoted by 2(
is closed for every
4.
INTERIOR AND BOUNDARY.
interior for
2(
the interior of
2(,
and that
2(
li1.
is closed if and only if 2( =
A point x of a subset
2(
It is easy to see that 2(.
in a topological space is called
if 2( is a neighborhood of x. The set of all interior points of 2( is called 2(
and is denoted by
2(0.
Let 2( be a subset of the topological space X. The interior 2(0 of the set 2( is an open set, and moreover, 2(0 is the largest open set contained in 2(. The set 2( is open if and only if 2( = 2(0. The set of all points of 2( which are not limit points for X\2( is exactly 2(0. The closure X\2( coincides with X\2(°.
THEOREM 2.3.
The boundary of a subset
2(
of a topological space X is the set of all points
x E X which are interior points neither for 2(, nor for X\2(. Clearly, the boundary of 2( coincides with the boundary of X\2(. For example, in an antidiscrete space the boundary of any nonempty subset is the whole X. In contradistinction, in a discrete space, the boundary of any set is empty. In the set ffi. of real numbers with the usual topology, the boundary of an interval (open, closed or half-open) consists of the endpoints of the interval, while the boundary of the set of rational numbers (as well as the boundary of the set of irrational numbers) is the whole R The relations between boundary, closure and interior are given in the next theorem. Let 2( be a subset of a topological space X and let b(2() be the boundary of 2(. Then: b(2() = 2( n (X\2() = 2(\2(0,
THEOREM 2.4.
PRELIMINARIES
8 X\b(~)
lit =
~
= ~o u (X\~)O,
u b(~) and
~o
= ~\b(~).
A set is closed if and only if it contains its boundary; a set is open if and only if it is disjoint of its boundary. A set ~ is called dense (or everywhere dense) in a topological space X if lit
=
X.
The space X is separable if X has a countable dense subset. A separable topological space satisfies the first axiom of count ability, but not necessarily the second one. On the other hand, a topological space satisfying the second axiom of count ability is separable.
5.
INDUCED TOPOLOGY.
Let (X, T) be a topological space and
The induced topology 0: on
~
~
a subset of X.
is, by definition, the family of all intersections of the
0: if and only if it = QJ n ~ for some QJ E T. The topological space (~, 0:) is called a subspace of the space (X, T). Note that if (~, 0:) is a subspace of (X, T) and (3,!3) is a subspace of (~, 0:) then (3, jJ) is a subspace of (X, T). elements of
T
with the set
THEOREM 2.5.
set in
~.
Thus, it
E
Let (X, T) be a topological space,
(~,
0:) a subspace of it, and
~
a
~.
Then: (a) the set
~
is o:-closed if and only if it is the intersection of ~ with some
T-closed set; (b) a point y E ~ is an o:-limit point for ~ if and only if it is a T-limit for~; (c) the o:-closure of the set ~ is the intersection of the set ~ with the T-closure of~.
§3. Convergence. Directedness 1. The topology of a space can be also described using the notion of convergence. Let
CD,
~) be a directed set (see §1) and
X a topological space. A directedness on X is a mapping S: 1) -> X defined on the whole 1). The value of a directedness S at a point dE 1) is often denoted by Sd, and the directedness itself is denoted by {Sd IdE 1)}. A directedness S belongs to a set ~ starting with a moment m E 1) if Sd E ~ for all d ~ m. A directedness S on the space (X, T) converges to a point SEX if for any T-neighborhood of s, there exists m E 1) such that S belongs to that neighborhood starting with m. For example, if X is the discrete space, then the directedness S
Convergence. Directedness
9
converges to a point s if and only if starting with a certain moment the values of
S coincide with s. On the contrary, in an antidiscrete space any directedness is convergent to every point of the space. Thus, a directedness may be simultaneously convergent to a lot of different points. The notions of limit points and closure of a set are characterized in terms of convergence as follows. THEOREM 3.1.
Let X be a topological space. Then:
(a) A point s is a limit point of a set 2t in 2t\{s} which is convergent to s.
~
X if and only if there is a directedness
(b) A point s belongs to the closure of a set 2t (s EN) if and only if there is a directedness in 2t which is convergent to s.
(c) A set 2t is closed in X (2t = N) if and only if no directedness contained in 2t is convergent to some point of X\2t. A directedness {Td IdE
1)}
is called a subdirectedness of a directedness
{Se leE IE} if there exists a mapping /-l: 1)
----+
IE such that
a) T = So /-l, or equivalently, Ti = S/-'i for each i b) for each e E IE there is a dE
1)
E 1);
such that if j E
1)
and j
~
d then /-lj
~
e.
A point s of a topological space is a limit point of a directedness S if and only if there exists a subdirectedness of S convergent to s. THEOREM 3.2.
2. For an important class of topological spaces, the convergence can be characterized
in terms of sequences. A sequence is a mapping defined on the set of natural numbers, i.e., a di-
rectedness for which the directed set is the linearly ordered set N
=
{1, 2, ... }. A
subsequence is a subdirectedness with /-l mapping N into N (i.e., 1) = IE = N). In other words, a subsequence is an infinite part of a sequence. Note that a sequence may have subdirectednesses which are not subsequences. In a topological space satisfying the first axiom of countability the convergence is completely described in terms of sequences. THEOREM 3.3.
Let X be a topological space satisfying the first axiom of countability.
Then: (a) A point s is a limit point of a set 2t if and only if there is a sequence in 2t\{a} which is convergent to s.
PRELIMINARIES
10 (b) A set
situated in
1.2(
1.2(
is open if and only if any sequence convergent to a point of 1.2( is
starting with a certain moment.
(c) If s is a limit point of a sequence then the sequence contains a subsequence
which is convergent to s. 3. We return now to the case of general topological spaces. A point s is called an
w-limit point of a subset 1.2( in a topological space X if there is a sequence in 1.2(\ {s} which is convergent to s. A set 1.2( is sequentially closed if it contains all its w-limit points. Any closed set is sequentially closed, but there exist non-closed, sequentially closed sets. 4. Some important properties of topological spaces are expressed in terms of sepa-
rability. Roughly speaking, separability means the possibility of separating different points or points from sets. For a later use, let us mention the following definition. A topological space is called a Hausdorff space if any two distinct points of the space have two disjoint neighborhoods. Hausdorff separability of a space is characterized as follows.
A topological space is Hausdorff if and only if no directedness on this space converges to two distinct points.
THEOREM 3.4.
5. TOPOLOGICAL PRODUCT OF SPACES. On the direct product of a family of topo-
logical spaces (see §1) it is possible to define a canonical topology. Let us consider first the case of two topological spaces (X, a) and (~, ,8). The family of all direct products of the form U x QJ, where U is an a-open set in X and QJ is a ,8-open set in ~
forms a base for a topology
T
on X x~;
T
is called the product topology on X x~.
A subset mJ of X x ~ is open in the product topology if and only if for any element
(x, y) E mJ there exist two open neighborhoods U and QJ, of x and y, respectively, such that U x QJ ~ mJ. The space (X x ~,T) is called the topological product of the spaces X and ~. It is straightforward to extend this definition to any finite set of coordinate spaces. Consider now the case of an arbitrary family of topological spaces. Let 1.2( be an indexing set and, for any a E 1.2(, let (Xa, Ta) be a topological space. In §1 we defined the direct product Xa of the sets Xa as the set of all mappins x on 1.2( such
II
aE'.2l
Convergence. Directedness that Xa E X a, for any a E
m.
Let us define a topology T on
11
II Xa by taking as its aEQ(
base the family of all finite intersections of sets of the form p a- 1 (11), where 11 is an arbitrary Ta-open set in Xa and Pa is the projection onto Xa· The space
(II
Xa'T)
aEQ(
is called the topological product of the spaces Xa. The next two theorems give two answers to the natural question about some properties of the factors which are inherited by their product.
The topological product of Hausdorff spaces is a Hausdorff space. Each coordinate projector maps open sets onto open sets.
THEOREM 3.5.
Let X a , a E m, be the arbitrary family of topological spaces satisfying the second axiom of countability. Their topological product satisfies the second axiom of countability if and only if all Xa for a E m but a countable subset of m, are anti discrete.
THEOREM 3.6.
Other properties of topological products will be discussed below. 6. CONTINUOUS MAPPINGS.
topological space
(~,,8)
A mapping
f from a topological space (X, a) into a
is called continuous if the preimage of any ;3-open set in
~
is an a-open in X. It is easily seen that the composition of continuous mappings is continuous. Let f: X
f
----t
~
I mis also continuous.
continuous on
mS;;; X. Then the restriction I mis continuous is said to be
be a continuous mapping and A mapping f for which f
m.
The continuity of a mapping is characterized as follows.
Let X and ~ be topological spaces and f: X following assertions are equivalent:
THEOREM 3.7.
----t
~
a mapping. The
(a) The mapping f is continuous. (b) The preimage of any closed set is closed.
(c) For each point x E X, the preimage of any neighborhood of f(x) is a neighborhood of x. (d) For each point x E X and any neighborhood 11 of the point f(x), there exists a neighborhood m of the point x such that f(m) S;;; 11.
(e) For any directedness 8 in X, convergent to a point s, the composition f08 is convergent to f (s ) .
PRELIMINARIES
12
(f) I(m) S;;; 1(21) for every 21 S;;; X.
(g) 1-1(fJ3) S;;; r1(fJ3) for every fJ3 S;;; A mapping
1 from
~.
a topological space X into a topological space
~
is called
continuous at the point x E X if the preimage of any neighborhood of I(x) is a neighborhood of x. It is plain that
1 is continuous on a set if and only if is continuous
at each point of that set. Let us mention a useful property of continuous mappings with values in a topological product of spaces. THEOREM 3.8.
A mapping 1 from a topological space into a product space
is continuous if and only if the composition Pa
0
1 is continuous for every a E
II Xa aE2l
21.
7. A homeomorphism (or a topological isomorphism) of the topological spaces (X, a) (~,,8) is an isomorphism 1 of the sets X and ~ (see §1) such that both 1 and its inverse 1- 1 are continuous. Two spaces are called homeomorphic if there exists a homeomorphism between them. For example, all isomorphic discrete spaces are homeomorphic and all isomorphic antidiscrete spaces are homeomorphic. The identity mapping is a homeomorphism; the inverse mapping of a homeomorphism is an homeomorphism. Plainly, the composition of any two homeomorphisms is a homeomorphism. A property of a topological space, inherited by all spaces homeomorphic with it, is called a topological invariant. A property defined only in terms of elements and of the topology of a space is in fact a topological invariant. For example, the first and the second axiom of count ability and the Hausdorff separability are topological invariants. and
8.
COMPACTNESS.
A family 'Y of subsets of a topological space is called a cover of a
set 21 if 21 S;;; UO.1 I U E 'Y}. A cover 'Y is called open if each set U E 'Y is open. A subcover of a cover 'Y is a subfamily of'Y which is a cover itself. A topological space is called compact if every open cover of the space contains a finite subcover. A subset 21 is compact if it is compact in the induced topology. A set with compact closure is called precompact (or relatively compact).
A topological space X is compact if and only if any directedness in X has a limit point.
THEOREM 3.9.
A topological space X is called sequentially compact if every sequence in X
Metric spaces contains a subsequence convergent to a point sEX.
13 For a topological space X
satisfying the first axiom of count ability, sequentially compactness is equivalent with the fact that every countable open cover of the space contains a finite subcover. 9. COMPACTNESS AND SEPARABILITY. A closed subset of a compact space is compact.
The converse is not generally true: any nonempty proper subset of an antidiscrete space is compact but not closed. This situation does not occur in Hausdorff spaces.
is a compact set of a Hausdorff space X and x E X\21, then x and 21 have disjoint neighborhoods.
THEOREM 3.10. If21
Consequently, a compact subset of a Hausdorff space is closed.
For any disjoint compact subsets 21 and IJ3 in a Hausdorff space there exist disjoint neighborhoods.
THEOREM 3.11.
Let us consider now mappings on compact spaces.
Let f be a continuous mapping from a compact space X onto a topological space Z). Then the space Z) is compact. Moreover, HZ) is Hausdorff and f is an isomorphism, then f is a homeomorphism. THEOREM 3.12.
Consequently, a real functional continuous on a compact set 21 is bounded and attains its greatest and its smallest value on 21. We close this paragraph with a result concerning the product of compact spaces. THEOREM 3.13 (A. N. Tihonov).
The topological product of compact topological
spaces is compact.
§4. Metric spaces 1. An important class of topological spaces is the one consisting of those spaces with
the topology defined using a distance function (or a metric). A metric on a set X is a nonnegative real-valued functional d defined on the direct product X x X which satisfies (for x, y, z E X) the following conditions (the axioms of a metric):
a) d(x,y)
= d(y,x);
b) d(x, y) c) d(x, y)
+ d(y, z) ~ d(x, z)
(the triangle inequality);
= 0 if and only if x = y.
PRELIMINARIES
14
The pair (X, d), where X is a space and d is a metric on X, is called a metric
space. For (x, y) E X, the number d(x, y) is called the distance between x and y (or the d-distance). Let r > O. The set {y I d(x, y) < r} is called the open r-ball centered at x; the set {y I d(x,y) :s;;r} is called the closed r-ball centered at x. The family of all open balls provides a base for a topology T on X, called the metric topology, or the topology induced on X by the given metric. According to this definition any closed r-ball is closed in (X,T). For example, let X = C and d(x,y) = Ix - YI; d is a metric on C called the usual metric on complex numbers. The topology induced on C by the usual metric coincides with the usual topology. Let X be a set, and set d(x, y) = 0 if x = y and d(x, y) = 1 if x#- y. Then d is a metric on X and the open r-ball centered at any point x coincides with {x} if r :s;; 1 and with X if r > 1. It follows that the topology induced by d is discrete. The closed r-ball centered at any point x is {x} if r < 1 and X if r ~ 1. The open I-ball centered at x is {x} and the closed I-ball centered at x is X.
2.
METRIC SPACES ARE HAUSDORFF
(see §3). By the distance from a point x to a set
2l we mean the number inf d(x, a). aE'J.
The closure of a set the space at distance zero from Qt.
THEOREM 4.1.
Qt
in a metric space X is the set of all points of
Consequently, for any x E X the one-point set {x} is closed.
Any metric space is normal, i.e, any two disjoint closed sets have disjoint neighborhoods.
THEOREM 4.2.
From Theorems 4.1 and 4.2 it follows: THEOREM 4.3.
Any metric space is Hausdorff.
Any metric space satisfies the first axiom of countability. A metric space satisfies the second axiom of countability if and only if the space is separable.
THEOREM 4.4.
From Theorems 4.4 and 3.3 it follows that the convergence in a metric space can be described completely in terms of sequences.
A sequence {sn I n E N} in a metric space (X, d) is convergent to if and only if the sequence of real numbers {d( Sn, s) I n E N} is convergent to zero. THEOREM 4.5.
S
3. Is the topological product of metric spaces a metric space? In other words, is
15
Metric spaces
it possible to define a metric on a product of metric spaces such that the topology induced by this metric coincides with the product topology? Theorems 4.4 and 3.6 imply a negative answer in the case of arbitrary families of factors. However, in the case of countable families the next result holds. THEOREM 4.6.
Let {(Xn,d n )
In
E N} be
a sequence
of metric spaces. For any
x = {Xn I Xn E Xn, n E N} and Y = {Yn I Yn E X n , n EN}, let d(x, y)
=
LT n min{l,
=
dn(xn, Yn)}. Then the functional d is a metric on the product of
i=l
the spaces X n , n E N, such that the corresponding metric topology coincides with the product topology. 4. Consider now the following general problem: what are those properties a topolog-
ical space (X, r) must satisfy, such that there exists a metric on X which induces the topology r? If this is the case, r is called metrizable. In order to answer the question we first need some definitions. A family 'Y of subsets of a topological space is called
locally finite if any point of the space has a neighborhood disjoint from all but a finite set of elements of 'Y. The family 'Y is called discrete if every point of the space has a neighborhood disjoint from all but at most one element of the family 'Y. It is plain that a discrete family is locally finite. Finally, the family 'Y is called (T-Iocally finite
((T-discrete) if 'Y is the union of a countable set of locally finite (discrete) subfamilies. THEOREM 4.7 (Metrizability criterion). The following three conditions are equiva-
lent for a topological space: (a) The space is metrizable.
(b) All one-point sets are closed, the space is regular and has a (T-Iocally finite base. (c) All one-point sets are closed, the space is regular and has a (T-discrete
base. 5. COMPLETENESS. Let (X, d) be a metric space. A sequence {sn I n E N} is called Cauchy (or fundamental) if d(sn, sm) -+ 0 for n, m -+ 00, i.e., for any c > 0 there is
e E N such that d(xn' xm)
< c for all n, m >
e.
A metric space is called complete if
all its Cauchy sequences are convergent. THEOREM 4.8. In a metric space, any metric convergent sequence is Cauchy. 1£ a
Cauchy sequence has a limit point, then it converges to that point.
PRELIMINARIES
16
THEOREM 4.9. A closed subspace of a complete metric space is complete. A com-
plete subspace of a metric space is closed. By Theorem 4.6, a countable topological product of metric spaces is metrizable. We consider now the problem of its completeness. THEOREM 4.10. The product of a countable set of metric spaces is complete if and
only if each coordinate space is complete. A sequence in a product space is fundamental if and only if each of its projections on a coordinate space is fundamental. 6. COMPACTNESS AND MONTEL SPACES. In a metric space the compactness of a set is equivalent with its sequentially compactness. More precisely, we have: THEOREM 4.11. A subset ~ of a metric space is compact if and only if every sequence
in
~
contains a subsequence convergent to a point in
~.
The next definition allows us to give another compactness criterion for subsets of a metric space. A set if for any point x E
~
1)1
in a metric space (X, d) is called an c-array for a set
there exists a point y E
such that d(x, y)
S;;; X
< c.
~
in a metric space X is compact if - and (when X is complete) only if-for any c > 0 there exists a finite c-array for~.
THEOREM 4.12.
A subset
~
(Hausdorff). A set
1)1
~
of a topological space X is called bounded if there exist a point
x E X and a number r > 0 such that ~ is included in the r-ball centered at x. From Hausdorff Theorem above it follows that a compact subset of a metric space is bounded. Consequently, Theorem 4.1 implies that any precompact subset of a metric space is bounded. In particular, any convergent sequence is bounded. A metric space is called a Montel space if all its bounded subsets are precompact. A simple example of a Montel space is given by the space C of complex numbers, with the usual metric. The space C n , the topological product of n copies of C, is also a Montel space.
§5. Spaces of mappings 1. We consider now spaces of mappings from a set X into a fixed topological space
!D
and some special topologies on such spaces. Let:F be a family of mappings from
Spaces of mappings
17
X into !f). It follows from the definition of the direct product (see §1) that F is a subset of the space !f)X = !f). The topology induced on F by the direct product
II
xEX
topology is called the topology of pointwise convergence (coordinatewise convergence,
simple convergence) or, shortly, pointwise topology. A directedness {Jd IdE i"} in !f)X is convergent to J E !f)X if and only if {Jd(X) IdE i"} converges to J(x) for every
x E X. If!f) is Hausdorff or regular, then the space F with pointwise topology has the same property. Let F(x) implies the following. THEOREM 5.1.
= {J(x) I J E F}.
Tychonoff Theorem (Theorem 3.13)
A family F of mappings from a set X into a Hausdorff space !f) is
compact in pointwise topology if and only if the next conditions are satisfied: (a) the family F is closed in !f)X in the direct product topology; (b) for every x EX, the set F( x) is precompact in !f). Condition (b) above is a criterion for the precompactness of the family F in the pointwise topology. 2. The definition of pointwise topology does not depend on any topology existing on X. In fact, X plays in that definition merely the role of an indexing set. More sofisticated topologies on F can be constructed using topologies on X. Thus, let X be a topological space and (!f),p) a metric space. A directedness {Jd IdE i"} in!f)X is said to be uniformly convergent on a set 2l ~ X to a mapping
J E !f)X if for any £ > 0
there exists £ E i" such that for d E i" with d;;: £ we have p(fd(a),j(a)) <
£
for all
a E 2l. We denote by C(X,!f)) the family of all continuous mappings from X into !f).
The topology of compact convergence (uniform convergence on compacts) on C(X,!f)) is defined as follows: a directed ness {Jd IdE i"} is convergent to J E C(X,!f)) if the directed ness is uniformly convergent to J on each compact 2l ~ X. A family F
~
C(X, !f)) is called equicontinuous at a point x if for every
there exists a neighborhood it of x such that p(f(y), J(z)) <
£
£
>0
for all y, zEit and for
allJEF Pointwise topology and topology of compact convergence coincide on any equicontinuous family.
THEOREM 5.2.
Theorems 5.1 and 5.2 immediately imply: THEOREM 5.3.
The following two conditions are sufficient for the precompactness
PRELIMINARIES
18
of a family F ~ C(X, ~): (a) F is equicontinuous;
(b) for every x
E
X, F(x) is precompact in
~.
The next theorem will be used in Chapter II below. THEOREM 5.4. Let {Fn
In
E
E N} is convergent for any x' E
{Fn
In
E
N} ~ C(X,~) be such that the sequence {Fnx' In E X', where X' is a dense subset of X. Then the sequence
N} is convergent in the topology of compact convergence to some F
E
E C(X,~).
3. Let X
= £
be a compact topological space and
f
E C(£, ~).
The functional
p(f(x),O) is continuous on £ for every fixed 0 E~. By Theorem 3.12 this functional 00. Thus, we can define on C(£,~) a metric p by the
is bounded: p(f(x),O) (: cf < formula:
p(f,g) = supp(f(x),g(x)). xE.c
The metric
p is called the
A family F
uniform metric.
~ C(£,~)
is called uniformly bounded if it is bounded as a subset
of the metric space (C(£,~),p), i.e., there exists 9
E C(£,~)
and r > 0 such that
supp(f(x),g(x)) (:r « (0) for all f E F. xE.c
Let now
~
= 9Jt be a Montel space. Then the uniform bounded ness of a
family F from C(£,9Jt) is equivalent to its precompactness in the pointwise topology. Moreover, if the family F is equicontinuous, then, by Theorem 5.2 the pointwise topology coincides on F with the topology of compact convergence, and the last one clearly coincides with the topology induced by the metric
p. Hence, by Theorem 5.1,
we obtain a criterion of precompactness for a family Fin the metric space (C(£, 9Jt), p) (or, equivalently, in the space C(£,9Jt) with the topology of compact convergence): THEOREM 5.5. A subset F of the space C(£, 9Jt) is precompact in the topology of
the uniform metric if and only if F is uniformly bounded and equicontinuous. Theorems of this type (i.e., about precompactness of families of continuous mappings) are usualy credited to Ascoli, or Arzela-Ascoli. Arzela is quoted here because the case £ = [0,1] and 9Jt = lR. - with the usual metric - is due to him. A consequence of Theorem 5.5 is:
An equicontinuous family of mappings from a compact space £ into a Montel
Linear topological spaces
19
metric space 9J1 is itself a Montel metric space relatively to the uniform metric.
§6. Linear topological spaces 1. LINEAR SPACES. Let X be a set and let lK be the field IR. of real numbers or the
field C of complex numbers. The set X is called a linear space (or a vector space) provided that two operations called addition of elements in X and multiplication by scalars are given, satisfying the following axioms (x, y, Z E X, and a, (3 ElK): I. Addition
+ y = y + x; (x + y) + z = x + (y + z);
1) commutativity: x 2) associativity:
3) the existence of a unique element 0
E
X such that x + 0
= x;
4) for any x E X there exists a unique element (-x) such that x Instead of x
+ (-y)
+ (-x) = o.
we write x - y. The element 0 is called the zero element,
or the zero, of the space X; the element -x is called the opposite of x. II. Multiplication by scalars 1) associativity of multiplication: a((3x) 2) distributivity: (a 3) 1· x
+ (3)x = ax + (3x,
= (a(3)x;
a(x
+ y) = ax + ay;
= x.
A linear space is called real or complex according to the field IR. or C which the scalars are from. For example, the set IR. (C) relatively to the usual addition and multiplication is a real (complex) linear space. Here lK is IR. (C) itself. The following are some consequences of the axioms of a linear space: 1.
o· x
=
0 (the first zero is the real number zero, while the second one is the
zero vector); 2. (-1)·x=-x;
3. a· 0 = 0;
4. if ax = (3x and x
-I- 0,
then a = (3; if ax = ay and a
-I- 0,
then x = y.
Two linear spaces are linearly isomorphic if there exists an isomorphism of the underlying sets (see §1) which preserves the algebraic operations: if x f-+ x' and y f-+ y', then x + y f-+ x' + y' and ax f-+ ax'. 2. A set <E of vectors in a linear space X is linearly independent if for any finite subset {Xl, ...
,xn } of <E the equality
alXl
+ ... + anxn =
0 implies
al
= ... = an = O.
PRELIMINARIES
20
Otherwise, the set IE is called linearly dependent. In a linearly dependent set IE =I- {O} there exists an element which is a linear conbination of a finite number of elements in IE, different from it. A linear basis, or a Hamel basis, of a linear space X is a linearly independent subset IE with the property that for each x that x = a1X1
+ ... + anXn
E
X, there exists {Xl,'" , Xn} <;;; IE such
for some aI, ... ,an
E
IK. Any linear space X has a basis,
and any two bases are isomorphic. If a basis of X contains n elements, the space X is called n-dimensional. If X has no finite basis, then it is called infinite dimensional. A nonempty set £ of vectors is called a lineal if from {Xl, ... , Xn} <;;; £ it follows that a1 X2
+ ... + anXn
E
£ for any
aI, ... ,an E
IK. A lineal clearly contains
the zero vector. 3.
DIRECT SUM.
Let X be a linear space and let
lineals in X. If each vector X
X
£1,"" £m
E X has a unique representation of the form
= Xl + ... + Xm ,
Xi E £i, i
=
then we say that X is the direct sum of the lineals m
£1-i- ...
-i-£m or X = L m
the direct sum
-i-£i.
be a finite family of
1,2, ... ,m, £1,""
£m,m and
It is easy to see that in this case
i=l
n
£i
i=l
we write X
= {O} and that m
L -i-£i is linearly isomorphic with the direct product II £i, where the i=l
i=l
linear operations in the last space are defined coordinatewisely: for X = (Xl, ... , xm) and Y = (Y1,"" Ym), put X + Y = (Xl + Y1,···, Xm + Ym) and ax = (ax1, ... , axm ), where a E IK. Analogously, the direct sum of the linear spaces Xl, ... , Xm is the direct m
product
II Xi with the coordinatewise-linear operations given. An example of direct i=l
sum is the n-dimensional space en of vectors X = (Xl, . .. ,xn ), Xi n
i.e., en
E
e,
i
= 1, ... ,n,
= L-i-C. i=l
Let T be a topology on a linear space X. A natural assumption which enables an effective use of both the algebraic and topological structures, is the continuity of linear operations on X with respect to the topology T. If this is the case, then X is called a topological linear space (TLS). 4.
TOPOLOGICAL LINEAR SPACES.
Here are some definitions needed in what follows.
21
Normed spaces
For A E lK and II ~ X, ill is the set {Au I u Ell}. A set II ~ X absorbs a set 9Jt ~ X if there exists Ao E lK such that 9Jt ~ ill for all A with IAI ;;::: IAo I. A set
9Jt
~
X is called bounded if it is absorbed by any neighborhood of 0 in X. A mapping
f
from a topological space X into a topological space
proper if the preimage
f-
1 (IX)
~
is called
of any pre compact set 2t ~ ~ is precompact in X.
A mapping 9 from a linear topological space X into a topological space
compact if the image f(2t) of any bounded set 2t
~
X is precompact
in~.
~
is called
A compact
continuous mapping is called completely continuous. We return to the problem of metrizable topologies. In TLS case, the metrizability theorem of §2 can be reformulated more accurately as follows. THEOREM 6.1.
A Hausdorff TLS is metrizable if and only if it has a countable
neighborhood base for zero. A TLS is called locally bounded if the zero vector has a bounded neighborhood. THEOREM 6.2.
Any locally bounded HausdorffTLS is metrizable.
§7. Normed spaces 1. In a topological linear space, the linear structure is compatible with the topology.
This compatibility is more at hand if the topology is induced by a norm. A norm on a linear space X is a real-valued functional, usually denoted by II II, which verifies the conditions (x, y E X, a ElK):
1) Ilxll ;;::: 0; Ilxll = 0 if and if x = 0;
2) Ilx + yll ~ Ilxll 3) Ilaxll
+ Ilyll
(the triangle inequality);
= lal . Ilxll·
A linear space X with a given norm II lion it is called a normed space and is denoted by (X, II II). In a normed space, the formula p(x, y) a metric.
= Ilx - yll defines
This metric induces on X a topology called the norm topology.
norms II III and II 112 on X are said to be equivalent if there exist
CI,
C2
Two
> 0 such that
cIilxlll ~ IIxl12 ~ c211xlll for every x E X. A normed space which is complete in norm topology is called a Banach space. EXAMPLES. 7.1. The n-dimensional vector space
en is a Banach space relatively to
PRELIMINARIES
22
any of the following norms: n
n
(1)
Ilxll =
Lllxil12;
Ilxll
(2)
i=l
=
IXil; I"; i"; n max
Ilxll = L IXil·
(3)
i=l
The norm (2) is called the Tchebycheff norm. 7.2. The space !.m of all bounded real or complex sequences x
the norm
Ilxll = sup , IXil
=
(Xl, ... , X n , ... )
with
is an infinite dimensional Banach space.
7.3. The space Lp[a, b] of all complex-valued functions x(t) defined on an interval
[a, b] ~ lR and such that Ix(t)IP (p ~ 1) is absolutely integrable on [a, b], with the norm
Ilxll
=
b ) lip
!
(
Ix(t)IPdt
,
is an infinite dimensional Banach space. 7.4. Consider now the infinite dimensional linear space of all functions x(t) defined on an interval [a, b], continuous on [a, b], and having continuous derivatives up to an order k inclusively. On this space we define a norm by
Ilxll = max {max Ix(t)l, max Ix'(t)I, ... , max Ix(kl(t)I}. tE [a,b] tE[a,b] tE [a,b] Thus we obtain a Banach space denoted by Ck[a, b]. 2. A normed space is a metric space, so all properties of metric spaces mentioned
above are true for normed spaces. For convenience, we remind some of them. One-point sets in a normed space are closed, normed spaces are normal, so Hausdorff. A normed space satisfies the first axiom of countability. The second axiom of count ability is satisfied if and only if the normed space is separable. The norm topology can be described using sequences (a sequence {sn I n E N} is norm convergent to S if and only if IIsn - sll ----> 0 as n ----> 00). Any normed space X is isometrically isomorphic with a dense subset of a Banach space~. The space ~ is called a completion of X. A subset Q( of a normed space X is bounded if and only if there exists a number M > 0 such that Iiall ~ M for all a E Q(. Let us introduce now a few definitions:
23
Normed spaces The segment joining the points
(1 -
Xl
and
X2
is the set of elements of the form
~
X is called convex if Xl, X2 E Qt imply that the segment joining Xl and X2 is completely contained in Qt. It follows easily that in a normed space any r-ball (open or closed) is convex. y =
t )Xl
+
tX2,
0:( t
:(
1. A set Qt
A lineal ,c, closed in the norm topology, is called a subspace of the normed space. Any subspace is obviously convex. Every finite dimensional lineal is a subspace. THEOREM 1.1.
A normed space is a Montel space if and only if it is finite dimen-
sional. 3.
Let (X, II II) be a normed space, and let X
TOPOLOGICAL DIRECT SUM.
=
n
=
I: +'ci be the decomposition of the linear space X in a direct sum of lineals 'ci, i = i=l
= 1, ... , n.
=
Xl
+ ... +
This means that each vector X n , Xi
E 'ci, i
= 1, ... , n.
X
E X has a unique representation
If the coordinate projections n
are continuous in the norm topology, the direct sum
Pi:
X =
X ---. 'ci
I: -t-'ci is called the topological i=l
direct sum of the lineals 'ci. THEOREM 1.2.
Let X be a Banach space and assume that the lineals 'ci, i
=
n
=
1, ... , n are subspaces. If X
= I: +'ci then the
direct sum is a topological direct
i=l
sum.
4. We close this section considering the connection between real and complex normed
spaces. Let X be a real normed space and let i denote the imaginary unit: i2 = -1. Set iX
=
{ix I X E X} and form the topological direct sum X+iX
=
{x+iy I x, y EX}.
Define a norm in X + iX by Ilx + iY11 = vllxl12 + IIYI12. Thus, X + iX is a complex normed space, called the complexification of X. On the other hand, a complex normed space can be represented in many cases as a direct sum of two real normed spaces. Thus, let X be a complex normed space with a semilinear involution on it, i.e., a bijection g: X ---. X such that g(x + y) = g(x) + g(y), g(ax) = ag(x) and g(g(x)) = X for all x, y E X, a E Co An element X E X with g(x) = X is called real, and an element v E X with g( v) = -v is called pure imaginary. Then any element X E X can be uniquely represented in the form X = U + iv where u is real and v is pure imaginary, namely u
=
!(x + g(x)), v
=
~(x - g(x)). Therefore X
= 3-t-i3, where
PRELIMINARIES
24
3=
{u E X
I g( u) = u}.
So the complex space X is a direct sum of two real linear
spaces. If the involution g is continuous, then
3 is
a subspace of X and therefore (by
Theorem 7.2) the considered direct sum is topological. For example, consider the complex space en of vectors x = (Xl, ... ,xn ), Xi E i = 1, ... , n. As an involution g on en take the complex conjugation: g(x) =
E
e,
=
x (= (Xl, ... , Xn)). Therefore en may be considered as a 2n-dimensional real space.
§8. Linear operators and functionals 1. LINEAR OPERATORS.
By an established tradition, we will use, as a rule, the
term "operator" for a mapping between normed spaces (see §1). Also as a rule, the operators will be denoted by capital Latin characters. The terms "function" and "functional" will have the previously established meanings. Let X and
!D be normed spaces, both real or complex.
An operator A: X
is called linear if its domain D(A) is a lineal of X and A(ax + (3y)
=
aA(x)
~
!D
+ (3A(x)
a,(3 E K A linear operator A is bounded if IIAxl1 ~ M < 00 for all X E D(A) with Ilxll = 1. In this case, the (finite) number sup IIAxl1 is xED(A),lIxll=1 called the norm of the operator A and is denoted by IIAII. for any X,Y E D(A),
THEOREM 8.1.
A linear operator is continuous if and only if it is bounded.
EXAMPLES. 8.1. Consider the space en (Example 7.1) with any of the norm (1)-
(3), and a rectangular matrix (aik) of order n x m, i E {1, ... , m}, k E {1, ...
. . . , n}, aik
E
C. The equalities n
Yi = l:::aikxk,
i
=
1, ... ,m
k=l
define a linear continuous operator A: en ~ em (y = Ax for x = (Xl'"'' Xn) E E en, Y = (Yl, ... ,Ym) E em). Note that in a finite dimensional normed space any linear operator is continuous. 8.2. On Lp[a, b] (Example 7.3) with p = 1, let us consider the integral operator given by
J b
yet) =
K(t, s)x(s)ds,
a
25
Linear operators and functionals where K(t, s) is a continuous function on the square a
~
t, s
~
b. This is a linear and
bounded operator from Lp[a, b] into itself. 8.3. Consider the space Ck[a, b] (Example 7.4) and defin~ a differential operator by dk +1 y(t) = dtk+l x(t).
This operator is defined on the dense lineal of Ck[a, b] consisting of all functions which have continuous derivatives on [a, b] up to the order k + 1 inclusively and takes values in Cora, b]; it is linear but it is not bounded. 2. The set
into
~,
L(X,~)
of all continuous linear operators defined from the whole space X
where X and
~
are normed spaces, becomes a normed space with the linear
operations:
1) (A
+ B)x =
Ax + Bx,
2) (aA)x = a(Ax), (for A,B if
~
E L(X,~),
x
E X, a E JK)
and the norm
IIAII
=
sup Ilxll=l
IIAxll.
In addition,
L(X,~) is a Banach space, too. The 3) and BE L(X,~) is usually called the
is a Banach (i.e., complete) space, then
composition AoB of the operators A E
L(~,
product of the operators A and B, and is also denoted by AB. We have
IIABII ~ IIAII·
'IIBII, thus AB E L(X,3). If X
= ~,
we write L(X) instead of L(X, X).
The convergence of a sequence of bounded linear operators in the norm of the space
L(X,~)
is called the uniform convergence, and the corresponding topology is
called the uniform operator topology. We consider also a weaker topology, called the
strong operator topology. The convergence of a directedness {Ad IdE :D} <:;:; L (X, ~) in the strong operator topology is defined as the convergence of any directedness {AdX IdE :D} <:;:;~, for all x EX. This topology coincides with the topology induced on L(X,~) by the product topology of ~ = ~x, where we endow ~ with the
II
norm topology. A sequence {An In E N} <:;:; L(X,~) is a Cauchy sequence with respect to the strong operator topology if {Anx I n E N} is a Cauchy sequence in ~ for every x E X. THEOREM 8.2. If X and !V are Banach spaces, then any Cauchy sequence with respect to the strong operator topology in L(X, !V) has a limit.
PRELIMINARIES
26 3.
INVERSE OPERATORS.
Let X and
~
be normed spaces and A: X
----+
~
be a linear
operator. If there exists an operator A-I defined on R(A) with values in D(A) such that
then A is called invertible (see the general definition of the inverse relation in §1); A-I is called the inverse of A. Clearly (A- 1)-1 = A, and A-I is linear. A linear operator A is invertible if and only if from x E D(A) and Ax
=
0 it
follows that x = O. The following three theorems deal with the problem of existence and continuity of the inverse operator. In the first two of them, the continuity of the operator is not assumed. THEOREM 8.3. Let A: X
----+
~
be a linear operator satisfying the condition
IIAxl1 ? cllxll for any x E X, where c is a positive constant. Then A-I exists, is bounded, and II A - 1 1I:::; c- l . Let A: X ----+ ~ be a linear operator such that A-I exists and is continuous; let B be a continuous operator on D(A) such that IIBII < IIA- 1 11- 1 . Then the operator C = A + B has continuous inverse C- 1 : R(A) ----+ D(A) and
THEOREM 8.4.
THEOREM 8.5 (Banach). A continuous linear isomorphism of the Banach spaces X
and
~
is a homeomorphism.
A linear isomorphism between two normed spaces, which is also a homeomorphism is called a linear homeomorphism. THEOREM 8.6. Let n E N be fixed. All n-dimensional normed spaces are linearly
homeomorphic with IR n , so all of them are linearly homeomorphic. Consequently, all norms on a given finite dimensional space are equivalent. 4. LINEAR FUNCTIONALS. Let X be a normed space, ,c a lineal of X and f a linear functional defined on'c. The following theorem shows the possibility of extending the
Conjugate space. Conjugate operator functional
27
f on the whole space X without increasing its norm. For real spaces this
theorem was proved by Hahn and Banach; it was extended to the case of complex spaces by G. A. Suchomlynoff.
Any linear functional f, defined on a lineal £ of a normed space X, can be extended on X with the preservation of the norm, i.e., there exists a linear functional F on X such that
THEOREM 8.7 (Hahn-Banach-Suchomlynoff).
1) F(x)
2)
f(x) for all x E £;
=
IIFII = Ilfll·
Let 0 =I- Xo be an arbitrary element of the normed space X. Then there exists a linear functional f defined on X such that
COROLLARY 8.1.
1)
Ilfll = 1,
2) f(xo)
Ilxoll·
=
Let X be a normed space, £ a lineal in X, and let Xo be a point in X at the distance d > 0 from £ (d = inf Ilx - xolU. Then there exists a linear COROLLARY 8.2.
functional f defined on X such that 1) f(x) 3)
0 for x E £;
=
2) f(xo)
Ilfll =
xE,c
=
1;
d- 1 .
One of the basic results in Banach Space Theory is next theorem.
Let {x n I n E EN} be a sequence of vectors in (a Banach space) X such that the numerical sequence {(xn,J) I n E N} is bounded for each continuous linear functional f on X. Then
THEOREM 8.8 (Banach-Steinhaus principle of uniform boundedness).
Ilxnll::;; M < 00. §9. Conjugate space. Conjugate operator 1. Let X be a normed space. The space of all continuous linear functionals on X is
called the conjugate space of X and is denoted by X*. Thus, X* = L(X, lK), where lK is IR or C depending on the fact that X is a real or a complex space. Since IR and C are complete, X* is a Banach space. EXAMPLES. 9.1. Using Example 8.1 for m = 1, it follows that for each linear func-
PRELIMINARIES
28
n
tional f on en there exist n complex numbers al,"" an such that f(x) = for all x
=
f can be identified with (al,'" ,an)
(Xl,." ,xn ) E en. Thus,
L akXk, k=l
E en. Ac-
cording to the definition of the norm on L(X, Z)), the norms (2) and (3) from Example
7.1 induce in the conjugate space the norms (3) and (2), respectively; the norm (1) induces the same norm, so (en,
11111)* = (en, II lid·
9.2. For the space Lp[a, b] described in Example 7.3, the conjugate space is Lda, b], p
where q =
--. p-1
2. A normed space is called strictly convex if from x, y E X and
it follows that either
X
= AY, or y =
Ilxll + Ilyll
Ilx - yll ~ c > 0 it follows
Let X be a strictly convex normed space and x, y
Ilxll = Ilyll > 0 and (x, I) = (x, f) = Ilxli. Then x = y.
=
px, A, p E lK. A strictly convex space is called
uniformly convex if from x, y E X, Ilxll = IIYII = 1 and Ilx + yll :;;; 2 - 8 for a certain 8 = 8(c) > O. THEOREM 9.1.
Ilx + yll
(y, I) for
a certain f
E
X* such that
E
that
X such that
Ilfll =
1 and
3. Let X be a normed space and X* its conjugate space. Since X* is a normed space,
we may construct X**
=
(X*)*, and so on. Let f E X*. The expresion f(x) for a
fixed x E X can be considered as a continuous linear functional Fx (in the variable f) on the space X*: Fx(f) = f(x). (Based on these considerations,' we will often use the notation (x,I) - see §1 - for the value of f at x.) In addition, IlFxll = Ilxll. Thus, the space X is isometrically linearly isomorphic to a subspace of X**. Up to this isomorphism we can consider that X
~
X**. If this isomorphism is onto X** the space
X is called reflexive. If X is nonrefiexive, all spaces in the sequence X, X*, X**, X***, ...
are different. For instance, the spaces in Examples 7.1 and 7.3 are refiexive, and the spaces in Examples 7.2 and 7.4 are nonrefiexive. 4. Let X and Z) be normed spaces and A: X
----+
Z) a linear operator densely defined
in X (D(A) is dense in X). For any y* E Z)*, consider the linear functional on X given by (Ax, y*), x E X. If there exists an element x* E X* such that (Ax, y*) = = (x, x*) for all x E X, then one says that the conjugate operator A * is defined at y* and A *y* = x*. All these y* form the domain D( A *) of A * (the case D( A *) = {O} may occur), so A*:~J* ----+ X*. If, in particular, A E L(X,Z)), i.e., the operator A is defined
Weak topology and reflexivity
29
on X and is continuous, then A* E L(!D*,X*) and IIA*II = IIAII. For instance, the conjugate of the operator A in Example 8.1, given by the matrix (aik), i E {I, ... , m}, k E {I, ... , n} is the operator A * given by the matrix (aki) -
i.e., the transposed
complex-conjugate matrix. 5.
ORTHOGONALITY.
sion (x, f) -
ORTHOGONAL COMPLEMENT.
the value of
f
X* at x E X -
E
Let us return to the expres-
and consider it as a semilinear
functional in the second variable when the first one is fixed, i.e.,
(x, ah + 1312)
=
a(x, h) + f3(x, h), h, 12 E X*, a,13 E IK. As a consequence, we obtain a sesquilinear functional (linear in the first variable and semilinear in the second) which maps X x X* in K This functional (x, f) is called the scalar (or inner) product of x E X and f E X*. =
The elements x E X and
f
X* are called orthogonal if (x, f)
O. The sets 9.n c X and 9.n* ~ X* are orthogonal if (x, f) = 0 for any x E 9.n and f E 9.n*. The set 9.n~ of all elements in X* orthogonal to 9.n ~ X is called the orthogonal complement of 9.n. The set ~9.n* of all elements of X orthogonal to a set 9.n* ~ X* is called the *-orthogonal complement of 9.n*. E
=
For any 9.n ~ X (9.n* ~ X*), its orthogonal (*-orthogonal) complement is a subspace of X* (X) and ~9.n~ = din 9.n (~9.n*~ = din 9.n*), where din 9.n is the smallest closed subspace of X containing 9.n.
THEOREM 9.2.
§10. Weak topology and reflexivity A normed space X carries, along with the norm topology, topologies induced by linear functionals. The weak topology on X is defined as fol1. WEAK TOPOLOGY.
lows: a directedness {Xd IdE 1)} is weak convergent if the numerical directednesses
{ (Xd, f) IdE 1)} are convergent for every f E X*. In general, the weak topology is weaker than the norm topology. However, in the case of finite dimensional spaces these topologies coincide. THEOREM 10.1.
In a finite dimensional space the weak topology coincides with the
norm topology. We return to the general case of normed space. THEOREM 10.2. If a
sequence {xn
In
E
N} is weakly convergent to
XQ,
then the
PRELIMINARIES
30
sequence is norm bounded. In addition
2. Let X
=!D*
be the conjugate space of a normed space
!D.
The ultraweak (*-weak)
topology on X is by definition the topology induced on X by the topology of the direct product
II lK.
Thus, a directedness {Xd
yElI)
directedness {(y, Xd)
IdE 1)}
IdE 1)}
is ultraweakly convergent if any
is convergent for each y
E!D.
In other words, the
ultraweak convergence is the pointwise convergence. THEOREM 10.3. The ultra weak topology is weaker than the weak topology, and, in
the case of a reflexive space, these two topologies coincide. The closed unit ball of the space X* (the conjugate of the normed space X) is compact in the ultra weak topology. This ball is compact in the weak topology if and only if the space is reflexive.
THEOREM 10.4 (Alaoglu-Bourbaki).
3. We consider now the action of a linear operator between normed spaces endowed
with the weak topology.
and!D be normed spaces. A linear operator A: X -+ !D is bounded if and only if is continuous with respect to the weak topologies on X and !D.
THEOREM 10.5. Let X
Recall that an operator A: X
-+
!D
is completely continuous (see §6.4) if a) is
continuous and b) the image of any bounded set in D(A) is a precompact set in
!D.
For a linear operator, condition a) follows from condition b). Indeed, by b) the image of the unit ball of X is bounded in !D and hence the operator A is bounded, and by Theorem 8.1 it is continuous. Thus, the next result holds. THEOREM 10.6. A linear operator A: X -+ !D is completely continuous if and only if the image of any bounded set in D(A) is precompact in !D. THEOREM 10.7. The range of a completely continuous operator is separable.
An operator A: X -+ !D is called very continuous if it maps any weak convergent sequence in X into a norm convergent sequence in !D. THEOREM 10.S. Any completely continuous operator is very continuous. If X is
reflexive, the converse is also true. If X is the conjugate space of a certain normed
Hilbert spaces
31
space then the completely continuity of the operator A is equivalent to the fact that A maps any ultra weakly convergent sequence into a norm convergent sequence. Note that in a finite dimensional space X any linear operator is completely continuous. A linear operator in a Banach space is called finite dimensional if its range is a finite dimensional space. Obviously, any bounded finite dimensional linear operator is completely continuous. We conclude this section with some conditions for metrizability of weak and ultraweak topologies.
Any bounded subset of a normed space is metrizable in the weak topology. If the space is the conjugate of a normed space, the same is true for the ultra weak topology.
THEOREM 10.9.
The weak and the ultraweak convergence in any bounded set of a normed space can be characterized in terms of sequences. COROLLARY 10.1.
§11. Hilbert spaces 1. Let Sj be a linear space. A functional (.,.) defined on the product Sj x Sj is
called a scalar product (or an inner product) if it satisfies the following conditions (x, y, z E Sj, A ElK):
a) (x,y)
=
(y,x) (the bar denotes the complex conjugation);
b) (x+y,z) = (x,z) + (y,z); c) (AX, y) = A(X, y); d) (x, x);;:: 0; (x,x) = 0 if and only if X = o. A space Sj with a scalar product is called a pre-Hilbert space. Using the scalar product, it is possible to define a norm putting Ilxll = (x,x)!. If Sj is complete with respect to this norm, then it is called a Hilbert space. EXAMPLES. 11.1. In the space
en the formula n
(x, y)
=
L
XiYi'
i=l
for x
= (Xl,oo.,X n )
(t IXiI2)
1
'2,
and y
= (Yl,oo.,Yn),
defines a scalar product. Then
Ilxll =
i.e., the norm given by this scalar product is the norm (1) in Example
PRELIMINARIES
32 7.1. The space
en with this scalar product is a Hilbert space.
11.2. In the linear space of all real or complex sequences x =
(Xl,""X n , ... ) for
00
which
2: IXi 12 is finite, with the linear operations defined coordinatewise, consider i=1
the scalar product 00
(x,y) = 2:XiYi, i=1
where y
= (Yl, ... , Yn, ... ).
We obtain a Hilbert space, called the space £2.
11.3. The space L 2 [a, b] (the case p
=
2 in Example 7.3) is a Hilbert space with
respect to the scalar product
J b
(x, y)
=
x(t)y(t)dt.
a
2. The sets VJt and 91 in a Hilbert space 5) are called orthogonal if (x, y)
= 0 for all
x E VJt and y E 91. The set VJt..L of all vectors in 5) orthogonal to VJt is called the orthogonal complement of the set VJt in 5). We put VJt..L..L = (VJt..L)..L. As above (see Theorem 9.2), the next result holds: THEOREM 11.1. Let 5) be a Hilbert space. For any VJt ~ 5), its orthogonal comple-
ment VJt..L is a linear subspace of 5) and VJt..L..L
=
din VJt.
The direct sum of a subspace £ of a Hilbert space 5) with its orthogonal complement £..L is called an orthogonal direct sum and is denoted by £ EB £..L . THEOREM 11.2. If 5) is a Hilbert space and £ is an arbitrary subspace of 5), then 5)
=£
EB £..L. The decomposition 5) = £EB£..L produces a pair of complementary projections
PI, P2 (P2 = I - PI) on the subspaces £ and £..L, respectively. These projections are
called orthogonal projections (since their ranges are orthogonal). 3. Let ~ be an indexing set. A system of vectors {e a
Ia
E ~}
in a Hilbert space 5) is called orthonormal if (ea,eb) = bab, where bab are the Kronecker symbols: bab is 1 if a = band bab is 0 for a -I b. An orthonormal system {e a I a E ~} is called complete if there are no nonzero elements in 5) orthogonal to all vectors of the system,
Hilbert spaces
33
i.e., if x E Sj and (x, ea ) = 0 for all a E I.2l then x = O. For any orthonormal system
{e a I a
E
1.2l} and each x
E Sj
the set of nonzero numbers of the form (x, e a ) is at most
countable and, with a slight change of notation, we can write 00
L
I(x, ei)1 2 ~
Ilx11 2 .
i=l
The last relation is called the Bessel inequality. If the orthonormal system {e a I a E 1.2l} is complete, then we have the relation 00
L
I(x, ei)1 2
=
Ilx11 2 ,
i=l
called the Parceval equality. A complete orthonormal system is called a basis of Sj. If the Hilbert space Sj is separable, then all orthonormal systems are at most countable and if, in addition, Sj is infinite dimensional, then all bases are countable. 4. Theorem 8.6 asserts that all normed spaces of a fixed finite dimension are homeo-
morphic. More is true for Hilbert spaces. THEOREM 11.3.
All Hilbert spaces of a fixed finite dimension are isometrically
linearly isomorphic. All real (complex) infinite dimensional separable Hilbert spaces are isometrically linearly isomorphic with the real (complex) space £2. 5. The next theorem shows that any Hilbert space is isometrically semilinearly iso-
morphic with its conjugate space. THEOREM 11.4 (F. Riesz). Let Sj be a Hilbert space and f E Sj*. Then there exists
a vector xf
E Sj
such that
Ilfll = Ilxfll
and f(x) = (x,xf) for each x
E Sj.
Riesz Theorem implies that any Hilbert space Sj is selfconjugate, i.e., the conjugate space Sj* coincides with Sj up to an isometrically semilinear isomorphism; consequently any Hilbert space is reflexive. 6. If A is a densely defined linear operator in Sj (i.e., D(A) is dense in Sj), then it
is possible to define the conjugate (adjoint) operator A* as in §9.4. So the adjoint
A* verifies (Ax, y) = (x, A*y) for x E D(A) and y E D(A*). If A = A* then A is called selfadjoint; in this case (Ax,y) = (x,Ay) for all X,y E D(A), and D(A) = = D(A*). Selfadjoint operators are very important in various fields of mathematics and in applications. We will discuss them in more detail in Chapter IV.
Chapter 1 Differential calculus in normed spaces §1. The derivate and the differential
of a nonlinear operator Let X and
~
be normed spaces over the field lK (of complex or real numbers).
We consider an operator F defined on an open set ::D <;;; X and with values in
(D(F) =::D, R(F) <;;;
~).
Accordingly, we write F:::D
----> ~
~
(see Chapter 0, §1).
Let Xo be an element in ::D. DEFINITION 1.1. Suppose that for every hEX the limit
lim F(xo
+ th) - F(xo) = 8F(xo, h)
t->O
t
(1.1 )
exists, where t E lK is chosen such that Xo +th E ::D. Then the limit 8F(xo, h) is called the first variation of the operator F at the point Xo E ::D, and the operator F is said to be 8-differentiable at the point Xo. Suppose that 8F(xo, h)
=
Ah, where A is a continuous linear operator defined everywhere on X and with values in ~ (A E L(X, ~)). Then the operator F is said to be Gateaux differentiable (G-differentiable) at Xo. The operator A above is called the Gateaux derivative of F at Xo, and is denoted by A = F'(xo). The variation 8F(xo, h) is called then the Gateaux differential and is denoted by DF(xo, h). DEFINITION 1.2.
DIFFERENTIAL CALCULUS IN NORMED SPACES
36
ZJ) g(xo, ZJ)
The next example shows that the class V(xo, variation at Xo is strictly larger than the class
of operators having the first of Gateaux differentiable at
Xo operators. EXAMPLE 1.1. Consider the operator
F: ffi.2
----+
ffi. given by
= (hi, h2) -I- 0, we have
Obviously, for each h E ffi.2, h
F(thl' th2) - F(O, 0) t
i.e., 8F(0, h) exists, but it is given by a nonlinear operator (8F(0, h)
= Fh)
and there-
fore F is not Gateaux differentiable. The class F(xo, ZJ) of so-called F-differentiable operators is even smaller. DEFINITION 1.3. An operator F: 1) ----+
ZJ
is called Frechet differentiable (F-differen-
tiable) at a point Xo if there exists an operator A E L(X, ZJ) such that for all h with
Xo
+ h E 1) we have F(xo
+ h) -
F(xo) = Ah + w(xo, h),
(1.2)
where w(xo, h) satisfies the condition lim IIhll--->O
°
Ilw(xo,h)11 = Ilhll .
(1.3)
In this case the expression Ah is called the Frechet differential of F at Xo and is denoted by Ah
= dF(xo, h).
Clearly, an operator which is Frechet differentiable at Xo is continuous at this point and also Gateaux differentiable. Generally speaking, the converse is not true. EXAMPLE 1.2. Let F: ffi.2 ----+ ffi. be the operator defined by the equalities
{
F(xl, X2) = 1 if X2 = xI, Xl F(Xl,X2) = 0 otherwise,
-I-
°
37
Derivative and differential
This operator is not continuous at (0,0) and therefore is not Fhkhet differentiable. However, for any hE ]R2, h
(h 1 ,h 2 ), we have
=
for sufficiently small t. Consequently, the first variation 8F(0, h) exists; this is the Gateaux differential of F at 0. So, in general, the inclusions F(xo,!D) However, the next result is true.
~
9(xo,!D)
~
V(xo,!D) are strict.
Let F: 1) ~ !D be an operator which is Gateaux differentiable at any point of a neighborhood II of the point Xo E 1), and such that its Gateaux derivatives are the values of an operator-valued function THEOREM 1.1.
F': II ~ L(X, !D) which is continuous on ll. Then F
E
F(xo, !D) and DF(xo, h) = dF(xo, h).
<J From hypotheses, the limit (1.1) exists and
Choose h such that Xo
+ th E II for all t
w(xo, h) = F(xo For each y* E !D* -
E
8(xo, h) = F'(xo)h = DF(xo, h).
[0, 1] and consider the expression
+ h) -
F(xo) - DF(xo, h).
the conjugate space of!D (see Chapter 0, §4) -
(w(xo, h), y*) = (F(xo
+ h) -
we have:
F(xo), y*) - (DF(xo, h), y*).
Consider now the real-valued function
'P(t) = Re(F(xo
+ th), y*).
We show that 'I' is differentiable on (0,1). Indeed, 'I'
'(t) = lim 'P(t + 6.t) - 'P(t) = 6.t
Llt_O
= lim Re / F(xo + th + 6.th) - F(xo + th) ~~
=
\
Re(DF «xo
~
,y
*) =
+ th), h), y*).
Applying Lagrange formula to the function 'P(t) we obtain
IRe(w(xo, h), Y*)I = 1'1'(1) - cp(O) - Re(DF(xo, h), Y*)I =
DIFFERENTIAL CALCULUS IN NORMED SPACES
38
= Icp'(T) - Re(DF(xo, h), Y*)I = IRe(DF(xo
~ IIDF(xo
where
T
+ Th, h) -
DF(xo, h)lllly*11 ~
+ Th)
- DF(xo, h), Y*)I ~
IIF'(xo + Th)
Ily*ll,
- F'(xo)llllhll
E [0,1]. By Hahn-Banach-Suchomlynoff Theorem (see Chapter 0, §8) we may choose
the functional y* such that (w(xo,h),y*)
= Ilw(xo,h)11 and Ily*11 = 1. Then
Hence, using also the continuity of F' on 11, it follows that the equalities (1.2) and (1.3) hold. Therefore, the continuous Gateaux differentiability on a neighborhood 11
<:;;;
:D
implies Fnkhet differentiability on the same neighborhood, and, moreover, Gateaux and Fnkhet derivatives coincide on 11. • THEOREM 1.2.
in h (llhll
Let F be an operator such that the equality (1.1) holds uniformly
= 1) and 8F(xo, h) = DF(xo, h). Then F
E F(xo, ~).
<J Under our assumptions, for all h with Ilhll = 1 and for each
exists 8 >
°such that the inequality
II ~ (F(xo + th) -
°
holds for any < It I < 8. For each hI with
F(xo)) - DF(xo, h)11 <
IlhI11 < 8 there exists
and Ilhili = t. We have Ilw(xo,hdllllhIII- I < was arbitrary. •
an h with Ilhll E,
E
> 0, there
E
th which is equivalent to (1,3), since E =
1 such that hI
=
REMARK 1.1. Note that in Theorem 1.2, in contradistinction with Theorem 1.1, the
condition of Gateaux differentiability on a neighborhood of Xo is not required. REMARK 1.2. In the case of X being a complex space, the notion of first variation can be slightly relaxed, asking the existence of the limit in (1.1) only for real values of t. Under the conditions of Theorems 1.1 or 1.2, this is enough for obtaining the existence of the limit (1.1) for any complex t. However, in the general case this problem is still open.
Lagrange formula and Lipschitz condition
39
§2. Lagrange formula and Lipschitz condition Let X and
~
be normed spaces, 1) a set in X and F an operator from 1) into
~.
F satisfies the Lipschitz condition (with the constant we have
DEFINITION 2.1. The operator
e) on
1),
if for every Xl, X2 E 1)
(2.1)
Let 1) be a convex set in X and F: 1) ---> ~ an operator which is Gateaux differentiable on 1) with IIF'(x)11 ~ e for all X E 1). Then F satisfies the Lipschitz condition (2.1).
THEOREM 2.1.
and 0 ~ t
Since 1) is convex, it follows that tXI + (1 - t)X2 E 1) for any Xl, X2 E 1) ~
1. Therefore, the scalar function
'P(t) = (F(tXI + (1 - t)X2), y*) is well-defined on [0,1] and .continuous for any y* E ~*. We show that 'I' is differentiable on (0,1). Indeed
'P(t
+ ~t) - 'P(t) ~t
= /
+ (1 - t)X2 + ~t(XI - X2)) - F(txI + (1 - t)X2)
F(txI
~t
\
,y
*)
.
Therefore 'P'(t) exists and 'P'(t) = (F'(tXI + (1 - t)X2)(XI - X2), y*). By the mean-value theorem we have:
11'1'(1) - '1'(0)11 ~
sup
1'P'(t)l,
O~t~l
or, equivalently,
~
sup
1
I(F(xd - F(X2),y*)1 ~ (F'(tXI + (1 - t)X2)(XI - X2), Y*)I
~
O~t~l
~
sup O~t~l
1lF'(txI + (1 - t)x2)llllxl - x211 Ily*ll.
Hence, using also Hahn-Banach-Suchomlynoff Theorem we obtain the so-called Lagrange formula:
IIF(xd - F(X2)11:(
sup O~t~l
1IF'(tXI
+ (1 - t)x2)lIllxl - x211,
40
DIFFERENTIAL CALCULUS IN NORMED SPACES
which, together with
IIF'(x)11 ~ £for
all x
E 1),
leads to (2.1). ~
REMARK 2.1. Theorem 2.1 remains true if the condition of Gateaux differentiability
is replaced with the uniform boundedness of the first variation:
118F(x, h)11 ~ £llhll,
hEX.
