Linear Functional Equations. Operator Approach

187 Pages·1996·9.089 MB·English

by  Anatolij Antonevich (auth.)

Operator Theory
Advances and Applications
Vol. 83
Editor
I. Gohberg
Editorial Office: T. Kailath (Stanford)
School of Mathematical H.G. Kaper (Argonne)
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Linear Functional
Equations.
Operator Approach
Anatolij Antonevich
Translated from the Russian by
Victor Muzafarov and Andrei lacob
Birkhauser Verlag
Basel . Boston . Berlin
A. Antonevich
Dept. of Mathematics and Mechanics
Byelorussian State University
F. Skoryny4
Minsk 50
Byelorussia 220050
OriginaIly published in 1988 under the title "Lineynye funktsional'nye uravneniya.
Operatornyi podkhod" by Universitetskoe Publishers, Minsk.
© 1988 Universitetskoe Publishers, Minsk.
1991 Mathematics Subject Classification 47B48
Library of Congress Cataloging-in-Publication Data
Antonevich, A. B. (Anatolii Borisovich)
[Lineinye funktsional'nye uravneniia. Operatornyi podkhod.
English]
Linear functional equations. Operator approach / Anatolij
Antonevich ; translated from the Russian by Victor Muzafarov and
Andrei Iacob.
p. cm. - - (Operator theory, advances and applications ; vol. 83)
Includes bibliographical references and index.
ISBN-13:978-3-0348-9851-5
I. Linear operators. 2. Functional equations. 3. Functional
analysis. I. Title. II. Series: Operator theory, advances and
applications; v. 83.
QA329.2.A5813 1996
519'.925~c20
Deutsche Bibliothek Cataloging-in-Publication Data
Antonevic, Anatolij:
Linear functional equations: operator approach / Anatolij
Antonevich. Transl. from the Russ. by Victor Muzafarov and
Andrei Iacob. - Basel; Boston; Berlin: Birkhiiuser, 1996
(Operator theory; Vol. 83)
ISBN-13:978-3-0348-9851-5 e-ISBN-13:978-3-0348-8977-3
DOl: 10.1007/978-3-0348-8977-3
NE:GT
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the
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© 1996 Birkhiiuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland
Softcover reprint of the hardcover 1st edition 1996
Printed on acid-free paper produced from chlorine-free pulp. TCF
00
Cover design: Heinz Hiltbrunner, Basel
ISBN-13:978-3-0348-9851-5
987654321
CONTENTS
Preface .................................................................. vii
Chapter O. Introductory Material 1
Chapter 1. Functional Operators
1. Weighted Shift Operators. Boundedness Conditions .................. 17
2. The Operator Approach. Model Examples ........................... 27
3. Discussion of the Model Examples Discontinuity
of the Spectral Radius .............................................. 36
4. Operators Generated by a Finite Transformation Group ............. 38
5. Spectral Radius of a Weighted Shift Operator ....................... 42
Chapter 2. Banach Algebras Generated by Functional Operators
6. Extension of Operator Algebras by Operators
that Generate Automorphisms ...................................... 49
7. The Isomorphism Theorem for C*-Algebras Generated
by Dynamical Systems .............................................. 53
8. Corollaries of the Isomorphism Theorem ............................. 64
Chapter 3. Invertibility Conditions for Functional Operators. L2-Theory
9. Characterization of Invertible Operators by Means of
Hyperbolic Linear Extensions ....................................... 77
10. Discretization as an Orbital Approach to Study Invertibility ......... 90
11. Operators with a Convex Rationally Independent System of Shifts ... 96
12. Algebras of Type C*(A, G, Tg) in the Case of an
Arbitrary Group Action ............................................. 102
Chapter 4. Functional Operators in Some Special Function Spaces
13. Weighted Shift Operators in Spaces of Smooth Functions 107
14. Functional Equations in a Space of Periodic Distributions 119
v
vi CONTENTS
Chapter 5. Applications to Some Classes of Equations
and Boundary Value Problems
15. Functional-Differential Equations of Neutral Type ................... 125
16. The Symbol of a Functional-Partial Differential Operator ............ 128
17. Nonlocal Boundary Value Problems for Elliptic Equations ........... 136
18. Equations with Periodic and Almost Periodic Coefficients ............ 141
19. Boundary Value Problems with Data on the Entire Boundary
for the Equation of a Vibrating String.......... .. . ..... ............. 146
20. Index Formulas ..................................................... 151
Comments and Bibliographic Information ................................ 159
References........ . ............ . . . . . . ............ . . . . . . . ..... . . . ......... 165
Index.................................................................... 177
PREFACE
In this book we shall study linear functional equations of the form
m
bu(x) == Lak(X)U(Qk(X)) = f(x), (1)
k=l
where U is an unknown function from a given space F(X) of functions on a set
X, Qk: X -+ X are given mappings, ak and f are given functions. Our approach
is based on the investigation of the operators given by the left-hand side of equa
tion (1). In what follows such operators will be called functional operators. We
will pay special attention to the spectral properties of functional operators, first
of all, to invertibility and the Noether property.
