Eigenvalue Distribution of Compact Operators

256 Pages·1986·7.449 MB·English

by  Prof. Dr. Hermann König (auth.)

OT16
Operator Theory: Advances and Applications
Vol. 16
Editor:
I. Gohberg
Tel Aviv University
Ramat-Av iv, Israel
Editorial Office
School of Mathematical Sciences
Tel Aviv University
Ramat-Aviv, Israel
Editorial Board
A. Atzmon (Haifa) I Kailath (Stanford)
J. A. Ball (Blacksburg) H. G. Kaper (Argonne)
K Clancey (Athens, USA) S. I Kuroda (Tokyo)
L.A. Coburn (Buffalo) P. Lancaster (Calgary)
R. G. Douglas (Stony Brook) L. E. Lerer (Haifa)
H. Dym (Rehovot) M. S. Livsic (Beer Sheva)
A. Dynin (Columbus) E. Meister (Darmstadt)
P. A. Fillmore (Halifax) B. Mityagin (Columbus)
C. Foias (Bloomington) J.D. Pincus (Stony Brook)
P. A. Fuhrmann (Beer Sheva) M. Rosenblum (Charlottesville)
S. Goldberg (College Park) J. Rovnyak (Charlottesville)
B. Gramsch (Mainz) D. E. Serason (Berkeley)
J. A. Helton (La Jolla) H. Widom (Santa Cruz)
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M.A. Kaashoek (Amsterdam)
Honorary and Advisory
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T. Kato (Berkeley) B. Sz.-Nagy (Szeged)
S. G. Mikhlin (Leningrad)
Springer Basel AG
Hermann Konig
Eigenvalue Distribution
of Compact Operators
1986 Springer Basel AG
Prof. Dr. Hermann Konig
Mathematisches Institut
U niversităt Kiel
Olshausenstrasse 40-60
0-23 Kiell
CIP-Kuntitelaufnahme der Deulschen 8ibliothek
(Operator theory ; VoI. 16)
ISBN 978-3-0348-6280-6 ISBN 978-3-0348-6278-3 (eBook)
DOI 10.1007/978-3-0348-6278-3
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No part of this publication may be reproduced, stored in
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owner.
© Springer Basel AG 1986
Origina1ly published by Birkhauser Verlag, Basel in 1986
Softcover reprint of the hardcover 1s t edition 1986
TO JUTTA
7
P R E F A C E
In this book some methods from the geometric theory of Banach
spaces are used to prove asymptotic estimates for the eigenvalues
of certain compact operators, in particular, of integral operators.
I have tried to make these notes self-contained and readable by
any mathematician or student with basic knowledge of functional
analysis. In particular, this book might be useful as a text for
a seminar.
Propositions, theorems etc. are referred to (uniquely) by the
number of the subsection they appear in. E.g. (theorem) 2.a.6
designates the theorem in Subsection 6 of Section a of Chapter 2.
It is possible to study the main applications to eigenvalues
of integral operators (in Chapter 3) after understanding the main
two theoretical eigenvalue estimates (2.a.6 and 2.b.1 in Chapter2)
as well as some results on interpolation (in Section 2.c).
Clearly, some basic facts (presented in Chapter 1) are also needed.
It is my pleasure to record my gratitude to several mathe
maticians for valuable comments and proofreading, in particular
to M. Defant, H. Jarchow, C. SchUtt, M.A. Sofi and F. Zimmermann.
I also express my thanks to A. Pietsch to whom many results in
this book are due. Introducing the "Weyl numbers", he simplified
the proofs of several results which made a more concise presen
tation possible. Moreover, I had a chance to see the manuscript
of his forthcoming book treating similar (as well as other) topics.
Further, I am grateful to the editor of this series, I.C. Gohberg,
for valuable suggestions and to the Birkhauser-Verlag for
publishing this book. Finally, and in particular, I express my
thanks to Mrs. K. Giese who carefully typed the manuscript.
KIEL, October 1985 HERMANN KONIG
9
T A B L E 0 F C 0 N T E N T S
INTRODUCTION . • . . . . . . • • . . • . . . • . • . . . . . . . . . . . • • . . • • . . . . • . . . . . . 11
NOTATIONS AND CONVENTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1. BANACH SPACES AND OPERATORS ..........•...............•. 17
1.a. Riesz Operators.................................. 18
1.b. Singular Numbers of Operators •••.•.•.••.•....••.• 26
1.c. Classical Banach Spaces ..••••.••••.••••••.•••••.• 41
1.d. Operator Ideals and s-Numbers •.•..•..•.••••.•.... 56
2. EIGENVALUES OF OPERATORS ON BANACH SPACES ........•....• 77
2.a. Weyl's inequality in Banach Spaces ••••••..•...••. 78
2.b. Eigenvalues of p-Summing and Nuclear Operators ... 89
2.c. Interpolation of Operator Ideals ..•.......•...•.. 111
2.d. Estimates of Eigenvalues by Singles-Numbers •.•.• 129
3. EIGENVALUE DISTRIBUTION OF INTEGRAL OPERATORS •.....•... 143
3.a. Kernels Satisfying Summability Conditions •..•...• 145
3.b. Sobolev Spaces and Interpolation •.•..•••••.••.... 160