We close this section with the definition of the most general notion of differentiability needed in this book. DEFINITION 2.2. An operator F: 1) ~ !l) is said to be weakly differentiable at Xo E 1)
iffor any y* E !l)* and any hEX, the scalar-valued function i.ph(t) is differentiable (in the usual sense) at t there exists an element y( xo, h)
E
=
= (F(xo + th), y*)
O. When, in addition, for a given h
!l) such that i.p~ (0) = (y( xo, h), y*), then y( xo, h) is
called the weak variation of the function F at Xo along direction h, and is denoted by
8wF(xo, h). The class of operators having weak variation along all directions hEX is denoted by Vw(xo, !l)). Further, if 8wF(xo, h) = Ah where A E L(X, !l)), then the operator F is called weakly Gateaux differentiable; the set of these operators F is denoted by Qw(xo, !l)). If, moreover, lim i.ph(t) - i.ph(O)
t
t-+O
= (Ah
,y
*)
is uniform relatively to all h with Ilhll = 1, then the operator F is called weakly Fhkhet differentiable and the set of these operators is denoted by Fw(xo, !l)). The following inclusions are obvious:
V(xo,!l)) <;: Vw(xo,!l))
(2.2)
Q(xo,!l)) <;: Qw(xo,!l))
(2.3)
F(xo,!l)) <;: Fw(xo, !l)).
(2.4)
(Let us remark that in the literature weakly differentiable operators are often called Gateaux differentiable -
see, for example, Vainberg [1], [2]. In our opinion, the
terminology from Definition 2.2 seems to be more natural and appropriate. A reason for this is that it is based on the notion of weak convergence in Banach spaces. More than this, we will use below the well-known notion of weak analiticity, which is strongly related to the notion of weak differentiabily and which is also based on the notion of weak convergence.)
Fh§chet differentiable operators
41
REMARK 2.2. If the space ~ is finite dimensional, all relations (2.2)-(2.4) become
equalities, because the strong and weak convergences are equivalent. We will see below in Chapter III that this is also the case when X and
~
are infinite dimensional
if we assume, in addition, that X is a complex space and the weak variation exists on a neighborhood of Xo. REMARK 2.3. In fact, the proof of Theorem 2.1 uses a weaker hypothesis. The same
arguments lead to the next more general result:
Let F be an operator defined on the convex set :D such that the weak variation 8w F(x, h) exists at any x
E
:D and, for all hEX, we have
Then the conclusion of Theorem 2.1 is true.
§3. Examples of Frechet differentiable operators EXAMPLE 3.1 (Urysohn operator). Let K(t, s, x) be a function defined on a ~ t, s ~ ~
b,
Ixl ~ r, such that K(t, s, x) and K~(t, s, x) are everywhere continuous.
Then the
integral operator F given by
J b
Fx(t)
=
K(t, s, x(s))ds
(3.1)
a
Ilxll
r of the space e[a, b] of all real-valued continuous Ilxll = sup Ix(t)l. The operator F takes values a';;; t,;;; b in e[a, b] and is Fn3chet differentiable on the open ball Ilxll < r; moreover, for any Xo E e[a, b] with Ilxo I < r and for every h E e[a, b] we have: is well-defined on the ball
~
functions on [a, b] with the norm
b
(F'(xo)h)(t)
=
JK~(t,s,xo(s))h(s)ds.
(3.2)
a
Indeed, first of all notice that the operator defined by (3.2) is linear and contin-
DIFFERENTIAL CALCULUS IN NORMED SPACES
42
uous on
era, b].
Further
+ h) -
F(xo) - F'(xo)hll
+ h(s)) -
K(t, s, xo(s)) -
IIF(xo
J
~
b
~
a
IK(t, s,xo(s)
sup
a~t~b
K~(t, s,xo(s))h(s)1 ds ~
b
~ IlhllJ a
for a certain e
= e(h)
sup a~t~b
IK~(t, s,xo(s) + eh(s)) - K~(t, s, xo(s))1 ds
E [0,1]. Hence, the continuity of the function K~(t,
s, x) in the
variable x implies that lim
IIhll~o
J
Ilh11111F(XO + th) -
F(xo) - F'(xo)hll
~
b
~
lim
Ilhll~O
a
sup
a~t~b
IK~(t, s,xo(s) + eh(s)) - K~(t, s, xo(s))1 ds = o.
Note that the Urysohn operator (3.1) can be also considered on the space
O[a, b] of all complex-valued continuos functions; then F is Frechet differentiable on O[a, b] if the kernel K(t, s, x) has continuous partial derivatives with respect to the variable x in the disk
Ixl < r of the complex plane.
EXAMPLE 3.2 (Hammerstein operator). A particular case of the operator (3.1) is
the Hammerstein operator defined by the equality b
Fx(t) =
J
K(t, s)f(s, x(s))ds
(3.3)
a
where the functions K(t,s), f(s,x) and
f~(s,x)
are continuous with respect to all
variables for a ~ t, s ~ b, Ixl ~ r. Example 3.1 implies that the Frechet derivative of F is given by
J b
(F'(xo)h) (t) =
K(t, s)f~(s, xo(s))h(s)ds.
(3.4)
a
EXAMPLE 3.3 (Nemytzki operator). Suppose that the functions
are continuous with respect to all variables for a ~ s is the superposition operator
~
b,
Ixl ~ r.
f(s, x) and
f~(s,
x)
Nemytzki operator
(3.5)
Lemmas about differentiable operators defined on the ball
Ilxll :( r
43
of the space C[a, bJ (O[a, b]). The operator (3.5) is Frechet
differentiable and
(p'(xo)h) (s) = f~(s, xo(s»h(s).
(3.6)
Indeed, the operator p'(x o ) defined by (3.6) is linear and continuous on the space
C[a, bJ (O[a, b]) and
:( II~II
sup a~8~b
:( II~II :(
1I~IIIIP(xo + h) -
p(xo) - p'(xo)hll :(
l1(s, xo(s) + h(s»
- f(s, xo(s» -
sup
If~(s, xo(s) + ()h(s»
sup
If~(s,xo(s)
a~8~b a~8~b
for
Ilhll
-+
0, (()
=
+ ()h(s»
-
f~(s, xo(s»h(s)1
:(
f~(s, xo(s»llh(s)1 :(
- f~(s,xo(s»I-+
°
()(h) E [0,1]).
EXAMPLE 3.4. Let
ft(Xl,"" x n ), ... , fm(Xl, ... , x n ) be functions defined on a do-
main in the Euclidean space lEn = {x I x = (Xl, ... , x n )} and assume that ft,···, fm and all their first order partial derivatives are continuous. Then, the operator
is Frechet differentiable on the considered domain, and
F'(x)h =
8ft(x) ( 8Xl
:
(3.7)
8fm(x) 8Xl In the right hand side of (3.7) we have the product of the Jacobi matrix of F and the vector h
=
(h l , ... , h n ).
§4. Lemmas about differentiable operators The next lemma follows immediately from the properties of a linear operator.
Let A E L(X,~) be a continuous linear operator. Then for any X E X, the operator A is Frechet differentiable at x (A E F(x, ~») and
LEMMA 4.1.
A'(x)
== A.
DIFFERENTIAL CALCULUS IN NORMED SPACES
44
Let X, ~ and 3 be normed spaces. Suppose that F: X ----t ~ is an operator defined on a neighborhood of a point Xo E X and that G: ~ ----t 3 is defined on a neighborhood of Yo = f(xo) E~. Let T = GF be the composition of F and G. IfF E 9(xo,~) (F(xo, ~)) and G E 9(yo, 3) LEMMA 4.2 (The derivative of a composition operator).
(F(Yo,3)) then T E 9(xo,3) (F(xo,3)) and T'(xo) = G'(Yo)F'(xo).
(4.1)
(The derivatives in (4.1) are understood as Frechet or Gateaux derivatives, depending on the class the operators F and G belong to. The right hand side of (4.1) is the usual product of linear operators.)
hEX and u E
WI(XO, th) = F(xo W2(YO, tu) = G(yo
+ th) + tu)
~,
define
- F(xo) - F'(xo)th, - G(yo) - G'(Yo)tu,
where WI and W2 satisfy the conditions lim IlwI(XO, th)11 = 0 t->O
Further, take u
t
lim Il w 2(YO, tu)11 = '
= F'(xo)h + CIWI(XO, th)
t->O
o.
t
and consider the difference
T(xo + th) - T(xo) = GF(xo + th) - GF(xo) = = G(F(xo) + F'(xo)th + WI(XO, th)) - G(yo) = = G(yO + tu) - G(yo) = W2(YO, tu) + G'(yo)tu = = tG'(yo)F'(xo)h + G'(YO)WI(XO, th) + W2(YO, tu). Consequently lim IIT(xo t->O
+ th) - T(xo)11 = G'(yo)F'(xo)h t
(4.2)
a relation which is equivalent with (4.1), and which proves that T E 9( Xo, 3). When
F E F(xo,~) and G E F(Yo,3), the equality (4.2) holds uniformly with respect to hEX, Ilhll = 1, and therefore (using Theorem 1.2) T E F(xo,3). ~ The previous two lemmas have a simple but important consequence. COROLLARY 4.1.
Let F
E Q(xo,~)
and A
E L(~, 3)
(AF)'(xo) = AF'(xo).
be two operators. Then (4.3)
Partial derivatives
45
The equality (4.3) shows that a linear operator "passes through" the derivation sign. Rewriting (4.3) using the differential symbol, we have
D(AF) = ADF
(4.4)
which means that the differentiation operator commutes with any linear operator. For example, the Hammerstein operator (3.3) can be represented as the composition of the linear operator K, defined by the formula b
Kx(t)
=
J
K(s, t)x(s)ds
a
with the Nemytzki operator (3.5), namely F = Kp. Then, the relations (4.3) and (3.6) imply (3.4).
Let F E g(xo,!D) be an operator and assume that there exists an operator G:!D ----t X defined on a neighborhood U of the point Yo = F(xo), and satisfying the relations (4.6) GF = 1'1), FG = Iu COROLLARY 4.2.
where 1) is a neighborhood of Xo (we write in this case G = F- l ). If F- l then the linear operator F'(xo) has a continuous inverse and
E
g(yO, X),
(4.6)
The equality (4.6) follows immediately from (4.1) and (4.5).
~
REMARK 4.1. For a complex space X, Corollary 4.2 is true under weaker conditions
(see Chapter IV below). The finite dimensional case of this result is a theorem of Osgood (see, for example, B. Shabat [2]). REMARK 4.2. In Chapter IV it will be shown that, for complex spaces, the converse
of Corollary 4.2 is also true: if the operator F'(xo) has a continuous inverse, then the operator F is locally invertible and the equality (4.6) holds.
§5. Partial derivatives In this paragraph we suppose that the space X is decomposed in a direct sum X = Xl -i-x2 (see Chapter 0, §7). Assume that F is an operator defined on a neighborhood
DIFFERENTIAL CALCULUS IN NORMED SPACES
46
of a point a = (aI, a2), ai E Xi, i = 1,2, with values in a normed space ~, such that FE V(a, ~). Choose the vectors h(l) = (hI, 0) and h(2) = (0, h 2), where hi E Xi, i = 1,2. The expressions 8h1 F(a) = 8F(a, h(l)) and 8h2 F(a) = 8F(a, M2)) are called the variations of the operator F at the point a in the directions of the spaces Xi, or the
partial variations of F. In other words
(5.1)
(5.2) Note that both limits exist by Definition 1.1.
(:F(a, ~)), then the expressions (5.1) and (5.2) are linear in hI and h2' respectively, and we call them the partial differentials in the In particular, if F E
9(a,~)
sense of Gateaux (Frechet). In this case we will use the notations
and the expressions F~i (a) E L(Xi' ~), i = 1,2, are called the partial derivatives of the operator F at the point a (in the sense of Gateaux or Frechet, respectively). THEOREM 5.1. If FE 9(a,~) (:F(a, ~)) then the partial derivatives exist and
(5.3) As we have already mentioned, the existence of partial derivatives follows immediately from Definitions (1.1)-(1.3). The same definitions imply (5.3), if we <J
notice that
Xi, i
= 1,2.
F~i (a)
is the restriction of the linear operator F' (a) on the subspace
~
In general, the converse is false: the mapping F in Example 1.1 has partial derivatives with respect to both Xl and X2 - namely F~l (O)hl == 0 and F~2 (0)h2 == 0 - , but this mapping does not have a Gateaux derivative. However, the next result is true. THEOREM 5.2. Let F: X
-+ ~ be an operator defined on a neighborhood 1) of a point a E X and X = XI-tX 2 • Assume that F has partial Gateaux derivatives at every point of 1), and that these derivatives are operator-valued mappings, continuous on
47
Multilinear operators. Duality. Homogeneous forms ~.
Then F E F(a,~) and the equality (5.3) holds, hence F'(x) is also a continuous mapping from ~ into L(x, ~). (a) and F~2 (a) are the partial Frechet derivatives of F at a. Consider an arbitrary vector h = (hi, h 2), such that a + h E ~. Then
By Theorem 1.1, the operators
F~,
F(a+h)-F(a)
=
= F(al + hi, a2 + h2) - F(al + hi, a2) + F(al + hi, a2) - F(al' a2) = = F~2 (al
where
W2
+ hi, a2)h2 + w2(al + hi, a2, h2) + F~, (ai, a2)h l + Wi (ai, a2, hi)
= o(llh 2 11),
Since
Wi
F~2: ~ ----+
= o(llhd)· L( X, ~) is continuous in the uniform operator topology, we
have
Then
where
W3
= o(llhll). Finally, we obtain
where w(a, h) ately.
=
o(llhll). From this, all the assertions of the theorem follow immedi-
~
REMARK 5.1. Theorem 5.2 is true in the case of a complex space
x under a weaker
hypothesis: it is sufficient to require the existence of the partial Frechet derivatives on a neighborhood of a point a, without asking their continuity in the operator topology. (The finite dimensional case of this result is a theorem of Hartogs - see, for instance, B. Shabat [2].) REMARK 5.2. The conclusions of Theorems 5.1 and 5.2 can be immediately extended
by induction to the case when x is a topological direct sum of n normed spaces Xl, ... , x n . In this case, formula (5.3) becomes
F'(a)h
=
F~, (a)h l
+ ... +
This is called the formula of complete differential.
F~n (a)h n
.
(5.4)
DIFFERENTIAL CALCULUS IN NORMED SPACES
48
§6. Multilinear operators. Duality. Homogeneous forms Consider the space X = Xl + ... +Xn' where Xi are normed spaces, i = 1, ... , n. DEFINITION 6.1.
An operator G: X
----* ~,
everywhere defined on X, is called a
multilinear (n-linear) continuous operator, if G is linear in each of the variables Xi E Xi, i = 1, ... , n - by fixing the others - , and if G is continuous on X.
E
X ----* ~ is linear in each of the variables Xi E Xi, i = 1, ... , n - by fixing the others -, then G is a multilinear continuous operator if and only if (6.1)
THEOREM 6.1. If an operator G:
for all
X
= (Xl, ... ,xn ), where 0 ~ M < 00.
Let G be a multilinear continuous operator. Then for all >'1, ... ,An > 0 we
have
(6.2) and there exists r > 0 such that IIG(Yl,"" Yn)11 ~ 1 if IIYil1 ~ r, for all i = 1, ... , n. For Xi E Xi, choose Ai > 0 such that IIAilxil1 = r. From (6.2) we infer (6.1) with M=r- n .
Conversely, suppose that (6.1) is fulfilled with a nonnegative constant M <
00.
We will prove the continuity of the operator G at an arbitrary point a = (al,"" an) E
EX. For any
X
= (Xl, ... ,xn ), we have
G(X) - G(a) = G(Xl,"" Xn) - G(al,"" an) = = G(Xl,'" ,Xn ) - G(al,X2,'" ,Xn ) + G(al,X2,'" ,Xn)-G(al' a2, X3,"" Xn) + G(al' a2, X3,"" Xn) - ... '" - G(al,"" an-l, Xn) + G(al, ... , an-l, Xn) - G(al,"" an) = G(Xl - al,X2,'" ,Xn ) + G(al'X2 - a2,X3, ... ,Xn ) + ... ... + G(al,"" an-l, Xn - an).
(6.3) =
Hence, we obtain the inequality
IIG(x) - G(a)11 ~ M {llxl - alll·llx211· '" '1lxnll + Iialil ·llx2 - a211·llx311· ... ... . IIXnl1 + ... + Iialil . '" . Iian-lil . IIXn - anll}· The boundedness of the product Ilxlll· ... 'lIx n II, for X close to a, implies the continuity of Gat a. ~
Multilinear operators. Duality. Homogeneous forms
49
REMARK 6.1. The proof of Theorem 6.1 implies that the inequality (6.1) holds if
one requires only the continuity of G at the point (0, ... ,0)
E
X. Therefore, the
continuity of a multilinear operator at the origin implies its continuity on X. REMARK 6.2. The equality (6.3) shows that a multilinear operator, which is sepa-
rately continuous in each of its variables, is continuous on X. DEFINITION 6.2. If G is a multilinear continuous operator, then the smallest number
M for which the inequality (6.1) is fulfilled is called the norm of G. Thus
IIGII
=
sup IIG(x1,oo.,xn)ll· Ilxill:::;:l, 1 :(i:(n
(6.4)
The norm introduced by (6.4) defines a structure of normed space on the set
L n (X 1, ... , Xn;~) of all multilinear continuous operators from X 1-+- ... -+-X n into ~. Let us consider in more detail the space L2 (X 1, X 2; ~) of bilinear continuous operators from Xl-+- X 2 into ~. Take G E L(X1' X2;~) and fix a point Xl E Xl. Definition 6.1 produces a linear continuous operator in the second variable X2, acting from X 2 into ~, that is, G x , = G(X1' .) is an element in L(X2' ~). We clearly have sup IIGx,ll:( Ilx,ll :::;: 1
IIGII.
(6.5)
Taking into account the linearity of G in Xl, the operator G can be viewed as
Xl into G X ! E L(X2' ~), i.e., GEL (Xl, L(X2' ~)). This inequality and (6.5) show that the spaces L(X1' X2;
a linear operator which maps Xl
sup IIGx,ll. IlxIil :::;: 1 ~) and L(X1,L(X2'~)) are linearly isometric.
Then
I Gil :(
E
Similar results are true for n
~
2.
Let 1 :( p :( n -1 be fixed. The spaces Ln (X 1, ... , Xn; ~) and Lp(X1' ... ,Xp; L n- p(Xp+1, ... ,Xn);~)) are linearly isometric; their linear isometry is defined by the equalities THEOREM 6.2 (on duality).
Due to the symmetric role of the spaces Xi, i for any arbitrary rearrangements of the indexes.
=
1, ...
,n, a similar result holds
DIFFERENTIAL CALCULUS IN NORMED SPACES
50
We consider now the particular case Xl
= X2 = ... = Xn = X. The space of
all multilinear continuous operators will be denoted in this case by Ln(X; !D). DEFINITION 6.3. An operator H: X
----t
!D
is called a homogeneous form of order n
if there exists an operator G E Ln(X;!D) such that
H(x) = G(x,x, ... ,x).
(6.6)
We state without proof the next results (see H. Cart an [1]). THEOREM 6.3. Let H: X
----t
!D
be a homogeneous form of order n. Then there exists
a unique operator G E Ln(X;!D) which is symmetric relatively to permutations of the
variables and which satisfies (6.6). Such an operator G is called the polar form of the operator H and is denoted by
H.
THEOREM 6.4. Suppose that the operator H is defined by (6.6), where G acts from
+...
Xl +Xn into!D and is linear in each of its variables. Then H is continuous if and only if its polar form H is continuous. The following two relations are basic properties of a homogeneous form H:
H(>.x) = >.n H(x),
x
E
X,
(6.7)
for each >. E lK, and
IIH(x)11 ~ IIHllllxlln,
x
E
X.
(6.8)
It is easy to see that IIHII is the smallest constant for which (6.8) holds. Thus, the number IIHII is called the norm of the operator H. The set of homogeneous forms of order n will be denoted by Ln(x; !D). Clearly, Ln(X;!D) is a linear normed space relatively to the norm IIHII.
§7. Higher order derivatives A way of defining the derivatives and differentials of higher orders is the following. DEFINITION 7.1. Let F: X ----t !D be an operator and suppose that the first variation of(x, hi) exists in a neighborhood of a point x, for any element hi EX. If for each
Higher order derivatives
51
fixed hI and h2 the limit (7.1) exists in the space Z), then it is called the second variation of the operator F and is denoted by 82F(x, hI, h2)' The n-th variations are defined by induction. Namely, assume that 8n -dx, hI,
... , hn-d exists in a neighborhood of a point x for any hI, h 2, ... , h n - I E X. If for each fixed hI, h2' ... ,hn the limit (7.2)
exists in the space Z), then it is called the n-th variation of the operator F and is denoted by 8n (x, hI"'" hn ). Suppose now that the operator F is Gateaux differentiable in a neighborhood 1)
of a point x. Then, the variation 8F can be considered (on this neighborhood) as
an operator acting from X into L(X, Z)). Indeed, the equality
8F(x, h) = A(x)h maps x E
1)
into the operator A(x) = F'(x) E L(X, Z)). If this operator is Gateaux
differentiable in Xa, that is, if the limit lim A(x + th 2) - A(x)
t-a
t
=
B(x)h 2
(7.3)
exists, then, obviously, the value of B(x)h 2 at an element hI is exactly the second variation 82F(x, hI, h 2) defined by (7.1). Thus, in this case, 82F can be considered as an element of the space L (X, L(X, Z))), or, according to Theorem 6.2, as an element of the space L 2 (X; Z)), i.e., as a bilinear operator (relatively to hI and h 2 ) acting from
x-i-x into Z). This operator is called the second Gateaux derivative and its value at an element (hI, h 2 ) E X+X is called the second Gateaux differential and is denoted by D2 F(xa, hI, h2) (= 82F(xa, hI, h2))' The n-th Gateaux derivative and the n-th Gateaux differential of the operator F are defined by induction, using the reccurent relations
DIFFERENTIAL CALCULUS IN NORMED SPACES
52
The n-th Frechet derivative and the n-th Frechet differential of the operator
F are defined using Definition 1.3. Sometimes, it is more convenient to use Theorem
1.2. Thus, if for a Gateaux differentiable operator F the equality (7.3) is uniformly fulfilled for h2 with IIh211
=
1, where B(x) E L(X,L(X,~)), then the operator F is
said to be twice Frechet differentiable.
(n + 1)-th variations of an operator F exist at a point Xo E E X and at each point of an open interval]xo, Xo + h[, such that
for all x E ]xo, Xo + h[, Ilhlll = ... = IIhn+III = 1, where M < 00. Consider the function 'P(t) = (F(xo + th), yO) defined on [0,1) where y* is a fixed, arbitrary element of~*. The function 'P is differentiable at each point of [0, 1), and
'P'(t)
=
lim /,;
I....l.t
Ll.t--+O \
(F(Xo+th+~th)-F(Xo+th)),y*)= = (8F(xo + th, h), yO).
Obviously, the second derivative also exists
"(t) 'P
=
lim 'P'(t
+ ~t)
~t
Ll.t--+O
- 'P'(t) =
lim ((8F(xo+th+~th,h)-8F(xo+th,h)),y*)
=
Ll.t--+O
= (8 2 F(xo
~t
+ th, h, h), yO).
Analogously, all the derivatives 'P(k) (t), k = 3, ... ,n + 1 exist on the interval
[0,1) and 1'P(n+l) (t)1 ~ MIIY*II. Using Taylor formula for 'P, we have
'P(t) = 'P(O) where
T
1 1 + 'P' (0) + -'P" (0)t 2 + ... + -'P(n) (O)tn +
2!
n!
1
(n+1)!
'P(n+1) (T)t n+1
'
E [0, t]. The last equality is equivalent to
F(xo
+ th)
=
F(xo)
1
+ 8F(xo, h)t + ,8 2 F(xo, h, h)t 2 + ... 2.
(7.4)
... + ~8nF(Xo, h, ... , h)t n + ( 1 )18n+I F(xo + Th, h, ... , h)tn+l. n. n+ 1 .
Relation (7.4) is called the Taylor formula for the operator F. In particular, if the operator F in Gateaux or Frechet differentiable up to the order n taking t
+ 1, then,
= 1 in (7.4) and denoting ~! 8kF(xo, h, ... , h) = Ak(xo, h), we have
F(xo
+ h) =
F(xo)
+ Al (xo, h) + A 2 (xo, h) + ... + An(xo, h) + w(xo, h)
(7.5)
Higher order derivatives
53
where Ai(x, .) are homogeneous forms of order i, 1:::;; i:::;; n, and (7.6) Note also that when the operator F is Fnkhet differentiable up to the order n
+ 1 on
a neighborhood of the point xo, then the representation (7.5) is uniformly
valid in h, Ilhll:::;; E, and the number M in (7.6) does not depend on h. In this case, the operator F is called Taylor differentiable and the expression Ai(XO, h) is called the Taylor derivative of order i, 1:::;; i :::;; n. The converse assertion is also true. If the operator F is differentiable up to the order n
+ 1 on the interval
[xo, Xo
+ h]
and admits the representations (7.5)-(7.6),
where Ai(xo, h) are homogeneous forms of order i, 1:::;; i:::;; n, then (7.7) where, of course,
;C is the polar form of the operator Ai,
1:::;; i :::;; n.
A proof of this assertion is contained, for example, in H. Cart an [1]. REMARK 7.1. The existence of the succesive derivatives up to the order n
+ 1 is an
essential assumption in the previous assertion. Here is an example. Consider the operator F defined by F(x)=xn+l sinx- n -
1,
x
E
IR, xi=- 0, and F(O)=O. The operator
F is Taylor differentiable at the point 0 up to order n, i.e., F admits the representa-
tion (7.5)-(7.6) (with Ai(xo, h) = 0, 1:::;; i:::;; n, and w(O, h) = h n+1 sinh- n -
1)
but F
does not have even the first order derivative at O. DEFINITION 7.2. An operator
F: X
----+
q) defined in a neighborhood of a point Xo
is said to be 6-analytic at this point if for any hEX there exists a number r(xo, h) such that the representation 00
F(xo
+ th) =
F(xo)
+L
Bn(xo, h)tn
n=l
holds for all t E lK with It I < r(xo, h), where the series in the right-hand side is normally convergent, i.e., 00
L
IIBn(xo,h)llltl n <
00.
(7.8)
n=l
The operator F is called (I-analytic (or Gateaux analytic) if the expressions Bn(xo, h) are represented as the values of some homogeneous forms from X into q), evaluated at hEX.
54
DIFFERENTIAL CALCULUS IN NORMED SPACES Let p(xo, h) be the convergence radius of the power series appearing in the
inequality (7.8). By Cauchy-Hadamard formula we have
p(xo, h)
1
(7.9)
= -==--r=====
lim Y/IIBn(xo,
n-+oo
h)11
The operator F is called F-analytic (or, simply, analytic) in the point Xo, if it is Q-analytic and for all hEX with
Ilhll = 1 there exists p > 0 such that
p(xo, h)
~
p,
where p(xo, h) is defined by (7.9). Obviously, if an operator F is Q-analytic and, a = n-+oo lim
then F is analytic and for all h with
Y/IIBnll < 00,
Ilhll < a-I = r
we have the representation
00
F(xo
+ h) = F(xo) + L
Bn(xo, h),
(7.10)
n=I
where the series in the right-hand side is uniformly convergent for all h with
Ilhll ( r <
< r. (Here IIBnl1 is taken according to §6.) We denote the set of all 8-analytic (respectively Q-analytic or analytic) by 'H6(XO,~) (respectively 'Hg(xo,~) or 'H(xo, ~)). Using the notations in §1, we have 'H6(XO,~) ~ V(xo,~) 'Hg(xo,~) ~ Q(xo,~) 'H(xo,~) ~
F(xo, ~).
For complex spaces all these inclusions are, in fact, equalities (see Chapter III below). For real spaces, these inclusions are strict. In this chapter we constantly pointed out the special role of complex spaces in the theory of differentiability. The properties of differentiable operators on complex spaces - mentioned in some remarks - lead to far-reaching consequences. For instance, some local features of the solutions of operatorial equations in complex spaces are global. However, the methods of differentiation are not enough for proving these results; it is necessary to combine them with methods from integral calculus.
Complete continuity
55
§8. Complete continuity of operators and
of their derivatives This section will be devoted to the following question: how does the property of complete continuity of a differentiable operator influence the properties of its derivatives? First, we need the following result.
Let X and ~ be Banach spaces. A linear operator A: X ---of ~ is completely continuous on X if and only if it is completely continuous on a bounded ball ti S;;; X centered at the origin.
LEMMA 8.1.
Necessity follows from the definitions (see Chapter 0, §§6 and 10). For
sufficiency, let::D be an arbitrary bounded set in X. Choose a sequence {Yn}nEl':I S;;; A::D, i.e., Yn = AX n where {Xn}nEl':I S;;;::D. Since the ball ti is centered at the origin, it absorbs ::D, i.e., there exists a number A > 0 such that >.i1 :2::D. This means that the sequence {X~}nEl':I' where x~ y~
=
= A-IXn , lies in ti, = AY~, and
Ax~, is pre compact in~. But Yn
precompact.
so the sequence {Y~}nEl':I' where so the sequence {Yn}nEl':I is also
~
EXAMPLE 8.1. Let X = ~ =
era, b]. Consider the linear integral operator A: X
---of
~
defined by b
J
Ax(t) =
K(t, s)x(s)ds
(8.1)
a
where K(t, s) is a complex-valued function, continuous for (t, s) E il = [a, b] x [a, b] (see also (3.3)). The operator A is completely continuous. Indeed, A is clearly continuous. According to Arzela-Ascolli theorem and to Lemma 8.1, for proving the precompactness of A it is sufficient to show the equicontinuity of any function y(t) E AlB, where lB is the unit ball in X. We have
J b
Iy(td - y(t2)1
~
IK(tl' s) - K(t2' s)llx(s)lds.
a
Since the function K(t, s) is uniformly continous on il, then for each c > 0 there exists
8 = 8(c) such that IK(tl' s) - K(t2' s)1 < c for any s E [a, b] and any tl, t2 with It I - t21 < 8. But Y = Ax with x E lB, so taking into account that Ix(s)1 ~ Ilxll = max Ix(s)1 ~ 1, [a,b]
E
[a, b]
DIFFERENTIAL CALCULUS IN NORMED SPACES
56
we obtain
ifltt-t21<8.
A defined by the formula (8.1) on the space X = .L 2 [a, b]. It can be shown that, in this case, if K(t, s) is continuous on D = [a, b] x x [a, b] then the operator A is completely continuous. (Actually, it is sufficient to assume that the kernel K(t, s) is a so-called Hilbert-Schmidt kernel, that is, EXAMPLE 8.2. Consider the operator
b
b
JJ
IK(t, s)1 2dsdt <
a
00,
a
see, for instance, M. A. Krasnoselskii [4].) The proof follows from similar arguments as above and from Cauchy-Buniakovski inequality: b
Iy(td - y(t2)1
~
J
IK(tl' s) - K(t2' s)llx(s)lds
~
a
,; (!IK('" ,) -K('" ')1' ds)'" (!I*)I'dS)'" ,;
£(b - a).
Using Holder inequality we can show that the operator (8.1) with a continuous kernel
K(t, s) is completely continuous on each .Lp[a, b] and, moreover, that it is completely continous as an operator from .Lp[a, b] into O[a, b]. The following result is obvious:
X,!D, 3 be topological spaces and let F: X operators defined and contino us on the domains ~ ~ X, II
LEMMA 8.2. Let
---+
~
!D and G:!D ---+ 3 be !D respectively, with
F(~) ~ ll. If one of
G F: X
---+
the operators F or G is compact, then the composed operator !D is completely continuous.
This property gives easy proofs for the complete continuity of many operators. EXAMPLE 8.3. Let
X = !D = O[a, b]. Consider the Hammerstein operator F: X
f
b
Fx(t) =
a
K(t, 8)f(s, x(8))ds,
---+
!D,
57
Complete continuity
where K(t,s) is continuous on the square on the whole domain of the variables a
~
s
n = [a,b] ~
x [a,b] and f(s,x) is continuous
b and x E C,
F is completely continuous on the ball ;D = {x EX:
Ixl ~ r.
IIxil ~ r}.
Then the operator
Indeed, it is sufficient
to notice that the operator F is the composition of the superposition operator f(s, .) and of the linear completely continuous operator A given by b
Au(t)
=
J
K(t, s)u(s)ds.
a
REMARK 8.2. The superposition operator acting on the space
O[a, b] is not com-
pletely continuous, excepting the trivial case when f(s, x) does not depend on x. However, the same operator may be completely continuous on other spaces. For example, a continuous superposition operator is completely continuous if the space on which it acts is a Montel space (see Chapter 0, §4). Note that Theorem 0.4.1 implies that an infinite dimensional normed space does not have this property. Let now X, ~ be arbitrary Banach spaces and ;D a domain in X. THEOREM 8.1 (M. A. Krasnoselskii). Let F:;D
operator on ;D.
= F'(xo): X <J
----+
----+
~
be a completely continuous
If F is Fnkhet differentiable at Xo E ;D then the operator A
=
~ is also completely continuous.
Suppose that A is not completely continuous. Consequently, the set of
values of A on the unit ball is not precompact. Then there exist a sequence {hn}nEN <;;; <;;; X, IIhnil = 1, and a number 8> 0 for which IIA(h n - hm)II > 38, n, mEN, n
Put w(xo, h) = F(xo + h) - F(xo) - Ah. Let Xn is chosen such that Xn E ;D for all n E N, and 1 - IIw(xo, phn ) I p
Then, from the equality
it follows that
1
=
Xo
+ phn, where the number p > 0
= - IIF(xo + phn ) - F(xo) - pAhnil < 8. p
-I- m.
DIFFERENTIAL CALCULUS IN NORMED SPACES
58
n, mEN, which means -
since the sequence {xn} nEN S;; :D is bounded -
operator F is not completely continuous.
that the
~
The following more general result can be proved by induction.
Suppose that the operator F: X ----> ~ is completely continuous on the domain :D and admits, about Xo E :D, the Taylor representation
THEOREM 8.2.
F(xo
+ h) =
F(xo)
+ A1(xo, h) + ... + An(xo, h) + w(xo, h)
where Ai(xo, h) are homogeneous forms (in h) of order i, i
=
1,2 ... , n, and
Ilw(xo, h)11 ~ Mllhll n + 1 where M < 00 does not depend on h. continuous as operators from X into~.
Then the forms Ai(XO, h) are completely
REMARK 8.3. Under the assumptions of Theorem 8.2, it is enough to suppose that
the forms Ai(xo, h) correspond to some i-linear, not necessarly continuous, operators Ai(Xo , h 1 , ... , h n ), in order to prove their continuity in h.
r> 0 such that for all h with Ilhll < r, we have IIF(xo+th)-F(xo)11 for all t, 0 ~ t ~ 1, we obtain
where ¢(xo, th) = F(xo
> 0 there exists
~ E.
Consequently,
+ th) - F(xo) - w(xo, th).
For any linear functional y* E ~* the functional 'P(t) = (¢(xo, th), y*) is represented as a polynomial of degree n in t, satisfying the estimate all t, 0 ~ t ~ 1, and all y*,
Ily* II ~ 1.
1'P(t)1 <
N for
Then, there exists a number l > 0 such that the
absolute value of each coefficient of the polynomial 'P(t) is less than or equal to iN, i.e.,
Since y* E
for all h,
~*
is arbitrary, we have
IIhll ~ r,
i.e., the operator Ai(xo, h) is bounded and thus, continuous. ~
Complete continuity EXAMPLE 8.4.
59
C<;msider the nonlinear Volterra type operator F given by t
J
Fx(t) =
K(t, 8)f(8, x(8))d8,
a
where the functions K(t, 8) and f(8, u) are continuous for a ~ t, 8 ~ b, u
E
C,
lui ~ r,
and f(8, u) has the representation n
f(8, u)
=
I: ai(8)ui + b(8, u), i=O
where sup a(,s(,b
Ib(8, u)1 ~ mluln+l.
Arguing as in the previous example, it is easy to show that the operator F is completely continuous on era, b] and, if the functions ai(8) are continuous on [a, b] then the operator F has a Taylor representation of order n about zero, with completely continuous homogeneous forms t
(Ai(O, h)) (t) =
J
K(t, s)ai(s)hi(s)ds,
1~ i
~ n.
(8.2)
a
Moreover, the continuity of the functions ai(s) is not necessary for proving the complete continuity of the operators in (8.2). To see this, it is sufficient to show that the operators (8.2) act from era, b] into era, b]. Indeed, by Remark 8.3 it follows that the functions ai (s) are bounded on the interval [a, b]. But then the value of the right hand side of (8.2) is a continuous function in t on the interval [a, b], which completes our argument.
Chapter II Integration in normed spaces §1. Riemann - Stieltjes integrals of vector-functions Let us remind that by a function we mean a mapping having the domain in the scalar field IK of real or complex numbers (see Chapter 0, §1). In this chapter, we will consider functions with values in normed spaces -
called vector-functions - , as
well as functions with values in the field of scalars, which are called scalar-functions. The symbol f(t) means either the function f itself, or the value of fat t; the precise meaning will follow from the context. From now on the term "vector-function" will be used in a broader sense, for any mapping
f: X
~ ~
with the domain in a finite
dimensional space and with values in a normed space. In this section we consider that IK = lR and
~
is a Banach space, i.e., a
complete normed space. DEFINITION 1.1.
A vector-function y defined on an interval [0',,8] is called a
vector-function of bounded variation, if there exists a number M < the inequality
00
such that
(1.1 ) holds for any choice of a finite number of disjoint intervals (O'i,,6i) in (0',,6). The number Var[y] = sup
2: l[y(,6i) i
of the vector-function y on the interval [0', ,6].
Y(O'i)[[ is called the complete variation
INTEGRATION IN NORMED SPACES
62
REMARK 1.1. If Y is a vector-function of bounded variation then the scalar-function 'Py'
(t)
= (y(t), y*)
is also of bounded variation for any y* E Z)*. The converse is also
true and was independently proved by Dunford [1] and Gelfand [1]: if Var['Py' (t)] ~ ~
M(y*) <
00
for any y* E Z)* with Ily*11 ~ 1, then yet) is a vector-function of
bounded variation. DEFINITION 1.2. Let y be a vector-function defined on the interval [a,;3] and 9 a
scalar-function with values in lR (or C) defined on the same interval. The expressions n
S,,(y,g)
= Ly(td [g(ai) - g(ai-d] i=l
and
n
S7r(y,g)
= Lg(t i ) [y(ai) - y(ai-d] , i=l
where rr is a partition a = ao ~ al ~ ... ~ an =;3 of [a,;3], ai-l ~ ti ~ ai, i = 1, ... , n, are called Riemann-Stieltjes integral sums. Define also Irrl =
max
I ~ i ~ n
lai - ai-II·
THEOREM 1.1. If one of the limits
exists in the strong, weak or ultraweak topology of Z) then the other also exists. These limits are called Riemann-Stieltjes integrals and are denoted respectively by ~
(R-S)
~
J
J
a
a
y(t)dg(t) and (R-S)
g(t)dy(t). Moreover, we have
~
(R-S)
J
y(t)dg(t)
= y(t)g(t) I~ - (R-S)
~
J
g(t)dy(t).
a
<]
The conclusions follow immediately from the identity n
S7r(y,g) = y(;3)g(;3) - y(a)g(a) - L9(ai) [y(tHI) - yeti)], i=O
(1.2)
63
Riemann-Stieltjes integrals
Suppose that one of the functions y or 9 is of bounded variation and the other one is continuous (for y the continuity is understood in the strong topology). Then both the integrals
THEOREM 1.2.
(3
(R-S)
J
(3
y(t)dg(t)
and
(R-S)
J
g(t)dy(t)
exist in the strong topology of the space !D. In addition, if A is a continuous linear operator acting from !D into a Banach space 3, then
A (R-S) and
A (R-S)
I
y(t)d9(t))
~ (R-S)
I
I
9(t)dY(t))
~ (R-S)
I
Ay(t)dg(t)
(1.3)
g(t)dAy(t).
(1.4)
Suppose that the vector-function y is continuous in the strong topology
> 0 there exists 8 > 0 such that Ily(tI) - y(t2)11::;; 10 as soon as Ih - t21 < 8. Choosing l7rll, 17r21 < ~, we have
and 9 is of bounded variation. Then for any
10
IISn1 - Sn211 ::;; Var[g]
·210.
Hence the first assertion of the theorem follows. Let us prove now the equalities (1.3) and (1.4). If the function y is continuous or of bounded variation, then the linearity and
the continuity of the operator A:!D -function h = Ay: [a,,6]
----*
----*
3 imply the same properties for the vector-
3. Therefore the integrals in the right hand sides of (1.3)
and (1.4) exist. Furthemore, using again the linearity of A, we have
Sn (Ay,g) == ASn(y, g), Sn (Ay,g) == Asn(y,g), from which, by taking the limit, we obtain (1.3) and (1.4).
~
For a vector-function y and a scalar-function 9 satisfying the conditions of Theorem 1.2, the following relations
COROLLARY 1.1.
{3
((R-S)
(3
J
J
a
a
y(t)dg(t), y* ) = (R-S)
(y(t), y*)dg(t)
(1.5)
INTEGRATION IN NORMED SPACES
64
~
((R-S)
j g(t)dy(t), y* )
~
(R-S)
=
Q
j g(t)d(y(t), y*)
(1.6)
Q
hold for all y* E !I) * , where the integrals in the right hand sides of (1.5) and (1.6) are the usual Riemann-Stieltjes integrals of scalar- functions. COROLLARY 1.2. If the
vector-function y is continuous on [a,!3] and 9 is of bounded
variation then
1) ~
(R-S)jy(t)d9(t)
~
Ily(t)II' Var[g]
sup
=
(1.7)
M;
Q~t~~
2) if a sequence (Yn)nEN of continuous vector-functions is uniformly convergent on [a,!3] to the function y in the norm topology of the space !I), then ~
nl~ (R-S) j
~
Yn(t)dg(t)
=
(R-S)
j y(t)dg(t)
(1.8)
where the limit in (1.8) is understood in the strong sense. The inequality (1.7) follows from the estimate IISrr(y, g) II (1.8) follows from a similar estimate, since <1
~
(R-S)
j Yn(t)dg(t) -
~
(R-S)
j y(t)dg(t)
~
sup
~
M. The relation
IIYn(t) - y(t)11 . Var[g].
Q~t~~
REMARK 1.2. Formula (1.2) is a generalization of the classical formula of integration
by parts. Taking get) == 1 in this formula, we obtain the analogue of the Newton-Leibniz formula ~
(R-S)
j dy(t) = y(;3) - yea).
J
(1.9)
~
Taking get) == t in the integral (R-S)
y(t)dg(t), we obtain a generalization of the
Q
classical Riemann integral.
65
Pettis integral
§2. Pettis integral and the connection with Riemann-Stieltjes integral The integral introduced by Pettis [1] in 1938, as a generalization of the Lebesgue integral, will play an important role in the following chapters.
Y(IJ) be a vector-function defined on a measurable set (5 with values in a Banach space ~, such that the scalar-function (Y(IJ), y*) is integrable in the sense of Lebesgue for each y* E ~*. If there exists an element z E ~ such that
DEFINITION 2.1. Let
(z,y*) = j(Y(IJ),Y*)dlJ (5
for all y*
E ~ *,
then z is called the Pettis integral of the function y and one denotes
z = (P) j y(lJ)dlJ.
(2.1)
(5
Note that in the case when
~
is reflexive the Pettis integral exists provided
the functions (y(IJ), y*) are measurable and integrable in the sense of Lebesgue for all
y* E ~* (see Hille and Phillips [1]). The situation we are interested in is the case when ~ is a complex space and (5 is a contour in C. In this case we study the existence of Pettis integral and its connections with Riemmann-Stieltjes integral. We take as (5 a contour, in C given by the equation, = { TEe : T = T( t), a:::; t:::; ,6}, where T(t) is a continuous function of bounded variation on [a,,6]. Let y be a strongly continuous vector-function defined on ,. Then the scalar-function 'Py' given by 'Py' (T) = (y( T), y*), y* E ~*, is Lebesgue integrable on , and j3
j
'Py'
(T)dT = (R-S) j
'Py'
(T(t)) dT(t)
=
a
~
j3
= j (y (T(t)) , y*) dT(t). a
On the other hand, the vector-function y = YOT is strongly continuous on the interval [a, ,6] and therefore Theorem 1.2 implies the existence of the Riemann-Stieltjes integral
J
J
a
a
j3
(R-S)
y(t)dT(t) = (R-S)
j3
y (T(t)) dT(t) = z
E~.
INTEGRATION IN NORMED SPACES
66
According to (1.5), for any y* E !D*, we have
J {3
(z,y*)
(Y(T(t)),y*)dT(t)
(R-S)
=
=
a
J
(y(T),y*)dT,
~
which proves that the function y is Pettis integrable on T (3
J
J
a
a
{3
z = (P)
y(T)dT = (R-S)
y (T(t)) dT(t).
Moreover, if 'Y is a smooth contour, i.e., the derivative T'(t) exists and is continuous on [a,,8], then we have
(P)
{3
J
y(T)dT
=
(R)
J
y (T(t)) T'(t)dT.
(2.2)
When no confusions arise, we will omit the symbol (P) in the Pettis integral, and we simply write
J
Y(T)dT.
§3. Antiderivatives of vector-functions. Integral representations 1. The symbol of differentiation in (1.2) is, generally speaking, only formal, since
-
as it follows from Theorem 1.2 -
formula (1.2) is true for some vector-functions
which are not differentiable; take, for example, a vector-function which has complete bounded variation. We will show in this section that the classical Newton-Leibniz formula has a direct generalization for vector-functions which are assumed to be differentiable. To begin with, we prove, as in the classical analysis, the differentiability of an integral with respect to a variable upper limit; this result is also interesting on its own. LEMMA 3.1. Let Y be a vector-function defined on an interval [a,,8j with values in a Banach space !D which is Riemann integrable on [a,,8j (see Remark 1.2) and continuous at a point v E (a, ,8). Then the vector-function f: [a,,8j ---> !D given by
J T
f(T) = (R)
y(t)dt
C<
(3.1)
Antiderivatives. Integral representations
67
is differentiable at v and f'(v) = y(v).
f(v
J
V+E
f(v) _ y(v) = ~(R)
+ E) E
E
(y(t) - y(v)) dt.
v
Then, according to (1.7), we have
II"-f('--v-+--.-:E)'----'=f-'..(V-'-) E
Y(V)II
~
sup
Ily(t) - y(v)ll·
V:( t:( V+E
The continuity of y( T) at v implies that the right hand side of the previous inequality tends to zero as
E
approaches zero, so the conclusion of the lemma follows. ~
Let get) be a vector-function defined on [a,,6] and continuously-differentiable on [a, ,6]. Then the formula
COROLLARY 3.1 (Newton-Leibniz formula).
(3
g(,6) - g(a)
=
(R)
J
g'(t)dt
(3.2)
a
holds.
f given by (3.1), where yet) = g'(t). Then f(,6) is
the integral (3.2). FUrthermore,
(J'(t),y*) = (g'(t),y*), for every y* E
!D*.
t E (a,,6),
Consequently, there exists a constant c(y*) such that
(J(t), y*) = (g(t), y*) + c(y*). Since f(a)
= 0,
(3.3)
we have
c(y*) = -(g(a), y*). Using this equality in (3.3) and taking t
(J(,6) , y*)
= ,6,
we have for each y* E
= (g(,6) - g(a), y*),
!D*,
INTEGRATION IN NORMED SPACES
68 and so (3.2) follows. ~
2. By analogy with the classical analysis, the vector-function
f defined by (3.1)
is called the antiderivative (primitive) of y. If y is continuous on [a,,8], then its antiderivative vector-function f exists and, being differentiable on [a,,8], it is continuous on [a,,8]. Then the function
II = f
has also a continuous antiderivative
h
on [a, ,8].
By induction, we obtain the existence of succesive continuous antiderivatives fm for each order m U:"(t)
= fm-l(t) for
m;?:2 and f{(t)
= f'(t) = y(t)). Moreover, h(t)
is an iterated integral: t
h(t) = (R) (a
~
s~
T
~ t ~
t
J
J
J
'"
'"
'"
f(T)dT = (R)
fm(t) = (R) ...
dT' (R)
y(s)ds,
,8). Similarly, for every m;?: 2, we have t
(a~tl ~
T
t1
trn
J
J
J
'"
'"
'"
dtm(R)
dtm-l ... (R)
y(tI)dtl
~tm~t~,8).
Let us show now that fm(t) has an integral representation as an one-dimensional integral. THEOREM 3.1. Let y be a vector-function continuous on
[a, ,8]. Then the vector-fun-
ction f m given by t
fm(t) = (m ~ 1)! (R)
J
(t - s)m-ly(s)ds
(3.3)
is the m-th antiderivative vector-function of y satisfying the conditions
fm(a) = f:"(a) = ... = fSnm-l) (a) = 0, fSnm)(t) = y(t). prove the theorem by induction on m = 1,2, .... For m = 1, the assertion follows from Lemma 3.1. Assume that the theorem is proved for the function fm-l (t). Consider its antiderivative vector-function
J t
F(t) = (R)
fm-l(T)dT;
'"
69
Antiderivatives. Integral representations we will show that F(t) coincides with the vector-function defined by (3.3). Indeed
F(t) = (R)
I dr (R) IT (r(;;;, _s)m-2 2)! y(s)ds. t
'"
'" ~*
According to Corollary 1.1, for any y* E
(F(t), y*) = (R)
we have
I dr (R) IT (r(m _S)m-2 2)! (y(S), y*) ds. t
'"
'"
By Fubini Theorem we can change the order of integration in the last integral, so we get
I t
(F(t), y*) = ·(R)
I t
(y(s), y*) ds (R)
'"
(r s)m-2 (m _ 2)! dr.
8
Computing the inner integral, we obtain
I t
(F(t), y*) = (m
~ 1)!
(t - s)m-I (y(s), y*) ds
'" whence, using also Corollary 1.1, it follows that
(F(t), y*) for any y* E
~*,
= (fm (t) , y*) ~
so the theorem is proved.
Formula (3.3) may be seen as expressing a function in terms of its derivatives of order 0,1, ... ,m - 1, evalued at a point, and the values of its m-th derivative on a neighborhood of that point. For a generalization of (3.3), we formulate the following initial value problem. Consider the differential equation of order m
F(m)(t) with the initial conditions at a point
0:
E
=
y(t),
(3.4)
IR
F(o:) = Yo, F'(o:) = YI, ... , F(m-I)(o:) = Ym-I,
(3.5)
INTEGRATION IN NORMED SPACES
70
where y is a vector-function, continuous on a neighborhood of the point a, and with values in
!D,
and Yo, ... ,Ym-l are some given elements in
!D.
Then the next result is
true. COROLLARY 3.2. The solution of the differential equation (3.4) with initial condi-
tions (3.5) has the form
J t
1
F(t) = yo+(t-a)Yl + ... + (m _ I)! (t-a)m-l Ym _1+(R)
(t - s)m-l (m _ I)! y(s)ds. (3.6)
Q
Note that the integral in (3.6) is a antiderivative of order m of y, which,
together with all its derivatives up to order m -1, have the value zero at the point a. Moreover, the sum of the other terms in the right hand side of (3.6) is a polynomial of degree m -1 (so its m-th derivative is indentically zero) which takes at a the given values Yo, . .. ,Ym-l· ~ Formula (3.6) is the Taylor formula for the function F, with the remainder in an integral form. By contradistinction with the usual Taylor formula, the integral in (3.6) gives an explicit form of the remainder, which turns out to be very useful in many cases. However, note that we impose a restrictive condition on F, namely, the continuity of its m-th derivative on a neighborhood of the point a. To conclude this section, consider a vector-function a neighborhood
n of a
be an arbitrary point in
f
that is continuous on
point (0 E C and takes values in a Banach space
!D.
Let (1
n and let 'Y be a smooth contour lying completely in nand
joining the points (0 and (1: 'Y = {( En: (= «(t), a ~ t ~ {3, «(a) = (0, «({3) =
(d·
Take the partial contour 'YT = {( E 'Y : (= «(t), a ~ t ~ T} and consider the integral
J
f()d( =
~((T)).
The last integral is a function of (
= «(T),
where ( runs on the
'YT
contour 'YT from (0 up to «(T). Further, according to (2.2) we have
J T
~((T)) =
f((t))('(t)dt = F(T),
Q
where F(T) is a primitive of the function y(T) ~((T)) =
F(T).
= f((T))('(T),
with F(a)
= o.
Thus
Integrals of operators in Banach spaces
71
Suppose that the vector-function iP is differentiable with respect to ( in a neighborhood of (1. Then, using the formula for the derivative of a composite function, we find -
for
T
sufficiently close to
f3 - that
Hence
In conclusion, if the integral
J
f(()d(
= iP((1,1')
'Y
is considered on smooth contours joining (0 and (1, and if iP is a vector-function differentiable at (1, then
and iP( (1, 1') does not depend on 1', that is the integral on any contour l' joining the points (0 and (1 gives the value of the antiderivative of the function
f at the point
(1. which is zero at (0'
§4. Integrals of operators in Banach spaces The theory of abstract integration of vector-functions has many similarities with the integration theory for scalar functions. For example, as soon as we prove the existence of the abstract Riemann-Stieltjes integral, its properties can be derived from the similar properties of the integral for scalar-functions, using Corollary 1.1 and Hahn-Banach type theorems. Similar considerations can be made for the theory of measurable vector-functions, the theory of Riemann or Lebesgue integrals, and others (see, for example, 1. Schwartz [1]). However, in what follows we will need only the results stated in §§1-3. The theory of integration of operators acting between linear topological spaces is richer and more complicated. First, it is natural to construct the integral for operators in such a way so that to generalize the corresponding notions for scalar or vector-functions. Further, it is necessary to take into account the usefulness for applications. One of the approaches to the definition of the integral of an operator
INTEGRATION IN NORMED SPACES
72
relies upon the integration of its finite dimensional sections (see E. Hille and R. Phillips [1]). Although this approach is quite natural, we found it rather inefficient in many cases. For examples, from the beginning there are difficulties with changing the variable in the integral. Further, this approach does not help in constructing the analogue of the logarithmic residue from finite dimensional theory, a fact which causes many troubles for the theory of fixed points of noncom pact operators. Having in mind the traditional schemes, we introduce in this section only two types of integrals of operators acting between spaces with special properties. 1. Let X and
3 be Banach spaces and let ~ = L(X, 3) be the space of linear operators
from X into 3. Let F: X ---;
x E
1)
~
be an operator defined on a domain 1), i.e., for any
the value F(x) is a linear continuous operator from X into
3, depending on x
as a parameter. DEFINITION 4.1. Assume that the vector-function f(t)
= F(x + th)h is integrable
on the interval [0,1]. The expression x+h
J
1
F(x)dx = (R)
J
F(x
+ th)hdt
(4.1)
0
x
is called the integral of the operator F on the interval [x, x
+ h].
All the properties concerning the linearity of integral (4.1) can be easily checked using the results of §l. Further, if X = 3, then
~ =
L(X, X) is a Banach
algebra with unit I (the identity mapping of X), and x+h
J x
x+h
Idx=
J
dx=h.
x
We will prove now that a Newton-Leibniz type formula holds for the integrals defined by (4.1), i.e., a formula which recovers the values of a differentiable operator from its derivative. THEOREM 4.1. Let X and
3 be Banach spaces and let G: X ---; 3 be an operator
defined on a neighborhood of a point Xo. Assume that G is Gateaux differentiable at any point of an interval [xo, Xo + h] included in the domain 1) of G, such that the derivative G' is continuous in the uniform operator topology on this interval. Then
73
Integrals of operators in Banach spaces the Newton-Leibniz formula
G(xo
+ h) -
G(xo) = xlhG'(X)dX
(4.2)
Xo
holds.
= G(xo
Consider the vector-function 9 defined on the interval [0,1] by get)
+ th).
=
Then 9 is continuously differentiable on [0,1] and the differentiation
formula for composite functions yields
g'(t) = G~(xo
+ th)h.
From (4.1) we have xo+h
1
(R)
JG~(xo +
J
th)hdt =
°
G'(x)dx.
Xo
The equality (4.2) follows from the previous one and from the relations
J 1
G(xo
+ h) -
G(xo) = g(l) - g(O) = (R)
g'(t)dt =
°
J 1
G'(xo
+ th)dt.
0
On the other hand, a perfect analogue of the Taylor formula with the remainder term represented by an integral of the form (4.1) fails to be obtained. More precisely, for an operator F which is m-times continuously differentiable on a neighborhood of the point Xo, an application of formula (3.6) gives the representation
F(xo
+ h) =
F(xo)
+ F'(xo)h + ... +
J
1
(m _ I)! F(m-l) (xo)h m- 1 +
1
+ (m ~
I)! (R)
(1 - t)m-l F(m) (xo
(4.3)
+ th)hmdt
o where the symbol F(k) (xo)h k denotes the value of the homogeneous form F(k) (xo) of order k, acting from Xk into ~, at the element (h, ... , h). The integral in the right hand side of (4.3) is to be understood as the integral of the vector-function (1 - t)m-l F(m)(xo + th)hm.
INTEGRATION IN NORMED SPACES
74
2. Consider now the case when X is a Banach algebra over the field IK and F: X
---->
X
is an operator defined on a neighborhood 1) of the point Xo. If S is a rectifiable contour in 1), i.e., S is defined by an equation x
=
x(t), a
~t ~
(3, where x(t) is a
continuous vector-function of bounded variation, then by definition we put
(T)jF(X)dX= lim tF(X(Ti))(X(O"i)-X(O"i-d), 17r1-->0.,= 1 s where
= a ~ 0"1 ~ •.• ~ O"n = {3, O"i-1 ~ Ti ~ O"i· The existence of this integral can be proved as in §l. Its linearity, an es-
0"0
timate of its norm, and the commutation of the integral with linear continuous operators acting from X into X follow too. This integral is called the generalized
Riemann-Stieldjes- Taylor integral, or shortly, the T-integral.
Chapter III Holomorphic (analytic) operators and vector-functions on complex Banach spaces In Chapter I we introduced the notions of 8-differentiability, Q-differentiability and F-differentiability of an operator F : X ----t ~ at a given point. For each of these notions we defined the classes of m-times differentiable operators, 1::::; m ::::; 00, as well as the class of analytic operators. In the general case, each of these classes is a proper part of the previous one. However, when X and ~ are complex spaces and F is differentiable on a nieghborhood of the given point, the above mentioned classes coincide. This gives the possibility to extend the well-established theory of complexdifferentiable operators, a theory with meany deep results. (We will see below that in many cases the study of the Frechet derivative of an operator at a point has not only local consequences, but also provides indications about the global features of the behavior of the given operator.) On the other hand, many operatorial equations in a real space can be studied by considering the complexification of the space and by extending the operator such that the given equation reduces to an equation in a complex space. In order to emphasize the important role of the complex spaces in the theory of differentiable operators, we will use in what follows, for the complex case, the term "holomorphic" instead of the term "differentiable". The definitions of holomorphic operators, holomorphic vector-functions, and so on, will be given below. There are many similarities between the theory of holomorhic operators and the theory of holomorphic vector-functions; the only differences appear when it is essential to suppose that either the domain or the range of the operator are in a finite dimensional space. Therefore we develop these two theories in the same general frame.
76
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
in complex and real sense. Cauchy-Riemann conditions
§1. Differentiability
In what follows, we suppose that X and
~
are complex normed spaces. Let F: X
-+ ~
be an operator which is differentiable at a point Xo, according to one of the definitions 1.1.1- 1.1.3 (we will say that F is complex-differentiable). Then, the operator F is also differentiable relatively to the field lR of real numbers, according to the same definitions (in the case of Definition 1.1.3, this means that the operator has a real-linear Frechet derivative). In this situation we say that the operator is real-differentiable, according to the given definitions. When an operator F is real-differentiable at a point Xo, the first variation of F will be denoted by 811~F(xo, h). THEOREM 1.1.
If an operator F: X
-+ ~
is continuously real-differentiable with
respect to one of the definitions 1.1.1 - 1.1.3 on a neighborhood :D ~ X of Xo, then the operator F is complex-differentiable at Xo if and only if its first variation 8R.F(xo, h) satisfies the condition (1.1)
where i is the imaginary unit.
The necessity ofthe condition (1.1) is obvious, since the variation 8F(xo, h),
being the derivative of the vector-function f()..)
= F(xo + )"h),
is homogeneous in )...
The conditions is also sufficient. Assume that (1.1) is verified. We represent ).. E
C as )..
=
T
+ iry,
where
T,
"I E R For a fixed h, the vector-function f()..) defined
above obviously has a continuous derivative with respect to
T
at the point )..
=
o.
It remains to establish the existence of the continuous partial derivative of f()..) at ).. = 0 along the direction iry. Indeed, this will actually prove the existence of the
complete derivative 1'(0) of the vector-function f()..), as an operator acting from the space C
= lR + ilR into
the spac.e
1'(0)
=
~
(see Theorem 1.5.2). But
lim F(xo
+ )..~) - F(xo)
=
8F(xo, h);
A--->O
since h is an arbitrary element, the existence of the first variation of the operator F at the point Xo will follow. For
fir/D)' we have fi (0) = .lim F(xo + i?h) 1)
-- ~. 1·1m F(xo 1 1)--->0
11)--->0
+ ry(ih)) "I
- F(xo) =
1"1
- F(xo) -- _." IvR. F( XO,l·h) - "vIFt F( Xo, h) .
77
Cauchy-Riemann conditions
To complete the proof of the theorem, notice that if the operator F is Gateaux (resp. Frechet) differentiable, then the variation OIRF(xo, h) is real-linear in h. By (1.1) it is also complex-linear in h, i.e., it is the Gateaux (resp. Frechet) derivative in the complex sense.
~
Assume now that the space QJ (see §4 of Chapter 0), i.e.,
~ =
~
is the complexification of a real Banach space ~
QJ + iQJ and the topology of
topology of the product QJ x QJ. Then, any operator F: X
---* ~
coincides with the
can be represented as
F = P + iQ, where P, Q: X ---* QJ. Suppose further that the operators P and Q are real-differentiable on a domain :D containing the point xo. Then the operator F is also real-differentiable, and 'F(
vIR
Xo,
h)
l'
=
1m
P(xo
t--->O,tEIR
+ th) -
P(xo) . + I . l'1m Q(xo t t--->O,tEIR = OIRP(xo, h) + iOIRQ(xo, h).