Since the set X, the space F(X), the mappings Qk and the coefficients ak are
arbitrary, the class of operators of the form (1) is very rich and some of its individ
ual representatives are related with problems arising in various areas of mathemat
ics and its applications. In addition to the classical theory of functional equations,
among such areas one can indicate the theory of functional-differential equations
with deviating argument, the theory of nonlocal problems for partial differential
equations, the theory of boundary value problems for the equation of a vibrating
string and equations of mixed type, a number of problems of the general theory of
operator algebras and the theory of dynamical systems, the spectral theory of au
tomorphisms of Banach algebras, and other problems. Some interesting examples
concerning the theory of functions of a real or complex variable, number theory,
probability theory and astrophysics were considered already in the XIXth century.
At present numerous groups of reserchers are involved in studies related to
functional operators, and a variety of mathematical tools are applied in the inves
tigation of one or another class of functional operators. On the author's initiative,
in the last 15 years (the present book was originally published in 1988) a group of
mathematicians from Minsk has elaborated a line of research based on the study
of Banach algebras generated by functional operators. Such a general approach
allowed one to obtain some new results and applications and to clarify the connec
tions between many earlier known facts. The contents of the book reflect mainly
the development of this direction. It should be noted that, parallel with these
investigations, the operator algebra generated by functional-singular integral op
erators on a contour in the complex plane was studied by mathematicians from
Odessa, and a number of similar results were obtained for this concrete algebra.
vii
viii PREFACE
It would be impossible to incorporate even the main results in all directions of
investigation of functional equations in a book of rather small volume, so, naturally,
many interesting studies remained beyond the scope of the book. We treat essen
tially only the general results connected with the invertibility and Noethericity of
functional operators as well as some typical applications to functional-differential
equations and boundary value problems. A complete list of references dealing with
functional operators might contain thousands of items. The bibliography given
here consists mainly of the works concerning the line of study developed in the
book and does not claim to be complete.
The book consists of six chapters. Chapter 0 contains introductory mate
rial. In Chapter 1, we consider boundedness conditions for functional operators,
discuss model examples that display special features of the class of operators un
der consideration, and obtain an expression for the spectral radius of a weighted
shift operator in terms of geometric means with respect to invariant measures.
Chapter 2 gives an exposition of the main facts of the theory of C* -algebras gen
erated by functional operators. This theory is connected with classical objects such
as representations of commutative C* -algebras, group representations, dynamical
systems; however, the problem does not reduce to describing these objects. Al
though the latter are well studied, we still have to deal with the most difficult
problem of the spectral connections between these objects, which in the general
case is not well posed. The central result that establishes such a connection is
the isomorphism theorem proved in §7. The main approaches to investigating the
invertibility of functional operators in L (X)-spaces are discussed in Chapter 3.
2
In Chapter 4, some phenomena occurring in the study of functional operators in
other spaces are considered by using e,xamples of spaces of smooth functions. In
Chapter 5 we give applications to functional-differential equations, pseudodifferen
tial and singular integral equations with shifts, nonlocal boundary value problems,
convolution-type equations and other problems. Among the results ofthis chapter,
the most important ones are the construction of a symbol of a functional-partial
differential operator and the analogue of the Lopatinskii condition for nonlocal
boundary value problems. In §20 we obtain index formulas for some of the opera
tors under consideration.
The book is addressed to a broad circle of mathematicians, first of all, to spe
cialists on functional analysis, differential and integral equations, and the theory
of boundary value problems. It is accessible enough to senior students in mathe
matics. Some topics treated in the book were included in special courses delivered
at the Faculty of Mathematics and Mechanics of the Belorussian State University.
We use an uninterrupted numeration of sections. Ends of proof are marked
by the symbol D.
The author wishes to express his gratitude to M. L. Gorbachuk, E. A. Gorin,
P. P. Zabreiko, A. V. Lebedev, and Va. V. Radyno for reading the manuscript and
making a number of useful remarks.
o.
CHAPTER INTRODUCTORY MATERIAL
In this book we use a number of concepts and facts from several mathematical
disciplines. For the convenience of the reader we give here the main definitions and
(without proofs) statements necessary to understand the further text. For details
the reader may use the special references given after each section.
