Extreme Eigen Values of Toeplitz Operators
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618
1.1. Hirschman, Jr.
Daniel E. Hughes
Extreme Eigen Values of Toeplitz Operators
Springer-Verlag Berlin Heidelberg NewYork 1977
Authors I. I. Hirschman, Jr. Washington University St. Louis, MO 63130/USA Daniel E. Hughes Gonzaga University Spokane, WA 99202/USA
AMS Subject Classifications (1970): 47-02, 47 A10, 47 A55, 47 B35
ISBN 3-540-07147-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-07147-4 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE
The asymptotic distribution of the eigen values of finite section Toeplitz operators as the section parameter increases to • has been known ever since the fundamental paper of
Szego, "Ein Gren~wertsat!
uber die Toeplitzschen Determinanten einer reellen positiven Funktion", Math. Ann. 76, 490-503 (1915).
In the last fifteen years interest has
been focused on the asymptotic behaviour as the section parameter increases to
~
of the very large and the very small eigen values.
The object of
the present exposition is to give a systematic account of one major portion of this subject, incorporating recent advances and discoveries.
TABLE Of CONTENTS CHAPTER I, 1. 2.
TABLE
Introduction
The first eigen value problem The second eigen value problem
CHAPTER II.
1
10
Hilbert Space Background-Small Eigen Values
1.
A perturbation problem
2.
The resolvent equation
20
3.
Spectral resolutions •• Convergence in dimension
24
4.
CHAPTER III.
16
28
The Fourier Transform Theorem
.. .....
1.
Spaces and operators
2.
7.
The application of the perturbation theory Convergence of ~A (t)i to ~A)i Convergence of f.A(t) to FA A Convergence of ~ (t) ~' f. A(t) to ~A) 1J on MA, Part I • Convergence of ~A (t)tf.A (t) to (~A)i Part I I The asymptotic formula, I
8.
The asymptotic formula, I I
3.
4.
5. 6.
31
..
-
on.Ji,
39 44 49
54
57 65 70
CHAPTER IV. 1.
The Fourier Series Theorem Spaces and operators
2.
Application
3. 4.
5. 6.
7. 8.
74
of the perturbation theory ' 1. Convergence of ~A(t)t to ~A) 2 Convergence of FA(t) to FA Convergence of Convergence of
~A(t)iFA(t)~o
78 83
.....
83
A)i A
~
_ f. , Part I •
A ;A (t)t;A (t)to (~A )i on J:l• The asympotic formula, I The asympotic formula, II
Part II
92
93 93 96
TABLE OF CONTENTS
TABLE CHAPTER V.
Hilbert Space Theory - Large EigenValues
1.
A perturbation theorem
2.
Convergence in dimension
CHAPTER VI.
97 102
The Fourier Series and Fourier Transform Theorems
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