+ th) -
Q(xo)
t
If condition (1.1) is fulfilled, then
OIRP(xo, ih) + iOIRQ(xo, ih) = = iOIRP(xo, h) - OIRQ(xo, h). Comparing the real and the imaginary parts, this infers the equalities
OIRP(xo, ih) = -OIRQ(xo, h) OIRQ(xo, ih)
=
(1.2)
OIRP(xo, h)
which are generalizations of the well-known Cauchy-Riemann conditions. Indeed, if
X = C = {AI A = T+i1]}, then P and Q can be considered as vector-functions oftwo real variables T,1] E IR. Writing Xo
TO + i1]o, h
= 5:
VIR
P( Xo, 'h) I =
lim t--->O,tEIR
For h
=
(D.T, 0)
E IR2,
=
D.T + iD.1], we obtain
we have
1m
P(xo
+ tih)
- P(xo) t P(TO - tD.1] + i(1]o + tD.T)) - P(TO + i1]o) t l'
t--->O,tEIR
78
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
On the other hand, the first equality in (1.2) implies, for the same h, that b~P(xo,
Q(TO
lim
ih)
=
+ tf:1T + i7]o) -
t-.O,tE~
-b~Q(xo,
Q(TO
h)
+ i7]o)
t
= 8Q 8T
I (
Xo= 70,1]0
).
Thus we obtain the first Cauchy-Riemann condition
8P
(1.3)
87] Analogously we obtain the second Cauchy-Riemann condition 8Q 87]
8P 8T'
(1.4)
(1.3) and (1.4) the condition that X is an one-dimensional space is not essential. It is sufficient to assume that X, like q), is the complexification of a real space i1. In this case, Theorem 1.1 has the following consequence. REMARK 1.1. Notice that in order to prove
Let il, QJ be two real Banach spaces and let P, Q: il x il --> QJ be two operators defined on a neighborhood of the point Xo = (Po, qo) E il x il. Assume that P and Q have continuous partial Frechet derivatives on that neighborhood. Then the operator F = P + iQ: X --> q), where X = il + ill and q) = QJ + iQJ, is Frechet complex-differentiable if and only if the equalities COROLLARY 1.1.
P;(xo) P~(xo)
= Q~(xo) = -Q~(xo)
(1.5)
are fulfilled. <J
The necessity can be established as above, in the same way in which we
deduced (1.3)-(1.4). For sufficiency, choose an arbitrary vector h
=
consider b~P(xo, ih). Notice that in il x il the vector ih has the form ih Using (1.3), we have
= b~P (Po, qo, (-f:1q, f:1p)) = = P~(xo)( -f:1q) + P~(xo)f:1p = -Q~(xo)f:1q - Q~(xo)f:1p = -b~Q(xo, h),
b~P(xo,
=
ih)
f:1p
+ if:1q and
=
(-f:1q, f:1p).
Cauchy-Riemann conditions
79
which is the first equality in (1.2). The second one can be proved analogously.
~
COROLLARY 1.2. If, in addition to the conditions of the previous corollary, one assumes that P and Q are twice continuously differentiable and satisfy the conditions (1.5), then
D.F = 0
and
D.P = D.Q = 0
where the symbol D. denotes here the Laplace operator, i.e., for any G: UxU --+
D.G = G~p
~x~,
+ G~q.
(In fact the requirement that P and Q are twice continuosly differentiable is superflous in the previous corollary; as we have already mentioned, the complexdifferentiability of F implies its infinite real-differentiability, and this means that P and Q are also real-differentiable of any order.)
At the end of this section we consider the notion of the complex-conjugate differential, just by mimicing the classical constructions. To this aim, for any element
x = P + iq in the space X = U + ill we introduce the conjugate element x this case, p
=
1
"2 (x + x),
1
q = 2i (x - x). Therefore, an operator F: U x U --+
gives rise to an operator F(x, x)
=
=
P - iq. In
!D formally
F(p, q) by replacing p and q with their expressions
in terms of x and x. (The problem is that
F
can not be considered as a function
of two independent variables, since the change of x implies a simultaneous change of x.) Further, if the operator F is Frechet differentiable on a neighborhood of a point
Xo = (Po, qo) (in the real sense), then, by the formula of the complete differential, we have where h
=
(D.p, D.q), so F'(xo)h = =
F~(xo) [~(h + h)] + F~ [;i (h -
h)]
~ [F~(xo) - iF~(xo)] h + ~ [F~(xo) + iF~(xo)] h = =
F~(xo)h + F/r(xo)h,
where F~(xo)h and F/r(xo)h are the formal derivatives of the operator F with respect to x and x, defined by
F~(xo) = ~ [F~(xo) - iF~(xo)] F/r(xo) =
~ [F~(xo) + iF~(xo)]
.
80
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES Conditions (1.5) for the operator F
conclusion: if the operator F: il x il
~ ~
=
P
+ iQ lead directly to the following
is Frechet differentiable in the real sense,
then it is Frechet differentiable in the complex sense as an operator from into ~
= m + im if and
only if F~(x)
x = il + ill
= o.
§2. The p-topology and holomorphy As we have already noted, the class of operators having the first variation, the class of Gateaux differentiable operators, and the class of Frechet differentiable operators form a strictly decreasing chain with respect to the inclusion. Nevertheless, from the point of view of the complex analysis, these classes have many properties in common. It is therefore interesting to consider a characteristic of these classes which permits to
recover these common properties in a unified manner, and, on the other hand, which permits to distinguish between these classes of operators. For the Gateaux and the Frechet differentiable operators, such a characteristic seams to be a "connection" between the domain of definition of the operator and the operator itself. More precisely, we consider, on the domain of an operator, the so-called p-topology, which is weaker than the norm topology; in this topology both classes of differentiable operators have the same properties. Note that the interest in this approach arises not only from its generality, but also from the possibility of obtaining new results. As an example see §6 of Chapter VI, where some degenerate problems for implicite Frechet differentiable operators are transferred to nondegenerate problems for operators differentiable relatively to the p-topology. Our presentation and terminology mainly follow E. Hille and R. Phillips [1]. 1. For a set 1) ~ x and for elements x E 1), hEx, we introduce the following notion: p(x, h) = sup{p : x
+ Th E 1), TEe, ITI::;; p}.
Clearly, p(x, h) ~ 0, for all x E 1), hEx and for any set We will need the following property of p(x, h). LEMMA 2.1. Let 1) be a set in
x.
Then for any x E
have p(X,A-1h) =p(x,h)IAI·
1) ~
1),
(2.1)
x.
hEX, A E C (A i=- 0), we
81
Holomorphy
The equality is obvious if p(x, h) = o. Assume that p(x, h) > 0 for some and hEX. Denote h1 = A- 1h and take an arbitrary Po, O
x E 1)
Po. Denoting t = T A-1, we have that x + th E 1) 1 for any t with It I < polAI- = Pl. Consequently, P1 ~ p(x, h), or Po ~ p(x, h)IAI· Hence p(x,hd = p(X,A- 1h) ~p(x,h)IAI. Conversely, let P2 ~p(x,h)IAI. We have x + Th1 E 1) for any T with ITI < P2 (since x + Th1 = X + th, where t = TA- 1 and
Then x
+ Th1
E 1)
for any
ITI <
It I < p(x, h)). Then P2 ~ p(x, hd = p(x, A- 1h). ~ DEFINITION 2.1. A set 1) is called p-open (one-dimensional open) if p(x, h)
any x E
> 0 for
and hEX.
1)
It is clean that any open set in the norm topology of X is a p-open set. The converse
is false in general. To see this it is sufficient to consider the set 1)
= 1) 1 \
1)2,
where
1) 1
is an open ball in
([2,
1) <:;; ([2
defined as
centered at the origin, and
On the other hand, the following easy result is true. PROPOSITION 2.1.
A set
1)
is open in the norm topology of X if and only if p(x, h)
for all x E 1), hEX,
Let 1)
1)
Ilhll
~
8(x)
>0
(2.2)
= 1.
be a bounded p-open set in X. Using (2.1) we define a convergence in
as follows:
{X"'}"'EQl in 1) is p-convergent to an element x E 1) iffor any positive integer N there exists ao E 2t such that p(x, x", -x) > > N for all a E 2t with a > ao. The topology defined by the p-convergence will be called the p-topology. DEFINITION 2.2. We say that a directed ness
From Lemma 2.1 and from relation (2.2) we have: COROLLARY 2.1. If the set 1) is open and the directedness
{X"'}"'EQl in
1)
is con-
vergent to x E 1) in the norm topology, then it is also p-convergent to x.
Ilx", - xii,
For the proof, it is sufficient to use Lemma 2.1 with h = x", - x, A = and Proposition 2.1. ~
82
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
DEFINITION 2.3. For any
Dp(x, h)
= {z
E
x E ::D the set
= x + Th, T E C, ITI < p, hEX, Ilhll = I} Z = x + Th, T E C, ITI:::;; p, hEX, Ilhll = I})
X
Z
(Dp(x, h) = {z E X
is called the open (closed) disk of radi us p, centered at x, and along the direction h. Clearly, a set ::D
~
X is p-open if and only if for each x E ::D it contains an
open disk of non-zero radius in any direction h, centered at x. Let us show that a closed disk Dp(x, h), where p < p(x, h) is compact in the p-topology of the p-open set ::D.
= x+T"h, IT"I : :; p, a
E
E~.
We choose a subdirectedness {T,B},BE23 , Q3 ~ ~ convergent to a point T* E C; we will show that the corresponding subdirectedness {X,B},BE23 , x,B = x + T,Bh is pconvergent to the element x* = x + T*h. First of all, since x* E Dp(x, h), Proposition 2.1 implies that p(x*, h) > 0 for any hEX,
Ilhll = 1.
Then, by Lemma 2.1 we obtain
Since for any N > 0 there exists f30 E Q3 such that IT* - T,BI < p(x*, h)N- 1 for all f3 E Q3, f3 > f30 (the element h is fixed), the relation (2.3) implies
and the proof is complete.
~
2. Let us turn now to the notion of holomorphy. An operator F:::D --+ ~ is called p-holomorphic in ::D (in short FE 1ip(::D , ~)) if for any x E ::D and hEX the vectorfunction f(T) = F(x + Th) is complex-differentiable in the disk ITI < p(x, h) of the complex plane C. DEFINITION 2.4. Let::D be a p-open set in X.
Thus, any p-holomorphic operator F has at each point x E ::D the first variation 8F(x, h). DEFINITION 2.5. An operator F:::D --+ ~ is called Q-holomorphic if it is p-holo~or phic and for any point x E ::D there exists a linear operator A(x) E L(X,~) such that 8F(x, h) = A(x)h. In this case we will write F E 1i(::D, ~).
Holomorphy
83
If the set 1:J is open, i.e., condition (2.2) is fulfilled for all x E 1:J, then a
Q-holomorphic operator is called F-holomorphic or, simply, holomorphic. Obviously if F E H(1:J,!I)) then F is Gateaux differentiable at any point
x E 1:J. We will see in §3 below that if an operator F is F-holomorphic, i.e., the set 1:J is open, then F is Frechet differentiable at any point x E 1:J (this explains the term "F-holomorphic operator"). Since the notion of holomorphy depends on the notion of convergence, we may naturally consider the notion of weak holomorphy. However, it is not necessary to do that, because - as we have already mentioned - the strong holomorphy follows from the weak holomorphy. This result was obtained for vector-functions by N. Dunford
[1]. We will prove here a sligt generalization. An operator F: 1:J
---t
!I) is called locally bounded on 1:J if for each x E 1:J there
exists a neighborhood it of the point x such that F is bounded on it n 1:J. We make the supplementary assumption that there exists a Banach space !I) * such that (!I)*)*
=
!I).
THEOREM 2.1. Let 1:J be a p-open set in X. An operator F, locally bounded on 1:J,
is p-holomorphic (Q-holomorhic) if and only if (y*, F(·))
E
Hp(1:J, C) (H(1:J, C)) for
each y* E !I) *. (Maybe it is useful to explain that the value of the operator (y*, F(·)) at x E 1:J is the number (y*, F(x)), i.e., the value of the functional F(x) E !I)
= (!I)*)*
at the
vector y* E !I)*.)
The necessity of the condition is obvious. Let us prove the sufficiency.
Consider (y*, F(·)) E Hp(1:J, C). This means that the function
=
(y*, F(X+Th))
is differentiable on the disk ITI < p(x, h) for any x E 1:J, hEX, y* E !I)*. We have to prove the differentiabily in the same disk of the vector-function f (T) = F (x + Th) for any x E 1:J, hEX. To this end, consider the sequence
(2.4) for tn ---t 0 (n ---t 00), where ITI, IT+tnl < P < p(x, h); we will show that this sequence is fundamental. Indeed, for any y* E !I) *' we have 1
tn - tm (y*,£(tn) - £(t m _ 1 _ [
tn - tm
))
=
+ tn) -
tm
-
84
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
=
_1 27ri
J
(,\ - T)('\ - T + t n )('\
-
T + tm )
d,\,
IAI=p
where p < p(x, h). The last equality is obtained by using the classical Cauchy integral representation and by decomposing the integrated function in partial fractions. Since F is locally bounded, the function
1,\1 = p by a
constant M(y*,p). Then
for all tn, tm with Itnl, Itml < p. Using the Banach-Steinhaus principle of uniform boundedness (Theorem 0.8.8), we obtain the inequality
where M(p) is a finite constant, so the sequence (2.4) is fundamental. Since
exists for any
!V is complete, the limit
T
with
ITI <
p(x, h). Thereby we proved that F
E
Hp('IJ, !V), i.e., the
variation 8F(x, h) exists. Assume now that (y*, F(·)) E H('IJ, q, that is, the variation 8(y*, F(· ))(x, h) (for any y* E !V*) is represented by a bounded linear operator (relatively to h), acting from X into
(y*, 8F(x, h)) = 8(y*, F(· ))(x, h) we infer that F
E
H('IJ, !V).
~
As a consequence, we prove now an analogue of the classical Weierstrass theorem on the completeness of the spaces Hp('IJ,!V) and H('IJ,!V) for the topology of weak uniform convergence on the closed disks in 'IJ. Let Fn E Hp ('IJ , !V) (resp. Fn E H('IJ, !V)) be a sequence of operators, where 'I) is a p-open set in X. Assume that the sequence Fn converges uniformly on the closed disks in 'IJ (in the weak topology of !V) to an operator F: 'I) -+ !V. Then
THEOREM 2.2.
FE Hp ('IJ , !V) (resp. FE H('IJ, !V)).
Cauchy integral theorems
85
The first assertion follows directly from the classical theorem of Weierstrass
and from Theorem 2.1. Indeed, for any fixed x E :D and hEX, the scalar functions
---->
From this and from the principle of uniform boundedness it follows that 8F(x, .) is linear and bounded, so FE H(:D, ~).
~
§3. Cauchy integral theorems and their consequences As in the classical complex analysis, the Cauchy integral theorems represent powerful tools in the study of holomorphic operators. A lot of subtle and interesting recent investigations concerning finite dimensional integral representations could be carried out for infinite dimensional spaces. Here we restrict ourselves to present the simplest results, which are enough for our further considerations. Let :D be a p-open set in a Banach space X.
Let F be a continuous operator acting from the set:D endowed with the p-topology into a Banach space ~ with the norm topology. Then F E Hp(:D,~) if and only if the equality THEOREM 3.1.
(P)
J
F(x
+ .\h)d.\ = 0
(3.1)
i>-i=p
holds for any x E:D, hEX and for any p < p(x, h).
(3.1) exists (see Chapter II, §2) and the equality
((P) J i>-i=p
F(x + Th)d.\, y* )
=
J i>-i=p
(F(x + Th), y*)d.\
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
86
holds for any y* E ~ * . Our theorem follows now from Theorem 2.1 and from the classical theorems of Cauchy and Morera (see, e.g., B. Shabat [1]) . • The analogue of the Cauchy integral representation formula can be obtained similarly. THEOREM 3.2. For an operator F E
21fiF(x
Hp(TJ, ~), the representation
J
+ Th) = (P)
F(x
+ '\h)('\
- T)-ld,\
(3.2)
I>-I=p
is true for any x
ETJ,
TEe
hE oX and
ITI < p < p(x, h).
with
In particular
J
21fiF(x) = (P)
,>.-1 F(x
+ '\h)d'\,
(3.3)
I>-I=p
that is, the integral in the right hand side of (3.3) does not depend on h if p is small enough, and it is equal to the value of the operator F at x multiplied by 21fi.
TJ, then it is p-analytic at each point of TJ, i.e., F admits a decomposition as a generalized power series COROLLARY 3.1. If the operator F is p-holomorphic and locally bounded on
00
F(x
+ Th) = ~ okF(x, h)Tk
(3.4)
k=O
which is norm convergent (in okF(x,h)
~)
ITI ~ p < p(x, h)
for
J
= ~(P) 2m
and
,\-k- 1 F(x+,\h)d,\.
(3.5)
I>-I=p
Indeed, the kernel (,\ - T)-l from (3.3) has the power series representation 1
Tk
--""''\-T -
00
L
,\k+1
k=O
which converges absolutely for all
T
with
ITI < 1,\1 = p.
00
~ ,\-(k+1)T k F(x k=O
+ '\h)
Then the series
Cauchy integral theorems
87
converges absolutely (in the norm of !D) for all x E 1) and A E C with IAI
~
p(x, h), because F(x + Ah) is bounded. It remains to show that the integral and the sum "commute". For any positive integer n, we have
J
(P)
IAI=p =
r(k+1)T k F(x + Ah)dA
t k=O
tTk(P) k=O
J
=
(3.6)
A-(k+1)F(x+Ah)dA.
IAI=p
For a fixed T, denote by Pn(A) the expression under the integral from the left hand side of (3.6). Then Pn(A) converges for n ---;
00
to F(x + Ah)(A - T)-l, uniformly for
all A with IAI = p. Taking into account that
J
(P)
F(X+Ah)A-(k+1)dA
IAI=p
J IIF(x+Ah)IIIA-(k+1)lldAI~ IAI=p
~
where M;?
~
sup
M. p-(kH)27fp
=
(3.7)
27fp-k,
IIF(z)1I and Dp(x, h) is the open disk centered at x along the
zEDp(x,h)
direction h (see Definition 2.3), it follows that the partial sums in the right hand side of (3.6) also converge for any fixed TEe with ITI < p. Thus our assertion follows from Corollary 11.1.2.
~
From (3.5) and (3.7) we obtain directly the next generalized Cauchy inequalities. COROLLARY 3.2.
x E
1),
If F E H(1),!D) and IIF(x)11 ~ M for all x E 1), then for any
hEX and any p ~ p(x, h) we have (3.8)
REMARK 3.1. Actually, from (3.5) it follows the sharper inequality
(3.8') where M(x, h) ;?
IAI
sup
IIF(x + Ah)lI, a fact which is useful in many cases.
~p(x,h)
Ok F(x, h) in (3.4) are called the Taylor coefficients of the operator F at the point x in direction h. Note that (3.4) can be seen as the REMARK 3.2. The expressions
88
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
Taylor series of the vector-function f(T) = F(x with
ITI < p(x, h).
+ Th)
for fixed x
E 1),
hEX and T,
Using the obvious relations k = 0,1,2, ... ,
where y* is an arbitrary element ofZJ*, we obtain that Ok F(x, h) has the representation
If the operator F is holomorphic, some other representations for Ok F(x, h) are
also valid. For example, the next result is true.
Let 1) be an open set in X. If an operator F E H(1), X) is locally bounded on 1), then F is indefinitely Frechet differentiable. If dkF(x, hI"'" h k) is the derivative of order k of F, then we have
THEOREM 3.3.
1
k! dkF(x, h, h, .. . , h)
=
k
0 F(x, h),
(3.9)
i.e., the coefficients Ok F(x, h) in the decomposition (3.4) are homogeneous forms in h for fixed x E 1) (see §6 of Chapter I). (Here, for convenience, we put 6° F(x, h) <J
= F(x).)
Since 1) is an open set, by Proposition 2.1, we have that p(x, h) ;;:, o(x) > 0
for all h with Ilhll = 1. Assume that the inequality IIF(z)11 ~ M is fulfilled for all points in the ball centered at x and with radius c(x). Then, for any u with Ilull ~ 0 = = min{o(x),c(x)}, and taking h = UT- I , Ilhll = 1, relation (3.4) implies that 00
F(x
+ u) = of(x, h)T + L
Ok F(x, h)Tk
+ F(x) =
k=2
= A(x)u + w(x, u) + F(x), where A(x) E L(X, ZJ). Using Cauchy inequality (3.8) and the well known formula for the geometric series, we obtain
89
Cauchy integral theorems for any u
E
X,
Ilull < ();
this estimate proves the Frechet differentiability of the
operator F. Hence, it follows that the vector-function F(X+TIh l + .. .+Tnhn): e n ---> ---> ~ ~ for any fixed hI, ... ,hn ~ has continuous partial derivatives with respect to TI, ... , Tn
in a polydisk it with a sufficiently small radius, i.e., it is holomorphic on
this polydisk with respect to each of its variables. By the fundamental theorem of Hartogs (see, e.g., Shabat [2]), the function (F (X
+ ~ Tkhk)
,y* ) is holomorphic
in the polydisk it with respect to all of the variables, and consequently it has partial derivatives of any order. By Theorem 2.2 it follows that the vector-function F(x
+
+ ... + Tnhn)
has the same property. So, it follows, in particular, the existence of the second order derivative +TIh l
The symmetry of d 2 F and its linearity in hI, implies the linearity in h 2 . Analogously, we have dkF(xI' hI"'" h k )
= aTI,
.~~, aTk F(x + TIhl + ... + Tk h k)I
T 1= ... =Tk=O'
(3.10)
for all k;;::: 2. Taking into account the uniqueness of the Taylor decomposition, we obtain equalities (3.9). ~ Let us remark that from (3.9) and (3.10) we can infer the following representation for the form {)k F (- , . ):
a
k k 1 () F(x, h) = k! aTI,"" aTk F(x
+ TIhl + '" + Tkhk) I Tl=' '=Tk=O
(3.11)
h 1=,,·=hk=h
(an analogue of Liouville Theorem). Let X and ~ be Banach spaces, F E 'H(X, ~), and assume that for a point x E X there exists a positive integer m > 0 such that
. COROLLARY 3.3
IIF(x +
h)11 ~ Mllhll m
(3.12)
90
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
for all elements hEX with sufficiently large norm. Then the operator F is a polynomial of degree ~ m, i.e., F is a finite sum of homogeneous forms in h. In particular, if the value IIF(x)11 is uniformly bounded for all x E X, then F(x) takes a constant value in !D.
Taking
T
= 1 in (3.4) (note that p(x, h) =
00), we have
00
F(x
+ h)
=
L b F(x, h), k
k=O
where, according to Theorem 3.3, bk F(x, h) are homogeneous forms of order k in h. The inequalities (3.7) and (3.12) imply
where p = IIhll is sufficiently large. For p k
---t
00, we obtain that b k F(x, h) = 0 for
> m, so our first assertion is proved. The second assertion follows from the first
one taking m = O.
~
§4. Uniqueness theorems and maximum principles The next theorem easily follows from representation (3.4).
Let 1) be a p-open set which is a C-star relatively to a certain point Xo E 1) (i.e., x E 1) implies Xo + >.(x - xo) E 1) for all >. E C with 1>'1 ~ 1). If a p-holomorphic operator F and all its Taylor coefficients equal zero in Xo, then F(x) == 0 on 1). THEOREM 4.1.
If the domain 1) is arbitrary, but F is holomorphic, then the next result is true.
Let 1) be a domain in X and let F E H(1),!D) be a locally bounded operator. IfF and all its derivatives are zero at a certain point Xo E 1), then F(x) == 0
THEOREM 4.2.
on 1).
Let
~
be the set of those points in 1) where F and all its derivatives are
equal to zero. Put
~n
= {x
E
1) : bn F(x, h)
= O}, n = 0,1,2, .... Since bn F(x, h)
are continuous on 1), all the sets ~n are closed. But ~ = n~n' hence ~ is also closed. We show now that ~ is open in 1). Let z E ~. By (3.9), we infer that b k F(z, h) == 0
Uniqueness theorems
91
for all hEx and k = 0,1,2, .... Let W be a ball centered at z and included in 1) (recall that 1) is open). By Theorem 4.1 we have F(x)
W
s: <1:, so <1: is open.
1).
Since Xo E <1:, it follows that <1: = 1).
== 0 on W. This means that
Thus we obtain that <1: is simultaneously closed and open in the connected set ~
From Theorems 3.3 and 4.1, we obtain:
Let 1) be a domain in x and let F E H(1), Z) be a bounded operator. IfF equals zero on a neighborhood of a point Xo E 1), then F(x) == 0 on 1).
COROLLARY 4.1.
One of the most fruitful features of the complex analysis is the maximum principle for the norm of a holomorphic operator. We state without proof the onedimensional version of this principle.
Let [2 be a bounded domain in C and 'P( T) a holomorphic function on [2 with values in C. If the expression I'P( T) 1attains its greatest value at a certain point a E [2, then 'P(T) is constant for all T E [2. If 'P(T) is holomorphic on [2 and continuous on [2, then I'P(T)I attains its greatest value at a certain point T E 8[2. THEOREM 4.3.
We consider now a vector-function f (T) with values in a Banach space Z) which is holomorphic on a bounded set [2
s: C.
Let us suppose that the expression
Ilf(T)11 (1If(a)11 = M. Consider the function 'Py*(T) = U(T),Y*I, where y* E Z)*; 'Py*(T) is also holomorphic on [2. Obviously, l'Py*(T)1 (1If(a)IIIIY*II. In particular, for any y* with IIY*II = 1, we have l'Py' (T)I (M. Let M(y*) = sup l'Py* (T)I; then M(y*) (M. From Hahn-Banach attains its greatest value at a point a
E [2,
Ilf(T)11
i.e.,
TErl
theorem it follows that there exists an element Yo
'PYjj(a) = U(a),Yol = Ilf(a)11 = M for all T E [2, i.e.,
=
point
bE
[2 so that
with
Ilyo II =
1, such that
M. Hence, by Theorem 4.3 it follows that 'Pyjj(T) (f(T),Yol
Let us show now that
E Z)*
Ilf(T)11
Ilf(b)11 < M.
=
=
(f(a),Yol
Ilf(a)ll.
=
(4.1)
Suppose it is not; then we can find a
But then l'Py* (b)1 < M, which is impossible.
Z) is strictly convex (see Chapter 0, §9), then from (4.1) and from the equality Ilf(T)11 = Ilf(a)ll, it follows that f(T) = f(a) If we assume, in addition, that the space
for all
T
E
[2.
Let us return now to the general case of a space X. Let F E Hp(1), Z), where 1)
is a p-open set in the Banach space X. Assume also that for a certain point a
E [l,
92
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
the inequality IIF(a)11 ;? IIF(x)11 for all x E 1) is true. Then the norm of the vectorfunction f(T) = F(a+Th):C
----* ~
attains its greatest value at the point T = O. From
our previous results it follows that Ilf(T)11 is constant for all
T
with ITI < p(x, h).
Since the vector hEX was taken arbitrary, the equality IIF(x)11 = IIF(a)11 is valid on any C-star set 2!J centered at the point a and completely contained in 1). Now, if the space ~ is strictly convex, then, from IIF(x)11 = IIF(a)ll, it follows that F(x) = F(a) for each x E 2!J. Moreover, if 1) is a domain, then, by Corollary 4.1, the equality
F(x) = F(a) follows for all x E
1).
Thus we obtained the following generalization of the maximum principle.
Let
THEOREM 4.4.
erator, where IIF(x)11
~
1)
be a p-open set in X and let F E Hp(1),~) be an op-
is a Banach space. If for a certain point a
E 1),
~
IIF(a)11 is fulfilled for all x E 1), then: 1) the norm of the operator F is constant on the C-star component 2!J(a) of
1), centered at a; 2) if, in addition, the space ~ is strictly convex, then F(x) E
the inequality
=
F(a) for x E
2!J(a).
COROLLARY 4.2.
continuous on
1)
If the set 1) is open and bounded in X and F E
H(1),~)
is
then the following equalities are true: sup IIF(x)11 xE:D
= sup IIF(x)11 = sup IIF(x)ll· xED:D
xE1)
REMARK 4.1. The assumption in the second part of Theorem 4.4 on the space ~ to be strictly convex is essential. Indeed, let us take ~ = C 2 with the Tchebysheff norm Ilyll = max{IYll,IY21} (y = (Yl,Y2) E C 2) and consider the vector-function
f(T) = (1, T) which is holomorphic on the unit disk ITI ~ 1. We obviously have Ilf(O)11 = Ilf(T)11 = 1 for any T with IITII ~ 1. On the other hand, f(T) i- f(O) for T
i- o.
§5. Schwarz Lemma and its generalizations Schwarz Lemma, which is so significant for the theory of functions of a single complex variable, plays an important role in the multi-dimensional and infinite-dimensional cases too, both for qualitative and quantitative aspects. The quantitative aspects arise in relationship with various estimates, such as, for example, the domains of
93
Schwarz Lemma
existence for the solutions of a functional equation, the rate of convergence of an approximation process, and so on. The classical version of Schwarz Lemma asserts the following:
Let
f
be a holomorphic function on the unit disk ..1
which transforms ..1 into itself, i.e., f(Ll) a)
b) Moreover,
If('\)1 :::; 1,\1, 11'(0)1:::; 1.
for any'\
E
<:;;;
..1. If f(O)
=
= {,\
E C
1,\1 < 1}
0, then:
..1;
c) the equalities in both a) and b) occur if and only if f('\) = 1'(0)'\. Each of the assertion a) and b) admits various generalizations. We consider some of them in the sequel. THEOREM 5.1. -+ ~
let F: X
Let X and
~
be Banach spaces, let
be a p-holomorphic operator
on~.
~
be a p-open set in X and
Assume that, for fixed x E
~
and
hEX, F admits the representation 00
F(x + Th)
=
L
(5.1)
Tk 8k F(x, h),
k=m
for a certain m
~
0 and all
TEe
with
ITI :::; p(x, h).
If Mp(x, h)
=
sup
IIF(z)ll,
zEDp(x,h)
where p:::; p(x, h), then
for all T with
ITI :::; p.
Take an arbitrary functional y* E ~* with Ily* II = 1 and consider the function
Then, according to the maximum principle, we infer that
for all
T
for all y*
with
ITI :::; p,
E ~*
with
or, equivalently
Ily* I
=
1.
94
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES We are done using Corollary 0.4.1 of Hahn-Banch Theorem.
~
In particular, an analogue of the assertion a) in the classical Schwarz Lemma follows from:
Let F be a Fhkhet differentiable operator on a ball 1) centered at zero. Suppose that F satisfies the conditions F(1) ~ 1) and F(O) = O. Then COROLLARY 5.1.
IIF(x) II : :; Ilxll
for all x
E 1).
Ilhll = 1 and observe that the E C with ITI : :; Ilxll and m = 1. ~
Take x = Th, where
representation (5.1) for all
T
operator F admits
The assertion b) can be also easily generalized for arbitrary Banach spaces. Namely, under the assumptions of Corollary 5.1, the inequality
IIF' (0) II : :; 1
(5.2)
is true. This follows directly from Cauchy inequality (3.8) for k = 1. The case of assertion c) is considerably more complicated. All depends upon how we understand, in the general case, the right hand side of the equality in assertion c), and upon the properties of the norm of the space X. If the mapping f'(O)>.. is understood as a linear operator applied to
>.., then
assertion c) is no longer true in the general case.
Indeed, consider the space C 2 , with the Tchebysheff norm, and the operator
F:C 2 -+ C 2 given by the formula F(z)
unit polydisk 11 = {z E C 2 ,
= (zl,zD,
Ilzll < 1} into itself. IIF'(O)II =
z
= (Zl,Z2) E C 2, which maps the
Clearly F(O)
= 0 and
II(~ ~)II = 1.
However, the operator F is obviously nonlinear.
~
If we assume, in addition, that the operator F'(O) is a linear isometry, assertion
c) becomes true in the general case, too. This follows from the next generalization of a theorem due to H. Cartan for the finite-dimensional case. (Apparently, the first infinite-dimensional version of this theorem was obtained in 1969 by L. Harris [1], who supposed that 1) is a ball. His proof heavily relies on this assumption which is a serious restriction in many cases.)
Schwarz Lemma
Let 1) be an arbitrary bounded domain in the Banach space X and be an operator satisfying the conditions
THEOREM 5.2.
let P E
95
7-l(1),!D)
(5.3) and P(z)
=
(5.4)
z,
for a certain point z E 1). If P'(z) = I (the identity operator on X), then P is the restriction of the operator I on 1), i.e., P(x) = x for all x E 1).
From (5.3) it follows that the operators pn = popn-l, n = 1,2, ... , pO
I, are defined on
too; they satisfy pn
1),
(5.4) it follows that pn(z)
=
E
=
7-l(1), X) and pn(1)) <:;; 1). Moreover, from
z and (pn)'(z)
= I. Therefore, the operators
pn admit
the representation 00
pn (x
+ h) =
z
+I h +
L
{yk pn (z, h),
mn ~ 2,
(5.5)
k=m n
for all h with
Ilhll < dist(z, 81)) =
inf
yEa'!)
liz - yll = r.
Let us show that {jk P(z, h) =0 for all h with Ilhll ~ r and all k=2, 3, .... Suppose this is not the case; then there exists a positive integer ml such that {jm, P(x, h) is the first non-identically zero form in the decomposition (5.5) of the operator P. Since the domain
1)
is bounded, from Cauchy inequality (3.8) we have
IW pn(z, h)11 for all n
=
~ M(k) <
00,
Ilhll ~ r,
1, 2, . . . .
A direct computation using induction proves the equality
whence II{jm'p(z,h)11 ~M(md·n-l,
n=I,2, ....
But this means that 8m , P(z, h) == 0, a contradiction. Therefore P(z + h) = = z+h for all h with Ilhll ~ r. By the uniqueness theorem we conclude that P(x) = x for all x E 1). ~
96
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
COROLLARY 5.2. Let X and ~ be complex Banach spaces and F an operator which
acts from the unit ball :Dx centered at' zero in X into the unit ball :D!D centered at zero in
~.
If FE H(:Dx, ~), F(O) = 0, and F'(O) is a linear isometry from X into ~
=
then F(x)
F'(O)x for all x E :Dx.
Denote A
inverse A-I:~
---+
= F'(O). Under our assumption, the operator A has a continuous = Ilyli. Hence H (:D x, ~). Moreover, By Theorem 5.2, we obtain p(x) = x for all x E :D x . Then
X, and since
IIAxl1
=
Ilxll,
we infer that
it follows that P = A-I F acts from :D X into :D X and P P'(O) F(x)
= A-I A = Ix. = Ap(x) = Ax .
IIA-Iyll
E
..
An operator F::D
:D is called biholomorphic if it is holomorphic and one-
---+
ta-one, and its inverse F- I is also holomorphic. COROLLARY 5.3. If an operator F is biholomorphic on the unit ball :D centered at
the zero element in X and F(O) = 0 then F(x) == Ax, x
E
:D, where A is a linear
isometry on the space X.
Consider the linear continuous operator B
from (1.4.1) we obtain B has a continuous inverse B
0
F'(O) =
=
(F- I ),( 0). Since F- I 0 F = I,
= I. This means that the operator
A-I. Moreover, from (5.2) it follows that
A
=
F'(O)
IIAII ::;; 1 and
IIA-III ::;; 1.
Thus IIAxl1 = Ilxll for any x E X, i.e., A is a linear isometry. Our assertion follows now from Corollary 5.2 ... The space X will be called a space with a generalized Krein-Phillips transform, or shortly a K-P-space, if for any point a in the unit ball :D centered at zero in X there exists a biholomorphic operator P a on :D such that Pa(a) = a. As examples of K-P-spaces we mention the space en with the Chebyshev or the Euclidean norm, any Hilbert space over the field e of complex numbers, the algebra of all linear operators on a Hilbert space, and others. In particular, for the one-dimensional space e, the operator P a has the form P (A) = A - a (5.6) a Aa - 1 ' where
a is the complex-conjugate of a.
Let X be a K-P-space and let F: X ---+ X be a biholomorphic operator on the ball :D centered at zero in the space X. Then the operator F admits the representation
COROLLARY 5.4.
(5.7)
Schwarz Lemma where A is a linear isometry on X, a is a point in
~,
and b = F(a).
Indeed, for any point a E ~ the operator tPbFtP;;l is biholomorphic on ~
and satisfies the condition tPbFtP;;l(O)
5.3.
97
=
O. Now our assertion follows from Corollary
~
Finally, let us remark that inequality (5.2) gives an estimate for the Fn§chet derivative of the operator F only at a point. As we have already noted in Chapter I, a uniform estimate for the derivative on the whole domain
~
does not exist. However,
Scwarz Lemma gives the possibility of estimating the Frechet derivatives uniformly on any closed subset of the
domain~.
The next result is implicitly contained in T.
Hayden and T. Suffridge [1]. THEOREM 5.2.
and let F
E 1i(~,
Let ~ be the unit ball centered at the zero element of the space X X) be an operator satisfying the condition
Then the inequality , Ilxll IIF (x)xll ~ 1 _ IIxl12
is fulfilled for any x E
Fix x
E ~
~.
and choose a linear functional x*
E
X* such that Ilx* II = 1 and
(F'(x)x, x*) = 11F'(x)xll. Consider the function f(>..) = (F(>"xllxll- 1 ), x*), which is holomorphic on .1 = {>.. E C : 1>"1 < I}. Obviously If(>..) I < 1 for>.. E .1. Since Ilxll < 1, we have f(llxll) = (F(x),x*) = a, where lal < 1. The functions
( =
..) = >.. - Ilxll >"llxll-1 and
'l/J(w) = w - f(llxll) wf(llxll) - 1 are holomorphic on .1 (see (5.6)), and the function g(() conditions of Schwarz Lemma: Ig(()1 < 1 and g(O) equivalently
f(>..) - f(llxll)
If(>..)f(llxll) -
1
I
~
I >.. -
=
o.
=
'l/J(f(
Consequently Ig(()1 ~ 1(1, or
Ilxll I >..llxll - 1 .
98
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
Whence, we obtain the inequality
I : : :; If(A)7(lfXlf) If(A)A-- f(llxll) Ilxll Allxll - 1 Taking the limit for A ----+
Ilxll, we
have 1 -If(llxIIW 1 _ IIxl1 2
,
If (1IxlJ)1 : : :; But
1'(A)
=
11·
: : :;
1 1-
Ilx112'
(F'(AXllxll- 1 )xllxll- 1 , x*), so 1'(llxll) = (F'(x)x,x*)llxll- 1 = IJF'(x)xllllxll- 1 ,
hence
,
IIF (x)xll : : :;
Ilxll Ilx112'
1_
§6. Uniformly bounded families of p- holomorphic (holomorphic) operators. Montel property Let
1)
be a p-open set in X. Until now, we did not use the fact that
1)
itself can be
considered as a topological space with the p-topology defined in §2. Recall that in the initial topology of the space X, a p-holomorphic operator can have discontinuities (see Example 1.1.2). However, for the p-topology the next assertion is true.
Any p-holomorphic bounded operator is p-continuous. We prove below a stronger result.
Let 1) be a p-open set and let F = {F"'}"'EQl, F", E 1i p (1) , ~), be a uniformly bounded family (with respect to the norm topology of~), i.e., IJF",(x) II : : :; :::::; M < 00, for all a E 21. Then F is equicontinuous as a family of operators acting from the set 1) with the p- topology into the space ~ with the norm topology.
THEOREM 6.1.
Given an arbitrary c: > 0, take N ~ max{2c:- 1 M, I} and let x and x' be such that p(x, x' - x) > N. Then, setting x' - x = (,h, where Ilhll = 1, (, E C, and using Lemma 2.1, we obtain <J
p(x, (,h) = p(x, h)I('I- 1 >
N
99
Montel property or
1(1 < N-1p(x, h) ::;:; p(x, h). It follows that Fa(x') = Fa (x
+ (h)
has the Taylor representation (3.4). Then, by
Theorem 5.1, we have
IlFa(x') - Fa(x)11 ::;:; IlFa(x + (h) - Fa(x)11 ::;:;
2MI(1 [p(x, h)r 1 .
(6.1)
As h = (-l(X' - x), we actually have
COROLLARY 6.1.
If the set 1) is open, then a uniformly bounded family F =
= {Fa }aE'2l of operators holomorphic on 1), with values in a Banach space~, satisfies a Lipschitz condition, uniformly with respect to a
p(x, h) ;? 8(x) > 0 (see
2.2),
Ilx - x'il : ;:; 8(x).
21.
where hEX,
IlFa(x') - Fa(x)11 ::;:; 2MI(1 [8(X)]-1 ::;:; for all x' such that
E
Ilhll
=
1, (6.1) implies
2M [8(x)r 1 Ilx - x'il
~
Notice that the last relation implies the local uniformly boundedness of the Frechet derivatives of the operators Fa, a E
2(,
too.
We return now to the general case. Let us consider again 1) as a topological space with the p-topology. On the set of all continuous mappings from 1) into
~
we
introduce the topology Tp(1) , ~) of compact convergence (i.e., the uniform convergence on compact subsets in the p-topology of 1), see §5 of Chapter 0). Using Theorem 6.1 and Theorem 0.5.4, we obtain the following analogue of the Montel property.
Let FM be a bounded subset of Hp(1), ~), i.e., FM c::;; {F E : IIF(x)ll::;:; M < 00 for all x E 1)} and assume that for any x E 1),
THEOREM 6.2. E Hp(1),~)
the orbit FM(X) = {y E ~ : y = F(x), FE F M } is a sequentially compact subset of ~. Then the family FM is sequentially compact in the topology Tp (1) , ~). In contradistinction with the classical version of Montel theorem (X = en, ~ = em) we assume in Theorem 6.2 the compactness of the orbits; this compactness follows, for em, from the boundedness of the family F M. The next example shows the necessity of our assumption. Let X = ~ = £2. Consider the REMARK 6.1.
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
100
family {Bd ~ I of linear bounded operators (hence holomorphic operators) defined on the set {x = (0, ... ,0,
£2 : Ilxll < R < oo} by the equalities Bkx = Xn ,·· .). Obviously IIBkXl1 < R for all X with Ilxll < R, k= 1, 2, ....
= (XI,""X n , ... )
Xl,""
E
'--v-'" k
However the sequence {Bd~l is not compact; it is enough, for instance, to consider the values at the point x = (1,0, ... ,0, ... ). Suppose now that
~
is the dual space of some Banach space and that
~
is
metrizable in the corresponding ultraweak topology (this is the case, for instance, when
~
is separable).
0), any bounded set in
Then, by Alaoglu-Bourbaki Theorem (see §1O of Chapter ~
is sequentially compact in the ultraweak topology. Con-
sequently, we can use Theorem 0.5.4 and Theorem 6.1 in order to obtain the next result. THEOREM 6.3.
Suppose that
bounded set FM
~ HpCD,~)
If the space
~
~
is metrizable in the ultraweak topology. Then any
is sequentially compact in the topology Tp (:.D , ~).
is reflexive, the ultraweak topology in Theorem 6.3 can be
replaced with the weak topology of the
space~.
Thus, Theorems 6.1-6.3 provide a
Banach-Steinhaus type theorem (see §8 of Chapter 0). (Note, once more, the similarity between the geometric properties of holomorphic bounded operators and those of linear continuous operators.)
Assume that ~ is equipped with the norm (resp. ultraweak) topology. Let {Fn}~=l' Fn E Hp(:.D, ~), be a sequence of bounded operators. Then the sequence {Fn}~=l is convergent, in the topology Tp(:.D, ~), to a bounded operator F E Hp (:.D , ~) if only if the following two conditions are simultaneously fulfilled: THEOREM 6.4.
1) the family
{Fn}~=l
is uniformly bounded on :.D;
2) the sequence {Fn(X)}~=1 is fundamental (resp. ultraweak fundamental) for all x E :.D' , where:.D' is a subset of:.D, dense in the p-topology. We conclude this section with two generalizations of Vitali Theorem.
Let:.D be a p-open set which is a C-star relatively to a point Xo E :.D, and let {Fn}~=l' Fn E Hp(:.D,~), be a sequence of operators, uniformly bounded on :.D. Assume that for any hEX, Ilhll = 1, the sequence {Fn}~=l is fundamental on the disk Dp(xo, h) ~ 1) in the norm topology of the space~. Then the sequence is fundamental on :.D with respect to the norm topology of~, and, consequently, it is THEOREM 6.5.
Mantel property
101
strongly convergent to some operator FE Ji p ('1) , ~).
Let u be an arbitrary point in '1).
we find an element hEX and A
E
C with
Using the properties of the set '1),
IAI <
1, such that Xo
+h
E '1)
and
U = Xo + Ah. Moreover, p(xo, h) ?: 1. Then the vector-functions fn(() = Fn(xo + (h) are holomorphic on the unit disk 1(1 < 1 and all of them are bounded on this disk, with respect to the norm of~, by a constant M < 1. By our assumptions, there exists d> 0 such that the sequence {fn}~=l is fundamental on the disk 1(1 ~ d. (From now on, the proof reproduces almost verbatim the classical one. However, we may need some estimates established in the sequel, so we present a complete proof.) By virtue of Theorem 5.4 the sequence {fn}~=l is uniformly convergent, i.e., for any given positive 8, there exists N(8) such that whenever n > N(8), we have Ilfn(() - fn+m(() II < 8,
for
1(1 ~ d,
and
m
= 0, 1, ...
From these inequalities and by Cauchy inequality (3.8) we obtain
where 8~m are the Taylor coefficients of the operator Fnm
=
Fn - Fn+m . Note that
we also have
uniformly with respect to k
= 0, 1,2, ....
Then, setting
IAI = r «
1), we obtain
00
IlFn(U) - Fn+m(u) I
=
Ilfn(A) - fn+m(A)11 ~
L
118~mF(xO' h)Ak II·
k=O Using the formula with remainder for the geometric series, we have
=
8(1 - ~)
-1
(1 - (~)
P)
+ 2MrP(1 _
r)-l.
HOLOMORHIC OPERATORS ON COMPLEX BANACH SPACES
102
Now, for any c > 0 and 0 < q < 1, taking p> In
[cC1- r)(l- q)(2M)-lJ
(lnr)-l
(6.2)
and
{) > cq(r - d)dP- 1(r P - dP)-l, we obtain IlFn(u) - Fn+m(u) II <
E
(6.3)
as soon as n > N({)). ~
For the case of ultraweak convergence, the proof of the analogous assertion can be sligthly simplified; it is enough to use Montel property. However, we will be mainly interested in the case of holomorphic operators.
!D is metrizabile in the ultraweak topology and let ~ be a domain in X. Let {Fn}~=l be a uniformly bounded sequence of operators holomorphic in ~. Suppose that {Fn}~=l converges in the ultraweak topology to an operator F E 1{(~I,!D), for any element of a certain open set ~' ~~. Then F admits a holomorphic continuation on the whole ~ and the sequence {Fn}~=l' converges to this continuation in the ultraweak topology, for each element of ~. THEOREM 6.6. Assume that the space
By Theorem 6.3, the sequence {Fn}~=l contains a subsequence {Fnk}k=l which is convergent in the ultraweak topology, for any element of ~, to a certain <J
F E 1{(~, !D). F I ~' = F.
F is a holomorphic continuation of the operator
operator
The operator
F, i.e.,
The uniqueness theorem implies that any other convergent
subsequence {Fn"J~=l has the same ultraweak limit
F.
The proof is complete. ~
In spite of the fact that the previous proof is very simple, it is not at all constructive; it provides no estimates analogous to those in (6.2) and (6.3), which are very useful in many applications (see §2 of Chapter VI below).
Chapter IV Linear operators
This chapter, as well as Chapter 0, is mainly auxiliary. However, compared with Chapter 0, the topics considered here are more special. Some of the subsequent results were presented only in research papers, but not in monographs. Therefore, our exposition in this chapter will be quite detailed, and will include proofs.
§1. The spectrum and the resolvent of a linear operator 1. CLOSED OPERATORS. Let X and ~ be complex Banach spaces and A: X ----+ ~ a linear operator (see Chapter 0, §8 for definition). Recall that the operator A is equally well described as the set of pairs (x, Ax), with x E D(A) ~ X. The operator A is invertible (i.e., A has an inverse operator A-I) if and only if its kernel N (A) = = {x E D(A) : Ax = o} equals {O}. In this case, the operator A-I is defined from R(A) into D(A); moreover, A-I Ax = x for all x E D(A) and AA-Iy = y for all y E R(A).
An operator A is called closed if it is closed as a subset of the topological product X x ~ (i.e., if Xn E D(A), Xn ----+ x and AXn ----+ Y for n ----+ 00, then x E D(A) and Ax = y). If A is closed and invertible, then obviously A-I is also closed. If an operator A is bounded and defined on the whole X, and if B is a closed operator, then the operator A + B ~ defined on D(B) ~ is also closed. THEOREM 1.1. A bounded linear operator A is closed if and only if its domain D(A)
LINEAR OPERATORS
104
is closed. A is bounded then Xn E D(A) and Xn ~ x for n Ax. Therefore A is closed if and only if D(A) is closed. ~
implies that AX n ~
~ 00
In particular, any operator in L(X, ~), i.e., any bounded operator defined on the whole X, is closed. The converse is also true; we give the result without proof. THEOREM 1.2 (Closed Graph Theorem). Any closed linear operator A: X ~ ~ with
the domain of definition D(A)
=
X is closed.
2. SPECTRUM AND RESOLVENT. In this subsection we will consider X =~. Let A be a linear operator on X. A point A E C for which the operator (A - AI)-1 exists and is bounded on the whole X is called a regular point of A. The set of all regular points of A is denoted by p( A). THEOREM 1.3. A linear operator A with at least one regular point is closed.
The complement of the set p(A) in C is called the spectrum of the operator A and is denoted by a(A). According to Theorem 1.3, the spectrum of a non-closed operator coincides with the set C of complex numbers and the set of its regular points is void. This is why, everywhere below is this section, we will consider only closed linear operators. Let A be a linear operator on X. The resolvent TA of A is defined on the set p(A) of its regular points by TA(A) = (A - AI)-1, A E p(A). If no confusion may arise, the subscript A in T A (A) will be omitted. From the relation
T(/L)(A - AI)T(A) - T(/L)(A - /LI)T(A)
= (/L -
A)r(/L)T(A),
(A,/L E p(A))
the so-called Hilbert identity follows: (1.1)
which implies the commutativity of the values of the operator-function T(A). THEOREM 1.4. The set p(A) of the regular points of an operator A is open and the
resolvent T(A) is an analytic function on p(A).
The spectrum and the resolvent of a linear operator which is defined and bounded on the whole X.
105
This means that JL E p(A) and 00
r(JL) = (J - (JL - A)r(A))-l r(A). It follows that r(JL) = L)JL - A)nrn+l(A), where n=O
the series is convergent for all JL satisfying analytic operator-valued function. ~ COROLLARY 1.1. The spectrum
IJL - AI < Ilr(JL)II- 1 .
Thus r(A) is an
(T(A) is closed.
<J The conclusion follows directly from Theorem 1.4 and from de Morgan formula (see §1 of Chapter 0). ~
Note that the conclusion of Corollary 1.1 is also true for non-closed operators, since in this case (T(A) = C. Thus, the spectrum of a closed linear operator A on a complex Banach space is a closed subset of C, which can be void. If the spectrum (T(A) is non-void, it has three parts: 1) (Tl (A) consisting of all those A E (T( A) for which A - AI has a non-zero kernel N(A - AI);
2) (T2(A) consisting of those A E (T(A) for which N(A - AI) = {O}, but there exists a sequence {xn}nEN such that Ilxnll = 1, n E N, and (A-AI)Xn ~ 0 as n ~ 00; 3) (T3(A) consisting of all A E (T(A) such that N(A - AI) = {O} and R(A - AI) = = R(A - AI). It is obvious that (Tl(A), (T2(A) and (T3(A) are mutually disjoint and (T(A) =
i=l
The elements of the set (Tl (A) are called the eigenvalues of A; if A E (Tl (A) then there exists x E X, x =f:. 0, such that Ax = AX. The set X.x of all eigenvectors x corresponding to an eigenvalue A, together with the zero element of the space X, form a subspace of X called the eigenspace of A. The dimension of the eigenspace X.x is called the multiplicity of the eigenvalue A. Consider now the following question: how does the spectrum of an operator looks like. When the operator is unbounded, its spectrum may be the whole complex plane cc. Let us take a bounded closed operator A. In this case, by Theorem 1.1, the domain D(A) is closed in X. From now on we will consider in this section that D(A) = X, i.e., A E L(X) (= L(X, X)). COROLLARY 1.2. The spectrum
(T(A) is non-void and bounded.
<J Assume, on the contrary, that p(A) =
cc.
Using the Hilbert identity for
A = 0 we obtain that r(JL) = JL- 1 r(0)(JL- 1 J - r(O))-l. It follows that IIr(JL) I ~ 0 as JL ~ 00, therefore the resolvent is bounded. Thus, the operator-valued function
LINEAR OPERATORS
106
r: C
L(X) is analytic and bounded everywhere on C. Therefore, the function for: C --t C is analytic and bounded for any f E (L( X)) *. By Liouville Theorem, for is constant for any f E (L(X))*, hence r()..) is constant. Since r( 00) = 0, we obtain r()..) == 0, which contradicts the definition of the resolvent. For 1)..1 > IIAII, using Theorem 0.8.4 we have that the operator A - )..J = --t
= -).. (I -
~A )
maps homeomorphic ally X onto X. Therefore the spectrum a(A) is
contained in the disk
1)..1:::;; IIAII·
~
LEMMA 1.1. The set a3(A) is open.
),,1) is closed and N(A - )..1) = {O}. By Theorem 0.8.5 the operator (A - )..1)-1: R(A - )..1) --t X is bounded. Using Theorem 0.8.4 we get that the operator 1+ ().. - f.L)(A - )..1)-1 is a homeomorphism from X onto X for any f.L E C such that I).. - f.L1 < II(A - )..1)-111- 1. <J According to the definition, if ).. E a3(A) then R(A -
It follows that A - f.LI
= (A - )..1)(1 + ().. - f.L)(A - )..1)-1)
is a homeomorphism from X onto R(A - )..1), i.e., f.L
E
a3(A). ~
For)" E a1(A), we consider the non-decreasing sequence of subspaces
{O}, N(A - ),,1), N((A - )..1)2), ... , N((A - )..1)k) , .... If there exists m ~ 0 such that N((A - )..1)m) = N((A - )..1)m+j) for any j E N, then the subspace N((A - )..1)m) is called the root subspace corresponding to the value ).., and the integer m is called the rank of )... Let P be a bounded projection on X, i.e., a bounded linear operator with the property p 2 = P, and assume in addition that P and A commute: PA = AP. Then the subspaces Xl = PX and X 2 = (1 - P)X are invariant for A, i.e., AX 1 ~ Xl and AX2 ~ X 2· Indeed, setting P 1 = P and P 2 = I - P, for any Xi E Xi, we have
Assume that the spectrum a(A) of the operator A E L(X) is represented as the union of two disjoint closed subsets a1 and a2. Then there exists a rectifiable, simple (i.e., without self-intersections) and closed curve 'Y such that a1 is contained inside the curve 'Y and a2 lies outside. In this case the next result is true. THEOREM 1.5. Suppose that the spectrum a(A) of the operator A admits a partition
(as described above) into two sets a1 and a2. Then the space X decomposes as a
The spectrum and the resolvent of a linear operator
107
topological direct sum X l +X2 of two Banach spaces Xl and X 2 which are invariant for A, and such that the spectrum of the restriction A I Xi coincides with ai, i = 1,2.
Set PI =
The operator PI is bounded and P'f
-~ /r(~)d~. 2m
(1.2)
= PI; indeed:
(The computations above use the Hilbert identity and the Cauchy Integral Theorem). Therefore PI is a projection, and so is P2 = I - Pl. Thus X = X l +X2 where Xi = PiX, i = 1,2. Further, for each A E p(A), we have Plr(A)
= r(A)Pl ,
hence API = PIA. Therefore, the subs paces Xl and X 2 are invariant for A. Set Ai = A I Xi, i = 1,2. One easily checks that rAi(A) = r(A) I Xi. It follows that P(Ai) ;2 p(A). In addition, p(AI) ;2 a2. Indeed, rA , (A)U = r(A)U = r(A)Pl U for all U E Xl, A E p(A). For any A tt. ,,/, using (1.2) and Hilbert identity (1.1), we have
1/
r(A)Pl = - - . 2m
1/
r(A)r(Od~ = - - .
2m
d~ . (r(A) - r(o) \ C A
-
(1.3)
.,
If A lies outside the contour ,,/, then (1.4)
Since the right hand side of (1.4) is holomorphic outside the contour ,,/, then r(A)Pl' and consequently rA, (A), has an analytic continuation, holomorphic outside "/. This analytic continuation is the resolvent of AI. Therefore p(AI) contains the exterior of the contour ,,/, so a(Ad ~ al. Analogously, from (1.3) it follows that r(A)Pl = r(A)
+ 2~i
Jr(~) 'Y
~~
A
LINEAR OPERATORS
108
if A lies inside the contour "y. Thus r(A)(I - Pd (= r(A)P2 ) has an analytic continuation, holomorphic inside T As above, one concludes that a(A2) ~ a2. On the other hand, a point A E a(A) does not belong to peAl) n p(A2). Indeed, if it does, then A E peA) since the operator rA , (A)Pl +rA 2 (A)P2 is an inverse of A - AI. It follows that a(Ai) = ai, i = 1, 2. ~
§2. Spectral radius A useful tool in studying the spectrum of an operator A E L(X) is the Neumann series 00
LA-HlA i ,
where AEC, Ai-O.
(2.1)
i=O
The convergence radius of this power series with operatorial coefficients is given by the formula (2.2) rCA) = lim \lIIAnll· n--->oo
The existence of the limit in (2.2) is based on the next arguments. <3 Set inf II An II;'; n~l
=
r. Let us show that lim II An II;'; ~ r, where the symbol n~~
lim denotes the upper limit. For each E > 0 there exists mEN such that II Am II ~ ~ r+ +E. For an arbitrary n E N we have n = pm + q where 0 ~ q ~ m - 1, pEN. Then
It is easy to see that pmn- l ----> 1 and qn- l ----> 0 as n ----> 00 (m is fixed), therefore lim IIAnll;'; ~ r + E. Since E > 0 is an arbitrary positive number, we have n--->oo
lim IIA n II;'; ~ r. Thus we proved the existence of the limit
n-+oo
lim \lIIAnll = inf \lIIAnll = rCA).
n-+oo
n
~
1
THEOREM 2.1. For any A E L(X), we have
rCA) = sup IAI.
(2.3)
AEa(A)
<3Bytheabovearguments,r(A)? sup IAI. Letusshowthatr(A)~ sup IAI. AEa(A)
Set r
=
AEa(A)
sup IAI. By Theorem 1.4, the resolvent rCA) is analytic outside the disk AEa(A)
Spectral radius
109
of radius r, so it expands there in a Laurent series convergent with respect to the operatorial norm. Using the uniqueness theorem, this Laurent series coincides with the Neumann series. Consequently, lim IIA-nAnll = 0 for IAI > r, hence, for any n-->oo
c > 0, there exists m (= m(c)) EN such that
IIAnl1 ~ (c+r)n
for all n ~ m. It follows
that r(A) = lim IIAnll~ ~ r. n-->oo
COROLLARY 2.1. If
For
IAI < r(A),
then the Neumann series diverges.
IAI > r(A)
the Neumann series is convergent in the operatorial norm and
= r(A).
By Theorem 2.1, Corollary 1.1 and Corollary 1.2, a point from
00
- LA1-iAi i=O
the boundary of the disk of radius r(A) belongs to the spectrum of A. Therefore, the number r(A) is called the spectral radius of the operator A, since r(A) is the radius of the smallest closed disk centered at the origin of the complex plane C, which contains the spectrum of the operator A. From the inequality IIAn I ~ IIAlln, n E N, it follows that r(A) ~ IIAII. It is not difficult to find an example of an operator A for which r(A) < EXAMPLE 2.1.
{el,e2},
IIAII.
Let X be a two-dimensional real or complex space with a basis
Ilelll = IIe211 = 1, and let
A be the operator given by the matrix
A=
(~ ~)
with respect to this basis. Since An = 0 for n
~
2, we obtain that r(A) = 0, while
IIAII ~ 1. The next related result is interesting. THEOREM 2.2 (Ya. B. Rutitzky). Let A E L(X). Then for any c
norm
II . II"
r(A) ~ <J
> 0 there exists a
on X, equivalent with the initial norm, such that
Choose n such that
From the inequalities
IIAII" ~ r(A) + c.
IIAnl1 ~ < r(A) + c,
set r
(2.4)
= r(A),
and let
LINEAR OPERATORS
110
it follows that the norms II . II and II . lie are equivalent on X. Further, IIAlle
=
sup IIAxll e
Ilxll e=l
=
sup ((r + c)n-11IAxll + (r + c)n-21IA 2xll + ... + IIAnxll)· Ilxll e=l
According to the choice of the integer n, it follows that IIAnxll:::;; (r+c)nllxll. Hence, after a rearrangement of the terms, we obtain: IIAlle = (r + c) sup ((r + ct- 11lxll + (r + c)n-21IAxll + ... + IIAn-1xll) = Ilxll e =l
= (r + c) sup Ilxll e = (r + c) II x ll e =l
which proves the right hand side part of inequality (2.4). By (2.3), rCA) does not change if we replace the norm II . lion X with the equivalent norm II . lie. Setting IIAnll e instead of IIAnl1 in (2.3), we obtain that
rCA) :::;; IIAlle. ~ We conclude this paragraph by considering a family (called an analytic operatorial sheaf- see, for instance, A. S. Markus [1]) of linear bounded operators A(A) on X, which depends analytically on a parameter A. Our goal is to study r(A(A)) = rCA) as a function of the parameter A. The next result is true.