Normed Spaces and Linear Operators. A seminorm on a vector space F over
a field K, where K = CorK = R, is a mapping p: F ~ R satisfying the following
axioms:
1) p(u) ~ 0, u E F;
2) p()..u) = 1)..lp(u), ).. E K, u E F;
3) p(u + v) ::; p(u) + p(v). °
If, in addition, the equality p(u) = implies u = 0, then p is called a norm, and is
usually denoted by II . II·
If {pi, i E A} is a family of semi norms on a vector space F, then the sets
of the form Vi ,i2, ... ,i ,r(UO) = {u : Pik(U - uo) < r}, where iI, i2, ... , in is an
1 n
arbitrary finite set of indices in A and r > 0, form a base of neighborhoods of the
point Uo E F in some topology on F. This topology is said to be generated, or
induced by the family of seminorms Pi. If the family of seminorms consists of a
single norm, then a base of neighborhoods of the point Uo is formed by the open
balls Vr(uo) = {u: Ilu - uoll < r}.
A vector space equipped with a norm is called a normed space. All the normed
spaces considered in this book are vector spaces over C. A complete normed space
is called Banach space. Any normed space F admits a completion, i.e., a Banach
space FI such that Fe FI, IluliF = lIullFl for u E F, and F is dense in Fl.
Let E and F be normed spaces. A pair (A, D(A)) consisting of a vector
subspace D(A) c E and a linear mapping A: D(A) ~ F is called a linear operator
from E to F. The subspace D(A) c E is the domain of the operator A. The
linear operator A is said to be bounded if there exists a constant C such that
IIAullF ::; ClluliE for all u E D(A). The smallest of such constants C is called the
norm of the bounded linear operator A and is denoted by IIAII.
If the domain D(A) of the bounded linear operator A is dense in E and the
space F is Banach, then there exists a unique extension of A to the entire space E,
i.e., there exists an operator AI: E ~ F such that Alu = Au for u E D(A), and
Al is linear and bounded. Moreover, IIAII = IIAIII. In particular, if a bounded
operator is defined on the normed space E, it can be extended to the completion
1
2 CHAPTER O. INTRODUCTORY MATERIAL
of E. In what follows we will consider operators that either are defined on the
entire space or admit such an extension.
By L(E, F) we denotethe set of bounded linear operators A: E -+ F defined
on the entire space E. This set is a normed space equipped with the operator norm.
If F is Banach, then L(E, F) is also Banach. For the case E = F the notation is
abbreviated to L(E, F) = L(E).
A set M in a metric space is said to be precompact if any sequence of points
from M has a Cauchy subsequence. An operator A E L(E, F) is said to be com
pact, if the image of any bounded set in E is a precompact set in F. The subset
JC(E, F) c L(E, F) of compact operators forms a vector subspace in L(E, F),
closed in the topology induced by the norm.
A bounded linear mapping of Eon C, i.e., an element of L(E,C), is called a
bounded linear functional on the normed space E. The space L(E, C) is called the
dual of E and is denoted by E' (or by E*).
In the space L(E, F), in addition to the topology induced by the norm, there
exist other natural topologies. The topology induced by the family of seminorms
{pu : Pu(A) = IIAull, u E E} is called the strong topology on L(E, F). The topology
induced by the family of seminorms {Puf : PUf(A) = If(Au)l, u E E, f E F'} is
called the weak topology on L(E, F).
The topology generated by the family of seminorms {pu : Pu(J) = If(u)l,
u E F} is called the weak *-topology on the dual space F'.
The operator A' acting from F' to E' by the formula (A' J)(u) = f(Au) is
called the conjugate or adjoint of the operator A E L(E, F).
The closed vector subspace Ker A = {u : Au = O} is called the kernel of the
operator A E L(E, F). The vector subspace R(A) = {v : 3u E E, v = Au} is called
the range or image of the operator A. The range of a bounded linear operator A
is also denoted by 1m A.
An operator A E L(F, E) is said to be invertible if there exists an opera
tor B E L(F, E) such that AB = IF, BA = IE, where IE is the identity operator
in E. The operator B is called the inverse of A and is denoted by A-I. Banach's
theorem on the inverse operator states that, if E and F are Banach spaces and
if Ker A = 0, R(A) = F, then A is invertible. The set of invertible operators is
open in L (E, F) in the norm topology.
The operator A' is invertible if and only if A is invertible.
An operator A E L(E, F) is called a Noether operator if the kernel Ker A is
finite-dimensional, the kernel Ker A' of the conjugate operator is finite-dimension
al, and the range R(A) is closed. The integer indA = dimKer A - dimKer A' is
called the index of the Noether operator A. (dimE denotes the dimension of the
finite-dimensional vector space E.) A Noether operator with index zero is called
a Fredholm operator. (In the literature one can also find another terminology: a
Noether operator in the sense of the definition given above is called a Fredholm
operator and no special name is used for the operators with index zero.)
Now let us indicate the basic properties of Noether operators.