3.c. s-Numbers of Sobolev Imbedding Maps .•..•....••.•. 178
3.d. Kernels Satisfying Differentiability Conditions .. 196
4. FURTHER APPLICATIONS 217
4.a. Trace Formulas 217
4.b. Projection Constants •••.•...•.•.•.•.••..••...•... 230
REFERENCES . . . . . . . . . • . . . . • . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
LIST OF SYMBOLS . . . . . . • • • . . . . . . . . . . • • . . . . . . . . . . • . . • • . . . . . . . . 257
SUBJECT INDEX . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . • . . . . . • . . . . . . . . 259
11
I N T R 0 D U C T I 0 N
The purpose of this book is to present asymptotic estimates
for the eigenvalues of certain types of (power-)compact operators
in general Banach spaces which were proved during the last decade.
For linear integral operators, it is a classical problem to relate
the order of decay of the eigenvalues to integrability or regu
larity properties of the defining kernel. While Fredholm, Schur
and Carleman treated continuous and, more general, Hilbert-Schmidt
kernels, Hille-Tamarkin [31] considered kernels having derivatives
belonging to suitable L -spaces, i.e. satisfying mixed diffe-
P
rentiability and summability conditions. Further results in this
direction were achieved and presented by Gohberg-Krein [23].
Lately much more precise estimates were obtained by Birman-Solom
jak who in their survey [7] also treat the case of weighted kernel
operators on unbounded domains.
As it turns out, the Lorentz sequence spaces l present a
p,q
natural framework to formulate precise results on the decay of the
eigenvalues Xn. Under various assumptions one proves that
(Xn) E lp,q. This distinguishes the case of p-th power summable
eigenvalues (p = q) from the one involving eigenvalues which are
of order of magnitude n-1/p(p < q = ""). The first (main) index p
defines the "power type" of decay while the second (minor) index q
serves to distinguish "logarithmic type" differences in the order
of decay of (X ). Already in 1931, the spaces l arose
n p,q
naturally in analysis when Paley [85] proved that periodic
functions f in LP (-rr,rr) with p'<2 have Fourier coefficients
1
in lp,p'• This is a special case of an eigenvalue result since
the Fourier coefficients of f are eigenvalues of the operator of
convolution with f •
My aim has been to show that methods from the theory of Banach
spaces may be used successfully to obtain quantitative estimates
12
on the decay of the eigenvalues of abstract classes of compact
maps, which, when applied to integral operators, yield extensions
of both the classical as well as new results. The general Riesz
theory of compact operators offers qualitative information re
garding the spectrum. Quantitative results for certain compact
maps in Hilbert spaces H became available in the late 1940's
when Schatten and von Neumann introduced the classes SP(H) of
operators having p-th power summable singular numbers and H. Weyl
[126] proved that their eigenvalues are p-th power summbable. The
singular numbers are a measure of compactness of the operator.
Using classical approximation theory, these numbers are easily
estimated for integral operators if regularity conditions are im
posed upon the kernel. Most of the results since 1949 are based
on Weyl's inequalities. Surveys on the SP(H)-classes can be found
in the treatises of Dunford-Schwartz II [20],Gohberg-Krein [23],
Schatten [115] and Simon [119].
I believe that the more general Banach space approach pre
sented here offers a larger degree of flexibility than the (by
now classical) Hilbert space method which relies heavily on
approximation theory. In the case of integral operators, it is
generally difficult to treat by the Hilbert space method kernel
operators (other than Hilbert-Schmidt) satisfying only summability
conditions. Nuclear operators in Banach spaces were studied by
Grothendieck [27] and Ruston [113]. The famous paper of Linden
[70] showed that Grothendieck-Pietsch's p-sum
strauss-Pe~czyftski
ming operators TIP constitute useful classes of Riesz operators
in general Banach spaces. They coincide with s2(H) on Hilbert
spaces H. The related ideals of (p,2)-summing maps np,2(p~2)
extend the SP(H)-classes to Banach spaces. In [96] Pietsch in
troduced the concept of s-numbers (sn(T)) of operators T in
Banach spaces, asanaxiomatic generalization of the singular num
bers of Hilbert space maps. The approximation numbers constitute
one example. Requiring these numbers to be p-th power summable
yields another useful class SP of compact operators in Banach
spaces. Asymptotically optimal results on the distribution of the