Let A(A) be an analytic operatorial sheaf defined C. Then the spectral radius rCA) is a subharmonic function.
THEOREM 2.3 (E. Vesentini [1]).
on a domain
1) ~
Using Cauchy Integral Formula, we have
J 2n
(A(A), I) for any 1 E L(X)*, A E Hence
1)
=
~ 2m
(A(A + ei'Pr, I) dcp,
o
and any circle of radius r centered at A, entirely lying in
1).
J 271"
IIA(A)II
=
1 sup I(A(A), 1)1:::;; -2 Ilfll=l 7f
IIA (A + ei'Pr,
1)11 dcp,
o
i.e., IIA(A)II is a subharmonic function. Since for any n E N the operatorial sheaf An(A) is also analytic, the function IIAn(A)11 is sub harmonic too. Therefore 1
IW(~) II'';; (2~ lIlAn(~ +e"r) II d"') ".; 2~ llIAn(H e"r) II' d",.
Resolvent and spectrum of the adjoint operator Taking the limit for n conclusion. ~
-+ 00
111
in the last inequality, we obtain the required
COROLLARY 2.2 (Maximum Principle). The spectral radius of an analytic operatorial sheaf attains its maximum only on the boundary of the domain :D.
§3. Resolvent and spectrum of the adjoint operator Let A be a linear operator densely defined on the Banach space X. Then (see §9 of Chapter 0) there exists the conjugate operator A * on X*. <J Let ( " .) be the canonical scalar product on X x X* (see §9 of Chapter 0). If A* f = 0 for a certain f E D(A*) then 0 = (x, A* 1) = (Ax,1) for all x E D(A), hence f E R(A)~. Therefore, from R(A) = X it follows that f = O. On the other hand, if y ~ R(A), then by Theorem 0.8.7 there exists a functional f E X* which satisfies the conditions fey) = 1 and f I R(A) = O. Consequently, (Ax,1) = 0 for all x E D(A) so f E D(A*) and A* f = O. Thus we proved the next result. LEMMA 3.1. If A is a linear operator on X with dense domain, then the operator
(A*)-1 exists if and only if R(A) = X. ~ Assume that the operator A is invertible and that X = D(A) = = R(A). Then (A*)-1 = (A- 1)*. In this case, the operator A-I is defined and bounded on the whole Xif and only if the operator A is closed and (A *) -1 is defined and bounded on the whole X*.
THEOREM 3.1.
<J The operator (A*)-1 exists by Lemma 3.1. The existence of the operator
(A-l)* follows from D(A-l)
=
R(A). If y E R(A) and f E D(A*), then
so R(A*H:D((A-l)*) and (A- 1)*A*f=f for all fED(A*). Therefore R((A-l)*)~ ~ D(A*) and A*(A-l)* f = f for all f E D((A- 1)*). Consequently (A-l)* ~ (A*)-I, and, due to the opposite inclusion, (A*)-1 = (A-l)*. If, in addition, the operator A-I is defined on the whole X and is bounded, then (A -1) * is also bounded, hence (A *) -1 is bounded. By Theorem 1.1 the operator A-I is closed and therefore (A*)-1 is closed, too. Conversely, if the operator (A*)-1 is defined and bounded on the whole X*, and if that the operator A is closed, then
LINEAR OPERATORS
112 we have
for all x E R(A) and
f
E
X*, i.e., the operator A-I is bounded.
~
Using these results and Theorem 1.1, we get:
Let A be a closed linear operator on X with D(A) = X. Then C : ~ E p(A)} and TA*(A) = TA(~)*.
THEOREM 3.2.
p(A*)
= {A E
(xo, (A* - AI)!) = ((A - ~I)xo,!) = 0,
for all f E D(A* - AI). But R(A* - AI) = X. Thus, Xo E .lX* = {O}, so the operator A - ~I has an inverse (A - ~I)-1 and both these operators are closed. By Lemma 3.1 we obtain that D ((A - AI)-I) (= R(A - ~I)) = X. This equality and Theorem 3.1 imply that ~ E p(A). ~ Using de Morgan formulas, we obtain:
Under the conditions of Theorem 3.2, the spectrum of the adjoint operator A* is the complex-conjugate of the spectrum of the operator A, i.e.,
COROLLARY 3.1.
O"(A*)
= {A
E C : ~ E O"(A)}.
Let us consider now an operator acting on a Hilbert space X = 5). In this case, the conjugate operator acts on 5), too, so the notion of a selfadjoint operator, i.e., a linear operator A with D(A) = 5), satisfying A* = A, makes sense. Then (Ax,y) = (x,Ay) for all x,y E D(A) (= D(A*)). By Corollary 3.1, the spectrum of a selfadjoint operator A is real, i.e., O"(A) <;;; JR., and its resolvent has the next property: THEOREM 3.3.
Let A be a selfadjoint operator and A =
0:
+ i{3
E
C, {3 =1=
o.
Then (3.1)
We have
Resolvent and spectrum of the adjoint operator
for all x
E
D(A), and thus (3.1) follows.
113
~
A remarkable property of selfadjoint operators is the existence of their spectral decomposition. We state the theorem concerning the spectral decomposition without any proof. First, we need a definition. Let Band C be two bounded selfadjoint operators on Sj. We shall write B ~ C if (Bx, XI ~ (Cx, XI for all X E X. THEOREM 3.4. Let A be a selfadjoint operator on Sj with the domain D(A). Then
there exists a family of orthogonal projections (see §11 of Chapter O)E).., -00 < ). < < 00, satisfying the properties: 1) E).. ~ EJ1 for any)., p with), < p; 2) {E)..} is left hand side strongly continuous, i.e., E)..-ax ---+ E)..x as a ---+ 0, a > 0, for any x E Sj; 3) E- oo = lim E).. = 0, E+oo = lim E).. = I; ).._-00
4) BE)..
)..-00
E)..B for any bounded operator B which commutes with A. (Here, a bounded operator Bon Sj is said to commute with A if x E D(A) implies Bx E D(A) and ABx = BAx.) An element x belongs to D(A) if and only if =
J 00
).2d(E)..x, XI <
00.
-00
For such an element x, we have
J
Ax =
-00
J
J 00
00
)'dE)..x
and
IIAxl12 =
).2d(E)..x, XI·
-00
b
00
The integral
)'dE)..x above is understood as the limit of the integral
a
---+ -00
and b ---+
J
)'dE)..x, for
a
-00
+00.
The family {E)..} mentioned in Theorem 3.4 is called the spectral decomposition of the selfadjoint operator A.
LINEAR OPERATORS
114
§4. The spectrum of a completely continuous operator Let X be a complex Banach space. Consider a completely continuous linear operator A defined everywhere on X. According to Theorem 0.10.6, the complete continuity of A is equivalent with the precompactness of the image AS of the unit ball S of X (={ x E Xj Ilxll ~ I}). As we noticed in §1 of Chapter 0, any linear operator A acting on a finite dimensional space is completely continuous. Its spectrum a(A) coincides with the point spectrum ap(A) = al (A), and contains a finite number (~ the dimension of the space) of eigenvalues, each of these, evidently, of finite multiplicity. The spectrum of a completely continuous operator on an infinite dimensional space has a very similar structure to the one of the spectrum of an operator on a finite dimensional space. First, we establish the next preliminary result: THEOREM 4.1.
The adjoint of a completely continuous operator is completely con-
tinuous. By Theorem 0.10.7, the range of a completely continuous operator A is separable. Let {fn}nEN be a bounded sequence in X*. We show that there exists a subsequence {fnj }j EN such that {A * f nj }j EN is a Cauchy sequence in X*. Let {Vk} kEN be a dense sequence in R(A). For each k, the scalar sequence {(Vk' fn) }nEN is bounded. Therefore, using the diagonal method, we can choose a subsequence {(Vk' fnj )}jEN which is Cauchy for each kEN. Since the sequence {vkhEN is dense in R(A), then {(v,fnj)}jEN is a Cauchy sequence for any v E R(A). The functional lim (v,fn) is linear in v on R(A). By Theorem 0.8.7, there <J
J --+(X)
J
exists an extension of this functional on the whole X, with the same norm. Denote this extension by f. Then, for all v E R(A), we have (v, fnJ ---> (v,!), i.e.,
(Ax,fnj)
--->
(Ax,!),
(4.1)
for all x E X. Let us show that A* fnj ---> A* f as j ---> 00. Assume that this is not the case. Then there exists a subsequence {he} eEN of the sequence {fnj - J} j EN, and a number 8, such that IIA*hell ~ 8> 0, for all e E N. Since IIA*hell = sup I(x, A*he)l, for each Ilxll=l e E N there exists a vector Xe E X, Ilxell = 1, such that I(Axe, he)1 ~ 8/2. From the complete continuity of A it follows that {Axe} eEN contains a Cauchy subsequence, so we may assume that {AxdeEN is a Cauchy sequence itself, i.e., for any e > 0 there exists m = m(e) such that IIAxe - Axsll < e for e, s > m. Then 8 "2 ~ I(Axe, Axs)1 ~ Me + I(Axs, he)l,
The spectrum of a completely continuous operator where M
sup Ilheli. Taking the limit for C ---->
=
eEN ~ ME.
we have 8 /2 contradiction.
Since
E
00
115
and a fixed s, and using (4.1),
is an arbitrary positive number, it follows that 8
=
0, a
~
Let A and B be two completely continuous linear operators on X. Then any linear combination of A and B is a completely continuous operator, too.
THEOREM 4.2. a)
b) If A is a completely continuous linear operator on X and C E L(X) then AC and C A are completely continuous operators.
c) If A = lim An in the norm topology of L(X) and An are completely n--->oo continuous linear operators on X, then A is completely continuous.
a) The assertion is obvious.
b) Since a bounded linear operator transforms bounded sets into bounded sets, the complete continuity of C A and AC follows from the definition of completely continuous operators. c) Let An E L(X), n E N, and An ----> A. If {Xj}jEN is a bounded sequence in X then each sequence {AnXj}jEN is precompact. Using the diagonal method we get the precompactness of the sequence {Axj} jEN' ~ The main theorem of this section deals with the structure of the spectrum of a completely continuous operator. In order to prove it, we need a preliminary result.
A bounded projection on a Banach space is a finite-dimensional operator if and only if it is completely continuous. LEMMA 4.1.
Let dim PX = 00. We define recurrently a sequence {Vn}nEN such that Vn E E PX and Ilvnll = 1;;::: Ilv n - vmll for n -I- m. Assume that the vectors VI, ... , Vk are already defined and set wt = din {VI, ... , Vk}. Then, there exists a vector u E PX such that u 1:- wt. Let wtu = din{vI, ... ,Vk,U}; obviuosly we have < d = inf Ilu - xii.
°
Choose a sequence {Xn}nEN such that Xn E wt and d
= n--->oo lim Ilu-xnll.
xE9J1
Since the space
wt is finite-dimensional, the sequence {Xn}nEN is bounded and, therefore, precompact. We may assume that Xn ----> Z E wt as n ----> 00; then d = Ilu-zll. Put Vk+1 = d-I(u-z). Clearly Ilvk+11l = 1 and }~Jn IIVk+1 - xii = }~Jn
II u -
z - dx d
II = dllu 1 - zll =
1.
LINEAR OPERATORS
116 Further
IIVk+1 - vkll ~ inf
xE9JI
Ilvk+1 - xii
~ 1,
for m = 1, ... , k. Thus there exists a sequence {vn}nEN satisfying all the above mentioned properties. Clearly none of its subsequences is convergent. On the other hand PV n = Vn , n E N, which rontradicts the complete continuity of P. Consequently,
dimPX <
00. ~
THEOREM 4.3. Let A E L(X) be a completely continuous operator. Then (T(A) is an at most countable set, with no limit points different from zero. Each A E (T(A), A -# 0, is an eigenvalue of A of a finite multiplicity, and "X is an eigenvalue of A * of the same multiplicity.
Let A be a limit point for the set of eigenvalues of the operator A. If we assume that A -# 0, then there exists a sequence {An}nEN of eigenvalues of A such that An -# 0 and An ----* A. Let Un be an eigenvector of A corresponding to the eigenvalue An, and let 9Jtn = clin{u1, ... ,un }. Since the vectors U1, ... ,Un , ... are linearly independent, the subspace 9Jt n- 1 is a proper subspace of 9Jt n and dim 9Jt n- 1 = dim 9Jtn - 1. Therefore (see the proof of Lemma 4.1) there exists a vector Vn E 9Jt n with Ilvnll = 1 and such that d n = inf Ilvn - xii = 1. Thus we obtain a sequence
xE9Jln~l
{Vn}nEN with Vn E 9Jt n , Ilvnll = 1 and dn = 1. Let us show that the sequence {A;;-l AVn}nEN contains no Cauchy subsequences. Indeed, for m < n we have A;:;-1 AVn - A;;,1 AVm
=
Vn - (A;;,1 AVm
-
A;:;-I(A - AnI)vn ) .
We can easily prove that the second term in the right hand side of the above equality belongs to 9Jtn - l . Therefore the norm of the right hand side vector is greater than or equal to 1. This means that the distance between any two terms of the sequence {A;;-l AVn}nEN is no less than 1. Consequently, none of the subsequences of this sequence is convergent. This contradicts the complete continuity of A, since the sequence
IIA;;-1vnll
is bounded:
IIA;;-1Vnl ~ (inf IAml) -1 < 00. mEN
Thus A = 0 is the only possible limit point for the set of eigenvalues of A. II. Let us prove now that the lineal R(A-),J) is closed if A -# 0 and A tJ- (Tp(A). Assume that (A - AI)X n ----* y for n ----* 00. We will show first that {xn}nEN is bounded. If this is not so, then we may suppose (choosing eventually a subsequence) that
Ilxnll
----* 00
as n
----* 00.
Set Un
= 11::11. Then Ilunll = 1 and (A - AI)Un
----*
o.
Due to the complete continuity of A we may assume that AUn ----* U, U -# o. Then Au = Au, which contradicts our assumption. Thus Ilxnll ~ c < 00, and we can suppose that AX n ----* x. Then Xn ----* A-I(y - x), hence y = A-I(A - AI)(y - x) E R(A - AI).
Normally solvable operators
117
III. Consider the set a' of all points A E C such that either A E ap(A) or A E ap(A*). By Theorem 3.2, a' ~ a(A). On the other hand C \ cr' ~ peA). Indeed, since R(A - AI) is closed for all A E C \ cr' it is enough to notice that N(A* - AI) = R(A - A1)l... So cr' = cr(A). It remains to show that the eigenspace corresponding to A E a(A) is finite-dimensional. By Lemma 4.1 it is sufficient to establish the complete continuity of the projection P onto this subspace. As in Theorem 1.5, consider the projection P
=
_(27ri)-1
Jr(Od~,
where
I
I is a circle centered at A and with a sufficiently small radius such that 0 is lying outside this circle. We have Ar(~) = I+~r(~), hence ~-1 Ar(~) = ~-1 I+r(o· Since Cld~ =
J
= 0, then P = -(27ri)-1
I
J
C 1Ar(~)d~. By Theorem 3.2, the operator
Ar(~) is com-
I
pletely continuous for any ~ E ,. By the same theorem, the projection P is completely continuous as a uniform limit of integral sums which obviuosly are completely continuous operators. From Theorems 3.2 and 4.1 it follows that cr(A*) \ {O} consists of eigenvalues of A* of finite multiplicity, and, moreover, cr(A*) \ {O} = {X : A E cr(A), A~O}. The range of the projection P* ~ the adjoint of P ~ is the eigenspace of A* corresponding to the eigenvalue A. Since the projections P and P* have the same rank, the multiplicities of A and X coincide. ~ (First Fredholm Alternative). If A is a completely continuous operator on X then, either (A - 1)-1 E L(X), or there exists a non-zero solution of the equation Ax = x.
COROLLARY 4.1
§5. Normally solvable operators Let X, ~ be Banach spaces and A : X --+ ~ be a linear operator with D(A) ~ X, R(A) ~~. Let N(A) (= Ker A) be the kernel of the operator A. Let us recall (see §9 of Chapter 0) that by 9)1l.. and l..9)1* we denote the orthogonal and the *-orthogonal complement of a set 9)1 ~ X, and 9)1* ~ X*, respectively. Assume that the operator A is densely defined, i.e., D(A) = X. Then the adjoint operator A* : ~* --+ X* exists and
THEOREM 5.1.
1)
N(A*) = R(A)l..,
2)
R(A) =
l.. N(A*).
(5.1)
LINEAR OPERATORS
118
If, in addition, D(A) = X and the operator A* is densely defined, then A 3)
N(A)
=-L
R(A*),
E
L(X) and
(5.2)
<J If D(A) = X then (see §9 of Chapter 0) the adjoint A* : ~* ----> X* exists and (Ax, I) = (x, A* I) for all x E D(A), f E D(A*). 1) Let f E N(A*). Then (Ax, I) = (x, A* I) = 0 for all x E D(A), hence f E R(A)-L. So N(A*) S;; R(A)-L. Conversely, let f E R(A)-L. Then (Ax, I) = 0 for all x E D(A), hence f E D(A*). Therefore the equality (Ax, I) = (x, A* I) holds and, because D(A) is dense in X, we obtain A* f = 0, i.e., f E N(A*). So R(A)-L S;; N(A*) and relation 1) is proved. By Theorem 0.9.2 the equality 2) follows from 1) taking the *-orthogonal complements. Assume in addition that D(A) = X and D(A*) =~. Let us show that the operator A is closed. Indeed, let {Xn}nEl":I C X, such that Xn ----> x and AX n ----> Y as n ----> 00. For any f E D(A*) we have lim (Axn' f) = lim (xn' A* f) = (x, A* I) and n---+oo
lim (Axn' f)
n--+oo
= (y, I).
So y
=
n---+oo
Ax and therefore A is closed. By Theorem 1.2 the
operator A is bounded. Consequently, its kernel N(A) is closed. The adjoint operator A* is bounded, too, and everywhere defined (see §4 of Chapter 0). We prove now equality 3). The inclusion N(A) S;; -LR(A*) can be obtained by arguments similar to those in the proof of equality 1). Let x E -LR(A*). Then (x, A* I) = 0 for all f E ~*. Therefore Ax = 0, i.e., x E N(A), hence -L R(A*) S;; N(A). Equality 4) follows from 3) using Theorem 0.4.8. ~ A densely defined linear operator A is called normally solvable if R(A)
=
-L N(A*).
(5.3)
By equality 2) in (5.1) we have: THEOREM 5.2 (Hausdorff). A densely defined operator A is normally solvable if and
only if its range is closed, i.e., R(A)
= R(A).
COROLLARY 5.1. If an operator BE L(X) is completely continuous then A = 1- B is normally solvable. <J If 1 E pCB) then R(A) = ~ (= R(I - B)) is closed. Assume that 1 E a(B). Then, by Theorem 4.3, we have that 1 is an eigenvalue of the operator B of a finite
Noether and Fredholm operators
119
multiplicity, i.e., the corresponding eigenspace Xl is finite-dimensional. Let PI be the projection onto Xl, defined by (1.2), corresponding to the obvious spliting of the spectrum: UI = {l} and U2 = u(B) \ UI· Then Xl = PIX and X = PIX + (I - PI)X. Setting X 2 = (1 - PdX and B2 = B I X 2 we obtain that 1 E p(B2). Therefore (I - B 2)X2 = X 2 and, consequently, R(A) = (I - B 2)X = (1 - B 2)X2 = X 2 is a closed subspace of X. ~ The next result follows from (5.3). THEOREM 5.3. If an
operator A : X
----+ ~
is normally solvable then the equation
Ax=y
has a solution, if and only if (y, 1)
=
0 for all solutions j of the equation A*j = O.
COROLLARY 5.2. If equation (5.1)
is solvable for any y E
(5.4)
(5.5)
has the only solution j = 0, then equation (5.4)
~.
§6. N oether and Fredholm operators The following classes of operators play an important role in the theory of linear equations with normally solvable operators. A normally solvable operator A : X ----+ ~ is called a Noether operator, or an N-operator, if both the spaces N (A) and N (A *) are finite-dimensional. For a Noether operator A, both n(A) = dimN(A) and m(A) = dimN(A*) are non-negative integers, called the number of zeros and the defect of the operator A, respectively. The integer X(A) = n(A) - m(A) is called the index of the operator A. A Noether operator A satisfying X(A) = 0, i.e., n(A) = m(A), is called a Fredholm operator, or an F -operator. Let A : X ----+ ~ be a Noether operator. If {gdk=l is a basis of IJ!(A*) then the condition of solvability for equation (5.4) can be written as a system of equalities:
(y,gk)
=
0,
k = 1,2, ... ,m.
If, in addition, A is a Fredholm operator then, using Corollary 5.2, we obtain the next
result, usually referred to as the Second Fredholm Alternative.
LINEAR OPERATORS
120
A : X ----t ~ is an F-operator then, either the equation (5.4) has a unique solution for any vector in the right hand side of that equation, or equation (5.5) has only the zero solution.
THEOREM 6.1. If
(5.3) the solvability of the equation (5.4) for any vector in its right hand side is equivalent with the equality ~ = J.. N(A*), which leads to N(A*) = {a}. From dimN(A) = dimN(A*) we obtain N(A) = {o}. On the other hand, let us assume that equation (5.5) has only the zero solution, i.e., N(A) = {a}. Then N(A*) = {a}. By (5.3) we obtain R(A) = ~, i.e., equation (5.4) has a solution for any vector y in its right hand side. Since N(A) = a such a solution is unique. ~ <J By
Suppose now that A E L(X, ~). If A is a Fredholm operator and, moreover, N(A) = {a}, then by Theorem 6.1 its range R(A) coincides with the whole space ~. By the Banach Inverse Mapping Theorem, the operator A is a linear homeomorphism between the spaces X and~. Assume, on the contrary, that N(A) =f. {a}. In this case the operator A can be perturbed by a finite rank operator K such that the corresponding perturbation B = A + K becomes a homeomorphism between the spaces X and ~. The construction of such a perturbation, which is given in the sequel, was obtained for the first time by E. Schmidt [1] in the case of integral operators. Let A E L(X,~) be a Fredholm operator, with n = n(A) ~ 1. Let {adk=l be a basis of N(A), and let {gdk=l be a basis of 5Jt(A*). By the Hahn-BanachSuchomlynoff Theorem we can choose two collections of elements Udk=l and {bdk=l in the spaces X* and ~, respectively, such that
(6.1) Define a linear operator K by the formula n
Kx
= 2)x, fi)bi ,
x
E
X.
(6.2)
i=l
The operator K acts from the space X onto the finite-dimensional subspace £ = clin{b 1 , ... , bn } ~ ~ and satisfies the conditions
Kai=bi,
i=l, ... ,n.
(E. Schmidt). Let A E L(X,~) be a Fredholm operator with n(A) ~ 1. Then the operator B = A + K, where K is defined by (6.2), has a bounded inverse
LEMMA 6.1
Projections. Splitable operators
121
operator B- 1 defined on the whole ~, i.e., B is a homeomorphism of the spaces X and~.
Assume that Bx = 0 for some x E X. Then, by (6.2), we have
<J
n
Ax
= -
(6.4)
2.Jx, fi)bi. i=1
From (6.4) and using the equality N(A*) = R(A).l (see Theorem 5.1) we obtain that (Ax, gk) = 0 for any gk E N(A*), k = 1, ... , n. Consequently n
O=L(x,fi)(bi,gk)
=
(X,fk),
(6.5)
k=l, ... ,n.
i=1
n
From (6.4) and (6.3) it follows that Ax = 0, i.e., x E N(A), and therefore x = L D:iai. i=1
By (6.5) we find D:k = (x, fk) = 0, hence x = O. So we conclude that N(B) = {O}. Let us prove now that R(B) =~. Let y E ~ be fixed. Consider the element n
f} = y - L(y,gi)bi . By (6.1) we have (f},gi) = 0, hence f} E .IN(A*) = R(A). This i=1
means that there exists an element x E X which satisfies f} = Ax. Set n
x=x+ L((y,gi)-(x,fi))ai. i=1
Then Ax
=
Ax, and using (6.3) we obtain
n
= f}
+L i=1
n
(x, fi)bi + L( (y, gi) - (x, fi) )bi = f} i=1
n
+L
(y, gi)bi = y.
i=1
Thus the equality R(B) = ~ is proved. Now the conclusion of our lemma follows from the Banach Inverse Mapping Theorem. ~
§7. Projections. Splitable operators 1. PROJECTIONS. We have already dealt with projections when we defined the direct sum of spaces (see §7 of Chapter 0). Recall that by a projection on a Banach space X
122
LINEAR OPERATORS
we mean a linear operator P defined on the whole space X, with the property p 2 = P. We will consider below only continuous projections which, for brevity, will be simply called projections.
If P is a projection then Q = I - P is a projection, too, since obviously Q is linear, is defined on the whole X, is continuous, and Q2 = (I - p)2 = 1- P = Q. The restriction of the projection P on its range R(P) = PX is the identity operator P on this subspace, so that, if P =1= 0 then IIPII ~ 1. Indeed, for any U E R(P) we find an element x E X such that u = Px and, therefore, Pu = P Px = p 2 x = Px = u. Thus R(P) is closed, i.e., it is a subspace in X. Any element x E X has a unique representation x = u + v where u = Px, v = Qx. Therefore the space X decomposes into the direct sum of subspaces R(P) and R(Q), i.e., X = R(P)+R(Q). By Theorem 0.4.2 this direct sum is topological. Thus any projection P gives rise to a decomposition of the space X into the topological direct sum X = X l +X 2 , where Xl = R(P) is the subspace onto which P acts, and X 2 = R(I - P) is the subspace along which the operator P projects the elements of X. It is clear that R(I - P) = N(P) and R(P) = N(I - P). The projections P and Q = I - P are called mutually complementary. According to the previous considerations, a subspace £ of a Banach space X is called topologically complementable if there exists a projection P on X such that R(P) = £. Generally speaking, not all the subspaces of a Banach space are topologically complementable. There are a lot of papers and monographs devoted to this topic - the existence of topological complements (see, for instance, M. Kadec, B. Mityagin [1]). However, in the case of a Hilbert space X = 5), any subspace £ has not only a topological complement, but also an orthogonal complement (see Theorem 0.11.2). More precisely, any proper subspace £ can be obtained as the range of a projection P, with minimal norm IIPII = 1, such that the subs paces R(P) and R(I - P) are orthogonal. Let us return now to the general case of a Banach space X. A projection P on X will be called proper if IIPII = 1. Since the finite-dimensional subs paces of a Banach space have always topological complements, they play an important role in some problems concerning projections. To be more specific, let X be a Banach space and (!; a finite-dimensional subspace of X, with dim (!; = n. Choose a basis {ai}f=l in (!; and define the linear functionals Ii, 1 ~ i ~ n, as follows. For u = alaI + ... +anan E (!; we put Ii(u) = ai, so that (ai, Ij) = Dij, i,j = 1, ... , n. Obviously the functionals Ii are well-defined on the subspace £ and are bounded. By Theorem 0.8.7 they admit extensions on the
123
Projections. Splitable operators whole X, denoted also by
Ii, i = 1, ... , n, with
the same norms. For x E X we set
n
Px
=
L(x, Mai.
(7.1)
i=l
Then the operator P is a projection on X, with R(P)
=
IE. Indeed,
n
=
L (x, li)ai = Px, i=l
for any x E X, hence p 2 = P. Further, it is clear that R(P) <;;;: IE. On the other hand, for each u = a1a1 + ... + ana n we have Pu = u. Therefore IE <;;;: R(P) and, consequently, IE = R(P). 2. SPLITABLE OPERATORS. Assume that P1 and E1 are two given projections on the Banach spaces X and ~, respectively. We denote
Then, any linear operator A : X
----> ~
defined on the whole X has the representation (7.2)
According to this representation, it is convenient to write the operator A in the matrix form (7.3) where Aij : £) ----> 9)1i is the restriction of the operator EiAPj on the subspace £j, = 1,2. The matrix representation (7.3) of the operator A is called non-degenerate if the operator A22 is a homeomorphism from the space £2 onto 9)12. Provided that this condition is fulfilled and, in addition,
i,j
(7.4) then the representation (7.3) is called regular.
124
LINEAR OPERATORS
An operator A : X -+ ZJ defined on the whole space X is said to be splitable if it admits a regular representation (7.3). THEOREM 7.1. A normally solvable operator A E L(X, ZJ) is splitable if and only if its kernel N(A) and its range R(A) have topological complements.
<J Suppose that the subs paces N(A) and R(A) have topological complements, i.e., there exist the projections Pi on X and El on ZJ such that R(Pd = N(A) and R(E1) = R(A). Set P 2 = I -Pi, E2 = I -E 1. Then formula (7.2) is true for all x E X. Since PiX E N(A), then E 1AP1x = 0, E 2AP1x = and Ax = E 1AP2x + E 2AP2x. Moreover, y = AP2x E R(A), so E 2y = 0, hence Ax = E 2AP2x. Thus, in our case, representation (7.3) has the particular form
°
A=
(0° 0) A22
(7.5) '
where A22 is the restriction of the operator E 2AP2 on R(I - Pd - the topological complement of the subspace N(A). It follows that R(A 22 ) = R(A), N(A 22 ) = {O} whence by the Banach Inverse Mapping Theorem we obtain that A22 is a homeomorphism. Obviously equality (7.4) is satisfied. So we proved that the corrditions of the theorem are sufficient. Conversely, assume the existence of the projections Pi on X and El on ZJ such that representation (7.3) is regular. Then the equation Ax = is equivalent to the system of equations { Al1P1X + A 12 P2X = (7.6) A 21 P 1X + A 22 P 2X = 0.
°
°
The second equation in (7.6), on its turn, is equivalent to (7.7) From (7.7) and (7.4) it follows that the first equation in (7.6) is an identity. Thus the kernel N(A) consists exactly of those elements which satisfy (7.7). Let us define a linear operator P on X by (7.8)
We have
125
Projections. Splitable operators
i.e., P is a projection (recall that A2"l : £2 -+ 9J1 2 and P l 9J1 2 then by (7.7) we obtain Px = (Pl + P 2)x = x, whence N(A) show that R(A) <;::; N(A). For each x E x we have
= {O}). If x <;::;
N(A), R(A). Next, let us E
Therefore, if we set Y = Px, then PlY = Plx and by (7.8) we find
From here and based on (7.7) it follows that Y E N(A) and, consequently, R(A) <;::; <;::; N(A). Due to the inverse inclusion we obtain that P is a projection onto N(A), and therefore N(A) has a topological complement. Assume now that Y E R(A), i.e., the equation Y = Ax has solutions. Equivalently, the system of equations {
Al1P1X + A 12 P2X = ElY A 21 P 1X + A 22 P 2X = E 2y
(7.9)
has solutions. The second equation in (7.9) gives
Replacing P2 x in the first equation of system (7.9) by the right hand side of the last equality, and according to (7.4), we obtain that R(A) consists exactly of those elements Y E ~ for which the relation ElY = A12A2"2l E 2y is fulfilled. Put E = = El - A12A2"l E l . As above, we get that E is a projection onto R(A). Consequently R(A) has a topological complement. ~ COROLLARY 7.1. Any Noether operator and, in particular, any Fredholm operator
A E
L(x,~)
is splitable.
At the end of this section we will establish, for a given operator A E L(x, ~), the relationship between the operator B = A + K considered in Lemma 6.1, where K is defined by (6.2), and the representation (7.3). Assume that A E L(x,~) is a Fredholm operator with n = n(A) ~ 1 and that {adk=l' {gdk=l are fixed bases for N(A) and N(A*), respectively. Let {fdk=l and {bdk=l be the biorthogonal systems corresponding to {ak}k=l and {gk}k=l (see §6). Set £n = clin{bl, ... , bn } and define the projection P l on x by (7.1), and the
LINEAR OPERATORS
126
n
projection El on !D by ElY
= L (y, gi)b i . Then the projection E2 = I - El maps !D i=l
onto R(A). Indeed, for any Y E !D we have n
ElY = L (y, gk)bk, k=l n
whence E2y
= Y - L(Y,9k)bk and k=l n
(E 2y,gj) = (y,gj) - L(y,9k)(bk,gj) = O. k=l
It follows that E 2 y E R(A). Moreover, from (5.3) and by the biorthogonality of the systems {bk}k=l and {gk}k=l' we have'cn n R(A) = {O}. Therefore!D = 'cn+R(A). Analogously, X = N(A)+9J1, where 9J1 consists exactly of those vectors x E X for which (x, Ii) = 0, i = 1, ... , n. By Theorem 0.4.7, the n-dimensional spaces 'cn and N(A) are homeomorphic. Consider now the operator B = A+K, where K is defined by (6.2). By (6.3), the operator KIN (A) yields a homeomorphism between the spaces N (A) and 'cn. It is easy to see that, relatively to the four projections P l , P2 , E l , E 2 , the operator B has the matrix representation
B =
(K~l A~2 )
,
(7.10)
where Ku = K I N(A) and A22 = A I 9J1 (see (7.5)). It follows that the operator B is an extension of the operator A22 on the whole space X. By Schmidt Lemma the operator B is invertible. Its inverse B- 1 has the matrix representation (7.11) Notice that representations (7.10) and (7.11) are not regular, since condition (7.4) is not fulfilled.
§8. Invariant subspaces In this section we consider operators A E L(X), i.e., bounded linear operators A : X --> --> X with D(A) = X. A subspace ,c ~ X is called invariant relatively to A if A,C ~ 'c.
Invariant subspaces
127
1. Let A E L(X). Assume that the kernel N(A) has a topological complement, i.e.,
X = N(A)+9J1, where 9J1 is a subspace of X. Let us consider the following question: when is the topological complement 9J1 of N(A) an invariant subspace relatively to A? If, in particular, the operator A is splitable, then, by Theorem 7.1, the existence of two decompositions of the space X follows: X
= N(A)+9J1,
X
= 'c+R(A).
When these two decompositions coincide, i.e., N(A) = ,c, R(A) = 9J1, then the subspace 9J1 is invariant relatively to the operator A and X = N(A)+R(A). Let now A be an arbitrary operator in L(X). Consider the following two sequences of lineals: (8.1) {O}, N(A), N(A 2 ), ... ,N(Ak ), ... , X, R(A), R(A 2 ), ... , R(A k ), ... .
(8.2)
Relatively to inclusion, the sequence (8.1) does not decrease and the sequence (8.2) does not increase, i.e., N(Ak) <:;;; N(Ak+l), and R(Ak) ~ R(Ak+l), for any k = 0,1, .... If there exists mEN such that
then we say that the operator A has a finite ascent, equal to m. If there exists r E N satisfying the conditions R(Ar)
= R(A r + 1 ),
and
R(Aj)
= R(Aj+l),
j
= 0, ... , r - 1,
(8.4)
then we say that the operator A has a finite descent, equal to r. It is easy to verify that condition (8.3) implies the equality N(Am) = N(Ak) for all kEN, k ~ m, and condition (8.4) implies the equality R(AI) = R(Ar) for all lEN, l ~ r. Moreover, as we will prove below, if a normally solvable operator has simultaneously a finite ascent m and a finite descent r then these integers coincide, i.e., m = r.
Assume that a normally solvable operator A E L(X) has a finite ascent m and a finite descent r. Then 1) the subspace R(Ar) is invariant relatively to A and the restriction A I R(Ar) is a homeomorphism; 2) the space X decomposes into the direct sum THEOREM 8.1.
LINEAR OPERATORS
128
3) the operator A has the same ascent and descent, i.e., m
=
r.
1) From R(N) = R(AT+l) and AR(AT) = R(N+ 1) we infer AR(N) = R(N). Let us show now that N(A I R(AT)) = {O}. Set A I R(N) = A. Let x E E R(AT) with Ax = o. Choose n E N such that n ~ max{m, r}. Then x E R(An), i.e., x = An z , where z E X. But An+l z = Ax = Ax = 0, whence z E N(An+l) = N(An). Therefore x (= An z ) = O. By Theorem 5.2 it follows that R(AT) is a subspace of X whence, by using Theorem 0.8.5, we obtain that A I R(AT) is a homeomorphism. 2) Let x be an arbitrary element of X. Put v = A ~T AT x and u = x-v. Then v E R(AT) and since ATu = ATx - ATv = ATx - AT A~T ATX = ATx - ATx = 0 we have u E N(AT). If we find other elements u' E N(AT) and v' E R(AT) such that x = u' + v' then <J
and since v' E R(AT) it follows that ATv'
= ATV', i.e., v' = A~T ATX = v. Conse-
= x - v = u and assertion 2) is proved. 3) Let us prove now that m = r. According to 2) any element x E N(AT+l) has the form x = u + v, where u E N(AT) and v E R(AT). Then 0 = AT+lX = = AT+lU+AT+lV and from AT+1U = 0 it follows that AT+lv = 0 hence, by virtue of 1), we obtain v = O. This means that x = u E N(AT) and therefore N(AT+l) = N(AT), i.e., m ~ r. On the other hand, if y E R(Am), i.e., y = Am x for some x E X then, representing x as x = u+v where u E N(AT), v E R(AT), we obtain y = Amu+Amv . Since N(Am) = ... = N(AT) then u E N(Am) hence Amu = 0, i.e., quently, u'
y
So m
~
= Amv =
r and the proof is complete.
Am+lA~lv E R(Am+l). ~
REMARK 8.1. If an operator A satisfies the conditions in Theorem 8.1 and A = I -B, then the space N(Am) is the root subspace for the operator B corresponding to the eigenvalue A = 1 (see §1) and the integer m itself is the rank of this eigenvalue. COROLLARY 8.1. Let B E L(X) and assume that the operator I - B is normally solvable. If A = 1 is an eigenvalue of rank 1 for the operator B and an eigenvalue of finite rank m* for the operator B* then the operator I - B is splitable and
x=
N(I - B)-t-R(I - B).
(8.6)
<J By Remark 8.1 the operator A = 1- B has the finite ascent m = 1 and the operator A* = 1- B* has the finite ascent m*. Therefore N ((A*)m*) =
129
Invariant subspaces =
N ((A *)m' +1). This equality and (5.1) imply that R (Am') = R (Am' +1). Conse-
quently, the operator A has the finite descent r ~ m*. By Theorem 8.1, the equality m = r = 1 holds. The decomposition (8.6) follows now from this last equality and (8.5) . • To conclude this subsection we give an example of a normally solvable operator with a finite ascent, but without a finite descent. EXAMPLE 8.1. Let SJ be a Hilbert space with an orthonormal basis {ei}~O' Define the linear operator A on SJ by the equalities: Aeo = 0, Aei = eH1, i)! 1. Then A E L(SJ), N(Ak) = clin{eo} and R(Ak) = clin{ek+1, ek+2, .. . }. Thus the operator A has the finite ascent m = 1 but it has not a finite descent. At the same time the operator A * has the finite descent r* = 1 but it has no finite ascent.
2. In this subsection we use a different approach to study the problem of decomposing the space X into a direct sum of the form (8.6). A key role in this approach is played by the next condition: (VB) The family of the iterations of the operator B is uniformly bounded, i.e.,
IIBkl1 ~ M < 00,
k = 1,2, ....
As a preliminary step let us prove the following result.
Assume that the operator BE L(X) satisfies condition (VB). Then the intersection of the kernel and the range of the operator A = I - B is equal to {O}, i.e.,
LEMMA 8.1.
R(A) n N(A) = {O}. <J
(8.7)
Consider the operators n
Bn
=n- 1 LB k ,
n=1,2, ....
(8.8)
k=l
From the equality BnA = n- 1
(~Bk - ~Bk+1)
=
n- 1(B - Bn+1) and by con-
dition (VB) it follows that for any y E R(A) the relation lim BmY
n--->CXl
=0
(8.9)
LINEAR OPERATORS
130
holds. If, on the contrary, y E N(A), then BnY = Y and equality (8.9) fails for O#YEN(A). ~ COROLLARY 8.2. If the operator BE
A
= J - B is Fredholm, then equality
L(X) satisfies condition (UB) and the operator (8.6) is true.
Since A is a Fredholm operator, the kernel N(A) is a finite-dimensional subspace of X. Let dimN(A) = n and let {adk=l be a basis of N(A). Using Hahn-Banach-Suchomlynoff Theorem and Lemma 8.1 we may find a system of vectors
Udk=l
~
x* with the properties: (8.10)
(y, !k) = 0,
k
= 1, ... ,n,
(8.11)
for any y E R(A). Condition (8.10) implies that the system of vectors Udk=l is linearly independent and from (8.11) together with (5.1) it follows that Udk=l is a basis of N(A*) (recall that dim N(A*) = n). The operator P defined by (7.1) is a projection from X onto N(A), and J - P is a projection onto R(A) = J..N(A*). ~ Let us return now to the more general case when the operator A is only normally solvable. For such an operator the next result is true.
L(X) satisfies condition (UB) and the operator A = J - B is normally solvable. Then equality (8.6) holds if and only if the limit in the strong operator topology (see §8 of Chapter 0) of the sequence of operators defined by (8.8) exists. The operator THEOREM 8.2. Assume that the operator B E
n
P
= n~oo lim Bn = n---+oo lim
is a projection onto N(A) and condition (UB).
IIPII : :; M,
n- 1 ' "
L.....t
Bm
(8.12)
m=l
where M is the same upper bound as in
Necessity. From (8.6) it follows that any element x E X can be represented
as
x = Y + z, where Y E R(J - B) = R(A) and z E N(J - B) = N(A). By this equality and from (8.8), (8.9) and condition (UB) it follows that IIBnl1 : :; M, n E N, and n
lim Bnx
n-+oo
= n-+oo lim n- 1 ' " (I - A)m z = z. ~ m=l
(8.13)
131
Invariant subspaces
As a consequence of the Uniformly Boundedness Principle, the operator P defined by (8.1) is a bounded linear operator with IIPII ~ M. According to (8.13) P is a projection onto N(A). Sufficiency. Assume that the operator P defined by equality (8.12) exists. Again as a consequence of the Uniformly Boundedness Principle, the operator P is linear, bounded and IIPII ~ M. For each Z E N(A) we obtain (see (8.13)) P(z) = z, hence N(A) ~ R(P). On the other hand, if u E R(P) then there exists v E X for n
which u
= Pv = lim
n- 1
n---i'OO
""'
~
Bmv. Then, by condition (UB), we have
m=l
Au = (I - B)u =
nl~ n-
1
(t,
Bmv -
t,
Bm+lv) =
lim n-1(Bv - Bn+lv) = 0,
=
n--+oo
hence u E N(A). Thus we obtain that R(P) = N(A) and that P I R(P) is the identity operator. Consequently, P is a projection onto N(A). Let us show that N(P) = R(A), a relation which clearly will complete the proof of (8.6). Let y E N(P). Then, by (8.12), for any c > 0 there exists n E N such that IIBnYl1 < c. We have y - BnY E R(A); indeed
y-BnY= (I-n- 1 tBm)y=n-1 t(I-Bm)y= m=l
m=l
n
= n-1(I - B)
L (I + B + B2 + ... + Bm-1)y = (I -
B)Yn,
m=l
n
where Yn = n- 1
L (I + B + B2 + ... + Bm-1)y. m=l
Since y - BnY E R(A) and R(A) is closed, taking into account that c was an arbitrary positive number, we obtain y E R(A), i.e., N(P) ~ R(A). The converse inclusion R(A) ~ N(P) follows directly from (8.9). ~ The arguments in our proof of Theorem 8.2 above are, actually, suitable modifications of the arguments used by K. Yoshida in the proofs of the Statistical Ergodic Theorem and its consequences (see, for instance, K. Yoshida [1]). This theorem asserts the following: REMARK 8.2.
Let B E L(X) be an operator satisfying condition (UB). Consider the operators B n , n E N, defined by (8.8) and assume that the sequence {BnX}nEN is weakly compact for x E X. Then this sequence is convergent in the norm topology of the space X.
132
LINEAR OPERATORS This theorem and Theorem 8.2 lead to the next simple but important result.
COROLLARY 8.3. Let X be a reflexive space. Assume that an operator B E L(X)
satisfies condition (UB) and the operator 1- B is normally solvable. Then the equality (8.6) is true. Some authors (see, for example, F. Riesz [1]) refer to Corollary 8.3 as the Statistical Ergodic Theorem.
Chapter V Nonlinear equations with differentiable operators §1. Fixed points. Banach principle 1. Let 1) be a topological space. Assume that F is an operator defined from 1) into 1),
i.e., (1.1)
DEFINITION 1.1. A point
z E
1)
is called a fixed point for the operator F if F(z)
= z.
Condition (1.1), which is refered to as the invariance of the set 1) for the operator F, enables us to define the iterations Fn of F, by the recurrent relations Fl = F, F n+1 = Fn 0 F, where "0" denotes the composition of operators. DEFINITION 1.2. 1) A fixed point z E 1) ofthe operator F is called locally attractive if there exists a neighborhood U ~ 1) of the point z, such that the iterations Fn(x) converge to z for all x E U. If U = 1) we shall say that z is an attractive (or globally attractive) fixed point. 2) The point z is called an s-fixed point (successful approximative fixed point) for the operator F in 1) if it is attractive uniformly on each closed and bounded subset of 1), i.e., the iterations Fn(x) converge uniformly on each closed and bounded subset of 1). 3) A fixed point z is called repulsive if the sequence Fn(x) converges to z if and only if x = z. 4) A fixed point z will be called neutral if it is neither locally attractive nor repulsive.
NONLINEAR EQUATIONS
134
DEFINITION 1.3. A fixed point z of F is called isolate if there exists a neighborhood
of this point which does not contain other fixed points for the operator F. REMARK 1.1. A locally attractive point z is an isolate fixed point. Moreover, a globally attractive point is unique. Indeed, assume that on a neighborhood ti of the point z the iterations Fn(x) converge to z for all x E ti. If we suppose that the operator F has another fixed point y in ti, then Fn(y) ----) Z. But y is a fixed point for the operator Fn for any n, since Fn(y) = FoFn-l(y) = F(y) = y. Hence z = y. We notice also that for a continuous operator F the convergence of the iterations implies
the existence of fixed points. A lot of work has been done to obtain criterions for the existence of fixed points and to study the properties of these points (uniqueness, isolation, attractiveness, and so on) for various operators. These criterions are frequently keystones in the theory of nonlinear operator equations. The theorems which establish conditions for the existence of fixed points are usually called fixed point principles. As a rule, the formulation of a fixed point principle starts with the condition of invariance (1.1). However, in many cases this condition is insufficient for the existence of a fixed point, and, even more, it is insufficient for the existence of an s-fixed point. Notice that the presence of an s-fixed point guarantees not only its uniqueness, but also the possibility of approximating it (by approximative computations). Let us remark that the development of the computer techniques enables one to approximate a fixed point by iterations even for rather complicated operators. At the same time the computer methods do not insure against accidental computational errors, which practically do not matter as soon as we know in advance that the iterative proccess converges uniformly.
One of the most effective fixed point principle is the Banach principle for contractive operators. Let (X, p) be a complete metric space with the metric p( " . ) (see §4 of Chapter 0). An operator F defined from a subset ~ <;;;; X into X is called a contractive operator or a contraction, if it satisfies a Lipschitz type condition with a constant q < 1, i.e.,
p(Fx, Fy)
~
qp(x, y),
x, y
E ~.
(1.2)
Obviously a contractive operator is continuous. THEOREM 1.1 (Banach principle for contractive operators). Let ~ be a closed
subset of (X, p) and let F be an operator defined on ~ which is a contraction and satisfies the invariance condition (1.1). Then the operator F has an s-fixed point in
135
Fixed points. Banach principle
::D. The rate of convergence for the iterations of the operator F to this fixed point is described by the relations
p(Fnxo, Fmxo) where m
~
~ Lp(xo, Fxo), 1-q
(1.3)
nand Xo is an arbitrary element in ::D.
<J As a matter of fact it is sufficient to prove inequality (1.3). This inequality implies that the sequence {xn}nEN, where Xn = Fn xo , is fundamental, and this property is uniformly fulfilled relatively to Xo in any closed and bounded subset of ::D. Therefore, by the completeness of the space X and the closedness of the set ::D, the sequence {x n }nEN converges to a point Z E ::D, uniformly relatively to Xo E ::D. Indeed, by (1.2) it follows that
Analogously,
By induction we get
p(xk,xk+d ~ qkp(xo,xd, for all k = 1,2, .... From these relations, using the triangle inequality and the formula for the goemetric series, we have
+ P(Xn+l' Xn+2) + ... + P(Xm-I, Xm) ~ ~ qn p(XO, xd(l + q + ... + qm-n-I) ~ 1 qn p(xo, xd.
p(xn' x m ) ~ p(xn, xn+d
-q
Setting now z = lim Fnxo, where Xo is an arbitrary element of ::D, we obtain that z n-+oo is an s-fixed point for the operator F in::D. ~ If we weaken slightly the conditions required for the operator F in Theorem
1.1, we obtain the next version of the Banach principle.
closed subset of a complete metric space X and let F be an operator satisfying the in variance condition (1.1). Assume that there exists an integer p ~ 1 such that the operator FP is a contraction on ::D. Then the operator F has a globally attractive, and, consequently, a unique fixed point in ::D. COROLLARY 1.1. Let::D be a
<J By Theorem 1.1 the operator FP has an s-fixed point z. Choose an arbitrary element Xo E ::D and consider the sequence {xn : n EN}, where Xn = Fnxo and
NONLINEAR EQUATIONS
136
N denotes the set
{O, 1,2, ... ,n, ... }. Let us show that this sequence converges to z. Indeed, the sequence breaks up into a finite number of subsequences
{FkP XQ
:
kEN},
{FkP X1
:
kEN}, ... ,
{FkP xp_ 1
:
kEN}.
Each of these subsequences is convergent to z. Now we show that z is a fixed point for F. (Notice that this does not follow from the convergence of the sequence {xn : n E E N}, since under our assumptions the operator F is not necessarily continuous.) We have FP F z = F FP Z = F z, whence it follows that F z is a fixed point for the operator FP. According to the uniqueness of the fixed point for FP we obtain that Fz = z. ~ REMARK 1.2. In the general case it is not true that an s-fixed point of the operator
FP, for certain p > 1, is an s-fixed point for F, too. However, as we shall prove below, this is true for holomorphic operators. In order to approximate a fixed point by iterations under the assumptions of Corollary 1.1 it is advisable, if it is possible, to start with an explicite computation of the operator FP and then to use its iterations. REMARK 1.3. The condition q < 1 in (1.2) is not necessary for the existence of s-fixed points. For example, consider the operator Fx = ax2, where 1/2 < a < 1, which maps the interval [0, 1] into itself and has the s-fixed point 0, though q = 2a > 1. Nevertheless, in the general case condition q < 1 can not be weakened by setting q ~ 1. For example, the operator Fx = x+a, a -I- 0, which maps IR into IR and satisfies condition (1.2) with q = 1, does not have fixed points. In this example the diameter of the set on which the operator F acts is infinite. However, there are examples of operators acting on domains with bounded diameter, in infinite dimensional spaces, which satisfy condition (1.1), or condition (1.2) with q = 1, and do not have fixed points (for more details see §2 below).
Let us return now to the concept of s-fixed points and state, without proof, a remarkable theorem due to P. R. Meyers [1], which explains the substance of this notion and represents the most complete converse of the Banach principle. (In connection with this result see also A. Levin and E. Lifsic [1].)
(P. R. Meyers). Let X be a complete metric space and 1) a closed subset of X. Assume that an operator F satisfies the invariance condition (1.1) and has an attractive fixed point z E 1), which is an s-fixed point in a neighborhood of z. Then z is an s-fixed point in 1) and, moreover, for any q with < q < 1 there exists a metric on the space X equivalent with the original metric, such that F satisfies the contraction condition (1.2) on :D with respect to this new metric. THEOREM 1.2
°
Although Meyers' result is merely an existence theorem, it shows that in some
Non-expansive operators
137
concrete problems, if we know a piori about the existence of an s-fixed point, then, for estimating the rate of convergence of the iterative process, it is convenient to try to find an equivalent metric, with respect to which F is a contraction. 2. One of the most important fixed point principles is the well known principle due to J. P. Schauder, which is a generalization of the finite-dimensional fixed point principle of Brouwer. THEOREM 1.3 (J. P. Schauder). Let F be an operator which maps a closed convex subset:D of a Banach space X into itself and is completely continuous on:D. Then F has at least one fixed point in :D.
All the proofs of this theorem we know about rely on subtle topological and geometric considerations. As a rule, these proofs use Brouwer Theorem, finite-dimensional approximations, and also some results from the degree theory for mappings and the vector fields theory (see, for instance, V. A. Trenogin [1], M. A. Krasnoselskii [1]). This is why we stated Schauder Theorem without proof. Obviously, neither Theorem 1.1, nor Theorem 1.3, are related to each other. Let us remark also that Schauder principle has no constructive features; it fails to indicate the number of fixed points, as well as methods for their approximation. In the following section we will consider a larger class of operators, which includes the class of contractive operators. At the same time, under the assumptions of Schauder principle, we will indicate some approximation methods for the fixed points.
§2. Non-expansive operators Let (X, p) be a metric space and :D a closed subset of X. DEFINITION 2.1. An operator F : :D strictly non-expansive, if the inequality
---+
p(Fx, Fy) is fulfilled for all x, y
for any x, y
E
:D, x
E
=1= y.
~
X is called non-expansive, respectively
p(x, y)
(2.1)
p(Fx, Fy) < p(x, y),
(2.2)
:D, respectively, if
NONLINEAR EQUATIONS
138
A large amount of work is devoted to the theory of non-expansive operators, either on metric, or on Banach spaces (see for example Z. Opial [1]). Here we treat only those aspects which are related to the fixed points of differentiable operators. As we have already noticed in Remark 1.2, the operators satisfying condition (2.1) may have no fixed points. In addition, let us present a few other examples.
2.1 (T. Hayden, T. Suffridge [1]). Let Co be the space of all sequences x = (Xl> ... , X n , ... ) of real or complex numbers which converge to zero, with the norm Ilxll = max IX n I· Define an operator F on the unit ball of this space by the equality
EXAMPLE
n
Fx =
(!, Xl, X2,·· .), for
any X = (Xl, ... , Xn , .. . ). Clearly IIFxII ,,;; 1 if IIxil ,,;; 1, i.e., F satisfies condition (1.1), and IIFx - Fyll = 11(0, Xl - YI,···, Xn - Yn," .)11 = IIx - YII, Le., F satisfies condition (2.1), too. At the same time, if we assume that there exists an element Z = (ZI, ... ,Zn, ... ) such that Z = Fz then, necessarily, Zl = Z2 = ... = = Zn = ... = i.e., Z ~ Co, a contradiction.
!,
We can show that condition (2.2), as well, is not enough for the existence of fixed points. EXAMPLE
2.2. Let X =
~
and :D
= [1, (0).
The operator F defined by Fx = X +
dearly does not have fixed points, while F(:D) jFx - Fyi
for all x, Y ~ 1,
X
= Ix -
Y+
~-
tl
=
~
Ix -
1
-
X
:D and
YII x Yx ; 11 < Ix - YI,
i=- y.
Related to the example above we easily notice that for any q, with 0 < q < 1, there exist x, Y E [1,(0) such that jFx - Fyi> qlx - YI. An analogous example can be produced on a ball with a finite radius in an infinite dimensional space, too. EXAMPLE 2.3 Let
F defined by Fx
1
+-n sin n.
=
X = Co and let :D be the unit ball in X. Consider the operator n-1
(YI, ... , Yn," .), where X
=
(Xl, ... , x n , ... ) and Yn
=
- - xn+
n We can easily prove that the operator F satisfies both conditions (1.1) and
(2.2) on:D. At the same time, F has no fixed poins in:D. Assume, on the contrary, n -1 1 that Z = Fz, where Z = (Zl"'" zn) E:D. This means that Zn = --Zn + - sin n. n n Therefore, Zn = sin n, which contradicts the definition of the space Co. All these examples show that the existence of a fixed point for a non-expansive operator requires some additional conditions.
Let (X,p) be a complete metric space, :D a closed subset of X, and F a strictly non-expansive operator on :D satisfying the invariance condition
THEOREM 2.1.
139
Non-expansive operators
(1.1), i.e., F(1») <:;: 1). If there exists at least one point Xo E 1) such that the orbit ~ Xo = {x E 1) : x = F n Xo, n E N} is precompact, then the operator F has a unique fixed point z E 1). Moreover, the sequence {xn : Xn = Fnxo, n E N} is convergent to z.
Next, let us notice that the operator F maps ~xo into ~xo. Indeed, for any y E ~xo there exists n E N such that y = Fnxo, so Fy = Fn+1xo, hence F(~xo) C ~xo. Since F is continuous, then E(Jxo) c Jxo, too. This implies that
a contradiction. The equality
p (z, Xni+l) = P (z, FXnJ :::;; p(z, Fy) + p (Fy, FXnJ < < p(z, Fy) + P (y, xnJ < p(z, y) - 8.
(2.3)
For any mEN, using (2.3) we obtain the inequalities
p (z, xni +m) = P (Fz, FXni+m-d < < p(z,xni+m-d < ... < p(z,y) - 8, which imply that y contradiction. ~
(i- z)
is not a limit point of the sequence {x nk
(2.4) kEN}, a
COROLLARY 2.1. If a strictly non-expansive operator F maps 1) onto a precompact subset of 1), then F has an attractive fixed point.
In this case the orbit
~x
of any point x
E 1)
is a precompact subset of 1). ~
NONLINEAR EQUATIONS
140
Let us notice the slight difference between conditions (2.2) or (2.1), and the contraction condition (1.2). In spite of this, the existence and the uniqueness of a fixed point do not necessarily imply its atractiveness. Moreover, a unique fixed point may be even repulsive. EXAMPLE 2.4. Let £2 be the Hilbert space of square sumable sequences over the field J[{ of real or complex numbers. Define the linear operator A : £2 ----+ £2 by the equality Aei = (1 - Di)ei+1' i = 1,2, ... , where {ei : i E N} is an orthonormal basis in £2, and {Di : i E N} is an arbitrary collection of real numbers satisfying the 00
conditions 0 <
Di
< 1 and
II (1 -
Di)
= D
> O.
i=l
Let X be the space L(J[{, £2) of all linear continuous operators from lK into n
£2. Set Fx
= Ax.
Clearly F : X
----+
X. For h E lK we have xh =
L Ai(h)ei, where i=l
Ai(h) E lK. Therefore,
IIFn xl1 2= IIAn xl1 2=
sup IIAnxh11
=
Ihl=l
i+n-1
00
sup
L
Ihl=l i=l
IAi(hW
II
00
(1 - Dj)2 < sup
L
IAi(h)12 = Ilx11 2,
Ihl=l i=l
j=l
for all n E N, i.e., IIFxl1 < Ilxll, for x condition (2.2) and has a fixed point z 00
-I-
O. Since F is a linear operator, it satisfies On the other hand, from the inequalities
= O.
00
it follows that the sequence {Fnx : n E N} converges to z
= 0 if and only if x = O.
The examples above show that the claim on precompactness in Theorem 2.1 and Corollary 2.1 is essential. If the considered space X is a Banach space and the set 1) is convex and bounded, then criterions for the existence of fixed points can be reformulated by slightly weakening the assumptions on F in Theorem 2.1. Let us recall (see §6 of Chapter 0) that an operator tJ> : 1) ----+ X is called proper on 1) if the preimage of any precompact subset of X is a precompact subset of 1). THEOREM 2.2. Let X be a Banach space, 1) a closed bounded convex subset of X,
and F : 1) ----+ 1) a non-expansive operator. Assume that the operator tJ> = I - F (where I is the identity mapping on X) is proper on 1). Then the operator F has at least one fixed point z in 1).
141
Non-expansive operators
<J Choose an arbitrary element Y E :D and consider the operators F k , k
=
= 1,2, ... , defined by the equalities
(2.5)
Since :D is convex, we obtain Fk(:D) S;;; :D. Moreover, by (2.5) we have
IIFkX - FkX'11
=
(1 - ~) IlFx - Fx'll:::;; (1 - ~) Ilx - x'll,
for any x, x' E :D and kEN, i.e., the operator Fk is a contraction on :D for any fixed kEN. By Theorem 1.1 there exists a fixed point x(k) in:D for the operator F k , i.e., FkX(k) = x(k). Consider the sequence
qJx(k)
= x(k) - Fx(k) =
( 1) 1- k
Fx(k)
1
+ kY -
Fx(k)
1
= k(Y - Fx(k)),
kEN.
Since:D is bounded, we obtain that qJx(k) ---- 0 as k ---- 00, whence, by our assumptions, it follows that the sequence {x(k) : kEN} is precompact in :D. This means that there exists a subsequence {x(k m ) : mEN} convergent to a point z E :D. Then qJz = lim qJx(km ) = 0, which is equivalent with the equality z = Fz. ~ m--->oo
2.2. Let X and:D be such as in Theorem 2.2 above. If a non-expansive operator F maps :D onto a precompact subset of:D, then there exists at least one fixed point for F. If, in addition, we assume that the space X is strongly convex (see §9 of Chapter 0) then, for any () with 0 < () < 1, and any Xo E :D, the sequence COROLLARY
xn
= ()Xn-l + (1 - ()FXn-l,
n E N,
(2.6)
converges to a fixed point for the operator F. <J The first assertion follows directly from Theorem 2.2 if we notice that the operator qJ = I - F is proper. Indeed, let {Yn : n E N} be a convergent sequence in qJ(:D). Then its preimages have the following form: Xn = FX n +Yn. Since {Fx n : n E N} is precompact it follows that {x n : n E N} is precompact, too, as requested. Let us prove now the second assertion. First of all observe that, by the well known Mazur Lemma (see, for instance N. Dunford and J. T. Schwartz [1]), the sequence (2.6) is precompact, since it lies in the closed convex hull of the union of the precompact set F(:D) and the point Xo. Denote by Y a certain limit point of this
NONLINEAR EQUATIONS
142
sequence and assume that y is not a fixed point for the operator F. Let z be a fixed point for the operator
F(())x
=
()x
+ (1 -
())Fx,
0 < () < 1.
(2.7)
Suppose now that X is strongly convex. Then liz - F(())YII
= liz - ()y - (1 - ())FYII = II()z - ()y + (1 - ())z - (1 - ())Fyll < (2.8) < ()llz - yll + (1 - ())IIFz - Fyll ~ liz - yll·
In the proof of (2.8) we supposed that the vectors Fz - Fy and z - yare not collinear. Hit is not so, i.e., Fz-Fy = A(Z-y), A > 0, then, since F is non-expansive, there are two possibilities: either A = 1 and then y is a fixed point for F, or A < 1 and then inequality (2.8) is obvious. It follows that 28 = liz - yll - liz - F(())YII > o. Further, following verbatim the computations in (2.3), (2.4), we obtain that y is not a limit point of the sequence (2.6) and, consequently, it is not a fixed point for the operator F(()). However, if y is a limit point of the sequence (2.6), i.e., there exists a subsequence {Xnk : kEN} convergent to y, from (2.6) we obtain
and, by induction, we get
for any mEN. This implies the convergence of the sequence of iterations Xn
= F(())Xn-l
which, actually, coincides with sequence (2.6).
~
REMARK 2.1. Let us emphasize that the second assertion in Corollary 2.2 does not at all mean that the fixed point z for F is an attractive fixed point for F(()). More precisely, if z is the unique fixed point for F, then it is an attractive fixed point for the operator F(()). Otherwise z can be a neutral point only.
Moreover, it can be shown that the set of all fixed points for a non-expansive operator on a strongly convex Banach space is a convex set. Consequently, if there exists, more than one fixed point, then there are no isolated points among them. REMARK 2.2. The requirement in Theorem 2.2 for 1) to be a convex set can be
removed and replaced by the weaker condition on
1)
to be a star-shaped set (for the
143
Fixed points for differentiable operators definition of the star-shaped set see Chapter III). In this case we have to set y (2.5).
= 0 in
REMARK 2.3. Notice that the first assertion in Corollary 2.2 follows, as a matter of fact, from Schauder principle. However, we preferred a straightforward proof which enables us to consider not only convex sets (see the previous remark).
§3. Fixed points for differentiable operators 1. The next result follows from Theorem 1.2.1 and the Banach principle (Theorem
1.1).
Let 1) be a closed convex set in a Banach space X and let F be an operator on 1) which is Gateaux differentiable at any point x E 1) (i.e., for any point x E 1) and any direction h there exists a neighborhood of 0 E ][( in which the vector-function F(x + th) is defined and differentiable). Assume that the derivative of F satisfies the condition (3.1) l = sup 11F'(x)11 < 1. THEOREM 3.1.
xE!)
BE
~ 1),
then there exists an s-fixed point z E [0, 1] the sequence
If F(1))
xn
= BXn-l + (1 - B)Fxn-l,
n
=
1),
for F. Moreover, for any
1,2, ... , Xo
E 1),
(3.2)
converges to z, with the rate of a geometric progression
Ilx n
-
zil
~
qn
--llxo - Fxoll, l-q
(3.3)
where q = B + (1 - B)l < 1.
Let 1) be a closed convex set in X and F : 1) that is Gateaux differentiable at any point x E 1), such that
COROLLARY 3.1.
sup xE!)
II(FP)'XII <
1,
---> 1)
an operator
(3.4)
NONLINEAR EQUATIONS
144
for a fixed pEN. Then the operator F has a unique attractive fixed point z E fl.
The differentiability of the operator FP follows from Lemma 1.4.2. In fact,
(FP)'x
=
(F')P x ,
(3.5)
where the right hand side of (3.5) must be understood as the p-th power of the linear operator F'. Now our conclusion follows from Theorem 1.2.1 and Corollary 1.1. ~ REMARK 3.1. We will show below that for a Frechet differentiable operator condition
(3.4) actually implies the existence of an s-fixed point. REMARK 3.2. Since for a Frechet differentiable operator the constant l in condition
(3.1) is the smallest Lipschitz constant in inequality (1.2.1) then, by Remark 1.3, it follows that this condition is not necessary for the existence of s-fixed points.
Namely, let z be a certain fixed point for a Frechet differentiable operator
F, lying strictly inside fl. Assume that II(FP),zll < 1 for some pEN. Then, by the continuity of F'(x) in the uniform operator topology, there exists a closed ball II centered at z and completely lying inside fl, such that sup II(FP)'xll < 1. Moreover, xEU
IIFP x -
zll
=
IIFP x - FPzl1 ~ sup II(FP)'yll
. liz - xii < Ilx - zll,
yEll
for all x E ll. Thus FP (II) C ll. By the continuity of F, there exists a closed neighborhood QJ ~ II of the point z such that F(QJ) ~ QJ. Then, according to Corollary 3.1, the point z is attractive in QJ. Moreover, if z is globally attractive in fl then, by Meyers Theorem (Theorem 1.2) z is s-fixed on the whole fl, regardless of how large the value of the norm of F'(x) on fl is. In particular, this explains why the point 0 is an s-fixed point for the operator F in Example 1.3, where F'(O) = O. ~ In Section 2 (see Corolary 2.1) we established the existence of an attractive fixed point y for a non-expansive operator mapping a closed set fl onto a precompact subset of it. For a differentiable operator the following version of Corollary 2.1 is true.
fl be a closed convex set in the Banach space X and let F : : fl -+ fl be an operator which is indefinitely Fnkhet differentiable in a neighborhood of each point x E fl. Assume that for some p, q E N the following conditions are fulfilled: 1) the operator FP maps fl onto a precompact subset of fl, and 2) II(Fq),(x)1I < 1 for all x E fl.
THEOREM 3.2. Let
Fixed points for differentiable operators
145
Then the operator F has an attractive fixed point.
Fpq satisfies all the conditions in Corollary 2.1. Indeed, on one hand, Fpq = (Fq)P and, consequently, II (Fpq)'(x) II ~ II(Fq)'(x)IIP < 1, for all x E :D, i.e., the operator Fpq is strictly non-expansive. On the other hand, Fpq = (FP)q and, consequently, the operator Fpq maps :D onto a precompact subset of :D. Thus Fpq <J The operator
has an attractive fixed point z E :D which, obviously, is an attractive fixed point for the operator F, too. ~ 2. A sufficient condition for the existence of an s-fixed point for an operator F is, for instance, condition (3.1). At the same time, if the Frechet derivative F'(z) of F at the fixed point z has norm 1, then it does not follow that z fails to be an s-fixed point. For example, on the two-dimensional space X, the linear operator A defined, relatively to the canonical basis, by the matrix
has norm 1 (recall that A' = A - see Lemma 1.4.1), yet the point z = 0 is an s-fixed point for A. Therefore, we would like to point out some criterions that characterize, at least locally, the uniformly attractive fixed points. The strongest sufficient - and, in the case of complex spaces, also necessary - condition which accomplishes this goal is given by r((FP)'(z)) < 1, (3.6) for some pEN, where r(A) denotes the spectral radius of an operator A (see §2 of Chapter IV). <J If (3.6) is fulfilled then, according to Theorem IV.2.2, we can introduce on the space X a new norm 1111*, equivalent with the original one, such that II (FP)'(z) 11* < < 1. By Remarks 3.1 and 3.2 above the next result follows.
Assume that F is a Fhkhet differentiable operator on an open set :D and z = F(z) E :D. IfF satisfies condition (3.6), then there exists a neighborhood U of the point z such that F(Ii) ~ Ii and z is an s-fixed point for the operator F on Ii. IE, in addition, the operator F is continuous on the closure :D of:D, F(:D) ~ :D, and z is an attractive fixed point in :D, then z is s-fixed. ~
THEOREM 3.3
Let:D be a closed convex set in X and let F : :D ---+ :D be an operator that is differentiable in :D and satisfies condition (3.4). Then F has an s-fixed point in :D. COROLLARY 3.2.
NONLINEAR EQUATIONS
146
The existence of a unique attractive fixed point z is a consequence of Corollary 3.1. By equalities (3.5) and (IV.2.2), it follows that the spectral radius of the Fh3chet derivative of F at the point z is less than 1. In order to conclude the proof, it remains to use Theorem 3.2. ~ <J
REMARK 3.3. Let us consider the linear equation
x
= Ax + y,
y E X, A E L(X, X),
(3.7)
where r(A) < 1. By the arguments above we can construct a new equivalent norm on X, relatively to which A becomes a contraction. Consequently, for any y E X the operator F defined by F(x) = Ax+y is a contraction, too. Therefore, using inequalities (3.3) we get an estimate, with respect to this new norm, for the rate of convergence of the succesive approximations to the solution of equation (3.7). Unfortunately, this estimate does not hold for nonlinear equations, in general. Estimates (3.3), under the conditions in Theorem 3.2 or Corollary 1.3.2 and with respect to a suitable norm, are true only when Xo is sufficiently closed to z. If, in addition, the operator FP subject to the conditions in Corollary 3.2 can be computed explicitly, then it is convenient to use its iterations to approximate the fixed point z. However, let us also point out the fact that if the F'rechet derivative of the operator F at the fixed point z is zero, then the iterations of F on a certain neighborhood of the point z converge to z, with the rate of a geometric progression of an as small as we please ratio. This remark can be strengthened if we assume the existence of higher order derivatives for the operator F. <J Namely, let us assume that the operator F is m-times continouously differentiable on the closure of::O, and F(k)(z) = 0 for all k = 1,2, ... , m - 1. Then, using one of the Taylor formulas 1.7.5 or 11.4.4, we obtain the inequality
where M
=
sup IIF(m) (x)ll, which is true for all x sufficiently close to z, let us say, for xE:D
those x lying in the ball centered at z and with radius 8 (8 < dist(z, 0::0)). Assume now that z is an attractive point for F. Then there exists an integer no such that Ilxno - zll < 8, where Xn = FXn-l, n = 1,2, .... Setting L = M/m!, for any n ~ no we get Ilx n - zll ~ Lllxn-l - zllm ~ L m +1 1Ix n _2 _ zllm2 ~ _ zllm 3 ~ . , . ~ LHm+m2+.+mn-no-l Ilxno _ zllm n- no ,
~ L m2 +m +1llxn _3
Some applications of fixed point principles
147
or (3.8) where Ll
=
"'-n.
~
The estimate (3.8) shows that, starting with no, the iterations converge with a rate of convergence better even than the factorial rate. Notice that, under these conditions, the sequence (3.2), for more slowly than for e = 0.
e > 0, converges
§4. Some applications of fixed point principles The fixed point principles offer the possibility to apply a lot of methods from functional analysis in different other mathematical fields. 1. EQUATIONS WITH COMPOSITION OPERATORS. Let \!: be a fixed Banach space over the field lK. Choose and fix a subset fl of][( and consider a set X of vector-functions
defined on fl with values in \!:, such that X has the structure of a Banach space. For example, if fl is a compact set and X is the space of all continuous vector-functions from fl into \!:, then we can set
Ilxll = sup Ilx(CT)II<E, aEll
i.e., X = C(fl, \!:). If X is the set of all vector-functions measurable in the sense of §1 of Chapter II and, at the same time, square summable on fl in the usual sense, then X becomes a Banach space, with the norm
IIxl1 2 =
J
Ilx(CT)11 2 dCT.
f}
Following the usual notation, we have X = L2(fl, \!:). If \!: is a Banach algebra, then it can be shown that X is a Hilbert space. Analogously we can introduce the spaces Lp(fl, \!:), and some other structures of Banach spaces. Let us return to the general case and let X be a given space of vector-functions defined on the set fl, with values in the Banach space \!:. Let f(·, .) be a fixed operator
NONLINEAR EQUATIONS
148
defined on the topological product n x IE with values in IE. If f( a-, x( a-)) E X then the operator f(', . ) defines a composition operator F : X --t X by the equality
Fx(·) = fC,xC)).
(4.1)
The operator (4.1) is an analogue of the N emytzki operator (see Example 1.3.3). As in the case of the Nemytzki operator, set 5) = {x E IE : Ilxll ~ R} and assume that f : n x 5) --t IE is a jointly continuous operator, such that the operator f~ (a-, x) exists and is jointly continuous on n x 5), too. Suppose next that X = G(n, IE). Then the operator F is Frechet differentiable on the ball 1) of radius R centered at the origin and
F'(x(·))hC) = f~(a-,x(·))hC)· Further, if
Ilf( a-, x) I
~
n x 5), and
R for all (a-, x) E
Ilf~(a-,x)llQ:
sup
< 1,
(4.2)
(u,x)EllxSj
then, obviously, FCD)
<:;;; 1)
and sup Ilxll
1IF'(x)11 < 1.
~R
Hence, by Theorem 1.1, the operator F has an s-fixed point. Thus we proved the next result. PROPOSITION 4.1. Let
f(', .) be an operator defined on the topological product
n x 5) and with values in 5) (with nand 5) as above). Assume that both f and its partial derivative f~, x E 5), are continuous on n x 5), and that condition (4.2) is fulfilled. Then the equation xCa-) = f(a-, x(a-)) (4.3) has a unique solution in the set of all continuous vector-functions from Moreover, the sequence
xn(a-) = f(a-, Xn-l (a-)),
n
= 1,2, ... , Xo
n
into 5).
E 5),
converges to xCa-) uniformly relatively to xo, and, for a fixed xo, uniformly on each compact subset of n. By replacing condition (4.2) with the slightly weaker conditions sup (u,X)EllxSj
Ilf~(a-,x)11 ~ 1
Some applications of fixed point principles
149
or Ilf~(u,x)11
<1
x Sj,
for all (u,x) E fl
we can easily reformulate, in this new setting, counterparts of the results in §§2,3. 2. EQUATIONS WITH HAMMERSTEIN TYPE OPERATORS. Keeping the notations introduced in Subsection 1 above, let us set n = [a, bj and X = C(fl, ~). Consider the operator
J
K(u, t)f(u, x(u))du,
=
(4.4)
a
where K(u, t) - the kernel of the operator
J b
K(u, . )f~(u, x(u))h(u)du
a
holds. Assume that for any u E [a, bj the operator f(u, .) maps the ball Sj = {x E ~ : Ilxll!( R} into the ball of radius Rl with center at the zero element of the space ~. Let
PROPOSITION 4.2.
L =
Ilf~(u, x)11
sup
,
(<7,x)E!1xSj
If
RIK(b-a)!(R { and LK(b-a) < 1
(4.5)
then the equation b
x(t)
=
J
K(t, u)f(u, x(u))du
(4.6)
a
has a unique solution in C(n,
J
~),
which is the uniform limit of the sequence
b
xn(t)
=
K(t, u)f(u, Xn-l (u))du,
a
n
= 1,2, ... ,
(4.7)
NONLINEAR EQUATIONS
150
where xo(t) is an arbitrary continuous vector-function from [a, b] into S), i.e., Ilxo(t)ll:::;
:::;R. The first inequality in (4.5) guarantees that IIPx(·)1I :::; R, for any x with IIxll :::; R whereas the second one implies sup IIp'(x(· ))11 < 1. It follows that the Ilxll:S;;R operator P is a contraction on the ball of radius R with center at the zero element of the space X = C(rl, ~). ~
Further, let us remark that the operator P is the composition of the operator
F associated to f as in Subsection 1, with a linear operator A, that is P = AF, where b
Ay(t) = j K(t,cr)y(cr)dcr. a
We have
b
IIAII =
sup jK(t, cr)y(cr) Ily( lll:S;; 1
:::; K(b - a).
a
This remark enables us to weaken slightly the second inequality in (4.5). More precisely, the next result is true. PROPOSITION 4.3. Assume that
{
R1K(b - a) :::; R, Lr(A)(b - a) < b - a,
(4.8)
where r(A) is the spectral radius ofthe operator A. Then, equation (4.6) has a unique solution in C( rl, S)), which is the uniform limit of sequence (4.7) for any Xo ( .) E E C(n,S)).
F, respectively. If the second condition in (4.8) is fulfilled, then, for any e > 0, we can choose pEN such that IIAPII :::; (r(A) + e)p. For a sufficiently small e we have L(r(A) + e)) < 1 and, consequently sup II (pP)'(x( ·))11 :::; sup IIAPII IIx(·lll:S;;R Ilx(·lll:S;;R
II (FP)'(x( . ))11 <
1.
To conclude the proof it remains to use Corollary 3.1. ~ Although it is practically easier to check condition (4.5), however condition (4.8) is a matter of principle. For example, it shows that if the spectral radius of A is zero, then it is enough to check only the first inequality in condition (4.5).
Some applications of fixed point principles
151
Let us consider a rather general situation. Assume that the equation
x = AFx+y
(4.9)
is given on a Banach space X, where F is a continuously differentiable operator on the closed ball f) of radius R and centered at the zero element of the space X, A is a linear continuous operator on X, and y E X. (Such equations are usually called equations of the second genre.) <J
By Theorem 3.1, if conditions
{ I All . M + Ilyll (R, IIAII·L < 1
(4.10)
are fulfilled, where
M
=
sup IIFxll, Ilxll :::;R
L =
sup Ilxll (R
IIF'(x)ll,
then equation (4.9) has a unique solution in f), which is the limit, in the strong sense, of the sequence (4.11) xn = AFxn-l + y, n = 1,2, ... , with Xo an arbitrary element of f). Conditions (4.10) are straightforward generalizations of conditions (4.5). Setting C = AF we obtain C 2 (.) = AF(AF(·)) and
(C 2 )'(x)
=
AF'(AF(x))AF'(x).
Therefore sup II(C 2(x))'11 Ilxll :::;R
(L21IAI12.
More generally, if Lr(A) < 1 then, using (IV.2.2), we obtain sup II(Cn)'(x)11 (LniIAnll (Ln(r(A) Ilxll :::;R for a sufficiently small c > 0 and a sufficiently large n Corollary 3.1, conditions { IIAIIM + lIyll ( R, r(A)L < 1
+ c)n < 1, E
N. Thus, according to (4.12)
NONLINEAR EQUATIONS
152
are sufficient for the existence of the solution of equation (4.9), and for the sequence (4.11) to converge to that solution. ~ REMARK 4.1. Let us emphasize on the fact that the first inequality in (4.12), which
means the invariance of the ball :D relatively to G + y, is a rather rough estimate, and for certain y and M it may not be fulfilled. Yet, taking into account that M is a function of R, if the second inequality in (4.12) is fulfilled, then we can select R such that, for any y of a sufficiently small norm, the operator G + y has an invariant domain. Indeed, we may assume that F(O) = 0 (otherwise we can consider the equation x = AF1 (x) + y, where F 1 (x) = F(x) - F(O), which is equivalent with equation (4.9)). Then 0 E X is a fixed point for the operator AF. If the second inequality in (4.12) is fulfilled, then, for any sufficiently small E > 0 there exist a norm 1/ . 11* on X equivalent to the original one, and a number r > 0 such that the operator AF is a contraction on the ball {x : Ilxll* < r}, with the constant q = R(A)L+E < 1. If y satisfies the condition
Ilyll* < (1 -
q)r,
then
IIG(x)ll* : : ; IIAF(x)ll* + Ilyll* = = IIAF(x) - AF(O)II* + Ilyll* < qllxll* + (1 -
q)r
= r.
Thus, the condition r(A)L < 1 is sufficient for the local solvability of equation (4.9). However, it is not necessary. (By "local solvability" we mean the existence of the solution for all y in a certain neighborhood of zero, and for a sufficiently small value of IIF(O)II.) With the same notations as above, assume that
{ IIAIIM + Ilyll : : ; r, r(A)L::::; 1.
Then we can try to construct a norm relatively to which the operator G is non-expansive. In what follows we restrict ourselves to the case when IIAII = r(A) (this is the case, for instance, if A is a self-adjoint operator on a Hilbert space). If at least one of the operators A or F maps :D into a compact set, then G = AF will have the same property. In this case we can use the results in §2. In particular, from Corollary 2.2 we have: PROPOSITION 4.4. Let (E
=
OCn
,
for some n E N (OC is the field of real or complex
numbers), and let f(u,u) be a vector-function defined for u E [a, b] and u E OC n ,
Some applications of fixed point principles
153
Ilull ~ R,
with values in OCn . Assume that f and its partial derivative f~ (a, u) are continuous on the whole domain of definition. Further, let K(a, t) be a symmetric scalar-function which defines a linear completely continuous integral operator A on the space L2 (D, OC n ), by the formula
J b
Ax(t) = (P)
K(t, a)x(a)da,
x
E
L 2 (D, OC n ).
a
If the conditions {
i'LM(b - a) i'LL(b - a)
are fulfilled,
+ max Ily(t)ll][(n tEla,
bJ
~ R,
~ 1
then there exists at least one solution of the Hammerstein type equation
of the second genre
J b
x(t) = (P)
K(t, a)f(a, x(a))da + y(t).
a
Moreover, the sequence of vector-functions
J b
xn(t) = OXn-l (t)
+ (1 - 0) (P)
K(t, a)f(a, Xn-l (a))da + y(t),
n
= 1,2, ... ,
a
converges to a solution of this equation, for any Xo with any 0 E (0, 1).
Ilxo(t)11 ~ R,
t E [a, b], and
3. NONLINEAR VOLTERRA TYPE EQUATIONS. As a particular case of equation (4.9) let us consider an equation of the form
J t
x(t) =
K(t, a)f(a, x(a))da + y(t),
(4.13)
a
where K and f satisfy the conditions in Proposition 4.2. The spectral radius of the linear operator A - the Volterra operator - defined by
J t
Ax(·)
=
K(·, a)x(a)da,
a
NONLINEAR EQUATIONS
154
is equal to zero. Indeed, for any function x(·)
E C(fl,~)
t
IIA2x(·)11
J
~K21Ix(·)II(t~a)2
K(t,a)Ax(a)da
=
we have
a
More generally, for any n E N we obtain
or, (4.14) Since the right hand side of the last inequality approaches zero as n r(A) =
lim VllAnl1
n-+oo
=
~ 00,
we get
o.
From condition (4.10) we have: PROPOSITION 4.5. If the
vector-functions K,
KM(b - a) + max
a~t~b
!
and y satisfy the condition
Ily(t)11
~
(4.15)
R,
then equation (4.13) has a unique solution, and the sequence t
xn(t) =
J
K(t,a)!(a,x n_l(a))da+y(t),
n
= 1,2, ... ,
(4.16)
a
where
Ilxo(t)11 ~ R,
converges to that solution. Moreover, the next estimate sup
Ilxll~R
II(Cn)'(x)11 ~ IIAnllL n ~ ~(KL(b - a))n n.
can be established analogously to estimates (4.14). These estimates imply that the sequence (4.16) converges to the solution of equation (4.13) with a factorial type rate, i.e.,
REMARK 4.2.
As we have already noticed, condition (4.15) requires some rather strong restrictions on vector-functions K, ! and y. However, according to Subsection 2 above, we can assert that equation (4.13) has a unique solution, for any K and M, if
155
Some applications of fixed point principles
y is sufficiently close to the element f(a, 0). Moreover, taking into account the specific form of the operator AF (it involves an integral over an interval with a variable upper limit), we can assert the following: for any y E C([2, ~), with Ilyll < R, there exists a number T E [a, b] for which equation (4.13) has a unique solution x E C([2r, ~), where [2r = [a, T]. Indeed, condition (4.15) can be replaced, in this case, by the following condition: (4.17) which is obviously fulfilled for ~ T '"
a
+ R - Ily(t)11
K,M'
In the case when R = 00, i.e., f(a, x) is defined on the whole [2 x ~ and satisfies all the conditions listed above, then equation (4.13) has a unique solution for any K" M < 00 and any y E C([2, ~). REMARK 4.3. Let us note also the following possible approach. Consider the operator G defined by the right hand side of equation (4.13). Then, it is easy to construct a norm on X, equivalent with the original norm, with respect to which G becomes a contraction. Indeed, for any T E [a, b] set
where m > 0 will be specified below. Clearly,
where
Ilxll r
is the usual norm of an element x E C([2r,
Ilxll r = If Ilxll; : :; Re- mr and Consequently,
max
a:(t:(r
~),
i.e.,
Ilx(t)llt.
Ilzll; : :; Re- mr , then x, z E .fj and
IIF(x) - F(z)llr :::;; Lllx - zllr.
t
IIG(x) -
G(z)llr =
max e- mt jK(t, a){f(a,x(a)) a:(t:(r
f(a, z(a))}da
a
t
:::;; K,L
max e- mt a:(t:(r
j a
Ilx(a) - z(a)lltda :::;;
:::;;
NONLINEAR EQUATIONS
156
t
~ ",L
max
a~t~T
jem(U-t)e-mUllx(O") - z(O")IIQ!dO"
~
a
t
~ "'Lllx -
zll; max jem(U-t)dO" aCU:;>
=
a
1 = "'L-(1 - e-mT)llx - zll;. m Therefore, setting m;;:: ",L, we obtain that G is a non-expansive operator relatively to the norm II II;. This remark enables us to specify not only a new estimate for the convergence rate of sequence (4.16), but also a new estimate for y, which guarantees that equation (4.13) is solvable. To this end, it is sufficient to repeat the arguments in Remark 4.1 for q = ",Lm- 1 (1 - e mT ) and m;;:: ",L. Then we get the following condition Ilyll; ~ (1 - q)r,
r =
Re- mT •
In particular, for m = ",L we obtain
4. THE CAUCHY PROBLEM FOR DIFFERENTIAL EQUATIONS IN A BANACH SPACE. The next result is a straightforward consequence of the results in the previous subsection.
As above, let [l = [a, b], let Q; be a Banach space, and 1(0", u) an operator from [l x Sj into Q;, where Sj is the open ball of radius R in Q;. Assume that both 1(0", u) and 1~(0", u) are continuous on [l x Sj. Then, for any element Yo E Sj, there exists a unique solution of the differential equation PROPOSITION 4.6.
X' (0") =
1(0", X (0")),
(4.18)
on a certain interval [a, T], which satisfies the initial condition
x(a)
= Yo.
(4.19)
In addition, if the norm Ilyoll is sufficiently small, or Sj = Q;, then the solution of the problem (4.18)-(4.19) can be continuously extended in a unique way to a solution on the whole interval [a, bJ. For the proof it is sufficient to replace problem (4.18)-(4.19) by the equivalent integral equation
t
x(t) =
J
1(0", x(O"))dO" + Yo,
a
Some applications of fixed point principles
157
which is a particular case of equation (4.13), at least for a finite-dimensional space \E. ~
REMARK 4.4. The integral operators appearing in the problems considered above are completely continuous. Therefore, the existence of the solutions of these problems follows from Schauder principle, under some less restrictive conditions: it is not necessary to require the differentiability of the operator; it is sufficient to have only the invariance of a suitable domain relatively to that operator. (As we have already observed, in this case, we can not guarantee the uniqueness of the solution.) However, in many practical cases, this does not lead to significant improvements.
5. EQUATIONS WITH ANALYTIC INTEGRAL OPERATORS. In connection with some problems in nonlinear mechanics, many authors, such as, for example, A. M. Liapunov, Lichtenstein, E. Schmidt, A. N. Nekrasov, H. H. Nazarov, and others (see M. M. Vainberg and V. A. Trenogin [1]) studied the Hammerstein equation in which the function f(O", u) is analytic in u. First, let us consider the simplest nonlinear equation that often occurs in a series of problems in electrotechnics:
J b
x(t) =
f(O", t)xm(O")dO" + y(t),
(4.20)
a
where the complex-valued functions f(O", t) and y(t) are continuous for a:::;: t, 0":::;: b, and m > 1. Since in many concrete problems the function y(t) has a well-determined meaning, it is important to find an estimate for the absolute value of y(t), which guarantees that equation (4.20) has solutions. In order to accomplish this we will consider the auxiliary, so-called, majorant numerical equation: (4.21) where N is a positive number. Since the function in the right hand side of (4.21) is concave upward for r > 0, equation (4.21) has only two positive solutions r* and r*, r* < r*, for m-1 (4.22) Ilyll < l = m rn-VmH; for Ilyll = l it has the unique positive solution ro 1 below).
=
r*
=
r*
= (mN)-rn"-l (see Figure
NONLINEAR EQUATIONS
158
r*
r*
r
Figure 1
Thus, if we set N = K,(b - a), where
Ilyll ~
1'0,
=
max
a ~ lI,t ~ b
11(0", t)l,
then for
m-l m
=-y'mK, (b -
(4.23)
a)
the operator F defined by the right hand side of (4.20) satisfies the following condition
IIF(x)11 ~ Nr*m + Ilyll
= r*,
for Ilxll ~ r*. According to Schauder principle, there exists at least a solution of equation (4.20) satisfying the estimate max Ix(t)l~r*.
(4.24)
a~t~b
Whenever y satisfies the stronger inequality (4.22), with N the operator F is a contraction on the ball Ilxll ~ r*. <J
=
K,(b - a), then
Indeed, under these conditions we have r* < ro, hence
IIF(xd -
F(X2) II ~ K,(b - a)
::;;; mK,(b -
Ilxi"(t) - x;"(t) II ~ a)r:,,-lllxl - x211 = qllxl - x211,
where q = mK,(b-a)r:,,-l < 1. Moreover, IIF(x)11 < r* for IIxll < r*. Therefore, under condition (4.22), equation (4.20) has a unique solution x(t), satisfying the estimate (4.25)
Some applications of fixed point principles
159
In addition, the sequence {Xn (t)} nEN given by b
xn(t) =
J
K(O", t)X~_I (t)dO" + y(t),
n = 1,2, ... ,
(4.26)
a
converges uniformly on the interval [a, b] to x(t), for any continuous function xo(t) which satisfies the condition
Let us point out the next important remark. REMARK 4.5. If condition (4.22) is fulfilled, then between the numbers r * and r*
there exists a "clearance" (see Figure 1), which guarantees the existence of at least one solution for equation (4.20) continuous on [a, b]. However, in this case, following the same arguments, we can not infer from condition (4.27) the existence of a unique solution for equation (4.20) satisfying the above mentioned estimate. To achieve this goal we have to use essentially the properties of analytic operators. REMARK 4.6. An upper-bound of the norm of y that guarantees the existence of a unique continuous solution can be obtained using the classical Cauchy-Goursat majorant series method (see V. A. Trenogin [1]). Using Cauchy inequalities (see §3 of Chapter III) for the coefficients ak(O"), we can estimate the Lipschitz constant L for the function f(O", u) on a certain disk of radius PI < P and centered at zero, as follows:
M P
(
~)
m-I
(m _ ~ (m _
1))
(1- ~f
Obviously PI can be chosen such that LK,(b-a) < 1. Now, assuming that PI is chosen and fixed, we can find a number it :( l such that, for
liyll :( it,
NONLINEAR EQf{ATIONS
160
equation (4.22) has two solutions r * and r* satisfying the relations
o < r * ~ PI ~ r*. Then, the operator F, defined by the right hand side of (4.22), maps the ballllxli ~ r* into itself and is a contraction on it. For example, assume that m = 2. Then the number PI must satisfy the quadratic inequality
<
1 . ",(b - a)
In the general case the numbers PI and h are rather difficult to be found.
§5. Implicit and invertible operators. Connection with fixed points 1. The arguments presented in this subsection may be highly useful in finding a way
to solve some concrete problems. Let X, !D and A be Banach spaces over the field IK. Consider an operator F : : X x A ---- !D defined on a domain 1) x D <;;; X x A, with values in !D. An implicit operator G : it---- 1), it c 1), is defined as a solution for the equation
F(x,).) i.e., for x
= G()'),
= 0,
(5.1)
the equality
F(G()'),).) == 0
(5.2)
holds for all ,\ E it. (The operator G defined in this way is not necessarily singlevalued.) If the operator G is defined on the whole domain D and satisfies equality (5.2) we say that the implicit function problem is globally solvable, or that the equation (5.1) is globally solvable. However, the global solvability does not always take place. For instance, if X = !D = A = IK then the equation x - ,\ 3 = 0 defines a global implicit operator G('\) = ,\3, which is single-valued on the whole A, while the equation x 2 + ,\2 = 0 defines a single-valued operator G(.>..) = 0 for.>.. = 0 only. We say that the implicit function problem is locally solvable, if it is an open subset on D. Quite
Implicit and invertible operators
161
often the local problem is formulated in the following way. If for a certain point (xo, AO) E ~ x n we have (5.3) F(xo, AO) = 0, then we are interested in the existence of an implicit operator G, defined on a neighborhood il of the point AO, and satisfying the condition
G(AO) = Xo.
(5.4)
If, in addition, such an operator G is continuous on il, then it is called a small solution of equation (5.1) subject to conditions (5.3), (5.4). Notice that in some certain cases the problem defined by (5.1)-(5.3), (5.4) may have a single-valued solution, while the more general problem (5.1)-(5.2) has a multi-valued solution. For instance, when X = !D = A = JR., the equation x 2 + A2 = 1 has on the interval [-1, 1] two continuous solutions: Xl(A) = "II - A2 and X2(A) = -VI - A2. Considering the same equation with AO
=
~,
Xo
= -
V;,
then problem (5.1)-(5.3),
(5.4) has the unique solution G(A) = X2(A) = -VI - A2. At the same time the additional conditions (5.3)-(5.4) do not always provide a single-valued branch of the implicit operator. For example, the equation x 2 - A = 0 with the initial condition x(O) = 0 has two solutions:
i.e., the operator G(A) is multi-valued. Therefore, in each particular case, we have to point out exactly the type of the implicit operator problem, and what we mean by a single-valued implicit operator. Consider now an operator clJ : X ---> !D defined on a domain ~ ~ X. Assume that on a certain domain ~ ~ clJ(~) there exists an operator If/ : ~ ---> ~ such that clJlf/ = I I ~, where I is the identity operator on !D. Then the operator If/ is called an inverse of the operator clJ on~. Obviously, finding an inverse of the operator clJ, is a particular case of an implicit operator problem. Indeed, the operator If/ is a solution for the equation clJ(x) = y, which is a particular case of equation (5.1) if we set A = !D and F(x, y) = clJ(x) - y. Consequently, in this case, we can speak about a local or a global inverse, as well as about a single-valued inverse, or an inverse satisfying the initial condition If/(yo)
= Xo,
(5.6)
NONLINEAR EQUATIONS
162
if (xo) = Yo· The converse is also true. Any implicit operator problem can be reformulated as an inverse function problem. Indeed, let us assume that equation (5.1) is given. Consider the spaces IE = X x il, .f) = ~ x il, and the operator : IE -+ .f) defined by the equality = (F,1 A ), where h is the identity operator on the space A, i.e., the operator maps each pair (x, >.) E ::D x D s;:: IE into (y, >'), where y = F(x, >'). Suppose futher that the operator has an inverse on a certain domain m s;:: s;:: ( IE x D), i.e., there exists an operator l}/ : m-+ IE such that
Since the equation e = h, where e = (x, >.) and h = (y, >'), is equivalent with the system { F(x, >.) = y, >.=>. then, obviously, the operator i.e., the equality
l}/
can be represented as
l}/
= (T, 1A), where T : ~
-+
IE,
F(T(y, >'), >.) == y is true for all (y, >.) E m. But this means that the operator G, with G(>') = T(O, >'), satisfies equality (5.2) for all >. E it = PrA m. Consider now the important particular case when X = ~. In this case equation (5.1) can be written as (5.7) x = P(x, >.). Thus, for each given >., the value G(>') of an implicit operator G is a fixed point for the operator PA = P(·, >.) in the domain ::D. Let us remark that there are many possibilities to rewrite equation (5.1) as in (5.7). For instance, consider the equation x 2 + >. - 5x = O. By adding x on both sides of this equation we get x = x 2 + >. - 4x. We can as well add _x 2
-
>. on both sides, and then divide by -5, to obtain
Finally, since x =I- 0 for>. =I- 0, we can rewrite the equation as 5x - >.
X=---,
x
Implicit and invertible operators
163
and so on. Apparently, it is impossible to give a general rule to find a suitable form (5.7) for equation (5.1). However, it obviously makes sense to consider the next two aspects: 1) the transformation of equation (5.1) into (5.7) has to be done such that the operator P( ., .) to have "good" properties (continuity, differentiability, and so on); 2) equation (5.7) must satisfy one of the fixed point principles. Let us point out also the following circumstance. In concrete problems the parameter A in equation (5.1), or the parameter y in equation (5.5), has well-determined physical meanings. Therefore, it is important to obtain estimates which characterize the domain of existence it of the implicit operator G(A), or the domain of existence mof the inverse operator tJt(y) (as we have already noted, these problems are equivalent). It may also turn out that the operator G = G(A) is multi-valued on the whole domain of existence it, but it is single-valued on some subset II ~ it. The points lying in it \ II at which the operator G has a multi-value (a ramification) characterize, as a rule, the critical cases (for more details see our considerations below). Therefore, here too, an estimate, as accurate as it can be given, which characterize the set ll, is sometimes more important than finding the operator G itself. The classical theorems on implicit operator-functions lack basically in providing such estimates, or give rather vague estimates. A typical method for obtaining estimates is to transform equation (5.1) into equation (5.7). Assume that this was done and, in addition, the operator P appearing in equation (5.7) is completely continuous for each fixed A E fl. If we can find in fl a ball it centered at some point AO E A, such that P(x, A) E f) for all (x, A) E f) x it, then, according to Schauder principle, there exists an implicit operator G(A) on the ball it satisfying equation (5.7) and, consequently, equation (5.1). The value of the radius of it gives actually the desired estimate. Sometimes this estimate can be improved by choosing another set i> ~ f), which is invariant relatively to the operator P A for (x, A) E i> X itl, where itl is a ball with a radius larger than that of it. We must point out that we have already followed this approach in Subsection 4 of §4. Indeed, let us return to equation (4.27): x = Fx + y, where F is a Hammerstein integral operator. The solution x = tJt(y) of this equation can be considered as an inverse of the operator P = I-F. Therefore, the estimates for Ilyll obtained there are nothing else but estimates for the domain of existence of the inverse operator tJt. Finally, let us remark that the problem concerning the fixed points of a certain operator S can be solved as well, by reducing it to an implicit operator problem. This can be done, as we will see below, by introducing a parameter in the right hand side of the equation x = S(x). To be more specific, let us assume that we have already
NONLINEAR EQUATIONS
164
x
constructed an operator P : ::£) x st ---7 which satisfies the equality P(x, >'9) = S(x) for a certain >'0 E st. In the case when, for some reasons, there exists an implicit operator G(x) defined on a certain set it 3 >'0, which satisfies equation (5.1) for F(x, >.) = x - P(x, >'), then the point x* = G(>.o) is obviously a fixed point for the operator S. If the set ::£) is a ball centered at the zero element of the space x, we can almost always choose the operator P as P(x, >.) = >'S(x), where>. E [0, 1] if is a real Banach space, or >. E C, 1>'1 :::; 1, if is a complex space. Taking into account the convexity of::£) we can also set P(x, >.) = >'S(x) + (1 - >')y, where y is an arbitrary element of::£) and>' E [0, 1]. In this case, if the operator G(>') exists on a set containing the interval [0, 1], then the point x* = G(1) is a fixed point for the operator S.
x
x
2. In the rest of this section we present some local theorems on implicit operators. A standard theorem which generalizes the classical case, can be formulated in the following way.
Assume that the operators F : x x A ---7 ZJ and F~ : x x A ---7 ZJ are defined and continuous on the set {(x, >.) E x x A : IIx - xoll :::; R, II>' - >'011 :::; T}, for some (xo, >'0) E x x A such that F(xo, >'0) = 0, and the linear operator A = F~(xo, >'0) is continuously invertible. Then there exist the numbers p, 8> 0, p:::; R, 8:::; T, and a unique continuous implicit operator x = G(>') satisfying equation (5.1) and condition (5.4), which is defined on the ball {>. E A : II>' - >'011 < 8} and takes values in the ball {x E x : IIx - xoll < pl. Moreover, if the operator FHxo, >.) exists, then the operator G is differentiable at the point >'0 and THEOREM 5.1.
(5.8)
P = I - A-I F. Obviously, the operator P acts from equation (5.1) is equivalent to equation (5.7):
x
x A into
x
and
x = P(x, >.) = x - A-I F(x, >.). Then we have Xo
= P(xo, >'0)
P~(x, >.)
=
and
I - A-I F~(x, >.)
= A-I
(F~(xo, >.) - F~(x, >.)).
In particular, P~(xo, >'0) = 0. Since the operator F~(·, .) is continuous at the point (xo, >'0), then for any 0 < q < 1 we can find two numbers /1, TJ > 0, with /1:::; T, TJ:::; R, such that the inequality
(5.9)
Implicit and invertible operators
165
is fulfilled for Ilx - Xo I :s; 'f} and I A - AO II :s; /J. In addition, taking into account the continuity of the operator F, we can choose a number 0 < v < T such that
IIA-III for
IIA -
AO I < v. Let us denote min{/J, v} =
IIF(xo, A)II
T.
:s; (1- q)'f)
Then
IIP(x, A) - xoll :s; IIP(x, A) - P(xo, A)II + IIP(xo, A) - xoll :s; sup IIP~(x, A)II Ilx - xoll + IIA-III IIF(xo, A)II :s; :s; Ilx-xoll::;; TJ 11>'->'011::;; T
(5.10)
The last inequality shows that for A with I A - Ao I < T the operator PC, A) maps the ball {x EX: Ilx - xoll :s; 'f}} into itself, and inequality (5.9) guarantees that the operator P(·, A) is a contraction on that ball. According to Banach principle, for any A with IIA - Aoll:s;; T there exists a unique solution x = G(A) of equation (5.7), hence, of equation (5.1), which satisfies the condition IIG(A) - xoll :s; 'f}. To get the first assertion of the theorem it remains to put p = 'f} and 8 = T. The continuity of operator G follows by some standard arguments. The proof of this fact, as well as the proof of the second assertion of the theorem, are left to the reader. Notice however that (5.8) can be obtained directly differentiating the identity F(G(A), A) == 0 with respect to A, for IIA - Aoll :s; 8. ~ The numbers 8 and p play an essential role in different applications. Therefore, it is desirable to find some estimates for 8 and p. To this end we need some additional assumptions. Suppose that the operators F~(·, . ) : X x A -+ !D and F(xo, . ) : A -+ !D satisfy the Lipschitz conditions with constants Land N, respectively, i.e., 11F~(x, A) - F~(xo,
(we considered here that the norm and
IIF(xo, A) for (x, A) E X x A with are fulfilled for
where m =
IIA-III.
Ao)11 :s; L(llx - xoll + IIA - Aoll) II(x, A)II
is equivalent with the norm
F(xo, Ao)11
:s; NIIA - Aoll,
Ilx -xoll :s; R, IIA - Aoll :s; T.
(5.11)
Ilxll + IIAII) (5.12)
Then inequalities (5.9) and (5.10)
NONLINEAR EQUATIONS
166
Solving this system of linear inequalities we obtain T) ,,::. -
1
'" mL
.
qmN ---=-----
mN + 1 - q ,
T ,,::. -
1
'" mL
.
q(1 - q) mN + 1 - q
~--'-----'--'--
T), T are functions of the parameter q, where q E [0, 1]. For q = and q = 1 we have T = 0. Therefore, the largest value of T is reached for some q* E (0, 1). It is easy to verify that
It is clear that the numbers
°
y'1 +mN q* = -v771=+=m=7N'+-vrm=N~ and T*
= T(q*) = q~~) T(q)
(5.13)
1 =
mN( y'1
(5.14)
+ mN + vmN)2·
Set also
y'mN T)*
= T)(q*)
=
(y'1
(5.15)
+ mN + vmN)mL·
Finally, we obtain the next result.
Assume that the conditions in Theorem 5.1 and inequalities (5.11), (5.12) are fulfilled. Then, on the ball
THEOREM 5.2.
11,\ - '\011::;; b = min{r, T*}, there exists a unique continuous implicit operator G('\) defined by equation (5.1) and relation (5.4), which satisfies the condition
IIG('\) - xoll ::;; p = min{R, T)*}, where the numbers T*, p* are defined by formulas (5.14), (5.15), respectively. Taking into account the arguments developed in Subsection 1, from Theorem 5.2 we straightforwardly obtain the next result.
Let 1) = {x EX: Ilx - xoll ::;; R} and let ~ is continuously invertible; 3) there exists a number L > 0 such that
COROLLARY 5.1.
1) -> ~
be an
Implicit and invertible operators for all Xl,X2 E::D. Then, on the ball
p
. { = mIn
m = {y E ~ 1,
mL( VI
:
Ily - yoll ~ p}, where
1
} ,
+ m + y'm)2
there exists a unique Frechet differentiable operator p-l : condition p-l(yO) = Xo and is an inverse ofp. Moreover, we have
IIp-l(y)11 ~ min{R, y'm(mL( for all y
E
167
m---; X which satisfies the
vm + 1 + y'm))-l},
m.
As we have already noticed, the transition from an inverse operator problem to an implicit operator problem is also possible. We illustrate this by an example involving analytic operators, for which we will indicate some other estimates. We will use the Cauchy-Goursat method of majorant series.
lPx
=
lPxo
+L
8k lP(xo, h),
k=l
where h = x - Xo, lPxo = A-lyo, 8l lP(xo, h) = A-lp'(xo)h = h, and 8k lP(xo, h) = = A- 1 8k P(xo, h) are homogeneous forms of order k. The equation Px = y is obviously equivalent to the next equation 00
h
+L
8k lP(xo, h) = z,
(5.16)
k=2
where z = A-l(y - Yo) E X and Ilhll ~ R. We will try to find a solution for this equation of the form 00
h
=
P(z)
=
L 8kP(O, z),
(5.17)
k=O
where 8kP(O, z) are homogeneous forms of order k. Substitute series (5.17) in equation (5.16) and equate the forms of the same order. As a result we obtain an infinite recurrent system from which we get all the forms 15kP(O, z), succesively. (Explicit
NONLINEAR EQUATIONS
168
formulas for 8k P(0, z) can be found, for example, in J. Orava and A. Halme [1J and A. P. Yuzhakov [2J.) If we show that the formal series (5.17) converges normally on some neighborhood {z EX: Ilzll < 8} of the point z = 0, then this will prove the existence of an analytic solution of equation (5.16), and the existence and the analyticity of the operator tfJ- I = l}/-I A-I: I{) --t X for all y with Ily - yoll < p = 8m-I, where m = lIa-III, as well. For the proof let us consider the numerical series 00
'P(()
=
L (kI1 8kl}/(XO, h)11 k=2
which converges for all (with 1(1 ~ R, ( E IK, Ilhll = 1. By Abell Theorem the series is convergent for all ( lying in the disk of radius R of the complex plane. Consequently, the function 'P(() is analytic on this disk. Let sup I'P(() I ~ M < 00. Then, by
1(1 ~ R,
(EC
Cauchy inequalities, we have
(5.18) and, consequently,
I'P(() I ~ M
f
1(l k R- k
~ M I~:
(1 _I~)
-1
= q(I(I).
k=2
Consider the auxiliary numerical quadratic equation
t=I(I,
t=1]+q(t), which, obviously, has the unique solution
satisfying the condition teO)
= O.
It is clear that for
11]1 ~ 8 =
1] with
R2 4(M
+ R)
the discriminant of this equation is greater than or equal to 11]IR ~ 0 and, therefore, the function t(1]) is analytic on 1] in the disk 11]1 ~ 8. Hence it can be represented as a convergent power series 00
t(1])
=
L
k=O
Ck1]k.
169
Implicit and invertible operators
Since the coefficients Ck can be also found from the recurrent relations obtained substituting the last series in equation (5.19), using relation (5.18) it is not difficult to get the estimates
From these estimates it follows that the series (5.17) converges normally for all z with Ilzll < D. This means that the operator 1[/-1 is analytic on the ball Ily - yoll < p = = om-I. Thus we proved the next result.
Let X, ~ be arbitrary Banach spaces over the field lK and : X ~ ~ an operator analytic on the ball ~ = {x EX: Ilx - xoll ~ R} for a certain Xo EX. Assume that the operator A = '(xo) : X ~ ~ is continuously invertible. Then, on the ball QJ = {y E ~ : Ily - yoll < p}, where Yo = (xo) and
THEOREM 5.3.
p M
= R2(4(M + R)IIA- 1 11)-1,
= IIA- 1 11
sup 11f:(kOk(xo,h)ll, 1(1 ~ R k=2
(Ee,
Ilhll=1
there exists an analytic operator -1 which is an inverse of the operator on this ball. ~ Considering now the related implicit operator problem we obtain:
Let X, ~, A be Banach spaces over the field lK and let F : X x A be an operator analytic in the domain
COROLLARY 5.2. ~ ~
{(x, >.) :
II(x, >.) -
(xo, >'0)11
Assume that the operator A = F~(xo, >'0) : X on the ball {>. E A : II>' - >'011 ~ p}, where
~ ~
~
~
R}.
is continuously invertible. Then
(5.20)
( A B = 0
F~ (xo, fA
Uo
00
>'0)) '
=
(xo, >'0),
h
=
(x - Xo, >. -
>'0),
and fA is the identity operator on A, there exists a unique implicit operator G(>.) satisfying the identity F(G(>'), >.) == 0 and the condition G(>'o) = Xo.
170
NONLINEAR EQUATIONS
REMARK 5.1. The method of constructing the inverse operator by formula (5.17),
described in the proof of Theorem 5.3, is called the method of undetermined coefficients. It was used more than once by different authors, in order to solve some concrete integral equations (see, for example, N. N. Nazarov [1] and A. I. Nekrasov [1]). It seems that this method in its general form was used for the first time by K. T. Ahmedov [1] for an implicit operator depending on a numerical parameter, and by J. Orava and A. Halme [1] for the inverse operator problem. Formulas (5.13)-(5.15) were established by V. A. Trenogin (see [1]). Using the same approach we can obtain some other, yet equivalent, estimates for T, 'f] if we set II(x,A)11
= max{llxll, IIAII}
or
In §5 of Chapter VI below we will establish some more precise - and more convenient for computations - estimates for the numbers p, 8, in the case of complex spaces.
Chapter VI Nonlinear equations with holomorphic operators Throughout this chapter we will consider complex Banach spaces only. In this case, many local features related to the solvability of equations with operators that are differentiable in the complex sense, turn out to be global.
§1. s-fixed points for holomorphic operators. A converse of Banach principle In this section we deal with differentiable operators and their fixed points that have the property of "succesful approximation", i.e., s-fixed points (see Chapter V). Let f> be an arbitrary bounded domain in X and F an operator holomorphic in f> (i.e., F is Frechet differentiable in the complex sense on a neighborhood of each point of f». Assume that F satisfies the invariance condition
F(f» <;;; fl.
(1.1 )
Suppose further that z E f> is a fixed point for F, that is, Fz = z. As we have already noticed, if F E C(f>, X), i.e., F admits a continuous extension on of> - the boundary of the domain f> - , and if the point z is an attractive fixed point for the operator F (see §1 of Chapter V), then, as soon as z is a local s-fixed point, it is a global s-fixed point too. Indeed, by Meyers Theorem (Theorem V.1.2), the next assertion is a consequence of our assumptions:
(*) for any q with 0 < q < 1 there exists a metric p on the set f>, equivalent to the metric given by the norm of X, relatively to which the operator F satisfies the
NONLINEAR EQUATIONS
172
Lipschitz condition with constant q, that is, p(Fx, Fy)
~
qp(x, y),
x, y E 1:>.
(1.2)
Thus the assumptions in Theorem V.1.1 - the Banach principle - are fulfilled. If inequality (1.2) is satisfied on some closed subset II ~ 1:> and F(ll) ~ ll, then we will say that the operator F is a q-contraction on ll. The following result holds.
Let F be a holomorphic operator on 1:>, satisfying condition (1.1), let zE1:> be a fixed point for F, and let A=F'(z) be the Fhkhet derivative of Fat z. a) The following assertions are equivalent: 1) Fn x -+ Z for any x E 1:> and the operator e iO I - A is normally solvable (see §5 of Chapter IV) for all () E [0, 27rJ, where I is the identity operator on X; 2) rCA) < 1, where rCA) denotes the spectral radius of the operator A; 3) there exist a number r > 0 and a norm II . II * equivalent to the original norm of X, such that F is a ql-contraction on the ball II,. = {x E 1:> : Ilx - zll* ~ r} for a certain ql with 0 ~ ql < 1; 4) there exist a number q2 with 0 ~ q2 < 1 and a metric equivalent to the metric given by the norm of X relatively to which the operator F is a q2-contraction on some neighborhood of the point z. b) If, in addition, F E C(1:>, X) and Fn x -+ Z for any x E 81:> then conditions 1)-4) are equivalent to condition (*). THEOREM 1.1.
EXPLANATIONS: 1. Under condition b) the implication 1) =} (*) is a global converse of the Banach principle with respect to a metric equivalent to the metric given by the norm of X. The implication 1) =} 3) is a local converse of that principle, but this time, with respect to a norm equivalent to the norm of X. In A. A. Ivanov's book [1] a construction of a metric such as in condition 4) above is given. In many concrete problems this construction turns out to be rather difficult. If we know already the spectral radius rCA), then it is very simple to find the norm II . 11* and the number r appearing in condition 3) (see Lemma V.3.1). This enables us to estimate the rate of convergence to z of the iterations {Fnx }nEN, starting with an arbitrary x E II,.. 2. The implication 2) =} 1) means that condition rCA) < 1 is, in fact, a global feature of the s-fixed point z (see §3 of Chapter V) and, consequently, it guarantees the uniqueness of the fixed point z on the whole 1:> and also the convergence of the iterations, for any x E 1:>.
Proof of Theorem 1.1. The implication b)&1)&2)=} (*) follows by the
173
s-fixed points for holomorphic operators
already mentioned Meyers Theorem. The implication (*) =} 4) is obvious. Therefore it is sufficient to prove only the equivalence of conditions 1)-4). Let us prove that 1) =} 2). Denote by An the Frechet derivative of the operator F n at the point z, i.e., An = (Fn)'(z), n ~ 1 (then Al = A). From condition Fn x - z -+ 0 and by Cauchy inequalities for the holomorphic operator Fnx - z (see Chapter III) it follows that lim An h
n--->oo
= 0,
for all hEX
(1.3)
(here the limit is considered in the strong sense). Moreover, An = An, where An is the n-th iteration of the operator A (see identity (3.5) in Chapter V). Further, it will be enough to show that the set [2 = {A E C : IAI ~ I} is contained in the resolvent set of the operator A. This will clearly imply that r(A) < 1. By (1.6) the set [2 does not intersect the point spectrum Ul (A) of the operator A (the set of eigenvalues of the operator A). Indeed, if A E [2 n ul(A) then there exists y E X, y =1= 0, such that Ay = AY, and consequently, Any = Any, whence lim Any =1= 0, which contradicts n--->oo
(1.3). Moreover, by the Uniform Boundedness Principle it follows that
(1.4) Therefore the sequence n=O
converges uniformly to the operator (AI - A)-l for all A with IAI have (AI - A)Smx = x - A-(m+l) AX, x E X.
> 1. If IAI = 1 we
It follows that the set R(AI - A) - the range of the operator AI - A - is dense in X. Taking into account condition 1) once again, we conclude that R(AI -A) = X. Since, on the other hand, A tI- Ul (A), it follows that the operator AI - A is continuously invertible. Thus r(A) < 1. The implication 1) =} 2) is proved. The implication 2) =} 3) follows from Theorem V.3.3, and the implication 3) =} 4) is trivial. It remains to prove that 4) =} 1). By 4) it follows that the iterations {Fnx} nEN converge to z uniformly on some neighborhood of the point z. Since the set 1) is simply connected, then, by Vitali Theorem, the above mentioned sequence converges to z for any x E 1). Let ll,.1 = {x E 1) Ilx - zll < rt}, where rl is sufficiently small. For any c with 0 < c < rl we can find n such that IlFn x - zll < rl - c for all x E ll,.1' Hence and by Cauchy inequalities it follows that
NONLINEAR EQUATIONS
174
or, equivalently, IIAnl1 < 1, which means that the operator eilt I - A is continuously. invertible for all () E [0, 27r]. Thus condition 1) is fulfilled. ~ COROLLARY 1.1. If for some p;:: 1 the operator FP has the s-fixed point z E 1),
then z is an s-fixed point for F, too. REMARK 1.1. By Theorem IIL8.4, in condition 1) above it is enough to require the convergence of the iterations {Fnx} nEl'l for x in a dense subset of the domain 1).
If for some pEN the operator FP is completely continuous, then, by Theorem 1.8.1, the operator Ap = AP is completely continuous, too. It follows that the spectrum of the operator A coincides with the point spectrum (}l (A). Thus we get the next
result.
n C(1), X) be (1.1). Assume that for some pEN the operator FP iterations {Fn x }nEl'l converge to a certain point z E is a dense subset of1), then z is an s-fixed point for COROLLARY 1.2. Let F E H(1), X)
an operator satisfying condition is completely continuous. If the 1) for all x E 1)' u 81), where 1)' the operator F.
Recall that there exist examples of operators that are not completely continuous but have completely continuous powers (see, for instance, L. V. Kantorovic and G. P. Akilov [1]). REMARK 1.2. Actually, inequality (1.4) follows from the invariance condition (1.1), and the boundedness of the domain 1) only. Indeed, fix a ball of radius R and center z completely contained in 1), and a ball of radius M with center at the zero element of X, completely containing 1). Then for all x with Ilx - zll ~ R we have
and, by Cauchy inequalities, it follows that
for all
h with Ilhll
~
R. Therefore
But this means (according to the proof of implication 1) =} 2)) that the spectrum of the operator A lies completely in the unit disk. Thus we have proved the next important result.
s-fixed points for holomorphic operators
175
LEMMA 1.1. Let F be an operator satifying condition (1.1) and let z E 1) be a fixed
point for F. If A = F'(z), then r(A) ~ 1, where r(A) is the spectral radius of the operator A. The equality r(A) = 1 holds if and only if z is not an s-fixed point. This assertion can be also considered as an analogue of Schwartz Lemma (see §5 of Chapter III). Assume further that the operator F is continuous on the closure of the domain 1) and satisfies the stronger invariance condition (1.5) In this case, as it will be shown below, the structure of the spectrum of the operator A = F' (z) is relevant only on the boundary of the unit disk of the complex plane, and not inside this disk. For simplicity we will consider that 1) is a ball in X centered at the point z = o. THEOREM 1.2. Let us assume, in addition to condition (1.5), that F(O) = 0 and
where al (A) is the point spectrum of the operator A (i.e., the continuous and residual spectra of the operator A lie completely inside the open disk .:1 = {>. E C. : I>' I < 1}). Then z = 0 is an s-fixed point for the operator F. Suppose, on the contrary, that z = 0 is not an s-fixed point for the operator F. Then, by Lemma 1.1, we get r(A) = 1, and, consequently, there exists a number () E [0, 27r] such that e iO is an eigenvalue of the operator A. It follows that we can find an element x E 81) satifying Ax = e iO x. Choose a linear functional 1 E X* such that (x, I) = 1, 11111 = 1, and consider the analytic function
.) = (F(>.x), I) = (A (h)
= >.eiO(x, I) +
(Q(>.x), I)
+ Q(>.x), I) =
= >.e iO +
(Q(>.x),I),
where Q = F - A and>' E C. with 1>'1 < 1. From (1.5) it follows that 11')II < 1, for all >. E c. with 1>'1 < 1, and .x) = 0 for all >. E c. with 1>'1 < 1. Hence I(Fx,1)1 = 1. But, on the other hand, by Cauchy-Bunyakowsky inequality and by (V.1.5) we have I(Fx, 1)1 ~ IlFxll . 11111 < 1, a contradiction which proves the theorem. ~
NONLINEAR EQUATIONS
176
If the operator F satisfies (1.5), F(O) = 0, and for a certain 1, the operator FP is completely continuous, then 0 is an s-fixed point for
COROLLARY 1.3.
pEN, P ~ the operator F.
Indeed, the spectrum of the operator AP
=
(FP)'(O) satisfies the condition
In particular we have
Therefore 0 is an s-fixed point for the operator FP. By Corollary 1.1 it follows that 0 is an s-fixed point for the operator F, too. ~ Thus, we see that the geometric conditions of invariance and strong invariance enable us to infer from some local features of an operator at a fixed point, certain global properties of that point such as uniqueness, succesful approximation, and so on. In addition, the condition of strong invariance allows us to weaken slightly the requirements imposed on the spectrum of the operator A. Let us note here that the strong invariance condition alone is not enough to guarantee the attractiveness of a fixed point. Indeed, in Example V.2.4 we considered a linear, therefore holomorphic, operator A satisfying the strong invariance condition
IIAxl1 < Ilxll,
x =1= 0,
and such that its iterations Anx converge to 0 if and only if x = O. A sufficient condition for the above mentioned matter is the so-called condition of uniform or strict invariance: F(::O) ~ ::0', (1.6)
-,
where::O is strictly contained in::O. Moreover, as we will show in the sequel (Theorem 2.2), condition (1.6) is also sufficient for the existence of a fixed point (and, at the same time, for its uniqueness and for its uniform approximation by iterations of the operator F at any element x E ::0).
Criterions for the existence of an s-fixed point
177
§2. Criterions for the existence of an s-fixed point and its extension with respect to a parameter Let X and A be complex Banach spaces. Assume that 1) is an open bounded set in X and fl is a finitely connected p-open set in A (see §2 of Chapter III). Consider an operator 1> : 1) x fl ---> 1) defined for any (x,..\.) E 1) x fl, such that 1>(.,..\.) E 1i(1), X) for all fixed..\. E fl, and 1>(x, .) E 1ip(fl, X) for all fixed x E 1) (Le., the operator 1> is holomorphic on x and p-holomorphic on ..\.). A solution of the equation
x(..\.) = 1>(x(..\.) , ..\.) is called an s-solution, if for any operator Xo : fl the sequence {xn(..\.)}nEN defined by Xn+l (..\.) =
1>(xn(..\.), ..\.),
---> 1)
(2.1) such Xo (= Xo (..\.)) E 1ip(fl, 1»),
n = 0, 1,2, ... ,
(2.2)
converges in the strong topology of X to x(..\.) uniformly on each compact subset of fl, and, for a fixed..\. E fl, uniformly relatively to those xo(..\.) with values inside 1); in other words, x(..\.) is an s-fixed point of the operator 1>(., ..\.), for each..\. E fl. Now let us state the main result of this chapter.
Let 1> : 1) x fl ---> 1) be an operator which satisfies the above mentioned conditions. If, for at least one value ..\.0 E fl there exists an s-fixed point x* for the operator 1>(., ..\.0), then there exists an s-solution for equation (2.1). Moreover, for any class of operators IU s:;; 1ip(fl,1») which is closed in the topology of uniform convergence on the compact subsets of fl, and contains the orbit ~ = {v n : V n +1 = = 1>(vn' ..\.), n> N ~ 1, Vo E 1ip(fl, 1»)} of a certain element Vo, we have x(..\.) E IU. In particular, it is always true that x(..\.) E 1ip(fl, 1»). THEOREM 2.1.
u (= u(· )) E 1ip (fl, 1») the composed operator 1>(u(-),·) belongs to the class 1ip(fl, 1»), too. For any x E 1), ..\. E fl, TEA and ( E admits the representation
00
(2.3) n=O
where, for any fixed..\. and x, an is a homogeneous form in T of order n, and the series in the right hand side of (2.3) converges normally for all ( E
an
1
= -2 . (P) 7l"1
J
1<:I=p
(2.4)
NONLINEAR EQUATIONS
178
On the other hand, since the operator t[J is holomorphic on x, it follows that for any fixed ( with 1(1 = p and any element hEX with a sufficiently small norm, a representation of the following form (x + h, A + (T)
= t[J(x, A + (T) + t[J~(x, A + (T)h + Q(x, h, A + (T)
(2.5)
holds, where IIQ(x, h, A + (T)II = o(llhll). Moreover, by the generalized Schwartz Lemma (see §5 of Chapter III) we get IIQ(x, h, A + (T)II ~ Mllhl1 2 , where, due to the boundedness of the set :D, the constant M can be chosen independently of A Eiland x E :D. In its turn, the linear part in (2.5) can be represented in the integral form
t[J~(x, A + (T)h = ~ 2m
J
(P)
C 2 t[J(x
+ th, A + (T)dt,
Itl=6
where the number 8 is small enough, and t E C. Hence it follows that the vector-function t[J~(x, A + (T)h is continuous on ( with 1(1 = p, for any fixed A E il, x E :D, hEX, TEA. Consequently, from (2.4) and (2.5) we obtain
where bn(x, A, T)h
J t[J~(x,
= ~ (P) 2m
A + (T)hc(n+l)d(
1(I=p
is a linear operator on h (b n E L(X)), and
Ilcn(x, h, A, T)II =
~ 2m
J
(P)
c(nH)Q(x, h, A + (T)d(
~
1(I=p
~ 2~ . MllhI1 2 27rp-n = o(llhll), which clearly means that the operator an (·, A, T) is Frechet differentiable on x for any fixed A E il, TEA. From the above proof it follows that the vector-functions an(u(A+(T), A, T) are holomorphic in the disk 1(1 < p(A, T) for any n = 0,1,2, .... Then, by Weierstrass Theorem (see B. V. Shabat [1]), we obtain that for any functional I E X* with 00
11111 = 1, the vector-function rp(() =
L (an(u(A + (T), A, T), f) is holomorphic in the n=O
Criterions for the existence of an s-fixed point
179
same disk. Notice that the last series converges absolutely by virtue of the normal convergence of the series (2.3), and
(U(A
+ (T), A + (T)), I).
By Theorem III.2.1 the vector-function <1>(U(A + (T), A + (T)) is also holomorphic in the disk 1(1 < p(A,T). Hence it follows that the operators {Xn(-)}nEl"l defined by (2.2) are p-holomorphic in f2, i.e., if Xo E Hp(f2, 1», then x n (·) E Hp(f2, 1», for any n = 1,2, .... Let us show now that the sequence {x n (· )}nEl"l converges uniformly on any compact subset of f2. Taking into account the structure of the set f2 and by the generalized Vitali Theorem it is sufficient to prove the convergence on any disk f2p(AO, T) <;;:; f2 centered at a point AO and with a non-zero radius p. Since the point x* is an s-fixed point for the operator <1>(., AO) : 1> ~ 1>, then, by Theorem 1.1, there exists a neighborhood II of the point x* on which the operator <1>( . , AO) is a contraction with a certain constant q (0 ~ q < 1), relatively to a norm II . 11* equivalent to the original norm of the space X. Moreover, we can find a ball of radius M with respect to the norm II . 11* which completely contains the domain 1>. Then, by (2.3) and the Cauchy inequalities (III.3.8), it follows that
Since the operators a n (·, AO, T) are holomorphic in 1>, using the estimates in Corollary III.6.1 we get
for all x, y in the ballllr with a sufficiently small radius r point x*. Then
>
0 and centered at the
(Xl
n=O
~
2Mr- I KJ,
qllx - yll* +
1 _ KJ,P
Ilx - yll*·
p
Clearly, for any ql with q < ql < 1 we can choose a number PI with 0 < PI < p, such that the right hand side of the last inequality does not exceed qIilx - yll*. Moreover, by Schwartz Lemma, taking into account that <1>(x*, >'0) = x*, we obtain
NONLINEAR EQUATIONS
180
for 1(1 ~ P2 = min{p(2M)-1, pd· r(1 - ql). Hence it follows that
+ (7) - x* II ~ ~ 11<1>(x, >'0 + (7) - <1>(x*, >'0 + (7)11 + 11<1>(x*, >'0 + (7) ~ qllix - x* II + r(1 - qd ~ r. 11<1>(x, >'0
x*11 ~
for any ( with 1(1 ~ P2 and x E 1.4. Thus, for all >. E Dp2 (>'0,7), i.e., >. = >'0 + (7, where 11(11 < P2, the operator <1>( " >.) maps the ball 1.4 centered at x* into itself, and <1>(.,>.) is a ql-contraction on 1.4. Consequently, the sequence {Xn(>')}nEN converges to a certain element x(>.) E ~, for all >. E D p2 (>'0, 7). Since Ilxn(>')11 ~ M, for all n = 0,1,2, ... , it follows that the sequence converges uniformly, so our claim is proved. By Theorem III.6.4 we have x n (-) E Hp(D, ~). It is clear that x(·) E sn whenever sn is a subset of Hp(D,~) closed in the topology of uniform convergence on the compact subsets of D, and containing the orbit ~ = {vn+l = <1>( V n , >.) : n > N} of a certain element Vo E Hp(D, ~). ~ REMARK 2.1. The last assertion of our theorem turns out to be interesting in the
following situation. Consider an open set Dl ::J D. The class sn = H(Dl'~) is contained in Hp(D, ~). Although the operator <1>(x, .) E Hp(D,~) is not necessarily holomorphic in D l , it is possible for the orbit ~ of an element Vo E H( D l , ~) to be contained in H(Dl' ~). Then, taking into account that the class sn = H(Dl'~) is closed in the topology of uniform convergence on compact subsets, we obtain that the solution x( >.) of equation (1.1) is holomorphic in D l . Some related examples will be presented below (see §6). As a consequence of Theorem 2.1 we obtain the next criterion for the existence of an s-fixed point for a holomorphic operator. THEOREM 2.2. Let ~ be a bounded domain which is (>star-shaped relatively to
oE
X. An operator F E H(~, X) satisfying the condition F(~) S;;; ~ has an s-fixed point Z E ~ if and only if there exist a subset i> s;;; ~, a number pEN, and a number c > 0 such that dist(FP(i», ai» =
inf _ Ilx - yll > c.
(2.6)
yEFP(1J) xE8i>
<J
Necessity. Let z E
~
be an s-fixed point for F in ~ and set r
=
inf Ilx-yli.
xE81J
Then, by the uniform convergence of the iterations F (x) to z it follows that for any c with 0 < c < r and any set i> completely contained in ~ together with its boundary, n
Criterions for the existence of an s-fixed point
181
there exists a number p such that IIFP(x) - zll < r - E, for all x E i). Clearly (2.6) is fulfilled for such E and p. Sufficiency. Consider the operator X, defined by the equality
..) = >"FP x , where fl = {>.. E C : 1>"1 < 1 +E}. By (2.6) we have ..) E :D for all (x, >..) E :D x fl and ..) of equation (2.1), for all >.. E fl. In particular, setting z = x(l) we obtain z = FP(z). It follows that z is an s-fixed point for the operator FP, whence, as above, we conclude that z is an s-fixed point for the operator F, too. ~ If, in particular, :D is the open ball of radius R centered at the zero element of X, then the following corollary provides a sufficient condition for the existence of an s-fixed point. COROLLARY 2.1. Let F be an operator holomorphic in :D which satisfies the strict invariance condition, i.e., there exists r < R such that
IlFxll ~ r, for all x with
Ilxll < R.
Then the operator F has an s-fixed point z E :D.
It seems that this result was obtained for the first time by M. Helve [1] in the finite-dimensional case and by C. Early and R. Hamilton [1] in the general case of a Banach space, using the generalized Poincare metric (see also L. Harris [4], T. Hayden and T. Suffridge [1], and K. Goebel and S. Reich [1]). The results in this section have various applications which will be listed in the sequel. Now we return to an example of an integral equation studied in the previous chapter. Let us consider equation (V.4.20):
J b
x(t)
=
K(u, t)xm(u)du + y(t).
a
As it was proved in §4 of Chapter III, for y (= y(t)) with relation
IIF(x)11
~
K,(b - a)rm
+ lIyll < r,
Ilyll <
m - 1 the mJmK,(b - a)
NONLINEAR EQUATIONS
182
is fulfilled for all x such that Ilxll ~ r < r*, where r* is the largest rooth of the equation r = Nr m + Ilyll, where N = K(b - a) and K = max IK(a, t)l· a,tE[a, bJ By Corollary .2.1 the integral equation above has a unique solution x(t), satisfying the condition max Ix(t)1 < r. [a, bJ Moreover, for any function xo(t) continuous on [a, bJ and such that max Ixo(t)1 ~ r, [a, bJ the sequence b
Xn+l(t) =
J
K(a, t)x~(a)da + yet),
a
n = 0,1,2, ... , converges to x(t) uniformly on [a, bJ. Let us emphasize once more that for r* < Ilxll, where r* is the smallest positive rooth of the equation Nr m + lIyll = r, the operator fails to be a contraction (see Subsection 4 in §4 of Chapter IV).
§3. Regular fixed points. Geometric criterions Let 1) be a bounded domain in X. In this section we will assume that the operator F, holomorphic in 1), admits a continuous extension on the boundary f)1) of the domain 1), i.e., FE H(1),X)nC(1),X). The invariance condition (1.1) is in this case equivalent to condition F(1)) ~ 1). (3.1) Example V.2.1 considered in the previous chapter shows that condition (3.1) above is not enough for the existence of a fixed point for the operator F (the operator in Example V.2.1 is holomorphic and, moreover, non-expansive). Therefore in the subsequent considerations it will be necessary to impose supplementary restrictions. 1. Let us start with a simple assertion.
Assume that the domain 1) is
THEOREM 3.1.
°
Ilxt -
F(xt) II = IltF(xt) -
F(Xt)11 ~ IIF(xt)11 It-1I,
Regular fixed points. Geometric criterions
183
by the boundedness of the operator F on :D, we obtain that the sequence {yt} = {(I - F) (Xt)} converges for t ---> 1. Then, under our assumptions, it follows that the set {xt} is compact. Since the operator F is continuous on :D we infer that any limit point of 'the set {Xt} is a fixed point for the operator F. ~ In particular, if the operator F is completely continuous, then the operator I - F is proper. Therefore, Theorem 3.1 represents an analogue of the Schauder principle for holomorphic operators acting on a ((>star-shaped domain (recall that the classical Schauder principle is stated for convex domains). However, like the Schauder principle, Theorem 3.1 above does not guarantee the uniqueness of a fixed point. Moreover, neither the isolation, nor the succesful approximation of a fixed point follows. Note also that, if we deal with assumptions weaker than the strict invariance condition (V.2.6), then, naturally, we have to replace the succesful approximation property with a weaker condition. A suitable substitute is the property of regularity. DEFINITION 3.1. Let z be a fixed point for an operator F which is Frechet differentiable in a neighborhood of z. The point z is called a regular point (or a proper point) for the operator F if 1 is not an eigenvalue of the operator A = F'(z). Otherwise the point z is called singular.
By Theorem 1.1 an s-fixed point for a differentiable operator on a complex Banach space is regular. For a completely continuous differentiable operator the regularity of a fixed point implies its isolation. Indeed, the equation
x
=
F(x)
can be written as z+ h
= z + Ah + Q(z, h),
where h has a sufficiently small norm and IIQ(z, h) II = o(lIhll). Since 1 is not an eigenvalue of the compact operator A, the operator I - A is continuously invertible and the last equation is equivalent to h
= (I - A)-lQ(Z, h) = t[J(h).
Taking into account the equality t[J'(O) =0 we obtain that h=O is an s-fixed point for the operator t[J on a certain ball with a sufficiently small radius. Hence the considered equation has the unique solution X=Z in some neighborhood of the point z.
NONLINEAR EQUATIONS
184
We will show below that in some special cases a regular fixed point for a not necessarily completely continuous holomorphic operator may be isolated as well. The next result introduces a geometric condition which guarantees not only the isolation of a fixed point for a non-compact operator, but also its uniqueness.
n C(1), X) be an operator which maps 1) into itself. Assume that for some p ~ 1 the operator FP is compact and has no fixed points on the boundary [)1). Then the operator F has a unique fixed point inside 1) and this point is regular.
THEOREM 3.2. Let F E H(1), X)
<J Let J be the set of all fixed points for the operator FP. Under our assumptions there exists a domain 1) 1 such that 1) 1 :J J and 1) 1 C 1). Consider on 1) 1 the vector field 1- tFn which, for t E (0, 1) and t close to 1, is homotopic to the field 1- Fn and has no zeroes on [)1)1 (see M. A. Krasnoselskii and P. P. Zabreiko [1]). By Theorem 2.2, for such t's the operator Ft = tFP has an s-fixed point Xt E 1). It follows that these points are regular. Since the winding number 'Y of two homotopic non-degenerate vector fields is the same, then
Therefore
Xt
E 1)1, for any
t
E (0, 1) close to 1, and, moreover
Hence the equality
is equivalent to the fact that the set J has but one point z which is regular. By uniqueness, the point z is a fixed point for the operator F, too (see §1 of Chapter V). Moreover, if we assume that z is not a regular point for the operator F, then there exists an element u =1= 0, U E X, such that Au = u, where A = F'(z). But then APu = (FP)'(z)u = u, a contradiction. ~ Thus we obtained an analogue of the extension to the boundary principle for a non-discrete analytic set (see M. Herve [1]): if the set of fixed points of a compact holomorphic operator has more than one point inside a domain, then this set contains points from the boundary of the domain, too. 2. For operators that are holomorphic in domains of a strongly convex space we may give up the compactness condition. Moreover, for a holomorphic operator defined on
Regular fixed points. Geometric criterions
185
a ball of a strongly convex space and mapping that ball into itself, the set of all its fixed points has an affine structure.
Let ~ be the unit ball - centered at 0 - of a strongly convex Banach space X, and F E H(~, X) n C(~, X) an operator satisfying condition (3.1). Let ~ be the set of all fixed points for the operator F. IfO E ~ then ~ n ~ = Q; n ~, where Q; is the set of all fixed points for the linear operator A = F' (0). THEOREM 3.3 (W. Rudin [1]).
<J Let z E ~, z -=I- O. Choose a point u E 8~ with Ilull = 1, such that z = tu for some t E (0, 1). Consider now the function 'P(A) = (F(AU), I), holomorphic on the unit disk .:1 = {A E C : IAI < I}, where f E X* satisfies the conditions (u, I) = 1 and I f I = 1 (such a functional does exist by a consequence of the Hahn-Banach Theorem). The function 'P satisfies, obviously, the conditions: I'P(A)I < 1, for all A E .:1, 'P(O) = 0, and 'P'(O) = (Au, I). Since 'P(t) = t -=I- 0, by a consequence of the one-dimensional Schwartz Lemma we obtain 'P(A) = A, for all A E .:1, hence 'P'(A) = (Au, I) = 1. Since the space X is strongly convex it follows that Au = u, and, therefore Az = z, i.e., z E Q;. Thus we proved the inclusion ~ n ~ S;;; Q;. Conversely, let z E Q; n ~, z -=I- O. Then, for some function 'P as above, we have
'P(A) = (A(Au), I)
+ (Q(AU), f) = A(U, I) + (Q(AU), f),
where Q(x) = F(x) - A(x) and Q'(O) = O. Then 'P'(O) particular, 'P(t) = t, or (C 1 F(tu), I) = 1.
=
1, hence 'P(A) == A. In
By the strong convexity of X we get that C 1 F(tu) = u, or Fz = z. ~ From the above proof it follows that if 0 is a regular point for an operator F satisfying the conditions of the theorem, then 0 is the unique fixed point inside ~. However, it is necessary to note that from Rudin's theorem we can not infer any relationship between the boundary points of the sets Q; and ~; there are examples when ~ n 8~ -=I- Q; n 8~. Nevertheless, an assertion analogous to Theorem 3.2 is also true in this case. Namely, the absence of fixed points on the boundary of the domain ~ implies the existence of a unique fixed point inside ~. COROLLARY 3.1. ~
n 8~
=
Let the domain
~
and the operator F be as in Theorem 3.3. If
0, then the point 0 is the unique fixed point for the operator F.
Consider the holomorphic operator ~ defined by the equality
NONLINEAR EQUATIONS
186
for any x E 8'1). From Theorem 1.2 it follows that A = 1 is not an eigenvalue of the operator P'(O), and this means that 0 is the unique fixed point for the operator P. But the operators F'(O) and p'(O), as well as the operators F and P, have the same fixed points, hence the assertion follows. ~ 3. In this subsection we consider strongly non-expansive operators, that is
IIF(x) -
F(y)11 <
Ilx - yll,
for all x -1= y E '1),
(3.2)
or non-expansive operators, that is
IIF(x) -
F(y)11
~
Ilx - yll,
for all x, y
E
'1),
(3.3)
acting on a complex Banach space X. THEOREM 3.4. Let '1) be the unit ball of a complex Banach space X, centered at
the origin, FE 1{('1), X) n C('1), X) an operator satisfying condition (3.1), and z E '1) a fixed point for F. If the condition a) the intersection between the spectrum O'(A) of the operator A = F'(z) and the unit disk of the complex plane
are fulfilled, then z is an s-fixed point for the operator F, and the sequence
xn
=
OXn-l
+ (1 -
converges to z uniformly on '1) for any 0
E
(3.4)
O)F(xn-d
[0, 1) and Xo
E
'1).
(See the theorems in §2 of Chapter V.) The operator F(O) = 01 + (1- O)F satisfies condition a), or both the conditions a) and b), simultaneously with the operator F. Indeed, from (3.2) or b) it follows that IIAII ~ 1, hence rCA) ~ 1. By Lemma IV.1.1 there are no points from 0'3(A) on the unit circle of the complex plane. Therefore, by a), on that circle we can find points from O'l(A) only. Let B(O) = (F(O))'(O), and choose an arbitrary t E [0, 211'J. <J
.
Then e 1t I - B(O)
= (1 - 0)
(e1 _ 00) e 0 I - A . Setting = l-=-e we obtain IAI it -
A
it -
~ 1. If
F satisfies condition a), then the operator AI - A is either a homeomorphism of X, or has a non-zero kernel. In the first case the operator B(O) - eit I is a homeomorphism
Regular fixed points. Geometric criterions
187
of X, hence eit is a regular point of it, and in the second case p, = eit is an eigenvalue of B(B). Thus it is enough to prove the uniform convergence to z of the sequence of iterations {Fn(XO)}nEN' Xo E :D, only. Let hEX be such that Ilhll = 1 and z = th for a certain t E [0, 1). Consider the vector-function A('\) = F'('\h). By Lemma 1.4.2, it is analytic in the open disk .:1 of the complex plane C. From conditions (3.2) or b) the inequality sup
IIF'(x)11
~1
XEiJ
follows for any closed convex subset jj <;;;:D. Therefore r(A('\)) ~ 1, for all ,\ According to the maximum principle for the spectral radius, if
r(A('\o)) < 1,
E
.:1.
(3.5)
for some '\0 E .:1, then r(A('\)) < 1, for all ,\ E .:1. Thus reACt)) = r(F'(z)) < 1. The uniform convergence of the sequence of iterations {Fn(xO)}nEN, Xo E :D, follows from this inequality and Theorem 1.1. Thus, it remains to prove that condition a) together with one of conditions (3.2) or b) imply (3.5). We show that we can set '\0 = 0, i.e., rCA) < 1. 1. Assume that conditions a) and (3.2) are fulfilled. Then the operator G defined by G(x) = F(x) - F(O) satisfies the inequality
IIG(x)11 < Ilxll,
0 -1= x E :D.
(3.6)
Observe that G'(O) = A and let us show that the operator A has no eigenvalues on the unit circle of the complex plane. Suppose, on the contrary, that the equality Au = eitu is fulfilled for a number e it , where t E [0, 2nJ, and a certain u E :D with u -1= O. By the Hahn-Banach-Suchomlinoff Theorem we can choose a functional f E X* such that Ilfll = 1 and (u,1) = Ilull = r < 1, and let us consider the function
1) = = '\e it + r- 1(H('\u), 1).
g('\) = (r-1G(,\u), =
(r- 1A('\u), 1) + (r- 1H('\u), f)
(3.7)
This function is analytic on the closed disk .:1, and, by (3.6), the inequality Ig('\)1 < 1 holds for all ). E .:1. From (3.7) it follows that g'(O) = e it . According to Schwartz Lemma we obtain g(,\) = '\e it . Then l(r-1G(u), 1)1 = Ig(l)1 = 1, an equality which contradicts (3.6). Hence the unit circle consists of regular points for the operator A, so that, by Corollaries IV.1.l and IV.2.2, we obtain rCA) < 1.
NONLINEAR EQUATIONS
188
2. Assume that conditions a) and b) are fulfilled. Then, for all x E :D and t E [0,27rJ, the inequality IleitG(x)11 :::;; Ilxll is true. By Theorem 3.3 this inequality and the strong convexity of the space X imply that the set of all fixed points for the operator e-itG which lie inside :D coincides with the set of all fixed points for the linear operator e-itA = e-itG'(O). Therefore, if eit E O"I(A), i.e., there exists v EX with Ilvll = 1 such that v = e-itAv, then AV = eitG(Av), for all A E .d. It follows that F(AV) = G(AV) + F(O) = AW + s, where W = e-itv and s = F(O). As in the first part of the proof, we choose f E X* with Ilfll = 1 such that (w, f) = Ilwll = 1, and consider the function 'ljJ(A) = (F(AV),I) which is analytic in.d. It is easy to see that 1'ljJ(A) I :::;; 1, for all A E .d. On the other hand
'ljJ(A) = A(W, I)
+ (s, I)
=
A + (s, I).
Hence IA + (s, f) I :::;; 1, for all A E .d, and, therefore, (s, I)
(w + s, I)
= (w, I) = 1.
= 0.
But then (3.8)
Since IIAw + sll = IIF(Av)11 :::;; 1, for all A E .d, taking the limit as A ~ 1 we obtain Ilw + sll :::;; 1. By Theorem 0.4.7, this inequality, (3.8) and the strong convexity of the space X imply that s (= F(O)) = 0, which contradicts b). As above, we conclude that r(A) < 1. ~ If the operator F is completely continuous, then, by Theorem 1.8.1, the operator A = F'(O) is completely continuous, too; so, by Theorem IV.4.3, condition a) is fulfilled. The converse is not true. Some examples of non-completely continuous operators for which the Fnkhet derivative is completely continuous are presented, for instance, in L. V. Kantorovic and G. P. Akilov [1]. Note also - as Example V.2.4 shows - that condition a) is essential. In the case of a reflexive space X it is sufficient to assume that the operator ei7) I - A is normally solvable for all "7 E [0, 27rJ, in order to accomplish condition a). This follows from the Statistical Ergodic Theorem of K. Yoshida (see §4 of Chapter IV). REMARK 3.1.
It is not difficult to observe that the assertions in Theorem 3.4 remain true if one replaces the operator A with the operator An, where An is a certain power of the operator A, n = 1, 2, .... In many cases this may be useful, as the next example shows. REMARK 3.2.
Let X = Tn be the space of all bounded sequences x = (Xl, X2, ... , of complex numbers, with the norm Ilxll = sup IXkl, and let :D be the open
EXAMPLE 3.1.
Xk, ... )
k
A priori estimates unit ball of X. Consider the operator F : :D where
1
2
--+
189
:D given by Fx = (Zl' Z2, ... , Zk, ... ) 3
= "2 X1 + 8' Zk = 0, for k = 2m + 1, m = 1,2, ... , = (k - 2)(k -1)- l xk_l for k = 2m, m = 1,2, ... Zl
Zk
It is easy to see that F satisfies condition (3.3) on :D and condition (3.2) inside :D.
Moreover, obviously, the point x* = (~,O,O, ... ) is a fixed point for the operator F. Further, the operator F is Frechet differentiable and the operator A = F'(O) is defined by Ax
=
(Pl,P2,'" ,Pk, .. .), where
Pk
=
°
= 2m + 1, m = 0,1,2, ... , = (k - 2)(k - 1)- l xk_l for k = 2m, m = 1,2, ... Pk
for k
The operator A is not compact but it is nilpotent, since A2 = 0; hence condition a) is fulfilled for the operator A2. Consequently, the sequence (3.4) converges to x* for any Xo E :D.
§4. A priori estimates and the extension of an s-solution to the boundary of the domain The criterions for the existence of a fixed point considered in the previous section were based, essentially, on the following approach. Given the operator F(·) : :D --+ :D, we first construct an operator pC, .) : :D x L\ --+ :D, where L\ = {A E '1 < I}, satisfying the conditions in Theorem 2.1, and then we take the limit x(>') --+ x* for >. --+ 1, where x(>') is an s-solution of the equation x(>') = p(x(>.), >.).
(4.1)
This second step is possible because of additional properties of the operator F such as, for instance, the properness of I - F or the compactness of F, and so on. In the theory of linear equations the method of extending a solution with respect to a parameter is well-known. Roughly speaking, the method consists in establishing a priori estimates of the possible solutions for equation (4.1) - with P a linear operator - , for all >. E [0, 1) (let say, upper bounds for the norm Ilx(>')II) and then proving that such a solution can be extended on the interval [0, 1] (see, for instance, V. A. Trenogin [1]).
190
NONLINEAR EQUATIONS
We will use an analogous approach for equation (4.1) when if> is a holomorphic operator. To this end, it is enough for our purposes, to establish a priori estimates of the solutions for a linear equation associated to the given operator if>. THEOREM 4.1. Under the conditions in Theorem 2.1 let [l be the open disk ,1 =
= {>. E C : IAI < I} and assume that the operator if>(., A) : 1)
--+ 1) is continuous on for any fixed A E ,1. Moreover, suppose that the partial derivatives of the operator if> satisfy the conditions: 1) 11if>~(x, A)II ~ L < 00, (x, A) E 1) x ,1; 2) there exists 0: with 0 ~ 0: < 1 such that for all possible solutions of the linear equation
1)
(I - A(A))z
=
y,
z, Y E X,
where A(A) = if>~(x(A), A), and x(A) is a possible solution for equation (4.1) on ,1, the a priori estimate
Ilzll ~ M(1-IAI)-"'llyll
(IAI < 1)
is true. Then the vector-function X(A) ~ a solution of equation (4.1) ~ is defined and continuous on the closed disk ,1 = {A E C : IAI ~ I} and satisfies on this disk the Holder-type condition
Ilx(A) - x(A')11 ~ KIA _ A' 11 -""
(4.2)
where
and R is the radius of a ball centered at the origin and containing the domain
1).
~(x(A), A) for a fixed ..\ E ,1 lies completely inside the unit disk. Therefore the operator J - A(A) is continuously invertible for each A E ,1. Then, from (4.1) it follows that
x'(A)
= [J -
A(A)l-lif>~(X(A), A),
and, from conditions 1), 2), we obtain
IIX'(A) II ~ ML(l-IAI)-"'.
A priori estimates Therefore
1
(P)
191
1
J
x' (rei'P)dr
:::; M L
o
J
(1
~rr )<>
ML I-a'
0
and, consequently, the limit 1
lim x (re i6 ) = (P)
r-+1
J
x' (rei'P) dr
o
exists and is finite for all 'P E [0, 27r]. Thus the vector-function X(A) is defined for all A E r = aLl. Let us show first that (4.2) is fulfilled for all A, A' E r, with a suitable constant K. This will prove also the continuity of X(A) on r. Without losing the generality, we may assume that larg A-arg A'I :::; 7r. Consider the vector-function X(A) = (2R)-l x (A). Using the relation
'P 2 - 'P' 7r Ie1'P . - e1'P. I , I'P - 'P, I :::; 7r Isin - I ="2 I
for all I'P - 'P'I :::; 7r, we obtain that it is enough to prove the inequality Ilx(A) - x(A')II:::; K1 largA - argA'1 1 -<>
(4.3)
with a suitable constant K 1. To this end we may assume that Iarg A - arg A'I < 1, since, on the contrary, (4.3) with K 1 = 1 is obvious. Let us represent the left hand side of (4.3) as
X(A) - X(A') = (P)
J
x'(()d(,
I
where l is the piece-wise smooth curve consisting of the line segments [A, tAl and [tA', A'], and of the arc which connects tA and tA' along the circle 1(1 = t = 1-Iarg A - arg A'I < 1. Then
Ilx(A) - x(A')11 :::;
J
(1Ix'(rA)11
+ Ilx'(rA')II)dr + a/Alt Ilx' (tei'P) II d'P argA argA '
J
argA
MLt(l_t)-<>d'P 2R
:::;
NONLINEAR EQUATIONS
192
Thus (4.3) is fulfilled for all A, A' E
r, with
KI =maX{1,
ML 2R
(_2 +1)}. 1- IX
This means that (4.2) is fulfilled, for the same A, A', with
We fix now the points A, X E r and consider the vector-function Xl (() analytic in the disk 1(1 < 1 and continuous on the closure of this disk. Since Xl (0) = 0, by Schwartz Lemma we obtain
= X((A) - X((A'),
It follows that for any r E [0, 1] we have
Ilx(rA) - x(rX)11 ~ r sup Ilx(~A) ~
rK 2 sup I~A
-
-
x(~X)11 ~
1~1=1
~A'II-<> =
K2r<>lr(A - A'W-<> ~
(4.4)
1~1=1
Finally, let us prove inequality (4.2) - with an appropriate constant K -, for all A, A' E ,1 lying on the same ray, that is, arg A = arg A'. In this case
Ilx(A) - x(A')11 ~
1>.'1
JIlx'
(te iarg >.) II dt
~
1>'1 1>.'1
J
(4.5)
MLdt
(1 - t)<>
1>'1
Now let A and X be arbitrary elements of ,1. Choose a point A" such that and arg A" = arg X. It is easy to show that in this case we have
IA"I = IAI
IA - Alii + IA" - AI ~ 31)' - )"1·
A priori estimates
193
From inequalities (4.4) and (4.5) it follows that Ilx(A) - x(A')11 ~ Ilx(A) - X(A")II
~ K21A -
+ Ilx(A") -
+ _2a1 MLIA"
A"1 1 -a
x(A')11 ~
_ A'1 1-a
-0:
~ 31- a 2a max {K2' 1 ~a 0: ML} IA ~ 3 (~) a K21A _
A'1 1-a
~
~
A'11-a.
~
Setting K = 3(2/3)a K2 we get (4.2).
COROLLARY 4.1. Let F E H(::D, X) n C(::D, X), where::D is a C-star-shaped domain in X, and assume that for all possible solutions of the equation
B(A)Z where B(A)
= y,
AE
[0,
1], Y E X,
(4.6)
= 1- AF'(x) and x is a fixed element of::D, the a priori estimate Ilzll ~ 1'IIYII
(4.7)
is true. Then the operator F has a fixed point x* which can be approximated by the sequence Xt of the s-fixed points for the operators tF with t E [0, 1), and the rate of convergence is determined by the estimate
Ilx* - xtll ~ ~(1 - t), where ~ = (3/2)71" R . max{2, 31'} and R is the radius of a ball centered at the origin and containing ::D. Moreover, if the point x* belongs to ::D, then it is regular.
Indeed, the existence of the point x* E ::D and the estimate of the convergence rate follow straightforwardly from Theorem 4.1 if we set tP(x, A) = AFx, L = R, M = l' and 0: = o. If x* is an interior point of ::D then we can define the operator B = B(l) = 1- F'(x*). By estimate (4.7) and by the extension with respect to a parameter principle for the solutions of a linear equation (see V. A. Trenogin [1]), equation (4.6) has a unique solution for x = x*, A E [0, 1] and y E X. Consequently, the operator B is continuously invertible and A = 1 is not an eigenvalue of the operator F'(x*). ~
As an example, let us consider the integral equation
JL 1
x(t) =
0:
00
o n=2
Kn(t, s)xn(s)ds
+ (1 -
o:)y(t),
(4.8)
NONLINEAR EQUATIONS
194
where Kn(t, s) and y(t) are continuous complex-valued functions on [0, 1] satisfying the conditions ly(t)1 ~ t,
°t ~
~ 1, and 00
L
IKn(t, s)1 ~ 1.
n=2
We write equation (4.8) in the operatorial form
x = aF(x)
+ (1 -
a)y,
where F is the operator defined by the integral in the right hand side of (4.8), satisfying, obviously, the conditions
FE H('IJ, x) n C('IJ, x),
F('IJ) where
x = 0[0,
1] and 'IJ =
<;;:;
{x
'IJ,
F(O)
Ex: Ilxll
Let "Y
=
=
(sup Ilxll::;; 1
f
°
and F'(O) = 0,
= max Ix(t)1 < I}. O::;;t::;;l
n IKn(t, S)X n (S)I)-l
n=2
It is obvious that for all a E [0, "y) the operator t:Px = aF(x) + (1 - a)y acts from 'IJ into 'IJ and satisfies 11t:P'(x)11 ~ a"Y- 1 < 1. According to a consequence of the Banach principle it follows that equation (4.8) has a unique solution x(t) with IIx(t)11 < 1. Let us also show that - independently of the value of"Y - if a E (~, 1], then equation (4.8) has a unique solution. Indeed, using Schwartz Lemma, we obtain 11t:P(x)11 ~ allF(x)11
+ (1 -
a)llyll ~ allxl1 2
+ (1 -
a)llyll,
for any x with Ilxll ~ r ~ 1. Note that for any r E [(1 - a)a- 1 , 1] with 1/2 < a ~ 1, there exists q E [0, 1) such that ar2 + (1 - a) < qr. Hence 11t:P(x) II < qr, for all x with Ilxll ~ r. So our assertion follows straightforwardly from Theorem 2.2. In addition, the estimate Ilx* II < (1 - a)a is true. Thus we proved that equation (4.8) has solutions for a E [0, "Y) u (~, 1]. Finally we show that when "Y = 1/2 there exists a solution for all a E [0, 1]. To this end we consider the operator G(·, .) defined by G(A, x) = aF(x) + A(l - a)y, where A E L1 = {>. E C : IAI < I}. Obviously, G(L1 x 'IJ) <;;:; 'IJ and the operator G(O, .) = aF has the s-fixed point 0
195
Local inversion of holomorphic operators
for any a E [0, 1]. Let us show that if "I = 1/2, then the operator G ( " .) and the ssolution X(A) of equation (4.1) satisfy the conditions in Theorem 4.1 for any a E [0, 1]. Condition 1) is obviously fulfilled. We prove now condition 2). Together with the operator G consider the real-valued function f(s, t) = as 2 + (1 - a)t, s, t E [0, 1]. One of the solutions for the quadratic equation s = f(s, t) is the function
s(t)
=
1 - J1 - 4a(1 - a)t 2a '
relatively to which we assert that
Ilx(A)11 ~ s(t), Indeed, set XO(A)
= A(l - a)xo
IAI ~ t < 1.
(4.9)
and
Xn +l(A)=G(A,X n (A)),
n=0,1,2, ... ,
where Xo is an arbitrary element of :D. Obviously Ilxo(A)11 ~ s(t), for IAI ~ t, and assuming that Ilxn(A)11 ~ s(t) for IAI ~ t, then by induction we obtain
Taking the limit for n
~ 00
we get (4.9). Hence it follows that for IAI
=
t we have
III - A(A)II = III - G~(A, x(A))11 ?: 11 - IIG~(A, x(A))111 = =
?:
11 - allF'(x(A))111 ?: 11 - 2allx(A)111 ?:
11-1 + J1- 4a(1-
since 4a(1 - a) ~ 1, for a E [0, 1). By Theorem 4.1, the limit lim X(A) 'x->l
of equation (4.8). For a
=
a)tl
= x*
?:~,
exists and, obviously, x* is a solution
1, equation (4.8) has the solution x*
=
0.
§5. Local inversion of holomorphic operators and a posteriori error estimates Let us consider the equation
4>(X) =
°
(5.1)
NONLINEAR EQUATIONS
196
associated with an operator P : X -+ X, and assume that Xo E X is an approximate solution of this equation. There are a lot of methods for determining the value of xo, but many of them involve computational difficulties. Therefore in practice it is important to know - as accurate as possible - a posteriori estimates for the errors appearing in the approximate computation of Xo. (See M. A. Krasnoselskii [2], [3], and L. V. Kantorovic and G. P. Akilov [1].) The notion of error estimate, needs a more precise definition. We will state a definition introduced by M. A. Krasnoselskii.
error estimate for an approximate solution Xo of equation (5.1) is an arbitrary domain m c X containing xo, provided that this equation has at least one solution x* in m. Let J be the set of all solutions for equation (5.1). The error of an approximate solution Xo is defined by DEFINITION 5.1. An
8(xo) = inf Ilxo - xii· xEJ
Let us assume that the operator P is Frechet differentiable in a neighborhood of the point Xo. Without restricting the generality, we may set Xo = 0; on the contrary, equation (5.1) can be always replaced by the equivalent equation p(xo + h) = 0, and one has to look for the exact solutions of the last equation in a neighborhood of the approximate solution ho = O. The value P(O) E X is called the unbinding of equation (5.1). In many computing proccesses the point Xo = 0 is considered to be a sufficiently "good" approximate solution of equation (5.1) if the norm IIp(O)11 = E of the unbinding is small enough. Generally speaking, this works well in two cases: either in connection with a certain problem, when we are interested only in an approximate equation p(x) ~ 0 rather than equation (5.1), or, when the error 8(0) of the approximate solution Xo approaches 0 as E approaches O. Let us note that for a continuous operator P the converse is always true, namely, E -+ 0 for 8(0) -+ O. Therefore we are interested in estimating the value of the error 8(0) in terms of the norm E = IIp(O) II. To this end we represent the operator P as
p(x) = p(O)
+ p'(O)x + w(x),
where w(x) = o(llxll). In this section we are dealing with the so-called regular case, when the operator A = p(O) is continuously invertible. Then equation (5.1) is equivalent to
x=Q(x)+y, where Q(x)
= -A-1w(x) and y = _A-lp(O).
(5.2)
Local inversion of holomorphic operators
197
Since the operator F = I - Q satisfies the condition F'(O) = I, then, by the Local Inversion Theorem, there exists an operator G defined for all y E X with a sufficiently small norm, which is a solution for equation (5.2). Our aim is to obtain estimates - as precise as possible - for the domain of existence and continuity of the operator G. These estimates will enable us to find solutions for the above considered problem. Throughout this section we will assume again that X is a complex Banach space. To begin with, let us suppose that the operator Q is defined and differentiable on the ball 1) centered at Xo and with radius R (clearly the operator F = I - Q has the same properties). The next theorem is useful in obtaining the above mentioned estimates.
Let
THEOREM 5.1.
IIQ(x)11 : :;
x E X.
Suppose that the function 'Ij;(t) = t - O. Then 1) the operator G = F- 1 is defined and differentiable on the ball QJ centered at the origin and with radius r; 2) the estimate IIG(y)11 : :; 'Ij;-l(llyll) is valid for all y E QJ; 3) the operator G can be obtained as the limit of the sequence of operators
Go(Y) == 0,
Gn+1(y) = y + QG(y),
n
= 1,2, ....
Assume, by induction, that the operator G n is defined and differentiable on the ball QJ and satisfies the relation
(5.3) By the recurrence rule, the operator G n +1 is also defined and differentiable on QJ and satisfies an analogous estimate. Indeed, since the function
IIGn+1(y)11 : :; Ilyll + IIQGn(y)11 : :; Ilyll +
Then the operator G radius r, where
F- 1 is defined on the ball
=
QJ
with center at the origin and
R R2} r = min { -, M and M = sup IIQ(x)ll· 2 4
Ilxll
Moreover,
~ min { R, ~ } . IIYII, Ilyll < r,
IIG(y)11
and the sequence {Gn}nEN, with Go(Y) == 0 and Gn+l(Y) = y+QGn(y), n = 1,2, ... , converges to the operator G for all y E QJ. In the finite-dimensional case this result was obtained by A. P. Yuzhakov using the theory of logarithmic residues (see [1], [2]).
then
R
R
2
2
R2(2M)-1,
·"(R) >- R - M >- R - - = -.
"'"
"'"
Let us return now to the initial equation (5.1) and assume that the operator lJ> is differentiable on the ball ofradius R with center at the origin, and let m = IIA -111. From the already obtained results it follows that if the unbinding lJ>(0) satisfies the estimate E: =
11lJ>(0) II ~ 2~ min{1, R(2M)-1},
where
M
=
m· sup 11lJ>(x) - lJ>' (O)x - lJ>(0) II, IIxll
Local inversion of holomorphic operators
199
then the error of the approximate solution Xo satisfies the relation 8(0) ~
E .
This obviously implies that 8(0)
m . R· min{l, R(2M)-1}.
-+
0 as
E -+
o.
REMARK 5.1. It is worth to emphasize the qualitative aspect of the estimate found in Corollary 5.1. In the usual estimates obtained for arbitrary Banach spaces (see, for instance, M. A. Krasnoselskii [2], [3]) the constants rand M are related - as a rule - by a condition of the form ar + (3M ~ R, a, (3 ~ O. Whence it follows that for the existence of a unique solution of the equation (5.2) both rand M are necessarily bounded (at least by the number R). In our case it is clear that if M is sufficiently small, then r may be chosen as close to R/2 as we want, and, conversely, if r is sufficiently small, then the solution of equation (5.2) exists for arbitrary large values of M. We illustrate these remarks by an example.
Consider the Hammerstein equation b
x(t) =
J
K(s, t)f(s, x(s))ds + yet),
(5.4)
a
where: a) K(s, t) is a continuous function on the rectangle [a, b] x [a, b], and b) the function f(s, t) is analytic with respect to u in the disk lui < R of the complexe plane for a fixed s E [a, b], is continuous in s E [a, b] for a fixed u, and satisfies the conditions
f(s,O)
=
f~(s,
0,
0)
=
o.
Some conditions for the existence of a unique solution of equation (5.4) were obtained in §4 of Chapter V, using the Banach principle. They were {
liM(b - a) + Ilyll ~ Ro LIi(b - a) < 1,
where
Ro < Ii
=
max
a ~ s,
t
R,
~ b
IK(s, t)l,
NONLINEAR EQUATIONS
200 M =
max
a~s~b
If(s, u)l,
and
L
=
sup
If~(s, u)l·
a~s~b lui ~Ro
lul~R
These inequalities lead to the next relations
Ro M< ",(b-a)'
L
<
1 ",(b - a ).
Since equation (5.4) is a particular case of equation (5.2), from Corollary 5.2 we obtain a different condition for the existence of a unique complex-valued solution of equation (5.4), i.e.,
IlyIIM",(b -
a) < Rmin
{~, ~},
(5.5)
where, as a matter of fact, the constant L does not appear, and the constant M may be as large as we want if the norm Ilyll of y is sufficiently small. Let us note also that condition (5.5) is simpler than condition (V.5.20) obtained in §5 of Chapter V using the Cauchy-Goursat method of majorant power series. To conclude this section, we state a consequence of the already obtained estimates, based on the so-called covering theorems. These theorems play an important role in the finite-dimensional complex analysis (see Goluzin [1], Shabat [3], Lebedev
[1]). COROLLARY 5.2. Let F be a bounded analytic operator defined on some domain 1)
and such that F'(xo) is continuously invertible at least for one point Xo E 1). Further, let {Fn}nEN with Fn E H(1), X), be a sequence of analytic operators which converges uniformly to the operator F on some neighborhood of the point Xo. Then there exist a domain mand an integer no such that
§6. Single-valued small solutions in some degenerate cases Let X, A and !il be complex Banach spaces. Consider an operator F, with values in !il, defined and Frechet differentiable on a neighborhood of the point (0,0) E X x A. Assume that
F(O, 0)
=
0.
(6.1)
Single-valued small solutions If on some neighborhood fl by the equation
c A
201
of 0 E A there exists an operator x = X(A) defined
F(X(A), A)
= 0,
(6.2)
and such that X(A) --> 0 as A --> 0, then x is called a small solution of equation (6.2). Whenever the linear operator F~(O, 0) is continuously invertible (the nondegenerate case), in view of the Implicite Function Theorem, a local implicit operator x = X(A) exists and, obviously, it is a small solution due to its continuity. In the degenerate case, when 0 is a point in the spectrum of the operator F~(O, 0), some other possibilities occur also: either there are no solutions at all, or there is a solution but it is not single-valued (in this case we have to deal with branches of the solution), or, finally, there exists a single-valued solution, but it is not small (as, for instance, when x(O) =J 0). These situations are generically called critical. They will be considered in detail in Chapter VIII. Here we will be concerned with criterions which give some sufficient conditions to avoid critical situations when the operator F~(O, 0) is degenerated, i.e., criterions for the existence of a single-valued small solution in the degenerate case. Unfortunately the criterions which will be presented below have merely a particular feature: they provide neither sufficient nor necessary conditions to handle a general non-critical situation in a degenerate case. However they are extremely important in specific problems, since any small solution is stable. 1. SMALL SOLUTIONS IN A GENERALIZED SENSE. THE FINITE-DIMENSIONAL CASE.
Let us note that, as long as we are concerned with stability, in many problems it is enough to check it only along all linear one-dimensional directions, i.e., to show that small perturbations of the parameter A along anyone-dimensional linear subspace yield small changes in the solution X(A). In this respect, the next notion of "generalized smallness" is useful.
X(A) of equation (6.2), where the operator F satisfies condition (6.1), will be called small in the generalized sense if for anyone-dimensional subspace IE <;;; A there exists a neighborhood II C IE of 0 E IE on which the vectorfunction X(A) is defined and satisfies the condition X(A) --> 0 as A --> 0, A E ll. DEFINITION 6.1. A solution
Obviously if an operator x( .) is Gateaux differentiable on a neighborhood of the point 0 and satisfies equation (6.2) and condition x(O) = 0, then it is a small solution in the generalized sense. (Therefore a small solution in the generalized sense may not be always continuous on a whole neighborhood of 0.) Of course when the space A is one-dimensional, a small solution in the generalized sense is also small
NONLINEAR EQUATIONS
202
in the usual sense and therefore the considered set-up is interesting only when the dimension of A is greater than l. The above remarks lead to the conclusion that a possible approach towards some criterions for the existence of small solutions in the generalized sense, in a degenerate case, is to look for such solutions not in the class of Frechet differentiable operators but in the class of Gateaux differentiable operators (or in the class of pholomorphic operators if X, A, ~ are complex spaces). In this respect, Theorem 2.1 turns out to be useful. Until now we used neither that the operator 1>(., .) is p-holomorphic in the variable oX, nor the p-holomorphicity of its fixed point x = x(oX) (see §2 of this chapter). In this subsection we will consider only the finite-dimensional case to illustrate how Theorem 2.1 can be used. Let 1 be an arbitrary function, holomorphic in a neigborhood of 0 E en, with 1(0) = O. For any subspace ~ ~ en, we will denote by !1o(f I ~) the multiplicity of 0 for the function 1 I ~ - the restriction of 1 to ~ (see L. A. Aizenberg and A. P. Yuzhakov [1]). Let A = en, X = ~ = e and assume that F is a function which is holomorphic in a neighborhood li l x li2 of the point (0,0) E en+! and which admits the following representation (6.3) F(x, oX) = P(oX)x + Q(x, oX), where P and Q are holomorphic in li2 ~
P(O) = Q'(O, 0) = Obviously, THEOREM
en
and lil x li2 ~
e
n
+ l , respectively, and
o. F~(O,
0)
= 0 and so we find
ourselves in a degenerate case.
6.1. Assume that the functions P and Q in (6.3) satisfy the condition
Then, for anyone-dimensional subspace <E c en there exist a neighborhood m~ <E of <E and a unique function x(oX), holomorphic in m, implicitely defined by equation (6.2) and such that x(O) = o. Thus x(oX) is a small solution of equation (6.2) in the generalized sense. Moreover, if the function
oE
l(oX) = Q(O, oX) [p(oX)]-2
(6.4)
is holomorphic on li2, then there exists a neighborhood E ~ en of the point 0 E en in which the function x(oX) is holomorphic, and, consequently, x(oX) is a small solution in the usual sense.
Single-valued small solutions <J Consider the function
tJ>(x, A)
tJ> : e n +1
=
---+
{ x0,
203
e defined by the equality
F(x, A) peA) , A i= 0 A = o.
Fix an arbitrary element h E en with Ilhll = 1. Taking into account the assumptions of the theorem we obtain the representation
tJ>(x, (h) = (kg (x, (h),
(6.5)
for all ( E e with 1(1 < p(h), where p(h) is sufficiently small, k = /-Lo(Q I em) - /-L;;::' 1, and the function g(x, (h) is holomorphic in ( for any fixed x E lit, i.e., the function tJ> is p-holomorphic in some p-open subset [l :3 0 (we can choose [l = (ti2 \ {A E en : : peA) = O}) U {O E en}). Thus equation (6.2) is equivalent to
x = tJ>(x, A) considered on the set [l. Further, from (6.5) it is obvious that for any h with Ilhll = 1 we may choose p(h) such that tJ>(x,(h) E ti2 for all ( with 1(1 < p(h) and any x E til. Therefore there exists a p-open subset [ll :3 0 in [l having the property that tJ>(x, A) E til for (x, A) E til x [l2. Since the point x = 0 is an s-fixed point for the operator tJ>(., 0) : e ---+ e (recall that tJ>(., 0) == 0), by Theorem 2.1 it follows that the last equation has a unique s-solution X(A) which is p-holomorphic in [ll. So the first assertion of the theorem is proved. For the proof of the second one let us remark that, by condition (6.4), the function Vi (A) = tJ>(0, A) = P(A)f(A) is holomorphic in some neighborhood of the point 0 E en. Therefore the functions Vk+1(A) = tJ>(VdA) , A) are also holomorphic in a neighborhood E of the point 0 E en. Thus the orbit of the element Vo == 0 (vo E H(E, C)) is completely contained in H(E, C). Hence x = x(·) E H(E, C) (see Remark 2.1). ~ and F(x, A) = A~x2 + (Ai - A2) x + A~. -A1A2, where (Al,A2) = A E e Obviously the function F admits representation (6.3) with peA) = Ai - A2, F~(O, 0) = 0, and satisfies the first requirement in Theorem 6.1. Therefore equation (6.2) has a unique small in the generalized sense solution X(A) which is p-holomorphic in some set [l <;;;; (:2, [l :3 O. We deduce easily this conclusion by a straightforward computation. Indeed, solving the quadratic equation F(x, A) = 0 EXAMPLE 6.1. Let A
= e2 ,
X
=!D = e,
2.
NONLINEAR EQUATIONS
204
and setting A = (h, where hE ((:2 with Ilhll = 1, we obtain two solutions, X2(A). One of these solutions is given by
Xl
(A) and
and, therefore, it is small in the generalized sense, while the second one satisfies the condition lim x2((h) = 00 and, consequently, it is not small. Let us note (->0
that Xl (A) is a continuous function on some p-open set [l E 0, whose intersection with ~2 is the set fl represented in Figure 2 and having the form of a helix.
Figure 2 EXAMPLE 6.2. For the same A, X and ~ let now F be the function given by
where g(A) is an arbitrary holomorphic function. Obviously F(O, 0) = 0 and F~(O, 0) = = 0, and, consequently, in this case, we may not use the classical Implicit Function Theorem. However, the function F satisfies both the first and the second requirements in Theorem 6.1. Hence the equation F(x, A) = 0 has a unique holomorphic solution X(A) such that x(O) = o. Indeed, solving straightforwardly this equation we obtain 00
X(A) = -(Ai - A2)
L
2n Ayn[g(A)t+l,
n=O
where the series in the right hand side converges uniformly on some neighborhood of the point A = o.
Single-valued sma,l1 solutions
205
2. SMALL SOLUTIONS FOR AN OPERATOR LINEAR IN THE PARAMETER. THE INFINITE DIMENSIONAL CASE. We return once more to a complex Banach space X. Now we will suppose that equation (6.2) has the next special form
Bx
=
AG(X)
+ y,
(6.6)
where: a) B : X ----+ X is a linear normally solvable operator such that (*) the kernel N(B) = ker B of the operator B is isomorphic with the linear space Sj = X~R(B) where R(B) is the range of the operator B, b) y is an arbitrary fixed element in R(B), c) G is an operator anaytic in the ball:D = {x EX: Ilx - zll < R}, where z is a solution of the equation Bz = y, and d) A E C. In this setting we have to find conditions for the existence of an analytic solution X(A) of the equation (6.6) satisfying the relation x(O) = z, and to indicate the domain of analyticity for this vector-function. Equations as (6.6) above arise in connection with a series of problems in nonlinear mechanics (see for example P. P. Rybin [1], [2], M. M. Vainberg [1] and M. M. Vainberg and V. A. Trenogin [1]). Moreover, equation (6.6) with an integral operator G, is related to the well-known Duffing equation, which has applications in electrotechnics. When N(B) = {O}, the operator B is continuously invertible and, consequently, we find ourselves under the conditions in the Implicit Function Theorem. When N(B) -I- {O}, it is convenient to assume that B is finitely degenerated, i.e., ind B = dim N(B) - dim N(B*) = 0 (B* is the adjoint of B). In this case, conditions for the existence of a unique analytic solution X(A) have been obtained for the first time apparently by K. T. Ahmedov [1]. To be more specific, according to the Nekrasov-Nazarov method, one looks for a solution represented by a power series whose convergence is established using a majorant numerical series. This approach enables one to indicate an estimate for the analyticity domain of the solution X(A) and this turns out to be important in many applications. Our aim in what follows is to provide conditions for the existence of a unique solution for equation (6.6) in the case of an arbitrary space N(B) which is isomorphic with the space Sj (this property is obvious if the operator B is finitely degenerate). These conditions are obtained by reducing our problem to finding an s-solution for an equation of type (2.1). This approach enables us to point out a different way for approximating X(A), and also to establish a more convenient and more precise estimate for the domain of analyticity.
NONLINEAR EQUATIONS
206
First of all we observe that by the Banach Theorem the operator B I X~N(B) - the restriction of B on the subspace X~N(B) - has a continuous inverse operator r : R(B) -> X~N(B). Setting x = z + h we write G(x) as G(z
+ h) = G(z) + Ah + Q(z, h),
(6.7)
where A is the Fnkhet derivative of the operator G at the point z and IIQ(z, h)11 =
= o(llhll)· Let us introduce the notations M(G) = sup IIG(x)ll; xE:D
M(Q)
= sup
IIQ(z, h)ll;
Ilhll
P : X -> Sj -
A=
A I N(B) -
the projection from X onto Sj;
the restriction of the operator A on N(B).
THEOREM 6.1. If A is an isomorphism from N(B) onto Sj, and G(z) E R(B), then
equation (6.6) has a unique analytic solution X(A) for all A such that
R . { dR d Rd 2 IAI < 8 = 211T1IM(G) mm 1, 2M(Q) , 211AII' 8M(Q)IIAII
}
'
(6.8)
where d = IIA-1 PII- 1. Moreover, X(A) can be approximated in the norm topology of X by the sequence {Xn(A)}nEN which is defined by the relations Xn(A)
= z + ArG(X n -1(A)) + Vn -1(A)),
Vn(A) = -[AAPArG(X n _1 (A))
+ APQ(z, Xn(A)
(6.9) - z)],
n = 1,2, ... , where XO(A) is an arbitrary element of X such that
Ilxo(A) - zll
=
min { R,
2M~Q)} ,
and VO(A) is an arbitrary element of N(B) such that
Ilvo(A)1I <
~f.
(6.10)
207
Single-valued small solutions
Setting x =·u+v, where u E X':""N(B), v E N(B), and projecting equation (6.6) on the subspaces Sj and R(B), respectively, we obtain a system of equations
(6.11)
PG(u+v)=O, Bu
= >"G(u + v) + y,
(6.12)
r
to the both sides of which is equivalent to equation (6.6). Applying the operator (6.12) and taking into account the equalities u = x - v and ry = z we obtain PG(x) x
= 0,
(6.13)
= z + >..rG(x) +v = FI(>..,x,v).
(6.14)
We perform now a few formal transformations, the substantiation of which will become apparent below. Let us replace h in (6.7) by >..rG(x) +v. Based on (6.14) we get G(x)
= G(z) + >"ArG(x) + Av + Q(z, >..rG(x) + v).
Applying the operator P to the both sides of the last equality, and taking into account the equalities (6.13), PG(z) = 0 and PAv = Av (recall that Av E Sj) we obtain 0= Av
+ >"PArG(x) + PQ(z, >..rG(x) + v),
or Av = -[>"PArG(x) +PQ(z,>..rG(x)
+ v)],
whence v
= _[>"A- I PArG(x) + A-I PQ(z, >..rG(x) + v)] = F2(>'" x, v).
Consider now the Banach space ~
= max{llxll, Ilvll}, for any w = (x,v)
E~.
(6.15)
=
X-t-N(B) with the norm given by Ilwll = Denoting F = (FI ,F2) we may rewrite the
system (6.14), (6.15) as w = F(>",w).
(6.16)
We show that the mapping F is defined and analytic in some domain in ~, and for all >.. E fl = {>.. E IC : 1>"1 < b} (see (6.8)) it satisfies the conditions in Theorem 2.l. The mappings Fl and F2 are defined and analytic if Ilhll (= II>..G(x)
+ vii) < r:::;; R.
(6.17)
NONLINEAR EQUATIONS
208
Consider the domain ::Dr = {w : IIx - zll < r, Ilvll < rc} included in !C. Inequality (6.17), for all w = (x, v), is obviously fulfilled under the following conditions: 0< c < 1 and
r(1 - c)
(6.18)
IAI < IlrIIM(G)'
Let us establish the existence of such an r = f > 0 and of a function 8(r) : [0, R] ----+ ----+ [0, 8] so that the mapping F leaves the domain ::Df invariant for A with IAI < 8(f). From (6.14) and (6.18) it follows that IIF1 (A,w)11 < r for any r with 0 < r < Rand IAI < p(r). Consider the inequality (6.19) It is not difficult to convince ourselves that inequality (6.19) follows from
IAI <
drc -
r2 M(Q)R- 2
IIArIIM(G)
=
tp(r),
for any r such that tp(r) > O. A direct computation shows that
{Rdc
(Rdc)2}
O~~R tp(r) = tp(r) = mm 21IArIIM(G)' 41IArIIM(G)M(Q) A
•
where f
= min { R,
,
2M~ Q) } .
Thus the mapping F is analytic and F(A, w) E ::Df for all w E ::Df and all IAI < 8(f) = min{p(f), tp(f)}. Clearly 8(f) depends also on c, and, as one can easily see, it attains its maximum value for c = 2- 1 , whence (6.8) follows. From (6.14) and (6.15) it is not difficult to remark that the operator F(O, . ) is a q-contraction on some neighborhood of the point Wo = (z,O) for a certain q with 0 < q < 1. By Theorem 2.1 we obtain an s-solution W(A) = (X(A), V(A)) of equation (6.16) and the iterative sequence Wn(A) = F(A,W n-1(A)) = (Xn(A),Vn(A)) which appears in formulas (6.9), (6.10). ~ To conclude this section let us show what the conditions in the theorem above mean in the case of a finitely degenerate operator B. Assume that N (B) has dimension m and ind B = O. Futher, let {V1, . .. , v m } be a basis of N(B) and let SJk = clin{Av1, ... ,Avk-1,Avk+1, ... ,Avm} be the linear hull of the elements AV1"'" AVk-l, AVk+1,"" Avm. Consider the spaces 'ck = = R(B)+-SJk which are proper subspaces of the space X.
Single-valued small solutions
209
The following conditions are equivalent: 1) the operator A = A I N(B) is an isomorphism from N(B) onto Sj; 2) d k = P(AVk' ~k) = inf IIAvk - xii> 0, k = 1, ... , m;
PROPOSITION 6.1.
XE£k
3) the equalities
are fulfilled for a certain basis
{'Ij;d~l
of the space N(B*) (B* is the adjoint of B).
<J From 1) it follows that the system of elements {Avd~l forms a basis of Sj. Assume that condition 2) fails for some k = 1, ... , m. Then there exists a non-zero element Y E R(B) such that
AVk = Y +
L if-k
1:::;; i
or
~
(XiAVi,
(Xi
E
m
m
Y = L;JiAvi, i=l
where ;Ji = -(Xi if i i- k, 1::;; i ::;; m, and ;Jk = 1, whence Y E Sj. This is impossible since Y i- 0 and Sj n R(B) = {o}. The implication 1) =} 2) is proved. Assume now that condition 2) is fulfilled. By a consequence of the HahnBanach Theorem (see, for instance, Chapter 0) there exist the elements 'lj;k E X*, k = 1, ... , m, such that (6.20)
('Ij;k, AVk)
=
1,
II'Ij;kll = d-,;l,
(6.21) (6.22)
for all k = 1, ... , m. From (6.20) it follows that 'lj;k E N(B*). Indeed, since ~k ::J R(B) then for any x E X the equality
holds, whence - in view of the fact that x is arbitrary - we have B*'Ij;k = O. Moreover, taking account of equality (6.21) we obtain that
NONLINEAR EQUATIONS
210
In order to prove 2) =? 3) it remains to establish that the system of elements {'¢d~l is linearly independent. Assume that
for some complex numbers 'Yk
E
C. Then for each 1 ~ i
~
m we obtain
Let us prove now implication 3) =? 1). To this end it is sufficient to show that, if condition 3) holds, then the system of elements {Avd~l is a basis of Sj. Indeed, from 3) it follows that the system {Avd~l is linearly independent. It remains to show that the elements AVi, i = 1, ... , m, lie in Sj. Assuming the contrary, we obtain that for some 1 ~ j ~ m the element AVj lies in R(B). From the Fredholm alternative it follows that AVj is orthogonal to the space N(B*), which is impossible since
for all j = 1, ... ,m.
~
Chapter VII Banach manifolds §1. Basic definitions Let Sj be a Hausdorff tolopogical space and let !D be a Banach space. DEFINITION 1.1. The space Sj is called a manifold modelled on !D, if for any point x E Sj there exist an open neighborhood ilx s;:;; Sj of x and a homeomorphism i.px : llx ----t s;:;; !D, where is an open set in !D.
mx
mx
If the space !D is the n-dimensional Euclidean space OC n , then Sj is called simply an n-dimensional manifold.
A pair (llx, i.px) as above is called a chart (sometimes, a local chart) of the manifold Sj. A family of charts {(llx,i.px)}x such that Ullx 2 Sj is called an atlas of x
the manifold Sj. We say that two charts (llx, i.px) and (lly, i.py) satisfy the conm dition of e _ compatibility, m = 0,1,2, ... ,00 (respectively, of O-compatibility), if whenever llx n lly f. 0, the mappings i.px 0 i.p:;/ : i.py(llx n lly) ----t i.px(ilx n lly) and i.py Oi.p;;l : i.px(llxnlly) ----t i.py(llxnlly) are of class em, m = 0,1,2, ... ,00 (respectively, of class 0 ~ the class of analytic mappings). DEFINITION 1.2.
If all the charts belonging to a given atlas are em-compatible, m = 0, 1,2, ... , (respectively, O-compatible), where m does not depend on the charts, then we say that the manifold is of class em (respectively, of class 0). A manifold of class eO is called a topological manifold. A manifold of class 0 is called analytic (if!D is a complex space, the manifold is called complex-analytic). 00
BANACH MANIFOLDS
212
The manifolds of class C m , m ) 1, are also called smooth manifolds. REMARK 1.1.
In connection with the above introduced definitions the following
questions arise. 1) Can we speak about an unique manifold if the space SJ is endowed with two different atlases? An answer to this question depends, of course, on the convention we agree upon. The advisability of a convention is determined by the applications which may occur later on. Usually two atlases on the same space SJ are called equivalent if any chart of one of the atlases is compatible (relatively to a fixed class C m , respectively 0) with any chart of the other atlas. In this case we say that both atlases define one and the same structure of a Banach manifold on the space SJ. 2) Are there manifolds which coincide as topological spaces but have different models? In other words, if SJ is a manifold with the model space !D, can SJ be a manifold with a model space !D' =I- !D? An answer to this question will be given in the sequel. However, even at this moment, it is obvious that, if this is the case, the spaces !D and !D' must contain some homeomorphic open sets. Any open subset of some Banach space X (in particular the space X) offers a simple example of a manifold. Indeed, for such a subset il S;;; X, the model space is the space X itself, and the atlas consists of one chart defined by the whole set il and the identity homeomorphism. These manifolds are smooth (of class C=) and even analytic. If X is the n-dimensional Euclidean space, then these manifolds are n-dimensional. Any subset of a space X consisting of isolated points is a zero-dimensional manifold. A more interesting example is offered by the circle {(x, y) E]R2 : x 2 +y2 = 1}. Indeed, let us choose the open cover of the circle given by the four semicircles {i4}[=1 obtained intersecting the circle with the half-spaces y > 0, x > 0, y < 0, x < 0, respectively. Take a point (x, y) with y > on the semicircle ill. The mapping 'PI which associates to (x, y) E ill the point x E]- 1, 1[ S;;; ]R1 is a homeomorphism. The inverse mapping 'Pi1 is given by 'Pi 1(x) = (x, v'1- x 2). The pair (il1,'Pd defines a chart. The charts (il2' 'P2), (il3, 'P3), (il3, 'P3) are obtained analogously. Thus the circle is a manifold with the model space ]R1, or, a manifold of dimension 1. Further, let us remark that the indicated atlas gives a smooth structure on the circle as a manifold. Indeed, consider, for instance, the neighborhood il12 = ill n il2 consisting
°
213
Smooth mappings
of that part of the circle which lies in the positive quadrant y > 0, x > 0. There are two homeomorphisms defined on ilj2, namely, 'PI : (x, y) ----* x and 'P2 : (x, y) ----* y. The overlap mapping 'PI 0 'P2I : Y f---7 X is given by the equality x = and is a homeomorphism from the intervaljO, 1 [ onto itself. Analogously we can show that the sphere is a smooth manifold of dimension 2. In the above mentioned examples it is worth to remark that the circle and the sphere are subsets of the spaces JR2 and JR3, respectively, which, on their turn, are also manifolds. We will be concerned with such aspects in §3.
+JT=Y2
§2. Smooth mappings The main reason for introducing the notion of manifolds consists in the possibility they offer to differentiate and integrate functions or mappings defined on "nonlinear" spaces. This is done by identifying locally these spaces with theirs models. Thus, for example, a function defined on a circle can be considered locally as a function given on an interval of the real line. Therefore it is convenient to think of a circle not as a set of points in the two-dimensional space JR2, but as a set of points which is identified locally with subsets of the one-dimensional space JR I . Let us give the precise definitions.
em,
m ~ 1, modelled on the space !D, and let f be a mapping from Sj into a Banach space 3. The mapping f is called m-times continuously Frechet differentiable at a point a E Sj if there exists a chart (lla, 'Pa) with a E lla <:;;; Sj and 'Pa : lla ----* l!1a <:;;; !D, such that the mapping f 0 'P;;I : : l!1 a ----* 3 is m-times continuously Frechet differentiable, in the usual sense, at the point b = 'Pa(a) E !D. DEFINITION 2.1. Let Sj be a manifold of class
For the correctness of this definition it has to be shown that the differentiability of f is independent of the choice of a compatible chart. Indeed, let (ll~, 'P~) with a E ll~ be another chart, compatible with (lla, 'Pa) for which the mapping f 0 'P;;I is m-times differentiable at the point b = 'Pa(a) E!D. Then the mapping f 0 'P';;I = = f 0 'P;;I 0 'Pa 0 'P';;I is m-times differentiable at the point b' = 'P~(a) E !D as the composition of two m-times differentiable mappings, since the mapping 'Pa 0 'P';;I is m-times differentiable by the compatibility of the charts. In particular, if!D = K n , then the differentiability of f means the existence of the partial derivatives of the mapping f 0 'P;;I, with respect to the variables (XI,X2, ... ,Xn ) E Kn, at the point Xa = ('PIa(a)"",'Pna(a)). Therefore the homeomorphism 'Pa is often called a local system of coordinates for the manifold Sj at the
BANACH MANIFOLDS
214
point a, and the corresponding neighborhood lla ~ Sj is called a local coordinate neighborhood. In other words, a chart in an atlas of the manifold Sj consists of a local coordinate neighborhood and a local system of coordinates. , m ~ 1, modelled on spaces lD and 3, respectively, and let 9 be a mapping from Sj into 9)1. The mapping 9 is called m-times Fn§chet differentiable at a point a E Sj if there exists a local system of coordinates '¢b at b = g(a) E 9)1, such that the mapping f = '¢b 0 9 : Sj ---> 3 defined on some neighborhood of the point a E Sj is m-times differentiable at a in the sense of Definition 2.1.
DEFINITION 2.2. Let Sj and 9)1 be two manifolds of class C m
In this case it is also easy to show that the differentiability of g, in the indicated sense, does not depend on the choice of the local system of coordinates '¢b. Moreover, we have the next obvious result. PROPOSITION 2.1. A mapping 9 : Sj ---> 9)1 is m-times differentiable at a point a E Sj
- in the sense of Definition 2.2 - if there exist two local systems of coordinates, 'Pa at the point a E Sj and '¢b at the point b = g(a) E 9)1, such that the mapping '¢b 0 9 0 'P;; 1 : lD ---> 3 is m-times Frechet differentiable at the point 'Pa (a) ElDin the usual sense. Analogously we introduce the notion of continuously differentiable mapping and the notion of analytic mapping (if the manifold is analytic) at a point, or, on some open subset, of a manifold Sj. DEFINITION 2.3. Let Sj and 9)1 be two manifolds modelled on the spaces
lD and 3,
respectively, and let 9 be an one-to-one mapping from Sj onto 9)1 such that both 9 and g-l are smooth mappings at any point. Then 9 is called a diffeomorphism between
the manifolds Sj and
9)1,
and the manifolds Sj and
9)1
are said to be diffeomorphic.
Let us note that the model spaces of some diffeomorphic manifolds coincide, up to a linear isomorphism. In particular, two finite-dimensional diffeomorphic manifolds have the same dimension.
§3. Submanifolds Let
9)1
be a manifold modelled on the space X and let 5j be a subset of 9)1.
DEFINITION 3.1. The set 5j is called a submanifold of 9)1 if for any a E 5j there
Submanifolds
215
exists a chart (f)a, a), f)a ~ 9J1, such that a(a) = 0 and a(f)a n Sj) = lU a n Ita, where lU a = a(f)a) and Ita is a linear subspace of X. A sub manifold Sj is called a direct submanifold if each of the spaces Ita has a direct complement in X, i.e., X = a(f)a n Sj) are open subsets in a fixed subspace !D ~ X , for all a E Sj, then Sj is a manifold modelled on !D. In this case, an atlas for Sj is provided by the family of charts {(ita, CPa)}aEfJl where ita = f)a n Sj and CPa is the restriction of the homeomorphism a to the set ita. The manifold 9J1 appearing in Definition 3.1 above is often called the ambient manifold of the submanifold Sj. The notion of submanifold plays an important role in a series of problems. The case when the ambient manifold is a Banach space is especially interesting. However, not any manifold Sj which is a subset of a manifold 9J1 must be, at the same time, a submanifold of 9J1 (see, for instance, J. Milnor [1]). Now, we consider this problem in the case when the ambient manifold 9J1 is a Banach space X. THEOREM 3.1. Let X be a Banach space and let Sj ~ X be a manifold modelled on
!D.
Assume that
!D
is a subspace of X with a direct complement in X, i.e.,
X where
3
= !D+3,
(3.1)
is a linear subspace of X. Then the following conditions are equivalent:
a) for any point a E Sj there exists a chart (ita, CPa) such that the homeomorphism'ljJa = cp;;-l : lU a ----; ita, lU a = CPa(ita ), is continuously Fhkhet differentiable as a mapping from !D into X and the linear operator P'ljJ~ (b) : !D ----; !D is continuously invertible, where P is the projection from X onto !D, and 'ljJ~ (b) is the Fhkhet derivative of'ljJa at b = CPa(a) E!D; b) for any point a E Sj there exist a neighborhood it~ ~ Sj of the point a, a neighborhood IU~ of the point c = Pa E !D, and a Fnkhet differentiable mapping f : IU~ ----; 3, such that
{XEX: x=u+f(u), uEIU~}=it~.
is sufficient to prove the implication a) =} b) only, since the inverse implication b) =} a) is obvious. Indeed, the chart (ita, CPa) where ita = ll'a and CPa is the inverse of the mapping 'ljJa(u) = U + f(u) (i.e., CPa is the projection mapping
BANACH MANIFOLDS
216
from il'a into~) satisfies all the requirements in condition a). More precisely we have PljJ~(b) = I I ~, where I is the identity mapping on X and b = c = Pa. Assume that condition a) is fulfilled. By the Inverse Mapping Theorem it follows that P'l/Ja : QJ~ ----+ ~ is a diffeomorphism on some neighborhood QJ~ s;;: ~ of the point b E ~, i.e., for all x sufficiently close to the point a and y sufficiently close to the point b, the equation
has the unique solution
y = g(Px) = g(u),
u = Px,
and the mapping 9 is differentiable on a neighborhood of the point c = Pa. Further, let ll~ = 'l/Ja(QJ~). Then the points of the set ll~ are exactly the points x E X for which the equality (3.2) Qx = Q'l/Ja(g(Px)) is fulfilled, where Q = I - P. Setting x = u + v, where u f(u) = Q'l/Ja(g(u)) the desired conclusion follows. ~
Px, v
Qx and
Assume that the hypotheses of Theorem 3.1 and one of the conditions a) or b) are fulfilled. Then S) is a direct submanifold of X. THEOREM 3.2.
----+
X defined on some neighborhood 1)a C X
h = tfJx = P(x - a) where
+ Qx -
f(Px),
(3.3)
f is given by the right hand side of (3.2). Setting u = Px, v = Qx, we have tfJ~(a) = P -
f~(Pa) and tfJ~(a) = Q. Since both the partial derivatives are continuous on some neighborhood of the point a, tfJ is differentiable on that neighborhood. Moreover, tfJ is a diffeomorphism, since its inverse mapping IJt = tfJ- 1 : X ----+ 1)a, defined by the equality x = h + f(P(h + a)) + Pa, is also differentiable on some neighborhood of the point h = O. Further, by (3.2) and (3.3) we obtain tfJ(a) = 0 and tfJ(x) = P(x - a) E ~ for all xES) n 1)a, i.e., by (3.1), S) is a direct submanifold of X. ~
The mapping lJta : ~ ----+ S) satisfying condition a) in Theorem 3.1 is called a local parametrization of the manifold S). The mapping lJta(u) = u+ f(u) appearing in condition b) of Theorem 3.1 gives also a parametrization of the manifold Sj, which, due to its special form, is called an explicit or a direct parametrization.
Submanifolds
217
In the case of a finite-dimensional space X any submanifold admits obviously a local parametrization given by a suitable permutation of the coordinates. Therefore, from Theorems 3.1 and 3.2 the next result follows.
Let X = OC N and let Sj t;;; X be a smooth manifold of dimension n ~ N. Then Sj is a submanifold of X if and only if one of the following conditions is fulfilled: a) for any point a E Sj there exists a chart (ita, <Pa) such that the homeomorphism 'l/Ja =
Xl =XI,X2 =X2,···,Xn =Xn ,
In many mathematical theories the notion of submanifold is more convenient and more useful than the abstract notion of manifold. Moreover, in the mathematical literature, the term "manifold" is often used to call a set which, according to our definition above, is a submanifold (see, for instance, H. Cartan [1] and J. Milnor [2]). Nevertheless, in applications, both notions are equally justified. To be more specific, let us notice that a mechanical system with a finite number of degrees of freedom can be considered as a finite-dimensional manifold, and for this description it is not necessary to use any ambient space. For example, a system with four degrees of freedom, which is given as a set of points in the three-dimensional space, can not be considered as a sub manifold of that space. However, it is possible to "embed" such a system in a certain abstract space, or a manifold, of a greater dimension using some one-to-one mappings, and this enables us, sometimes, to obtain a series of new properties of the considered system.
9J1 be two manifolds. An embedding of Sj into 9J1 is any mapping f : Sj --> 9J1 such that the image f(Sj) is a submanifold of 9J1, and f induces a diffeomorphism between Sj and f(Sj). DEFINITION 3.2. Let Sj and
For our purposes it is not necessary to develop here the theory of embeddings. We mention only, without proof, that any smooth finite-dimensional manifold can be smoothly embedded in some finite-dimensional Euclidean space.
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218
§4. Complex manifolds and Stein manifolds 1. As we have already mentioned, a manifold 5) is called complex if its model space is a complex Banach space. Of course, any smooth complex manifold 5) modelled on ~ can be considered as a real smooth manifold modelled on ~lft - the realification of the space ~ (in the finite-dimensional case, a complex manifold of dimmension n is a real manifold of dimension 2n). Further, it is clear that if the mappings
Note that in the finite-dimensional case it is enough to require the analyticity of one of the indicated mappings only, since its inverse is also analytic, by a theorem of Osgood.
In what follows we will be concerned mainly with complex manifolds, i.e., manifolds of class 0, modelled on complex Banach spaces. The notion of a Fn3chet differentiable mapping f from a complex analytic manifold 5) into a complex analytic manifold [)1 is introduced with respect to the previous Definitions 2.1 and 2.2. If such a mapping f is Frechet differentiable, in the complex sense, on neighborhoods of any point in 5), then the mapping f is called analytic (or holomorphic). The term "holomorphic mapping" is mostly used in the theory of complex manifolds. In what follows we will use it in order to keep the tradition.
In particular, if [)1 = C is the complex field, then f is called a complex-valued analytic function on 5). The set of all complex-valued analytic functions on 5) will be denoted by A(5)). Let F(5)) be an arbitrary set of functions defined on 5) with values in C. DEFINITION 4.1. For an arbitrary compact set IB
Q;F(j») =
{x
E 5) :
If(x)l::(
sup
c
If(y)1
5) the set
for all
f
E F(5))}
yEI!3
is called the F(5))-convex hull of the set lB. In particular, the A(5))-convex hull of a compact set IB holomorphic convex hull of the set.
~ 5)
is called the
219
Complex manifolds and Stein manifolds
( 4.1)
A manifold Sj is called F(Sj )-convex if i5 F(Sj) is a compact subset of Sj for any compact subset ® S;;; Sj. In particular, the A(Sj)-convex manifolds are called holomorphically convex. DEFINITION 4.2.
In order to give some examples of holomorphically convex manifolds let us consider first the case of a complex Banach space X. For any subset ® S;;; X its geometric closed convex hull co ® can be also defined as co® =
{x
EX:
Il(x)l::;:
sup li(y)1 for alli E yErB
x*}.
If ® is compact, then by a theorem of Mazur, co ® is compact, too. Since X* = = L(X,q S;;; A(X), by (4.1) we obtain that i5 A (Sj) C co®, hence i5 A (Sj) is compact for all compact subsets of X, and, consequently, X is a holomorphically convex manifold. Let now Sj S;;; X be a complex-analytic manifold modelled on ~, where ~ S;;; S;;; X is a subspace of X. Assume that the hypoteses in Theorem 3.1 and one of the conditions a) or b) are fulfilled. According to Theorem 3.2, Sj is a submanifold of X. Since the restriction to Sj of any analytic function on X is an analytic function on Sj, then using again (4.1) we have (4.2)
for any compact subset ® S;;; Sj, where i5 A(X) denotes the A(X)-convex hull of ® considered as a compact subset of X. Clearly, by (4.2) and since i5 A(X) is compact, it follows that if Sj is a closed subset of X, then Sj is a holomorphically convex manifold. Actually this last assumption can be replaced by a weaker one, taking into account the fact that the geometric convex hull of a compact subset of the Banach space X is closed. The easy proof of the next result is left to the reader. Let X and Sj be as above, with Sj not necessarily closed in X. Let :D S;;; X be a domain, convex in the geometric sense, such that Sj S;;; :D. If Sj is closed in :D with respect to the induced topology on :D, then Sj is a holomorphically convex manifold. LEMMA 4.1.
We will now define a new class of manifolds.
BANACH MANIFOLDS
220
DEFINITION 4.3. A complex-analytic manifold Sj of a finite dimension n, which is
countable at infinity, is called a Stein manifold if a) Sj is holomorphically convex; b) Sj is holomorphically separate, i.e., for any a, bE Sj with a I- b there exists IE A(Sj) such that I(a) I- I(b); c) for any a E Sj there exists a collection of n functions II,··· , In E A (Sj ) which provide a local coordinate system at a. Let us recall that a Hausdorff topological space Sj is said to be countable at infinity if there exists a countable family <5 1 , <5 2 , ... of compact subsets of Sj such that each compact subset of Sj is contained in some <5 j, j = 1, 2, .... THEOREM 4.1. Any complex-analytic manifold Sj which is a closed submanifold of
a Stein manifold M is a Stein manifold.
Conditions a) and b) are obvious, since the restriction to Sj of any function which is analytic in the whole manifold M is an analytic function in Sj. Let us prove condition c). Take an arbitrary point a E Sj. Since a E M, there exists a system of functions II, ... , 1m E A(M), m = dim M, which provide a local coordinate system for M at a. On the other hand, since Sj is a submanifold of M of dimension n ~ m, there exists a local chart CD, 'P) for M with 'P = ('PI, ... , 'Pm) and a E ~, such that
~
n Sj = {x
E
M
'Pn+dx) = ... = 'Pm(x) = O}.
Since the Jacobian det(8 I i
1-0,
)
8'Pj
i,j=l,
,m
then for a suitable choice of 1 ~ iI, ... , in ~ m we have
det (
:~: )
/L,v=l,. ,n
I- O.
The last relation shows that the restrictions of Ii! , ... ,lin to Sj provide a local system of coordinates for Sj at a. This completes the proof. • Since the space em, as well as any convex domain in em are Stein manifolds we have: COROLLARY 4.2. Every complex-analytic manifold Sj which is a closed submanifold
of em is a Stein manifold of dimension n ~ m.
Complex manifolds and Stein manifolds
221
Assume now that the ambient manifold for 5j is a convex domain 1) in an infinite dimensional Banach space X. In this case, we can not use Theorem 4.1 since 1) is not a Stein manifold (in particular 1) is not finite-dimensional). However the next generalization of Corollary 4.2 is true. Let 1) be a convex domain in a Banach space X (in particular, X), and let 5j be a finite-dimensional countable at infinity complex manifold, which is a submanifold of1) and is closed with respect to the induced topology on 1). Then 5j is a Stein manifold.
THEOREM 4.2. 1) =
The holomorphic convexity of the manifold 5j follows directly from Lemma 4.1. For the holomorphic separation of 5j it is enough to recall that X* = L(X,
where (el, ... ,en) is a fixed basis in \Ca, and consider the restrictions of these functionals to 5j. Since rank(II, ... , in) = dim5j, then Udi=l (~ A(5j)) provides a local system of coordinates for 5j at the point a. The theorem is proved. ~
Chapter VIII Non-regular solutions of nonlinear equations §1. Ramification of solutions.
Statement of the problem Let 1) and [l be domains in the Banach spaces X and A, respectively, and let F be an operator acting from the topological product Sj = 1) x [l into X. Assume that, for a given Ao E [l, the equation x = F(x, A) (1.1 ) has the solution Xo E 1), i.e., Xo = F(xo, Ao). The point (xo, Ao) E Sj is called a regular point of equation (1.1) if, for any A in some neighborhood it <;;;; [l of the point Ao, there exists a unique solution x(A) E 1) of equation (1.1) such that x(Ao) = Xo. Otherwise the point (xo, Ao) is said to be non-regular. There are two kinds of non-regular points. A non-regular point (xo, Ao) of the first kind is characterized by the following property: there exists a neighborhood it of the point Ao such that equation (1.1) has no solutions for any A E it with A i- Ao, and for A = Ao it has the unique solution Xo. Such points are called non-prolongation points. A non-regular point (xo, Ao) of the second kind has the property that in any neighborhood of the point Ao there exists A for which equation (1.1) has more than one solution. Such points are called ramification points. If on some neighborhood it of the point Ao the identity
Xo == F(xo, A) is true, and the point (xo, Ao) is a ramification point, then Ao is called a bifurcation point.
NON-REGULAR SOLUTIONS
224
Bifurcation points play an important role in different applications. For instance, in perturbation theory for elastic systems, bifurcation points define the so-called critical loadings; in wave theory the bifurcation points define characteristic flows in which the waves occur, and so on.
Let us consider some examples. EXAMPLE 1.1. Let
x=
A = C. Consider the equation
whose solutions are
X(A) =
~±
v'f=4):.
2 2 The point (0,0) is a regular point for this equation, since for all A sufficiently close to Aa = 0 the equation has the unique solution x(A) = (1 - VI - 4A)/2 close to Xa = 0 and satisfying the condition x(O) = o. At the same time (~, is a ramification point since for all A close to Aa = there exist two solutions
i)
i
and satisfying the condition EXAMPLE 1.2. Let
Xi
(i)
= ~, i = 1,2.
x = e[O, 1] and A = 1Ft Consider on x the equation 1
x(t) = j (AX(S) a
+ x2(s))ds.
We can easily see that this equation has exactly two solutions: xdt) == 0 and X2(t) == == I-A. If we consider as (xa, Ao) the point (0,0), then this point is not a ramification point since in the domain :D of all x E x such that Ilxll = max Ix(t)1 < ~, the [a, 1)
considered equation has, for any A < ~, the unique solution x( t) == o. But if we consider the point (0,1), then this is a ramification point: the solutions X1(t) and X2(t) "branch out" from it. Moreover, the point AO = 1 is a bifurcation point; setting 1
F(x, A) = j(AX(S) +x2(s)ds we obtain F(O,A) == 0 for all A E 1Ft a An important particular case of equation (1.1) is offered by an equation of the form x = AP(x), (1.2)
Equations of ramification
225
where the operator
When equation (1.2) has a non-trivial solution x* i- 0, x* E 1), for some A E OC, then, by analogy with the theory of linear operators, the number AO 1 is called an eigenvalue of the operator
(xo, Ao) with Xo
=
F(xo, AO)'
Then, by Theorem V.5.1 about implicit operators, the condition for A = 1 to be a regular point of the operator A = F~(xo, Ao) gives a possible answer to Problems 1 and 2. This condition is but a sufficient one. In §6 of Chapter VI we gave some examples which showed that the above mentioned condition is not a necessary one; in those examples the point (xo, Ao) was a regular point for equation (1.1), but the number A = 1 was a point of the spectrum of the operator A. By the same Theorem V.5.1, this last condition is necessary for solving Problems 3 and 4.
§2. Equations of ramification 1. THE GENERAL SETTING. Let (xo, Ao) be a given point satisfying equation (1.1), i.e., Xo = F(xo, Ao). Without loosing the generality, we may assume, and we will,
NON-REGULAR SOLUTIONS
226
= (0,0). (If this is not the case, then we can make the following changes: y = X-Xo, f-l = A- Ao, p(y, f-l) = F(y+xo, f-l+ Ao) -Xo, and can consider the equation
that (xo, Ao)
y
1)
= p(y, f-l)
which is equivalent to equation (1.1) and, moreover, satisfies p(O, 0) = 0.) Further, let us assume that the following conditions are satisfied: i) the operator F is Frechet differentiable in the domain Sj = 1) x il, where
<; X, il <; A; ii) 1 is an eigenvalue of the operator A = iii) the operator B
=
F~(O,
0); I - A is normally solvable and splitable (see §7 of
Chapter IV). Then the operator B admits a regular matrix representation
obtained using a quadruple of projections Pi, P2 = I - H, Ql, Q2 = I - Ql. We denote (as in Chapter IV) lEi = PiX, 1E2 = P2X, IE~ = Q1X, lEg = Q2X. Then equation (1.1) can be written as a system of equations
{
BllU + B 12 V = Q1P(U + V,A) B21 U + B 22V = Q2P( U + v, A),
where U = PiX, V = P2x, P(-, A) of (2.1) as
= F(-, A) -
(2.1)
A. Let us rewrite the second equation
and consider the point (u, v) E lEi xA as a parameter. Since the operator H~(O, 0, 0) (= Q2P~(0, 0, 0) = B 22 ) is continuously invertible, by Theorem V.5.1 about implicit operators the last equation has a unique solution
= B22 -
v
= R(u, A)
(2.2)
defined on some neighborhood of the point (0,0) E lEi X A and satisfying R(O,O) = O. By substituing the right hand side of (2.2) in the first equation of (2.1) we obtain the equation (2.3) feu, A) = 0, which defines the implicit operator u = U(A) on the subspace PiX. If there exists a (single- or a multiple-valued) solution U(A) of equation (2.3), then U(A) provides a solution X(A) of the initial equation (1.1) given by the formula
X(A) = U(A)
+ R(U(A), A).
(2.4)
227
Equations of ramification
Equation (2.3) is called the equation of ramification for equation (1.1). Using formula (IV.7.4) it is easy to see that the operator f has the form
and acts from the space lEI into IE~. Equation (2.3) like the initial equation (1.1) is degenerate, i.e., f~(O, 0) = 0 (this follows easily by a direct computation). Therefore the Implicit Function Theorem can not be applied in this case. However, sometimes this equation can be studied using methods from finite-dimensional analysis. Namely, if the subspaces lEI and IE~ are finite-dimensional, then equation (2.3) can be represented as a system with a finite number of equations and a finite number of unknowns. There are many well-known different methods for the search and the study of solutions for such equations: elimination methods, Newton diagrams, algebraic methods, and others (see, for instance, M. M. Vainberg and V. A. Trenogin [1]). The above described general setting allows us to find different equations of ramification, depending on the properties of the operator B and on the choice of the subspaces lEI, IE~, 1E2, lEg, and of the projections PI, P2 , QI, Q2 relatively to which the regular matrix representation of the operator B is obtained. LIAPUNOV EQUATION OF RAMIFICATION. Let us consider the case when lEI N(B) and lEg = R(B) (recall that B is a normally solvable operator). Then the operator B admits the matrix representation
2.
=
B _ -
(0 0) 0
B22
'
and system (2.1) becomes {
0 = (I - Q)p(u + v,),) B 22 V = Qp(u + v, ),),
where Q is a projection onto R(B), P is a projection onto N(B) and u v = (1 - P)x. Using the first of these equations, the second one can be written as
If v
= R( u,),)
Px,
is a solution of this equation, then the equation of ramification (2.3) is
f(u,),)
= (I - Q)p(u + R(u, ),),),) = o.
(2.5)
228
NON-REGULAR SOLUTIONS
Assume, in addition, that B is a Fredholm operator (see §6 of Chapter IV), i.e., dimN(B) = dimN(B*) = n. Let {'Pdf=l and Ndf=l be bases in N(B) and N(B*), respectively, and let bdf=l C X* and {zdf=l C X be two systems of elements biorthogonal to these bases, respectively. Moreover, we consider that bdf=l was chosen such that (v, "Ii) = 0, i = 1, ... , n, for any v E X~N(B). This can be done by Hahn-Banach Theorem. Then every element U E N(B) can be represented as n
U=
L (i'Pi,
(i E lK
i=l
(lK is, as usual, the field of scalars). At the same time the equality n
(1 - Q)x
=
L(x, ~i)Zi
i=l holds for the projection 1 - Q. Taking into account the linear independence of the elements {zi}f=l' equation (2.5) can be rewritten as a system of n equations with n unknowns (1,"" (n:
k
=
1,2, ... ,no Equations as in (2.6) above, corresponding to integral equations with analytic functions, have been considered for the first time by A. M. Liapunov in connection with the equilibrium figures of a fluid mass in a revolving motion. Therefore, equations (2.6) are usually called the Liapunov equations of ramification. The above described approach, which reduces the initial equation to the system (2.6), is called the Liapunov procedure. This procedure can be also effective in the study of the solutions for equations with fixed values of the parameter A (see §4 below). 3. SCHMIDT EQUATION OF RAMIFICATION. With the notations and under the assumptions of the previous subsection, we infer other equations of ramification using essentially Lemma IV.6.1 (Schmidt Lemma). According to this lemma, the Fredholm operator B can be represented as
B=C-K,
Equations of ramification
229
where C is a continuously invertible operator on X, and K is a finite rank operator of the form n
Kx
= L(X,!'i)Zi. i=1
Then equation (1.1) can be rewritten as n
Cx
= <.p(x,>..) + LWiZi,
(2.7)
i=l
where
Wi
=
(X,!'i),
i = 1, ... ,no
(2.8)
If in equation (2.7) we consider (WI, ... , Wn , >..) E OC n x A as a parameter, then, using the Implicit Function Theorem, we obtain a solution for this equation of the form
x = S(W, >..),
(2.9)
where W = (WI, ... , wn ). To determine w, we substitute (2.9) in (2.8). Thus we obtain the following system of equations
Wi=(S(W1, ... ,Wn ,>"),!'i),
i=1,2, ... ,n,
(2.10)
which is called the Schmidt system of equations of ramification. This procedure, in contrast to the ones described in the previous Subsections 1 and 2, has the advantage that the considered operators are defined on the whole space X, and it is not necessary to split the operator B or to use the projections PI, P2, Q1, Q2. At the same time, there exists a direct connection between the Liapunov and the Schmidt equations of ramification.
In order to establish this connection, we rewrite first equation (2.10) into n
a slightly different form. Set x
= U
+ v, where U
=
L (i'Pi i=l
E
N(B), and remark that
by the equalities ('Pi,!'j) = 8ij , i,j = 1, ... , nand (v,!'j) = 0, j = 1, ... , n, we have
Further, C'Pi
=
B'Pi + K'Pi
=
Zi and, therefore
NON-REGULAR SOLUTIONS
230
Consequently, equation (2.7) can be written as (2.11) Let v = R((l, ... , (n, A) be a solution of equation (2.11), considered as an implicit n
function of the parameter ((1, ... , (n, A). Substituting x =
L (iifi + R((l, ... , (n, A) i=l
in (2.8) we obtain for (j the equations (j
=
(~(iifi + R((l, ... , (n, A), 'Yj )
,
j
=
1, ... , n,
which are equivalent to the equations in (2.10). According to (ifi, 'Yj) = 8ij , i,j = 1, ... , n, we finally get
( R (~ (iifi, A)
,'Yj )
=
0,
j
=
1, ... ,n.
(2.12)
Let us show now that system (2.12) represents nothing else but the Liapunov system of equations of ramification. In the literature (see, for instance, V. A. Trenogin [1]) (2.12) is often called the Liapunov-Schmidt system of equations of ramification.
Indeed, since C* = B* + K*, then C*'ljJi the system (2.12) can be written as
=
"Ii, i
= 1, ... ,n, and, consequently,
It is easy to see that the space x~N(B) is the kernel of the operator K, i.e., Kv = 0 for all v E x~N(B). Therefore equation (2.11) is equivalent to the next equation
BV=P(U+V,A), where B is the restriction of the operator B to the subspace x~N(B). Hence it follows that R(u,A) = B22Ip(u+R(u, A), A) = R(U,A), where B22 and R(-,·) are the same as in Subsection 2. From (2.11) and (2.12) we get the system of equations
Equations of ramification for an analytic operator which coincides with the Liapunov system of equations of ramification (2.6).
231 ~
As we have noted above, the Liapunov system is degenerated, i.e., det
II ~~; (0, 0) II = o.
Therefore, if it has a solution ((A) = ((1 (A), ... , (n(A)), then the functions (i(A) can be either single-valued or multiple-valued. Their feature determines the feature of the vector-function U(A), and, consequently, the feature of the solution X(A) of the initial equation (2.1).
§3. Equations of ramification for an analytic operator. The problem of coefficients 1. Let us assume that the operator F in equation (1.1) is analytic with respect to
the arguments (x, A) in some neighborhood of (0,0) E X x A. As before we will use the notations A = F~(O, 0), B = I - A, and we will suppose that the operator B is splitable. We consider again the system of equations
Bv = (u + v, A)
(3.1)
feu, A) = (I - Q)(u + R(U(A), A)) = 0
(3.2)
equivalent to equation (1.1), where u = Px, v = (I - P)x, B is the restriction of the operator QB(I - P) to the subspace ~2 = (I - P)X, P is a projection onto N(B), Q is a projection onto R(B), and = F - A. By the Implicit Function Theorem, equation (3.1) has a solution v = R(u, A) on some neighborhood U x Q of (0,0), where U ~ N(B), U 3 0, Q ~ A, Q 3 O. This solution is analytic with respect to the variables (u, A) in the whole neighborhood U x Q. Consequently, the operator f (-, .) that defines the left hand side of the equation of ramification (3.2), is also analytic in U x Q. Moreover, from ~(O, 0) == 0 it follows that R~(O, 0) == 0, hence f~(O, 0) == o. This means that the operator f admits on the domain U ~ N(B) the representation 00
feu, A) = feu, 0) +
L
Li(U, A),
(3.3)
i=O
where Li(U, A) are homogeneous forms of degree i = 0,1,2, ... in u for a fixed A E Q, and analytic operators in A for a fixed u E U. In addition Li(u,O) == o.
NON-REGULAR SOLUTIONS
232 On the other hand
00
J(u,O) =
L
Hk(u),
(3.4)
k=2
where Hk are homogeneous forms of degree k = 2,3, .... The series in (3.3) and (3.4) have non-zero radii of absolute convergence. The operators Hk, k = 2,3, ... , and L i , i = 0, 1,2, ... , are called the coefficients of ramification for equation (3.2). If all the coefficients of ramification are zero, then Problem 3 in §1 about the number of "branches" has an obvious answer: this number is infinite. Indeed, equality (3.2) is identically fulfilled on il. Replacing u(>.) in equality (2.4) by an operator u : il -4 il, we obtain different small solutions for equation (1.1). The features of these solutions are determined by the feature of the chosen operator u( . ). In particular, for >. = 0, the point x = 0 is a non-isolated solution for equation x = F(x,O). Moreover, according to R~ (0,0) = 0 we obtain that the operator G = = I I N(B) + R(·, .) is a bijection from the set il <;;; N(B) onto the set J n 1), where 1) is a neighborhood of 0 in X and J is the set of all fixed points for the operator F(·,O) (see Problem 4 in §1). If B is a Fredholm operator, and not all the coefficients of ramification Hk, k = 2,3, ... , are equal to zero, then it can be shown that the equation of ramification has a finite number of small solutions, determined by the first operator Hk which is different from zero (see, for instance, M. M. Vainberg and V. A. Trenogin [1]). In practice, it turns out frequently that both the construction of the equation of ramification and the computation of its coeficients are either rather cumbersome, or lead to difficult to solve problems. That is why the equation of ramification is not easy to handle. In the above cited book there are examples in which the first two or three coefficients of ramification are computed by reccurent methods, but these computations require a lot of work. In §4 below we will show that if the operator F(·, 0) leaves invariant some neighborhood of the point x = 0, then all the coefficients Hk in the decomposition of J(u,O) are equal to zero.
§4. The description of the set of fixed points for an analytic operator In this section we are mainly concerned with Problem 4 of §1, in the case of a complex Banach space X. More precisely, we will study the set of fixed points for the analytic operator T = F(·, 0) acting from X into X, or, equivalently, the set of solutions of
The description of set of fixed points
233
equation (1.1) for a fixed value of the parameter. We show that under some natural conditions this set has a quite complete description. Let us assume that in the set where T is defined there exists a bounded domain :D ~ X which is invariant for T, i.e., T(:D)
~
:D,
(4.1)
and let J be the set of all fixed points for T in :D. If :D is a ball in a strictly convex space, then the set J has a very simple structure: either J is void, or it is an affine variety in X (see Rudin Theorem Theorem 3.3. in §3 of Chapter VI). We emphasize that the condition of strict convexity is essential in this respect. For example, the operator F acting on the space X = CC 2 and defined by F(x) = (Xl, O.5(XI + x~)), X = (XI,X2), leaves invariant the unit polydisk :D = {x
E
CC 2
:
IXil < 1, i = 1, 2},
but the set J is not linear. Moreover, the intersection of 1)1 and J, where 1)1 is the set of all eigenvectors of the operator F'(O) corresponding to the eigenvalue 1, has but one point, X = O. Thus, in the absence of the strict convexity, Rudin Theorem is not true. Nevertheless, as we will see below, the assertions analogous to some consequences of Rudin Theorem such as the regularity of an isolated point, the continuousness of the set of non-regular points, its connectivity, the approach to the boundary, and others, are also true for a larger class of spaces. Let T be an analytic operator on an arbitrary domain :D of a complex Banach space X satisfying condition (1.1), and let J be the set of all fixed points for T in :D.
Let 9]1 be a connected component of the set J containing a given point x* E J. If the operator cP = I - T is Fredholm on some neighborhood of the set 9]1, then: 1) 9]1 is a direct complex-analytic manifold of dimension n, where n = dimker(I - T'(x*)); 2) if the set :D is convex and J contains more than one point, then n > O. THEOREM 4.1.
Recall that a nonlinear operator cP is called Fredholm at a point x* if the linear operator cP' (x*) is Fredholm.
By assertion 1) in Theorem 4.1 it follows, in particular, that whenever x* is an isolated point in J, then n = 0 = dimker(I - T'(x*)), a conclusion which -
REMARK 4.1.
NON-REGULAR SOLUTIONS
234
by Banach Theorem - implies the continuous invertibility of the operator 1- T' (x*). This means that an isolated fixed point is regular. Moreover, from assertion 2) it follows that, in this case, x* is the only fixed point for T in :D. Thus, for an operator T which leaves invariant some convex domain :D, and has a Fredholm complement cp up to the identity operator, the next alternative is true: either a fixed point for T (if it exists) is unique and regular, or there is an infinity of fixed points and all of them are non-regular and non-isolated. THEOREM 4.2. Let:D be a convex domain in X. Assume that the operator T admits
a continuous extension on :D and the operator cP = I - T is proper. Then: 1) the closure J of the set J in :D is connected; 2) if cP is Fredholm on some neighborhood of J, then J is a Stein manifold,
and, moreover, if J contains more than one point, then
J n 8:D =I=- 0.
The proofs of both theorems are based on the following preliminary results. Let :D be an arbitrary bounded domain in X, J =I=- 0, x* E J, and 91 = = ker(I - A), where A = T'(x*). Assume that for some subspace IE <;;; 91 there exists a linear projection P onto IE, commuting with the operator A: PA
=
AP.
(4.2)
1.1 <;;; IE of the point 0 E IE, with values in ker P such that 1IK,(u)11 = o(llull). Then the identity
LEMMA 4.1. Let K, be an analytic operator defined on some neighborhood
PT(x*
is true, for all u E 1.1 such that x* <J
+ u + K,(u» == Px* + u + u + K,(u)
(4.3)
E :D.
Without loosing the generality we will consider that x*
=0E
X and that
1.1 is a ball centered at the point 0 E IE, such that u + K,( u) E :D for all u E 1.1. Let us fix an arbitrary u E 1.1 and consider the vector-functions
1(1::::;: I}, Tn = ToT n - 1 , n = 1,2, ... , and TO = I. Since where (E L1 = {( E C the operator T satisfies the condition T(:D) <;;; :D and the domain :D is bounded, then all the functions 'Pn are defined on L1 and there exists M < 00 such that II'Pn(()1I ::::;: M,
n
=
1,2, ....
235
The description of set of fixed points The functions 'Pn have the power series representations
where hn E X is the first non-zero coefficient - if such a coefficient exists representation of the nonlinear part of 'Pn, and mn :? 2. By the Cauchy inequalities (see §2 of Chapter III) we have
Ilhnll:::::::M,
in the
(4.4)
n=1,2, ...
From the conditions APx = Px and (4.2) we get
PAnx == Px,
n
1,2, ...
=
By these relations, a direct computation of 'Pn(() , based on an induction argument, gives
'Pd()
= PT((u + K((U))
= PA((u) + PAK((U) + PS((u + K((U)) where S
=
= (u
+ h1(m + ... , 1
= T - A, and 'P2(() = PT2((U
+ K((U))
= PT(A(((u
+ K((U)) + S((u + K((U)))
=
= PA 2((u)+PAK((U)+PAS((U+K((U))+PS((u+AK((U)+S((U+K((U))) = =
(u+2h 1 (m 1
+ ...
Thus we conclude that mn = ml and h n = n· hl' n = 1,2, .... By (4.4) this is possible if and only if hl = O. Consequently
and setting (
= 1 we
obtain (4.3). ~
LEMMA 4.2. Let 1> be a convex domain in X and let VJt be a connected component of the set~. Let us assume that for some neighborhood 11 ~ 1> of VJt n 1>, such that 11 n (~\ VJt) = 0 the operator clJ = 1- T is uniformly non-degenerate on the set r = 811 \ 81>, i.e., there exists p > 0 such that
IlclJ(x)II :? p,
(4.5)
NON-REGULAR SOLUTIONS
236 for all x E T. Then 9J1
=
J, i.e., J is connected.
Let y E :r be an element which runs through some compact set ~ S;;; 1). For any e: E (0,1] we consider the operator T (= T(e:,y)) defined by the equality
T(x)
=
(1 - e:)T(x)
+ e:y.
By the convexity of 1) and by condition (4.1) the operator strictly inside 1). Consequently, the equation
x = T(x)
(4.6)
T
maps the domain 1)
(4.7)
has a unique solution x (= x( e:, y)) which depends continuously on y E ~ and satisfies the relation (4.8) Ilx(e:,y) - Tx(e:,y)11 < e:M, where M is the radius of a ball which contains completely the domain 1). Choose now two arbitrary points a, b E J and set y( 0:) = (1 - o:)a + o:b, where 0: E [0, 1]. Then the vector-function x(e:, y(o:)) defined by equation (4.7) is continuous in 0: on the interval [0, 1] and satisfies the conditions
x(e:,y(O)) = a,
x(e:,y(l)) = b.
Now if we assume that J =I- 9J1, then choosing b E J \ 9J1 and a E 9J1 n 1) we obtain that there exists 0:0 E [0, 1] for which x(e:, y(O:o)) E T. Setting e: = pM- 1 , by (4.8) we get the inequality IIx(e:, Y(o:o))11 < p, which contradicts (4.5). Consequently our assumption is false, hence J = 9J1. ~
Proof of Theorem 4.1. We will consider that x* = 0 E:r. Let A = T'(O). From the relations Am = (Tm)'(O), by the boundedness of the domain 1) and from Cauchy inequalities it follows that (4.9) for a suitable constant M <
00.
From Corollary V.8.2 we obtain the decomposition
:r =
»1+91,
(4.10)
where »1 = ker(I - A) and 9l = Im(I - A). We use now the Liapunov procedure. Let P and Q be linear projections from :r onto the subspaces »1 and 9l, respectively. Then the equation
x = T(x),
The description of set of fixed points
237
for x of a sufficiently small norm, is equivalent to the system of equations
u=PT(u+v)
(4.11)
v=QT(u+v),
(4.12)
where u = Px and v = Qx. By the Implicit Function Theorem equation (4.12) has a unique solution v = t£(u) E 9l (= ker P), which is analytic in some neighborhood ti of the point 0 E 1)1. A direct computation shows that 11t£(u)11 = o(llull). By substituing this solution in (4.11), according to Lemma 4.1 we obtain an identity which is satisfied for all u E ti. In other words, the point
x=u+t£(u)
(4.13)
is a fixed point for the operator T, for all u E ti ~ 1)1. Now, if we choose a neighborhood D ~ X of the point 0 E X such that P D ~ ti ~ 1)1, then the homeomorphism 'ljJ: D ----+ X defined by the equality 1Jr(x) = X-t£(Px) gives a local chart of the domain ~ as a manifold, which satisfies the condition 1Jr(0) = 0 and, by (4.13), the condition 1Jr(9J1 n D) ~ 1)1, too. Thus, by (4.10) it follows that 9J1 is a direct sub manifold of X of dimension n = dim 1)1. The assertion 1) in Theorem 4.1 is proved. We prove now assertion 2). Assume that 0 E ~ is an isolated point in this set (n = 0). Then the operator 1> = 1- T is locally invertible, i.e., we can find the numbers ri,r2 > 0 such that for all y with Ilyll ~r2 the equation 1>(x) = y has a unique solution x = 1>-l(y) in the ball Ilxll ~ ri, which depends analyticaly on y and satisfies the condition x(O) = O. Then for all x with Ilxll = ri inequality (4.5) is fulfilled for a suitable p. By Lemma 4.2, we obtain that ~ = {O}, which contradicts the assumption in 2). ~
Proof of Theorem 4.2. 1) Let 9J1 be a connected component of the set ~. Choose a neighborhood ti c ~ which contains 9J1n~ and such that tin (~\9J1) = 0. Then 1>(x) =I- 0, whence, by the fact that 1> is proper, inequality (4.5) follows with a suitable p. Our assertion is a consequence of Lemma 4.2. 2) If we assume in addition that 1> is a Fredholm operator, then, according to Theorem 4.1, it follows that 9J1 is a finite-dimensional complex-analytic submanifold of ~. Again by the fact that 1> is proper we conclude that ~ is countable at infinity and thus, by Theorem VII.4.2, it is a Stein manifold. Finally, let us suppose J n a~ = 0. Then there exists a neighborhood ~i (~ ~) of the set ~ such that all the values of the operator 1> (= 1- T) on a~i are
NON-REGULAR SOLUTIONS
238
different from zero. Consequently, for
t
In Theorem 4.1 we used that
decomposition X
=
lJ1-i-91 (*). During the translation of this book the paper of
P. Mazet and J.-P. Vigue [1] appeared in which it is also proved that under condition
( *) the set
~ of the fixed points of the operator T can be represented locally as a
complex-analytic manifold. Some versions of this theorem have been proved earlier by E. Vesentini [2], [3] and J.-P. Vigue [2] in the n-dimensional case, by M. AbdAlla [1] for the product of Hilbert balls, and by D. Shoikhet [2] in the general case of Banach spaces (see also M. Herve [2] and D. Shoikhet [3]). In addition let us note that P. Mazet and J.-P. Vigue established also some other conditions which are equivalent to condition (*).
By Theorem IV.8.2 such a decomposition holds too in the case when the operator
Chapter IX Operators on spaces with indefinite metric §1. Spaces with indefinite metric Let us consider a complex Banach space 23 endowed with a norm 11·11, and decomposed in a topological direct sum
The bounded mutually complementar projections generated by this decomposition are denoted by P±, such that 23± = P±23, Pl = P± and P+ + P_ = I. We define a functional J" on 23 by the formula
J,,(x) =
Ilx+II" -llx-II",
1/
> 0, x
E
23, x± = P±x.
(1.1 )
We say that the functional J" introduces an indefinite metric on the space 23, which is also called a J,,-metric. The space 23 endowed with a J,,-metric is called, for brevity, a J,,-space. J,,-spaces appeared as a natural generalization of Hilbert spaces with indefinite metric, which we will briefly recall in what follows. Let us consider a complex Hilbert space 5) with the inner product (., .) (see §11 of Chapter 0). Let G be a bounded self-adjoint invertible operator on 5) (i.e., the operator G is defined on the whole space 5) and the inverse operator G- 1 exists). We define a sesquilinear form [., .] on 5), called the G-metric, by the formula [x, y] = (Gx, y), x, y E 5). If the operator G is indefinite, i.e., its spectrum on the real line lies on both sides of zero, then the G-metric is also indefinite: for a non-zero vector x E 5) the "scalar square" [x, x] can be positive, negative or equal to zero. Assume now that the operator G- 1
240
SPACES WITH INDEFINITE METRIC
is bounded. Then the space Jj endowed with the G-metric is called a Krein space. Denoting by E).. the spectral resolution of the operator G we set
o
P- =
J
dE)..,
-00
Then P_ and P+ are mutually orthogonal projections: pl = P± = PJ., P+ +P_ = I. Setting Jj± = P±Jj we obtain a decomposition of the space Jj in the orthogonal direct sum Jj = Jj+ EB Jj_. Let us introduce on Jj a new inner product (., . ) by the formula
(x, y)
=
[p+x, yj - [p_x, y],
x, Y
E Jj.
This new inner product is equivalent to the initial one. Indeed, if x E Jj, then
(x,x) = (G(P+ - P_)x,x)::;;
IIGII(x,x),
and, conversely,
(x, x)
= (GP+x, x) - (GP_x, x) ;?
;?Al(P+X,X) +A2(P-X,X);? min{..\I,A2}(X,X), where Al = inf{..\ : A E u(G I Jj+)} > 0 and -A2 = sup{..\ : A E u(G I Jj_)} < Relatively to the inner product ( ., .), the form [., .j can be written as
[x, yj = (Jx, y),
J = P+ - P_.
o.
(1.2)
The operator J satisfies the properties J* = J- 1 = J and is called the canonical symmetry of the considered Krein space. The form [., .j given by (1.2) is called the J-metric of the space, and the space Jj with the inner product (., . ) and the J-metric [ ., . J is called a J -space. Setting Ilxll = (x,x)! for each x E Jj, we have (1.3)
hence a Krein space is a Hilbert h-space, in which P+, P_ are orthogonal projections. Let us return now to an arbitrary Jv-space 23. A vector x E 23 is said to be positive, negative or neutral if it satisfies the condition Jv(x) > 0, Jv(x) < 0 or Jv(x) = 0, respectively. A vector x such that Jv(x) ;? 0 (respectively, Jv(x)::;; 0) is
Spaces with indefinite metric
241
called non-negative (respectively, non-positive). The set of all non-negative (respectively, non-positive) vectors in lB will be denoted by it+ (respectively, it_). By it++ (it __ ) we will denote the set of all positive (respectively, negative) vectors in ~, and by ito we will denote the set of all neutral vectors. Before introducing a new notion, let us recall that by a lineal in lB we mean any subset L ~ ~ such that AX + f..Ly E L, for all X, y ELand A, f..L E C. The term subspace is constantly used to call a lineal that is closed in the norm topology. A lineal L is said to be non-negative if the inequality Jv(x) ~ 0 is true for all X E L. Analogously we define the non-positive, positive, negative or neutral lineals. The non-negative or non-positive lineals are also called semi-definite lineals, whereas the negative or positive lineals are said to be definite. Among the definite lineals we distinguish the uniformly definite ones, i.e., those definite lineals L with the property IJv(x) I ~c(L)llxIIV,
for all X E L, where c( L) > 0 is a suitable constant. By the continuity of the functional J v , the closure in ~ of any semi-definite (uniformly definite) lineal is still semi-definite (uniformly definite). For definite but not uniformly definite lineals a similar property is no longer true, as the next example shows. We need a simple preliminary result. LEMMA 1.1. Let X be a Banach space with the norm II . II, and let M be a lineal in X on which a bounded linear operator T is defined. If there exists a projection operator PM from X onto the lineal M, with the norm IIPM II = 1, then the operator T can be extended on the whole space X, without increasing the norm, by setting the extension equal to 0 on (I - PM)X.
Recall (see Chapter 0) that a linear operator T defined on a lineal '1)(T) is said to be an extension of the operator T defined on a lineal '1)(T) if'1)(T)
Tx
=
c '1)(T) and
Tx, for all x E'1)(T).
Let T be the operator defined by the equalities Tx = Tx, x EM, and Ty = 0, Y E (1 - PM)X. Then T is an extension of T on the whole space X and for all z = x + Y E M-i-(I - PM)X = X we have
IITzl1 = IITxl1 = IITPMzl1 ~ IITII ·llzll, i.e., IITII ~ IITII· Since the inequality IITII ~ IITII is obvious, we get IITII = IITli· ~ EXAMPLE 1.1. Let lB be a Jv-space with lB+ infinite dimensional, and let x+ E lB+,
x_ E lB_ be two fixed elements such that IIx+1I = Ilx-11 = 1. We define the operator
242
SPACES WITH INDEFINITE METRIC
K+ : lin{x+} - lin{x_} by the equality K+x+ = x_. Then IIK+II = 1. It is wellknown (see V. P. Odinec [1]) that for the one-dimensional subspace M = lin{x+} there exists a projection operator PM from ~ onto M, with the norm IIPM II = 1. We extend K+ to an operator K+ defined on the whole space ~+ by K+ I (I -PM)~+ = = O. According to Lemma 1.1 we have IIK+II = 1. By a direct verification we obtain that the lineal £+ = (P+ + K+)~+ is a non-negative subspace of~. Moreover, £+ is infinite dimensional. Let [ c £ + be a lineal which is dense in £ + and such that x+ + x_ ~ [ (for example we can choose [ = ker'P, where 'I' is a discontinuous linear functional on £ +, with 'P(x+ + x_) =1= 0). Then the lineal [ is positive, whereas its closure "£ = £ + contains a neutral vector x+ + x_ =1= O. ~
§2. Angle operators We introduce now the angle operator of a semi-definite lineal, one of the most important objects of study in this chapter. Let £ + be a non-negative lineal of ~. Since £ + C R+, it follows that liP_xii:::; IIP+xll, for all x E £+. Consequently, there exists a well-defined linear operator K+: P+£+ - p_£+, given by K+(P+x) = p_x, x E £+. Equivalently, for all x = x+ + x_ E £+ with x± = P±x, we have x_ = K+x+. It is clear that IIK+II :::; 1. The operator K+ is called the angle operator of the lineal £ +. We define analogously the angle operator K_ : P_ £ _ - P+ £ _ of a non-positive lineal £_. Further we will study mainly the non-negative lineals. The corresponding results for non-positive lineals can be obtained analogously. If the lineal £+ is positive (i.e., £+ \ {O} C R++), then IIK+x+11 < Ilx+ll, for all 0 =1= x+ E P+ £ +. If, in addition, £ + is a uniformly positive lineal, then IIK+II:::; 1-,),(£+), where 0 < ')'(£+):::; 1 is a suitable constant. Indeed, by the definition of the uniform positivity of the lineal £ + we have Ilx+ Ilv -llx_llv ~ c( £ + )llxII V , for all x = x+ + x_ E £+, where c(£+) > o. Hence, and by the obvious relation Ilx+11 = IIP+xll:::; IIP+llllxll, it follows that
Angle operators
243
A semi-definite lineal is said to be maximal semi-definite provided that it is not a proper subset of another semi-definite lineal. The maximal definite, maximal uniformly definite and maximal neutral lineals are introduced analogously. Considering on the set of all semi-definite lineals the order relation corresponding to the inclusion, and using Zorn Lemma (see §1 of Chapter 0), we conclude easily that any semi-definite lineal is contained in a maximal semi-definite lineal. The analogous conclusions for definite or neutral lineals are also true. Let us consider the class 9Jt+ of all non-negative lineals L + such that P+ L + =
= 123+. Analogously, we define the class 9Jt_ of non-positive lineals. It is obvious that alllineals L + E 9Jt+ are maximal non-negative subspaces. The converse is, generally speaking, false. Namely, as Lemma 1.1 shows, the question whether a maximal nonnegative subspace L + belongs to the class 9Jt+ is related to the question about the existence of a projection operator of norm 1 from the space 123+ onto the subspace P+ L +. It is well-known (see §11 of Chapter 0) that in the class of all Banach spaces, the Hilbert spaces are the only ones which have the property that any subspace is the range of a projection operator of norm 1. Therefore, when 123+ is a Hilbert space, the set of all maximal non-negative subspaces coincides with 9Jt+. In the general case of a Jv-space 123, this is no longer true. (For more information about projections of norm 1 on a Banach space see V. P. Odinec [1].) However, as Lemma 1.1 shows, if dim 123+ ~ 2, then we have dim L + ~ 2 for any maximal non-negative subspace L + of 123. In what follows we will denote by 9Jt~ (9Jt~) the set of all uniformly positive (respectively, negative) subspaces L+ E 9Jt+ (L_ E 9Jt_). Later on we will use the next result. PROPOSITION 2.1.
Every vector x E.R+ (.R++) is contained in a subspace L+ E 9Jt+
(9Jt~). Let K+ be an angle operator corresponding to the one-dimensional subspace lin{x+} c .R+, where x+ = P+x. If x E .R++ then IIK+II < 1. As we have noted above, anyone-dimensional subspace is the range of a projection of norm 1. Therefore the operator K+ can be extended, preserving the norm, to an operator K+ defined on the whole space 123+. The subspace L+ = (P+ + K+)123+ satisfies the desired condition. ~
Let us consider a subspace L + in the class 9Jt+. For any such subspace L + we have an angle operator K+ : 123+ -> 123_, with IIK+II ~ 1, defined on the whole space 123+, such that
(2.1)
SPACES WITH INDEFINITE METRIC
244
or C+ = (P+ + K+)IB+. In order to determine the operator K+ we observe first the next simple fact. LEMMA 2.1. If
C is a non-negative lineal then the projection P+ maps C homeo-
morphically onto P+ C.
IIP+xll ~ lIP-xii· Therefore Ilxll IIP+xll ~ IIP+II Ilxll· ~
<J For any x E C we have
~
211P+xll.
On the other hand,
IIP+xll+llP-xll ~
~
Now let C+ E !.m+ be fixed. By Lemma 2.1 the operator P+ I C+, i.e., the restriction of the projection P+ to C +, has a bounded inverse (P+ I C +) -1; for all x in C+ the equality x = (P+ I C+)-l x + is fulfilled. Then x_ = P_(P+ I C+)-l x +. Consequently, the operator K+ in (2.1) has the form (2.2) We denote by K+ the closed unit ball of the space C (IB+, IB_) of all bounded linear operators acting from IB+ into IB_, and by K~ its interior. By (2.1) any K+ E K+ determines a subspace C+ in the class !.m+. Summing up all the considerations above we obtain the next result.
There exists a bijective correspondence between the sets !.m+ and K+ given by the relations (2.1) and (2.2).
THEOREM 2.1.
§3. Plus-operators We consider a linear operator A defined on a lineal ::D(A) of a Jv-space lB. DEFINITION 3.1. The operator
A is called a plus-operator if A(.R+ n ::D(A)) c .R+.
A is said to be strict if Jv(Ax) n ::D(A) , where p,(A) > 0 is a constant.
DEFINITION 3.2. A plus-operator
all x E ~
DEFINITON 3.3. The operator
~
p,(A)Jv(x), for
A is called J;; -expansive if (3.1)
for all x E ::D(A) n .R+. If the inequality (3.1) is fulfilled for all x operator A is said to be Jv-expansive. DEFINITION 3.4. The operator
E
::D(A), then the
A is called uniformly J;; -expansive if (3.2)
Plus-operators
245
°
for all x E :D(A) nR+, where ,(A) > is a constant. If the inequality (3.2) is fulfilled for all x E :D(A), then A is said to be uniformly Jv-expansive. DEFINITION 3.5. The operator A is said to be a Jv-isometry if
for all x E :D(A). A Jv-isometry is called a Jv-semi-unitary operator if :D(A) = 113, and a Jv-unitary operator if :D(A) = 113 and, in addition, R(A) (= A:D(A)) = 113.
In the general case, a plus-operator on a Jv-space 113 is unbounded. We give next an example of an unbounded Jv-expansive plus-operator A with :D(A) = 113. EXAMPLE 3.1. Let 113 be a Jv-space with 113+ infinite dimensional. Relatively to the decomposition 113 = 113++113_ we define the linear operator A on 113 by the matrix
° 0)
A = (A11
I
'
where A11 is an unbounded expansive operator on 113+, i.e., IIA11X+11 )! Ilx+11 for all X+ E 113+. Then the operator A is unbounded and
Our next aim is to establish a criterion for the continuity of a plus-operator. We need first a preliminary result. Let A be a plus-operator such that :D(A) n R++ -=I=- 0. Then the boundedness of the operator A is equivalent to the boundedness of the operator P+A.
LEMMA 3.1.
The necesity of the condition is obvious. In order to prove the sufficiency we assume, on the contrary, that A is unbounded. Then there exists a sequence {xn : n EN}, Xn E :D(A), such that Xn ----> and IIAxnl1 ----> 00 for n ----> 00. Since P+A is bounded we have P+Axn ----> 0, hence IIP-Axnll ----> 00. For any sequence {zm : mEN}, Zm E 113, with the property Zm ----> for m ----> 00, we clearly obtain that IIP-Axn + P-zmll ----> 00, for n,m ----> 00. Consequently, Jv(Axn + zm) ----> -00 for n, m ----> 00. Set Zm = m- 1 Ay, where y E :D(A) n R++ is fixed. Then there exists pEN such that Jv(Axn + zm) < 0, for all n, m)! p. But <J
°
°
SPACES WITH INDEFINITE METRIC
246
By the continuity of Jv there exists lEN, l ~ p, such that Jv(xn + (l/p)y) > 0, for all n ~ l. Put n = l. Then Jv(Xl + (l/p)y) > 0, and Jv(AXl + A(l/p)y) = = Jv (A(Xl+(l/p)y)) < 0, which contradicts the assumption that A is a plus-operator. Thus the bounded ness of P+A implies the boundedness of A. ~ THEOREM 3.1. Let
C
~
A be a plus-operator with the property that A(1J(A)
n Jt++)
\ {O}. If 1J(A) contains a uniformly positive lineal £ such that P+A12
C =
= P+A1J(A), then the boundedness of the operator A is equivalent to the boundedness of the operator P+A I £ : £
--+
IB+.
The boundedness of P+A I £ follows obviously from the boundedness of A. Conversely, let us assume that P+A I £ : £ --+ IB+ is a bounded operator. By Lemma 3.1 it is sufficient to establish the boundedness of P+A. Let d = IIP+A I £ II. Since P+A12 = P+A1J(A) it follows that for any z E P+A1J(A) with Ilzll > d there exists y E £ with Ilyll > 1 such that P+Ay = z. By the uniform positivity of £ we find E > 0 such that Jv(u) > 0 for any u in the E-neighborhood of the set M = {x : x E 12, Ilxll ~ I}. Let w E1J(A) be an arbitrary element with Ilwll < E. Then Jv(y - w) > 0 for any y EM. Since A(1J(A) n Jt++) C Jt+ \ {O} we obtain o i= Jv(Ay-Aw) = Jv(A(y-w)) ~ 0, hence P+Ay i= P+Aw, for all elements y EM. Consequently, there exists a number 8 > 0 such that IIP+Awll ::;; 8, for all w E 1J(A) with Ilwll < E, a condition which clearly implies the boundedness of P+A. As we have already noted, by Lemma 3.1 the proof of Theorem 3.1 is complete. ~ <J
Actually, under our assumptions, we can avoid the use of Lemma 3.1. Indeed, take as above w E 1J(A) with Ilwll < E, and y EM. Since Jv(y + w) > 0 we have y + wE 1J(A) n Jt++, hence 0 i= Jv(Ay + Aw) = Jv(A(y + w)) ~ o. It follows that
On the other hand we have
IIP_Awll = II(P_Ay + P_Aw) - p_AYII ::;; IIP_Ay + P_Awll
+ IIP_Ayll ::;; IIP+Ay + P+Awll + IIP_Ayll ::;; IIP+Ayll + IIP+Awll + IIP_AYII·
Therefore, taking into account that
IIP+Awll ::;; 8, we obtain
IIAwl1 ::;; IIP+Awll + IIP_Awll ::;; ::;; 211P+Awll + IIP+Ayll + IIP_AYII ::;; 28 + IIP+Ayll + IIP_AYII, for all w
E1J(A)
with
Ilwll < E,
and the boundedness of A follows.
::;;
Plus-operators
247
Theorem 3.1 enables one to obtain a simple and natural condition for the bounded ness of a strict plus-operator defined on the whole space 113. This very class of operators will be, mainly, the object of our further investigations.
Let A be a strict plus-operator defined on 113. If there exists £ ~ E E 9)1~ such that A £ ~ E 9)1+, then A is bounded.
THEOREM 3.2.
We show the boundedness of the operator P+A I £~ : £~ -.., 113+. Since £~ E 9)1~ we have Jv(x) ~ c(£~)llxIIV, for all x E £~, where c(£~) > O. Therefore
for all x E £~, where fJ(A) > O. Hence it follows that P+A(£~) is closed in 113+, P+A I £ ~ is one-to-one, and (P+A I £ ~)-l maps continuously the Banach space P+A(£~) onto the Banach space £~. By Theorem 0.8.5 the operator P+A I £~ is continuous, too. Applying Theorem 3.1 we obtain the bounded ness of A. ~ We will state next an obvious consequence of Theorem 3.2.
Let A be a strict plus-operator defined on the whole space 113. If All (= P+AP+) maps 113+ onto 113+, then the operator A is bounded. COROLLARY 3.1.
Further we will consider bounded plus-operators defined on the whole space 113. We need the following result.
A strict plus-operator A maps a uniformly positive subspace onto a uniformly positive subspace.
LEMMA 3.2.
Let £ + be a uniformly positive subspace. Then
where x E £+ and c
= fJ(A)c(£+) > O.
If z E A£+, then z
= nlim Axn , where ..... oo
Xn E £+. Since IIAxn - Axmll ~c~llxn - xmll, the sequence {xn : n E N} is fundamental in £ + and therefore Xn -.., x for n -.., 00, where x E £ +. This means that z (= lim Ax n ) = Ax E A£+, hence A£+ is a subspace. Hence and from n ..... oo
Jv(Ax)
~
cllxll v the uniform positivity of the subspace A£+ follows.
~
We consider now the action of a strict plus-operator on the subspaces belonging to 9)1+. By the defect number of a subspace £ (c J\+) we mean the cardinal number def £ = dim(SB+I P+ £) (note that by Lemma 2.1 P+ £ is a subspace of £ +i 113+1 P+ £ is the quotient of 113+ by P+ £).
SPACES WITH INDEFINITE METRIC
248
THEOREM 3.3. If A is a strict plus-operator, then the defect number def A £ + is
the same for all £ +
E
9J1~. .
--
0
1
2
By Lemma 3.2 we obtam A£+ = £+, for all £+ E 9J1+. Let £+, £+ E E 9J1~ and let K+, K~ be their angle operators. By the convexity of K~ the operator Kt = (1 - t)K+ + tK~ is the angle operator of some £~ E 9J1~ for any t E [0, 1]. We associate to the family {Kt} a new family of operators {WA(Kt) : WA(Kt) = = P+A(P+ + Kt), t E [0, I]}. Since A takes non-zero values on the set of positive vectors, all the operators W A (Kt) are P + -operators (we are using here the terminology in I. C. Gohberg and M. G. Krein [1]). By Theorem 7.1 in that book, for any t E [0, 1] we can find T > 0 such that for all s E [0, 1] with Is - tl < T the operators WA(K+) have the same defect index. We use the Heine-Borel Lemma and cover the interval [0, 1] with a finite number of open intervals, each of them corresponding to a constant value of the defect index. Therefore def A £ ~ = def A £ ~
!.
The possibility of extending Lemma 3.2 and Theorem 3.3 to the whole class 9J1+ is related to the values of the constant 1/. Namely, - as the next example shows - for 1/ ~ 1 this is impossible. EXAMPLE 3.2. Let 1)3
=
1)3++1)3_ be a two-dimensional Jv-space, where
1/
~ 1 and
1)3± = lin{e± : Ile±11 = I}. With respect to the basis {e+,e_} we define a linear operator A by the matrix
Then
i- 0
i.e., the operator A is Jv-expansive. But A( e+ - e_) = 0 and the vector e+ - e_ is neutral. When 1/ > 1 it is possible to extend Theorem 3.3 to the whole class 9J1+. We need the next result. LEMMA 3.3. Let A be a strict plus-operator acting on a Jv-space 1)3, with
Then there exists 8(A) > 0 such that IIAxl1 ;? 8(A)llxll, for all x
E
1/
>
l.
Jt+.
Assume the conclusion is false. Then there exists a sequence {xn : xn
E
Jt+,
= I} such that Axn ---; 0 for n ---; 00. We have Ilx+11 = IIP+xnll;? ~llxnll = ~. 1 l l 1 Set yn = (1 + An)vX+ + A;{x~, where A;{ = IIAx n ll- 2 , for Axn i- 0, and An = n, Ilxnll
otherwise. Then Jv(yn) ;? 2- V and
Jv(Ayn)
=
Jv (A ((1
+ An) ~
- At) x+
+ At Axn)
---; 0,
Symmetric properties for n ----)
00,
249
which contradicts the fact that A is a strict plus-operator.
~
Now using Lemma 3.3 we can extend Theorem 3.3 on the whole 9Jt+. 3.4. Let A be a strict plus-operator acting on a Jv-space Q3, with v > 1. Then the defect number def A L + is the same for all L + E 9Jt+.
THEOREM
<J
The proof is almost the same as the one of Theorem 3.3.
~
COROLLARY 3.2. If a strict plus-operator A on a Jv-space Q3, with v > 1, maps at least one subspace from 9Jt+ onto a subspace in 9Jt+, then A L + E 9Jt+, for all L + E 9Jt+.
§4. Symmetric properties of a plus-operator and its adjoint In the dual space Q3* of the Jv-space Q3 we define the sets oft:±: by
where P± are the adjoint operators of the projections P±. Clearly P± are projections on Q3* and Q3* = F+. Q3* + P': Q3* = Q3~ +Q3~. We introduce also Jt(; = oft~ n oft~. As in the case of the space Q3, for every lineal M ± C oft:±: there exists the angle operator Q ± :
: P±M± ----) P~M± such that IIQ±II ~ 1 and M± = {xl + Q±xl : xl E P±M±}. By 9Jt:±: we denote the class of all subspaces M ± C oft:±: such that P± M ± = Q3:±:. We single out also the class 9Jt:±:o of all uniform subspaces in 9Jt:±:. As in §2 above (Proposition 2.1) we establish easily that any vector x* E oft~ (x* E oft~) is contained in some subspace M + E 9Jt~ (M _ E 9Jt~), and any positive (negative) vector x* is contained in a subspace from 9Jt~o (9Jt~o). We are interested to describe the structure of the orthogonal complements of the subspaces in the classes 9Jt+ and 9Jt_. All the considerations below will be explicitly developed for subspaces in the class 9Jt+. The corresponding results for subs paces in the class 9Jt_ can be establish analogously.
If L + E 9Jt+ (9Jt~), then L ~ E 9Jt~ (9Jt~o). When the space Q3+ is reflexive, then for any positive subspace L + E 9Jt+ its othogonal complement L ~ is negative.
THEOREM 4.1.
<J If L+ E 9Jt+, then L+ = (P+ + K+)Q3+, where K+ E IC+. Further, we easily get that L~ = (P': + Q_)Q3~, where Q_ = -K.+, Q_ : lB~ ----) Q3~, and IIQ-II ~ 1. Hence L~ E 9Jt~. If L+ E 9Jt~, i.e., IIK+II < 1, then IIQ-II (= IIK+II) < 1, therefore L ~ E 9Jt~o. Assume now that lB+ is reflexive and let .c + E 9Jt+ be a
SPACES WITH INDEFINITE METRIC
250
positive subspace, i.e., IIK+x+11 < Ilx+11 for all 0 =1= x+ E lB+. Suppose that there exists x~ E lB~, with Ilx~11 = 1, and such that IIQ_x~11 = 1. Then IIK~x~11 = 1. Since the space lB+ is reflexive, we find a vector x+ E lB+ with Ilx+ II = 1 and such that (x+, K~x~) = 1, that is (K+x+, x~) = 1. From this relation and the inequality I(K+x+,x~)1 ~ IIK+x+11 it follows that IIK+x+11 = 1, which contradicts the assumption that C + is positive. ~ REMARK 4.1. The condition in Theorem 4.1 for lB+ to be a reflexive space can not
be dropped, as the following example shows.
x = (... , x - k, O}, where
.),
with the standard basis {en: nEZ, n
=1=
... ,
en = (... ,O, ... ,~, ... ,O, ... ). n
= £ I -+ £ I such that lB ± = £ I, and define an angle operator K + by 1 the equalities K+e;; = ~ne;:;+l' where ~n = 1 - ~' nEZ, n =1= 0, and e; are the elements in the standard bases of the spaces lB±. We easily see that IIK+x+1I < Ilx+11 for all 0 =1= x+ E lB+, hence the subspace C+ = (P+ + K+)lB+ is positive. Further we have lB* = m-+m, where m = £ i, and lBi: = m. Let z E lB~ be the vector with the coordinates Zk = 1, k E Z, k =1= O. Then, using the equalities K~e;; = ~ne~_l' we obtain IIK~zll = Ilzll· Therefore the subspace M_ = = (P~ - K~)lB~ contains We consider lB
Ct
the neutral vector z -
K~z =1=
O.
~
The next result follows from the equality -L£-L subspace £ ~ lB*.
= £, which holds for any
COROLLARY 4.1. If in Theorem 4.1 ~e replace the space lB by the space lB*, and instead of the orthogonal complements we consider the *-orthogonal complements, then the conclusions remain true.
We return now to the study of plus-operators. Recall that we are dealing only with bounded plus-operators defined on the whole space lB. DEFINITION 4.1. A plus-operator A is said to be a double plus-operator if A * Jt~
c
c
Jt~.
Of course, not every plus-operator is a double plus-operator. It is sufficient to consider on the two-dimensional Jv-space lB = lB+-+lB_, where lB± = = lin{e± Ile±1I = I}, the plus-operator A given - with respect to the basis
251
Symmetric properties { e+, e_} -
by the matrix
the operator A * is not a plus-operator. In the considered example the closure of A.c + for some subs paces .c + E 9J1~ does not belong to 9J1+. Our next aim is to formulate conditions under which a plus-operator is a double plus-operator. We need a preliminary result. For
10:1 < 1;31
Let x* be a negative element oIIB*, i.e., 1!Pi-x*11 there exists a subspace .c + E 9J1~ such that (.c +, x*) = {O}.
LEMMA 4.1.
<
IIP~x*ll.
Then
IIPi-x*ll. Set
.c += N( x *) n:.o++ I·III { (U
Then p+.c + = IB+ and Let x
nlB+, 0: E
C.
•
(.c +, x*) = {O}.
(y+, x*) } y+- (y_,x*)y- .
We show that
.c + E
9J1~.
= x+ + x_ = Z + 0: (y+ - ~~~: ::~ y_ ), where Ilx+11 = 1, We have I(x+, x*)1 = 10011(y+, x*)1 ~ IIPi-x*ll. Thus Ilx-11 = 10:11 ~~~:::~ I
THEOREM 4.2. If a plus-operator A
Z
E N(x*)n
~ IIP~x*ll· (y_,X*)-l < 1.
on
IB
satisfies the condition (4.1)
then A is a double plus-operator.
Let us consider first a positive element x* E Jt~ and assume that A * x* rt. rt. Jt~. Then IIPi-A*x*11 < IIP':'A*x*lI, i.e., A*x* is a negative element in IB*. By Lemma 4.1 there exists .c + E 9J1~ such that (.c +, A*x*) = {O}. Then (A.c +, x*) = = {O}, a relation which, by (4.1), contradicts a conclusion of Theorem 4.1. Therefore our assumption is false, hence A*x* E Jt~ for all positive elements of IB*.
Take now an arbitrary vector 0 =I- y* +P':'y* we have IIP+y~1I
>
IIP':'y~ll,
EJt~. Then for all y~ = (1 + ~) P+y* +
i.e., each y~ is a positive element. From the first
SPACES WITH INDEFINITE METRIC
252
part of the proof it follows that A *y~ E Jt~. Taking the limit as n
A*y*
----* 00,
we obtain
E Jt~. ~
If a plus-operator A has the property that A L + E 9J1+ for all L + E 9J1+, then, generally speaking, the adjoint operator A * fails to have the analogous property. Let 1)3 = 1)3++1)3_, where the space 1)3+ is reflexive and has a Schauder basis {ei : i E N}. We define a plus-operator A on 1)3 in the following way: on 1)3+ we put Ael = 0, Aei = ei-l, i = 2,3, ... , and on 1)3_ we set A == o. Then the operator A is bounded, and A L + E 9J1+ for all L + E 9J1+. For A * we have A*ei = ei+l, i = 1,2, ... , i.e., A* M + tf. 9J1~ for M + E 9J1~ However, if we add the next condition: "A maps the non-zero vectors from 1)3+ into positive vectors", then the property "A L + E 9J1+ for all L + E 9J1+" is symmetric. More precisely, we have: EXAMPLE 4.1.
1)3+ be a reflexive space and assume that A(I)3+ \ {O}) C Jt++. 9J1+ for all L+ E 9J1~, then A* M+ E 9J1~ for all M+ E 9J1~.
THEOREM 4.3. Let
If AL+ E
Let us consider a Jp-space 1)3, with p > 1. We introduce on 1)3 a new norm, equivalent to the initial one, by
1
Correspondingly, we introduce on 1)3* the conjugate norm Ilx* Ilq = (1Ix~ Ilq + Ilx~ Ilq) q where q
= ~1' and define a p-
,
Jq-metric on 1)3* by the relation
DEFINITION 4.2. A strict plus-operator A on the Jp-space 1)3 is called double-strict if A* is a strict plus-operator on the Jq-space 1)3*. DEFINITION 4.3. A strict plus-operator A is said to be focusing, if
for all x
E
Jt+, where i(A) >
o.
(Compare with Definition 3.4.)
Symmetric properties
253
A focusing operator A is said to be double-focusing, if A * is also a focusing plus-operator in the Jq-space 1l3*.
In the case of a Hilbert J-space Sj any strict plus-operator A with the property A .c ~ E 9J1+ is double-strict and JL(A) LEMMA 4.2.
= JL(A*)
for some
.c ~
E
9J1+
(4.2)
(see M. G. Krein and Y. A. Shmulian [1]).
In a Jp-space Il3 every strict plus-operator A with property (4.2) satisfies
the condition
(4.3) Assume that the conclusion is false. Then we find a sequence {x~ n E N} such that x~ E 1l3_, Ilx~11 = 1 and IIAl/A12X~11 -+ 1 for n -+ 00. Set an = = IIA1/A12x~11 and Xn = a~lAl/A12X~ -x~. Then Xn E.Ito (C Jt+) and
for n
-+ 00.
On the other hand, since the operator A is strict, by Lemma 3.3 it follows that IIAxl1 ~ 8(A)lIxll for all x E Jt+. Hence
So we obtained a contradiction.
~
Every focusing strict plus-operator A with property (4.1) on a reflexive Jp-space Il3 is double-focusing and double-strict.
THEOREM 4.4.
sentation
With respect to the basis {1l3+, 1l3_} the operator A has the matrix repre-
(~~~ ~~~),
where All = P+AP+, A12 = P+AP_, A21 = P_AP+, A22 = P_AP_. By Theorem 3.4, from condition (4.2) it follows that All : 1l3+ -+ 1l3+ is a homeomorphism. Relatively to the basis {1l3 ~, 1l3:'} the operator A * has the matrix representation
SPACES WITH INDEFINITE METRIC
254
Let us show that for the operator A * the condition
(4.4) analogous to (4.3), is fulfilled. Indeed, since A is focusing it follows that ::;; ,B11P+Axll, for all x E .It+, where ,B < 1. Hence we have
lIP-Axil ::;;
for all x+ E 113+ and any angle operator K+ E K+. Setting K+ = 0 and considering the adjoint operators we obtain IIAh II ::;; ::;; ,BIIAi111, a relation equivalent to (4.4). By the reflexivity of 113, the angle operators of the subspaces in OO1~ are exactly the adjoints of the angle operators of the subspaces in 001_. Next we show that
(4.5) where K _ E K _. Let us suppose, on the contrary, the existence of a sequence {K~ : n EN}, such that K~ E K_ and Ilk~*11 ---f 1 as n ---f 00, where k -n*
=
(A*12
Then we find elements y~
E
+ A*22 Kn*) -
(A*11
+ A*2 K1 -n*)-l .
113_ such that IIY~ II
= 1 and 8n = IIK~y~ II
---f
1 as
n ---f 00. Setting xn = k~y~ - 8ny~ we obtain Ilx+. II = Ilx~ II, i.e., xn E Jlo (c .It+). Hence - using the fact that A is focusing - and from (4.3) it follows that IIP_Axnll::;; (1 - i'(A))IIP+Axnll and IIP+Axnll ~ yn = k~y~ - y~, we obtain xn - yn ---f 0, whence
E
> 0, for n
~ no.
Setting
(4.6) Let M~ = (p~ (Ayn,
+ K~*) 1J3~.
M~) =
Then A* M~ = (p~
(yn, A* M~)
=
((p_
+ K~*) 1J3~.
Further
+ kr:.) yr:., (p~ + kr:.*) 1J3~) = {O},
which, by (4.6), contradicts condition M~ E OO1~; so (4.5) is proved. It remains to show that the plus-operator A* is strict. For any x* E .It~ we have: Jq(A*x*)
= Ilp~A*x*llq
-llp':A*x*ll q ~ (1- ,B*q)
II(Ai1 + A;lK:') x~llq =
= (1- ,B*)q IIAi1 (I + (Ai1)-1 A;lK:') x~llq ~ ~ Ilx~llq ~ ~Jq(x*),
Symmetric properties
255
Theorem 4.4 assures that any focusing strict plus-operator on a reflexive space, which satisfies condition (4.2), is collinear to a double uniformly -expansive operator, i.e., to an operator which is uniformly -expansive simultaneously with its adjoint. To conclude this section we take into consideration some conditions under which a Jp-expansive operator A is double Jp-expansive, i.e., both A and A * are J p-expansive.
J:
J:
Let A be a strict plus-operator on the Jp-space 23. Then the operators P_ ± AP+ and P_ ± P+A are continuously invertible.
LEMMA 4.3.
We consider the operators P_ ± P+A. Assume that Ilxnll = 1 and (P_± n ±P+A)x ----; 0 as n ----; 00. Then x~ ----; 0 and P+Ax n ----; O. Since Ilxnll = 1 we have Ilx+11 ;;:: a > 0 for n;;:: no. Further, P+Ax n = Anx+ + A12x~ and, since A12X~ ----; 0, then Anx+ ----; 0, too. On the other hand, by Lemma 3.3, we have IIA l1 x+ll;;:: 811x+11 ;;:: 8a > 0, for n;;:: no, a contradiction. Hence it follows that II(P-± ±P+A)xll ;;::~±llxll, for all x E 23, where ~± > O. By Theorem 0.4.5 we conclude that the operators (P- ± p+A)-l exist and II(P- ± p+A)-lll ~ ~±l. Assume now that (P_ ± AP+)yn ----; 0 as n ----; 00, where Ilynll = 1. Then y~ + P_Ay+ ----; 0 and P+Ay+ ----; O. As above, from Lemma 3.3 it follows that y+ ----; o. This means that P_Ay+ ----; 0, whence y~ ----; 0, too. Thus we obtain again that II(P+ ± AP+)yll ;;:: '/]±llyll for all y E 23, where '/]± > O. Hence the operators (P- ±AP+)-l exist and II(P- ±AP+)-lll ~'/]±l. ~ <J
LEMMA 4.4. If A
is a double-strict plus-operator on a Jp-space 23, then P+AP+23+
= 23+.
(4.7)
We notice that for a strict plus-operator on a Jp-space 23 condition (4.7) is equivalent to condition (4.2). <J By Lemma 4.3 the operator P_ + AP+ is a homeomorphism and therefore the lineal (P_ + AP+)23 is closed. If (P_ + AP+)23 =I- 23, then we find an element x* =I- 0 in 23* such that ((P- + AP+)23,x*) = {O}, that is (P::" + PtA*) x* = O. But A* is a strict plus-operator on 23*, hence, by Lemma 4.3, the operator P::" + P-i'-A* is invertible. It follows that x* = 0 ~ a contradiction. Thus (P_ + AP+)23 = 23, a relation which clearly leads to (4.7). ~
LEMMA 4.5.
Let A be a Jp-expansive operator on lB. Then the operator (4.8)
SPACES WITH INDEFINITE METRIC
256
is well-defined and
IIUxll p
~
Ilxll p
for all x E :D(U).
= (P_-P+A)y, where y E lB, and Ux = (P+-P_A)y. Hence and by Jp(Ay)? Jp(Y)
it follows that IIUxll~
=
IIP+yll~ =
+ IIP_Ayll~ ~
II(P- -
IIP-yll~
+ IIP+Ayll~ =
P+A)yll~ = ilxll~·
LEMMA 4.6. If a strict plus-operator A on the Jp-space lB satisfies condition (4.7),
then the operator U given by (4.8) is defined on the whole lB and the following inversion formulas are true: U A
= (P+ - P_A)(P- - p+A)-l
= -(P+ + P_U)(P- + p+U)-l
=
=
-(P_
+ AP+)-i(p+ + AP_),
(P- - U p+)-i(p+ - U P_).
(4.9) (4.10)
+ AP+)-i(p+ + AP_) + (P+ - P_A)(P_ - p+A)-l = = (P- +AP+)-i((p+ + AP_)(P_ -P+A) + (P- +AP+)(P+ -P_A))(P__ p+A)-l = = (P_ + AP+)-l(AP_ - P+A - P_A + AP+)(P- - p+A)-l = o. (P-
We establish now the first equality in (4.10), i.e., A By (4.8) we have
Ux
=
= -(P+ + P_U)(P_ + P+U)-i.
(P+ - P_A)(P- - p+A)-lX,
i.e., x = (P_ - P+A)z, z E lB and Ux relations
= (P+ - P_A)z. Hence we get the following
U(P_ - P+A)z
=
(P+ - P_A)z,
-P+U(p- - P+A)z
=
-P+z,
Invariant semi-definite subspaces
257
and
where J = P+ - P_. As it was already proved above (P_ - P+A)IB = lB. By Lemma 4.3 the operator (P_ - p+A)-l exists and is bounded. Therefore, by (4.8), P+(P_ - p+A)-lP+IB+ = P+UP+IB+ = IB+. We establish, as above, that (P_-P+U)fJ3 = IB and that the operator P_ -P+U is continuously invertible. Therefore (P_ - P+A)z = -(P_ - P+U)Jz and -P+Az = -(P_ - p+U)-lJz - P_z. On the other hand -P_Az = U(p- - P+A)z - P+z = -U(P_ - p+U)-lJz - P+z. Then
But
+ I)(P- - p+U)-l J = 1- (U + I)(P- + p+U)-l = + P+U) - (U + I))(P- + p+U)-l = -(P+ + P_U)(P- + p+U)-l.
1+ (U
= ((P-
So the equality A = -(P+ + P_U)(P_ + p+U)-l is established. Finally, the second equality in (4.10) can be proved analogously with the second equality in (4.9). ~
A Jp-expansive operator A is double Jp-expansive if and only if it satisfies condition (4.7).
THEOREM 4.5.
The necessity of condition (4.7) was established in Lemma 4.4. We prove its sufficiency. By Lemma 4.6 the inversion formulas (4.9)-(4.10) are true. By Lemma 4.5 it follows that IIUxil p ~ Ilxllp, x E lB. This means that IIU*x*llq ~ Ilx*llq, x* E IB*. Taking the adjoint operators in (4.9)-(4.10) we obtain
= (p~ - A * p t )-1 (pt - A * p~) = = - (pt + P~A*) (p~ + p t A*)-l ,
(4.11)
= (pt - P~U*) (p~ - p+.U*)-l = = _ (p~ + U*p+.)-l (P+' + U*P~).
(4.12)
U*
A*
From (4.11) we get x* Therefore
=
(p~
+ P+.A*) y*,
and, consequently, Jq(A*y*) ;?: Jq(y*).
~
y*
E
IB*, U*x*
= - (P+' + P~A*) y*.
258
SPACES WITH INDEFINITE METRIC
§5. The problem of invariant semi-definite subspaces In 1943 S. L. Sobolev concluded his work [1] "About the motion of a symmetrical top whose cavity is filled with a liquid" . The motion of the top is described by an equation of the form dR di=iBR+Ro,
R(t=O)=R(O),
(5.1)
where B is a linear operator and R is a vector in the phase (complex infinite dimensional) space. The solution of this equation is given by
J t
R
= e iBt R(O)
+
eiB(t-t,j R o (t1) dt 1,
o since R(t = 0) = R(O). In this case the operator B is considered to be bounded, so that, for sufficiently large absolute values of A E C, there exists the resolvent r A = (AI - B)-1. Then e iBt =
_1_. 27f1
J
eiAt r A dA,
I
where 'Y is a circle of a sufficiently large radius centered at 0 E C. A Hermitean form Q(R 1 ,R2 ) can be constructed on the phase space {R}. Its behaviour is determined by the quantity
where: a) k is the rotational momentum of the gravitational force and depends on the angle which expresses the deviation of the z-axis relatively to the vertical direction; b) w is the angular speed of the rotation of the top around its axis (the z-axis coincide with the symmetry axis of the top and the origin of the coordinate axes is considered at the fixed point); c) A1 is the innertial momentum of the shell relatively to the x- and y-axis associated to the top, and A2 is the innertial momentum of the liquid relatively to the same axes; d) C 1 and C 2 are the innertial momenta of the shell and of the liquid relatively to the z-axis; e) it and 12 are the distances between the fixed point and the centers of mass of the shell and of the liquid, respectively;
Invariant semi-definite subspaces
259
f) g is the gravitational acceleration; g) M
1
and M
2
are the masses of the shell and of the liquid, respectively.
If L > 0, then the form Q is positive definite, and if L < 0, then Q is an indefinite form with one negative square. The operator B is self-adjoint relatively to the form Q. Therefore, for L > 0 its spectrum is real and e iBt is a bounded operator. If L < 0 then the operator B can have a non-real spectrum. In this last case there exists a pair of complex-conjugate eigenvalues, coresponding to a pair of neutral eigenvalues. The presence - or the absence - of these non-real eigenvalues has a determining influence in the stability of the solution for equation (5.1). Extending Sobolev's investigations, L. S. Pontryagin published in 1944 his work [1], devoted to the study of self-adjoint operators relatively to a Hermitean form with K, < 00 negative squares. Let A be such an operator. Using some subtle algebraic methods the existence of a pair (1:- +, 1:- _) of invariant subspaces for A can be established, where 1:- + is a maximal non-negative subspace, and 1:- _ is a maximal non-positive subspace, with the following properties concerning the spectra of A I 1:- + and A I 1:- _: the eigenvalues in O"(A I 1:- +) and O"(A I 1:- _) are two by two complex-conjugate, and the imaginary parts of these eigenvalues are non-negative for 1:- + and non-positive for 1:- _. As a recognition of Pontriagyn's contributions, the Hilbert spaces with an indefinite metric having a finite number K, of positive or negative squares are called Pontriagyn spaces. Usually they are denoted by IIt<. By the end of the 40's and the beginning of the 50's, I. S. Iohvidov [1] using the Caley-Neumann transform, established the relationship between 1f-self-adjoint and 1funitary operators and proved that for a 1f-unitary operator U there exists a pair of invariant maximal semi-definite subspaces (1:- +, 1:- _) such that the spectra of U I 1:- + and U I 1:- _ lie outside and inside the unit disk, respectively. In 1950 M. G. Krein's paper [1] appeared, where a theorem on the existence of an invariant subspace for a plus-operator on a space IIK, was proved by an ingenious use of Schauder's fixed point principle. Later on the Iohvidov-Krein Theorem was extended by M. L. Brodskii [1] to the case of a plus-operator acting on a Banach J",-space ~ with dim ~+ < 00 Afterwards, the invariant subspace problem for the general case of an operator acting on a space with indefinite metric, which has both the number of positive and negative squares infinite, was considered. The axioms of such spaces were elaborated by M. G. Krein [3], and, in his honour, they are called Krein spaces. M. G. Krein achieved also an important step in solving the invariant subspace problem. In 1964 his paper [2] was published, in which, using once again a fixed point principle, he es-
SPACES WITH INDEFINITE METRIC
260
tablished the existence of an invariant maximal non-negative subspace £ + for a plusoperator A with a completely continuous "corner" P+AP_, and studied the spectrum of the restriction A £ +. On the other hand, in 1960 R. S. Phillips [1] considered the problem of extending a pair of semi-definite subspaces which are invariant relatively to some commutative operator algebra ~, to a pair of maximal semi-definite subs paces , by preserving the invariance relatively to~. To be more specific, this problem occurred in connection with the Cauchy problem for a dissipative system 1
under the condition that the real part of the matrix (a n i n j ) is non-positive for any finite collection {nd of natural numbers and where 00
L
laijl2
<
00
for every i E N,
laij 12
<
00
for every j
j=l 00
L
E
N.
i=l
This type of problems arise in the theory of small perturbations of the dissipative systems with infinitely many degrees of freedom. The precise statement of Phillips' problem is the following. Let 5) be a Krein space with the J-metric [', .], and let ~ be a commutative algebra of operators acting on 5), closed in the weak operator topology, and such that, together with any A E ~ the algebra ~ contains the operator AC, the adjoint of A relatively to the J-metric. Let (12 +, £ _) be a pair of subspaces which are invariant relatively to all operators in ~, such that £ ± C Jt± and [x, y] = 0 for all x E 12+, y E 12_ (i.e., 12+ and 12_ are J-orthogonal). The problem is to find out whether there exists a pair (12 ~ax, £ ~ax) of invariant relatively to ~ subspaces such that 12±ax E 9Ji±, 12± ~ 12±ax, and [12~ax, 12~ax] = {o}. Phillips reformulated this problem for a commutative group of J-unitary operators. In his paper [1] he stated the conjecture that the indicated extension of the subspaces is always possible. He proved that the conjecture is true only in the case of a finite-dimensional space 5). Soon after that, M. A. Naimark [1] proved that Phillips conjecture is true for commutative groups of J-unitary operators on a Pontriagyn space IlK' Afterwards, some advances were reached in a series of works (as, for example, V. S. Shulman [1], E. A. Larionov [1], [2], H. Langer [1], [2], V. A. Khatskevich [1]). On one hand the extension problem for invariant subspaces was solved for larger (even non-commutative)
Invariant semi-definite subspaces
261
families of operators acting, as above, on a Pontriagyn space IlK (and, sometimes, on IId. On the other hand, some restrictions were found for commutative groups of J-unitary operators on an arbitrary Krein space for which Phillips conjecture has a positive answer. In 1970 the work of J. W. Helton [1] appeared. There the existence of a maximal semi-definite invariant subspace was proved for any commutative group it of J-unitary operators on a Krein space which satisfies the next condition: there exists an operator U in it represented - relatively to the decomposition fJ = fJ+ EB fJ- by a matrix of the form
U=
(U~l ~2) +C,
where C is a completely continuous operator, and the spectra of Ul1 and U22 are disjoint, i.e., a(Ul1 ) n a(U22 ) = 0. Every group of 7r-unitary operators on a Pontriagyn space ilK satisfies this condition since it contains the identity operator
o) =
I-
(1+ 0
o -I-
)
+2P_
=
(-h 0
and either P + or P _ is completely continuous.
Based on Helton's result, V. A. Khatskevich [2] proved that Phillips conjecture is true for such a group it. Improving Helton's ideas, T. Ya. Azizov considered the so-called 1i and J( (1i) classes of double J -expansive operators and solved the invariant subspaces extension problem. His results in this direction are summed up in T. Ya. Azizov and 1. S. Iohvidov's book [1]. Other papers, known to us, in which the mentioned problem was solved for some families of operators acting this time on a Jv-space are: 1. S. Iohvidov [1], V. A. Khatskevich [3], H. Langer [3]. In its general form Phillips' problem is not yet solved. Moreover, the problem of the existence of a maximal semi-definite invariant subspace for an arbitrary Junitary operator on a Krein space fJ with infinite dimensional subs paces fJ+ and fJis unsolved too. In the next section we will apply the results of Chapters V and VI about fixed points for holomorphic operators to the problem of invariant subs paces for a plus-operator on a Jv-space.
SPACES WITH INDEFINITE METRIC
262
§6. An application of fixed point principles for holomorphic operators to the invariant semi-definite subspace problem Let 23 = 23++23- be a Banach space with the Jp-norm
where x = x+ + x_ E 23, X± E 23± and p > 1. We consider a bounded strict plusoperator A defined on the whole space 23 and satisfying condition (4.7): P+AP+23+ = = 23+. By Corollary 3.2 the operator A maps any subspace I: + E 9)1+ onto the subspace A I: + which belongs to 9)1+, too. Let now K+ E K+. Then the subspace 1:+ = (P+ +K+)23+ belongs to 9)1+, therefore AI:+ E 9)1+. By Theorem 2.1 we know that AI:+ = (P+ +K+)23+, where ~ -1 K+ = P_(p+ I AI:+) E K+. We have:
= (All + A 12 K+)23+, P_A23+ = (A21 + A22K+)23+.
P+AI:+
Therefore, setting K+
= FA(K+)
we obtain: (6.1)
Thus to each strict plus-operator A with property (4.7) it corresponds the linear fractional transformation FA: K+ ....... K+ given by (6.1). From Lemma 4.2 it follows that IIAll A1211 < 1. Therefore
FA(K+)
= (A2l +A22 K+) (I + A IlA 12 K+)-1 All =
= (A21 +A22K+)
C~o(-l)n (AIlA12K+t) All,
(6.2)
i.e., the mapping FA is holomorphic on K +. From (6.1) it follows that any fixed point K+ E K+ for the transformation FA is the angle operator for a subspace 1:+ = (P+ + K+)23+ E 9)1+, invariant relatively to A, and, conversely, if A I: + c I: + for some I: + E 9)1+ then the angle operator of the subspace 1:+ is a fixed point for FA. Hence, based on Corollary VI.2.1, we obtain: THEOREM 6.1. Let A be a focusing plus-operator with property (4.7). Then A has
in 9)1+ a unique invariant subspace I: ~ E 9)1~ satisfying A.c ~ = .c ~, and there exists a number r < 1 such that IAI :::; rlJ.t1 for all J.t E a(A I .c~) and A E a(A) \ a(A I .c ~).
An application of pixed point principles
263
Consider the transformation FA given by (6.1). From Definition 4.3 of a focusing plus-operator, it follows that
IJFA(K+)II ~ r, where r
= 1- 8(:)' 8(A)
is the number appearing in Lemma 3.3, and "'(
~ min{8(A),
"'((A)}. Thus 0 ~ r < 1. By Corollary VI.2.1, the transformation FA has an s-fixed point K~ E K ~. Therefore L ~ = (P+ + K~) 23 + (E 9J1~) is the unique invariant subspace of the operator A in 9J1+. The result on the spectrum of the operator A was established by A. V. Sobolev and V. A. Khatskevich in [1]. Their proof uses other methods than those in the present book. ~ According to Corollary 3.2 we have:
Let 23 be a Jp-space, such that 2L is the dual space of a normed space X, and let A be a strict plus-operator on 23 satisfying (4.7), such that its "corner" A12 is completely continuous. Then A has an invariant subspace L + E 9J1+, such that A L + = L + and COROLLARY 6.1.
Let tP A(Z, K+) = zFA(K+), where Z E L1 (L1 is the unit disk in the complex and K+ E K+. Obviously, tPA(Z, K+) ~ K+ for any Z E L1. If Z E L1 then tP A(Z, .) satisfies all the conditions in Corollary VI.2.1. Therefore there exists a unique fixed point K+ in K+, i.e., tP A(Z, K+) = K+. By Alaoglu-Bourbaki Theorem and by Theorem 0.2.19, the property of 2L to be the dual space of a normed space implies that the set K+ is compact in the ultraweak operator topology. The complete continuity of the operator A12 implies - by Theorem 0.4.15 - the convergence in the strong operator topology of the sequence (I + Ai} A 12 K.+) -1 (see (6.2)) for any sequence {K.+ : n E N} c K+ which is convergent in the ultra-weak operator topology. Therefore tPA(Z, .) is continuous in the ultra-weak operator topology. Let {zn : n E N} be a sequence of complex numbers which converges to Z = 1 and such that the sequence {K~n : n E N} is convergent in the ultra-weak operator topology to K+ E K+. Then tP A (Zn' K~n) = K~n ---.. K+ for n ---.. 00. But tPA (zn,K~n) ---.. FA(K+), whence K+ = FA(K+). By Theorem 6.1, for any n E N there exists rn < 1 such that IAI ~ rnlfll for all fl E O'(An I L n ), A E O'(An) \ O'(An I L n ), where Ln = (P+ + K~n)23+ and An = P+A + znP_. Hence, taking the limit for n ---.. 00, we obtain (6.3). ~
plane
q
SPACES WITH INDEFINITE METRIC
264
DEFINITION 6.1. We say that a subspace £ ±
exists £
±ax E 9J1± such that £ ± C;;; £ ±ax .
c .Yt± belongs to the class T± if there
COROLLARY 6.2. If an operator A satisfies all the conditions in Corollary 6.1, then
any subspace £ E T +, such that A £ = £, is contained in some subspace /:, + E 9J1+
which is invariant for A. <J Let us denote by K+ the set of all those K+ E K+ for which £
c (P+ +
+K+)!B+. Under our assumptions we have K+ =I 0. We can easily prove that K+ is convex and closed in the weak operator topology. From £ = A £ it follows that FA(K+) C K+. Repeating - for the set K+ - the corresponding arguments in the proof of Theorem 6.1 and Corollary 6.1, we obtain the existence in K+ of a fixed point K+ for the transformation FA. Then /:,+ = (P+ + K+)!B+ is the desired invariant for A subspace which extends £. ~
c .Yt± is called a dual pair (d.p.). If £± E 9J1± then (£+, £_) is called a maximal dual pair (m.d.p.).
DEFINITION 6.2. A pair (£ +, £ _) where £ ±
A dual pair is said to be invariant for an operator A if A £ ± C;;; £ ±. THEOREM 6.2. Any focusing strict plus-operator A on a reflexive Jp-space!B which
satisfies property (4.7) has a unique invariant m.d.p. (£~, £~). Moreover, £~ E E 9J1~, A £ ~ = £~, £ ~ + £ ~ = !B, there exists a number r < 1 such that 1)..1 ~ rlMI for all)" E u(A I £~), ME u(A I £~), and u(A) = u(A I £~) U u(A I £~). <J By Theorem 4.4, the operator A * is a focusing strict plus-operator satisfying condition (4.7). By Theorem 6.1, the operators A and A* have in 9J1+ and 9J1~ the unique invariant subspaces £ ~ E 9J1~ and £ ~* E 9J1~*, respectively, such that A£~ = £~ and A* £~* = £~*. Setting £~ = l..£~* we obtain A£~ c £~. By Theorem 4.1, we know that £ ~ E 9J1~. We show now that £ ~ + £ ~ = !B. Let K ± be the angle operators of £~. We have P++K~ +P_+K~ = I+K, where K = K~ +K~ and IIKllp < 1. By Theorem 0.4.6, the operator 1+ K is a homeomorphism from !B onto !B, whence (I + K)!B = !B, i.e., £ ~ + £ ~ = !B. From the last equality it follows that u(A) = u(A I £~) Uu(A I £~). Hence, by Theorem 6.1, the conclusions in Theorem 6.2 concerning the spectrum u(A) are true. ~
Let A be a double strict plus-operator on the reflexive Jpspace !B, such that A12 and A21 are completely continuous. Then any invariant d.p. (£+, £_) for which A£± = £± and £± E T± can be extended to an invariant m.d.p. (£~ax, £~ax). COROLLARY 6.3.
<J By Lemma 4.4, the operator A satisfies condition (4.7). By Corollary 6.2,
An application of pixed point principles the subspace
.c + can
265
.c r;'ax E 9J1+. + Q~.)I"2L ~ .c_. Since
be extended to an invariant for A subspace
R:: of those Q+ E K~ for which (P+ .c _ E T_, the set R:: is non-void, and from A.c _ = .c _ it follows that FA R::) c c R::. Hence, as in the proof of Corollary 6.2, we obtain the existence of a fixed Consider the subset
* (
point Q+,i.e.,FA*(Q+) = Q+. Setting -K_ = Q+ and get that A.c Illax c .c Ill ax and .c _ c .c Ill ax . ~~
.c Illax =
(P_ + K_)12L we
To conclude this section we establish a characteristic spectral property of a plus-operator with a focusig power.
Let A be a strict plus-operator on a reflexive Jp-space IB, satisfying property (4.7). Then a power A rn of A is a focusing operator if and only if there exists a m.d.p. (.c +, .c _) like the one in Theorem 6.2.
THEOREM 6.3.
Necessity. Let Am be a focusing operator for some m?': 1. It is easy to
see that FAm =
PA.
By Theorem
VI.2.2,
the transformation FA has an s-fixed
point K~ E K~. Therefore the subspace .c~ = (P+ + K~)IB+ is invariant for A. By Theorem 4.4, the operator A *m is focusing, too. Arguing as above we find an invariant for A * subspace .c ~* E 9J1~o. Setting .c ~ = l...c ~* we obtain the invariant m.d.p. (.c~, .c~). By Theorem 6.2, the spectrum O"(A) satisfies all the mentioned conditions. Sufficiency. Assume the existence of an invariant m.d. p. (.c ~, .c ~) such that 9J1~ and IAI ~ rlILI for all A E O"(A I .c ~), IL E O"(A I .c ~), where 0 ~ r < 1. Then IB = .c ~ +.c ~ and therefore any vector x E IB can be represented as x = y + z, where y E .c~, z E .c~. Put
.c ~
E
We have
+ 211P-11 IIAnzl1 IIAnyl1 211P+11 IIAnzl1 IIAnyl1
IIP_Anyll IIP+Anyll 1_
By Theorem
IV.2.2, we find
ml
EN such that
SPACES WITH INDEFINITE METRIC
266 where r ~ rl
< 1. Therefore sup
IIAnzl1
zE£':..,
Ilzll=l inf
---"----'-:---,;---,--:::----:-;- ----+ yE £':.. ,
IIAnYl1
0,
for n
----+ 00.
IIYII=l
Hence there exists a number m;;:: ml such that lm(x) ~ 'Y is focusing.
~
< 1, i.e., the operator Am
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ZAIDENBERG,
List of Symbols
8wF(xo, h) 8F(xo, h) DF(xo, h) dF(xo, h) Vw(xo, ~)
the first weak variation of the (nonlinear) operator F at a point Xo the first variation of the (nonlinear) operator F at a point Xo the Gateaux differential of the (nonlinear) operator F at a point Xo the Frechet differential of the (nonlinear) operator F at a point Xo the class of all (nonlinear) operators acting from a normed space x into a normed space ~ and having first weak variation at a point Xo the class of all (nonlinear) operators acting from a normed space x into V(xo, ~) a normed space ~ and having first variation at a point Xo Pettis integral (weak integral) the class of all p-holomorphic (holomorphic with respect to p-topology) operators F : 'j) ---> ~, where 'j) is a p-open set in x and x, ~ are normed spaces the class of all holomorphic operators F : 'j) ---> ~, where 'j) is an open 1t('j),~) set in x and x, ~ are normed spaces the spectrum of a linear operator A a(A) the spectral radius of a linear operator A r(A) an indefinite metric on a Banach space Jv(x) the set of all non-negative (non-positive) vectors in a Banach space with st+ (st_) indefinite metric st++ (st __ ) the set of all positive (negative) vectors in a Banach space with in definite metric 9)1+ (9)1_) the class of all maximal non-negative (non-positive) subspaces of a Banach space with indefinite metric
Subject Index
A
Freche differential 36, 213
Aizenberg, L. 202
Fredholm operator 120, 208, 233, 234
analytic manifold 233 analytic operator 53 Azizov, T. 261 B
G Gateaux differential 35, 53 G-metric 239 H
Banach principle 134 Banach Steinhaus type theorem 100 bifurcation point 233 Bijection 4
holomorphic function 83 holomorphic convex hull 218 Hausdorff space 12, 14 Hahn-Banach-Suchomlynoff theorem 27
C
Herve, M. 181, 184
Cauchy inequalities 87
HOlder condition 190
completely continuous operator 21, 30, 55, 144 converse of Banach principle 136, 173
I
Implicit Function Theorem 160, 164, 189, 203
D
invertible operator 169, 197, 198
defect number 247
injection 4
E
K
ergodic theorem 131 F
Krasnoselskii, M. 57, 137, 157, 170, 196, 199
focusing operator 252
Krein space 240, 259
280
SUBJECT INDEX
Krein, M. 240, 253, 259 L Liapunov equations of ramification 227 local chart 212
relativity compactness 12 Rudin, W. 185,233
s Schmidt equation of ramification 228 Schmidt lemma 120
M mapping 3 maximum principle for spectral radius 111 metric space 13 metrizability 15, 21, 200 Montel property 18, 23, 99
semi-definite lineal 241 Shabat, B. 200 Schauder principle 137 Schwarz lemma 93 small solution 201 smooth manifold 212
N
split able operator 129, 226
negative vector 240
Stein manifold 220, 234
neutral vector 240
strictly convex space 28
non-expansive mapping 136, 186, 226
submanifold 216
normally solvable operator 118
Suffridge, T. 138
o
T
operator
topological
- proper 182
- isomorphism 12
- g-analytic 53
- product 10
- 6-analytic 53
- space 5
- F-analytic 54
Trenogin, V. 137, 157, 170, 189
- Hammerstein 42, 45, 149, 157
V
- Nemytski 42
Vainberg, M. 137, 157
- p-holomorphic 202
Vesentini, E. 110, 182
- Urysohn 41
u
p
unbinding of an equation 196
positive vector 240
y
R
Yoshida, K. 131
regular fixed point 182
Yuzhakov, A. 200
Titles previously published in the series OPERATOR THEORY: ADVANCES AND APPLICATIONS BIRKHAuSER VERLAG
1. H. Bart, I. Gohberg, M.A. Kaashoek: Minimal Factorization of Matrix and Operator Functions, 1979, (3-7643-1139-8) 2. e. Apostol, R.G. Douglas, B.Sz.-Nagy, D. Voiculescu, Gr. Arsene (Eds.): Topics in Modern Operator Theory, 1981, (3-7643-1244-0) 3. K. Clancey, I. Gohberg: Factorization of Matrix Functions and Singular Integral Operators, 1981, (3-7643-1297-1) 4. I. Gohberg (Ed.): Toeplitz Centennial, 1982, (3-7643-1333-1) 5. H.G. Kaper, e.G. Lekkerkerker, J. Hejtmanek: Spectral Methods in Linear Transport Theory, 1982, (3-7643-1372-2) 6. e. Apostol, R.G. Douglas, B. Sz-Nagy, D. Voiculescu, Gr. Arsene (Eds.): Invariant Subspaces and Other Topics, 1982, (3-7643-1360-9) 7. M.G. Krein: Topics in Differential and Integral Equations and Operator Theory, 1983, (3-7643-1517 -2) 8. I. Gohberg, P. Lancaster, l. Rodman: Matrices and Indefinite Scalar Products, 1983, (3-7643-1527-X) 9. H. Baumgartel, M. Wollenberg: Mathematical Scattering Theory, 1983, (3-7643-1519-9) 10. D. Xia: Spectral Theory of Hyponormal Operators, 1983, (3-7643-1541-5) 11. e. Apostol, e.M. Pearcy, B. Sz.-Nagy, D. Voiculescu, Gr. Arsene (Eds.): Dilation Theory, Toeplitz Operators and Other Topics, 1983, (3-7643-1516-4) 12 H. Dym, I. Gohberg (Eds.): Topics in Operator Theory Systems and Networks, 1984, (3-7643-1550-4) 13. G. Heinig, K. Rost: Algebraic Methods for Toeplitz-like Matrices and Operators, 1984, (3-7643-1643-8) 14. H. Helson, B. Sz.-Nagy, F.-H. Vasilescu, D.Voiculescu, Gr. Arsene (Eds.): Spectral Theory of Linear Operators and Related Topics, 1984, (3-7643-1642-X) 15. H. Baumgartel: Analytic Perturbation Theory for Matrices and Operators, 1984, (3-7643-1664-0) 16. H. Konig: Eigenvalue Distribution of Compact Operators, 1986, (3-7643-1755-8) 17. R.G. Douglas, C.M. Pearcy, B. Sz.-Nagy, F.-H. Vasilescu, D. Voiculescu, Gr. Arsene (Eds.): Advances in Invariant Subspaces and Other Results of Operator Theory, 1986, (3-7643-1763-9) 18. I. Gohberg (Ed.): I. Schur Methods in Operator Theory and Signal Processing, 1986, (3-7643-1776-0)
19. H. Bart, I. Gohberg, M.A. Kaashoek (Eds.): Operator Theory and Systems, 1986, (3-7643-1783-3) 20. D. Amir: Isometric characterization of Inner Product Spaces, 1986, (3-7643-1774-4) 21. I. Gohberg, M.A. Kaashoek (Eds.): Constructive Methods of Wiener-Hopf Factorization, 1986, (3-7643-1826-0) 22. VA. Marchenko: Sturm-Liouville Operators and Applications, 1986, (3-7643-1794-9) 23. W. Greenberg, C. van der Mee, V. Protopopescu: Boundary Value Problems in Abstract
Kinetic Theory, 1987, (3-7643-1765-5) 24. H. Helson, B. Sz.-Nagy, F.-H. Vasilescu. D. Voiculescu, Gr. Arsene (Eds.): Operators in Indefinite Metric Spaces, Scattering Theory and Other Topics, 1987, (3-7643-1843-0) 25. G.S. Litvlnchuk, 10M. Spitkovskii: Factorization of Measurable Matrix Functions, 1987, (3-7643-1883-X) 26. N.Y. Krupnik: Banach Algebras with Symbol and Singular Integral Operators, 1987, (3-7643-1836-8) 27. A. Bultheel: Laurent Series and their Pade Approximation, 1987, (3-7643-1940-2) 28. H. Helson, C.M. Pearcy, F.-H. Vasilescu, D. Voiculescu, Gr. Arsene (Eds.): Special
Classes of Linear Operators and Other Topics, 1988, (3-7643-1970-4) 29. I. Gohberg (Ed.): Topics in Operator Theory and Interpolation, 1988, (3-7634-1960-7) 30. Yu.!. Lyubich: Introduction to the Theory of Banach Representations of Groups, 1988, (3-7643-2207 -1) 31. E.M. Polishchuk: Continual Means and Boundary Value Problems in Function Spaces, 1988, (3-7643-2217-9) 32. I. Gohberg (Ed.): Topics in Operator Theory. Constantin Apostol Memorial Issue, 1988, (3-7643-2232-2) 33. I. Gohberg (Ed.): Topics in Interplation Theory of Rational Matrix-Valued Functions, 1988, (3-7643-2233-0) 34. I. Gohberg (Ed.): Orthogonal Matrix-Valued Polynomials and Applications, 1988, (3-7643-2242 -X) 35. I. Gohberg, J.W. Helton, L. Rodman (Eds.): Contributions to Operator Theory and its Applications, 1988, (3-7643-2221-7) 36. G.R. Belitskii, Yu.!. Lyubich: Matrix Norms and their Applications, 1988, (3-7643-2220-9) 37. K. Schmudgen: Unbounded Operator Algebras and Representation Theory, 1990, (3-7643-2321-3) 38. L. Rodman: An Introduction to Operator Polynomials, 1989, (3-7643-2324-8) 39. M. Martin, M. Putinar: Lectures on Hyponormal Operators, 1989, (3-7643-2329-9) 40. H. Dym, S. Goldberg, P. Lancaster, MA. Kaashoek (Eds.): The Gohberg Anniversary Collection, Volume I, 1989, (3-7643-2307-8) 41. H. Dym, S. Goldberg, P. Lancaster, MA. Kaashoek (Eds.): The Gohberg Anniversary
Collection, Volume II. 1989. (3-7643-2308-6)
42. N.K. Nikolskii (Ed.): Toeplitz Operators and Spectral Function Theory, 1989, (3-7643-2344-2) 43. H. Helson, B. Sz.-Nagy, F.-H. Vasilescu, Gr. Arsene (Eds.): Linear Operators in Function Spaces, 1990, (3-7643-2343-4) 44. C. Foias, A. Frazho: The Commutant Lifting Approach to Interpolation Problems, 1990, (3-7643-2461-9) 45. J.A. Ball, I. Gohberg, L. Rodman: Interpolation of Rational Matrix Functions, 1990, (3-7643-2476-7) 46. P. Exner, H. Neidhardt (Eds.): Order, Disorder and Chaos in Quantum Systems, 1990, (3-7643-2492-9) 47. I. Gohberg (Ed.): Extension and Interpolation of Linear Operators and Matrix Functions, 1990, (3-7643-2530-5) 48. L. de Branges, I. Gohberg, J. Rovnyak (Eds.)" TopIcs in Operator Theory. Ernst D. Hellinger Memorial Volume, 1990, (3-7643-2532-1) 49. I. Gohberg, S. Goldberg, M.A. Kaashoek: Classes of Linear Operators, Volume I, 1990, (3-7643-2531-3) 50. H. Bart, I. Gohberg, M.A. Kaashoek (Eds ): Topics in Matrix and Operator Theory, 1991, (3-7643-2570-4) 51. W. Greenberg, J. Polewczak (Eds.): Modern Mathematical Methods in Transport Theory, 1991, (3-7643-2571-2) 52. S. PrOssdorf, B. Silbermann: Numerical Analysis for Integral and Related Operator Equations, 1991, (3-7643-2620-4) 53. I. Gohberg, N. Krupnik: One-Dimensional Linear Singular Integral Equations, Volume I, Introduction, 1992, (3-7643-2584-4) 54. I. Gohberg, N. Krupnik: One-Dimensional Linear Singular Integral Equations, Volume II, General Theory and Applications, 1992, (3-7643-2796-0) 55 R.R. Akhrnerov, M,I. Kamenskii, A.S. Potapov, A.E. Rodkina, B.N. Sadovskii: Measures of Noncompactness and Condensing Operators, 1992, (3-7643-2716-2) 56. I. Gohberg (Ed.): Time-Variant Systems and Interpolation, 1992, (3-7643-2738-3) 57. M. Demuth, B. Grarnsch, B.W. Schulze (Eds.): Operator Calculus and Spectral Theory, 1992, (3-7643-2792-8) 58. I. Gohberg (Ed.): Continuous and Discrete Fourier Transforms, Extension Problems and Wiener-Hopf Equations, 1992, (ISBN 3-7643-2809-6) 59. T. Ando, I. Gohberg (Eds.): Operator Theory and Complex Analysis, 1992, (3-7643-2824-X) 60. P.A. Kuchrnent: FloquetTheory for Partial Differential Equations, 1993, (3-7643-2901-7) 61. A. Gheondea, D. Timotin, F.-H. Vasilescu (Eds.): Operator Extensions, Interpolation of Functions and Related Topics, 1993, (3-7643-2902-5)
62. T. Furuta, I. Gohberg, T. Nakazi (Eds.): Contributions to Operator Theory and its Applications. The Tsuyoshi Ando Anniversary Volume, 1993, (3-7643-2928-9) 63. I. Gohberg, S. Goldberg, M.A. Kaashoek: Classes of Linear Operators, Volume 2, 1993, (3-7643-2944-0) 64. I. Gohberg (Ed.): New Aspects in Interpolation and Completion Theories, 1993,
(ISBN 3-7643-2948-3) 65. M.M. Djrbashian: Harmonic Analysis and Boundary Value Problems in the Complex Domain, 1993, (3-7643-2855-X) 66. V. Khatskevich, D. Shoiykhet: Differentiable Operators and Nonlinear Equations, 1993, (3-7643-2929-7)
BIRKHAU5ER
I. Gohberg I S. Goldberg
Basic Operator Theory 1981. 304 pages. Softcover. 3rd printrun ISBN 3-7643-3028-7 Basic Operator Theory provides an introduction to functional analysis with an emphasis on the theory of linear operators and its application to differential and integral equations, approximation theory, and numerical analysis. A textbook designed for senior undergraduate and graduate students, Basic Operator Theory begins with the geometry of Hilbert space and proceeds to the spectral theory for compact self-adjoint operators with a wide range of applications. Part of the volume is devoted to Banach spaces and operators acting on these spaces. Presented as a natural continuation of linear algebra, Basic Operator Theory provides a firm foundation in operator theory, an essential part of mathematical training for students of mathematics, engineering, and other technical sciences.
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MATHEMATICS
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Victor Khatskevich and David Shoiykhet Differentiable Operators and Nonlinear Equations
The need to study holomorphic mappings in infinite dimensional spaces, in all likelihood, arose for the first time in connection with the development of nonlinear analysis. A systematic study of integral equations with an analytic nonlinear part was started at the end of the 19th and the beginning of the 20th centuries by A. liapunov, E. Schmidt, A. Nekrasov and others. Their research work was directed towards the theory of nonlinear waves and used mainly the indetermined coefficients and the majorant power series methods, which subsequently have been refined and developed. Parallel with these achievements, the theory of functions of one or several complex variables was gradually enriched with more significant and subtle results. The present book is a first step towards establishing a bridge between nonlinear analysis, nonlinear operator equations and the theory of holomorphic mappings on Banach spaces. The work concludes with brief exposition of the theory of spaces with indefinite metrics, and some relevant applications of the holomorphic mappings theory in this setting. In order to make this book accessible not only to specialists but also to students and engineers, the authors give a complete account of definitions and proofs, and also present relevant prerequisites from functional analysis and topology.
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Birkhauser Verlag Basel · Boston · Berlin
ISBN 3-7643-2929-7 ISBN 0-8176-2929-7
