Extreme Eigen Values of Toeplitz Operators

150 Pages·1977·1.517 MB·English

by  I. I. Hirschman Jr., Daniel E. Hughes (auth.)

Lecture Notes ni
Mathematics
Edited yb .A Dold and .B Eckmann
816
.I.I Hirschman, .rJ
Daniel .E Hughes
emertxE negiE seulaV
of Toeplitz Operators
galreV-regnirpS
Berlin Heidelberg NewYork 1977
Authors
..II Hirschman, .rJ
Washington University
St. Louis, MO 63130/USA
Daniel .E Hughes
Gonzaga University
Spokane, WA 99202/USA
AMS Subject Classifications (1970): 47-02, 47A10, 47A55, 47 B35
ISBN 3-540-07147-4 Springer-Vertag Berlin Heidelberg New York
ISBN 0-387-07147-4 Springer-Verlag NewYork Heidelberg Berlin
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PREFACE
The asymptotic distribution of the eigen values of finite section
Toeplitz operators as the section parameter increases to w has been
known ever since the fundamental paper of Szego, "Ein Gren~wertsat~
~ber die Toeplit~schen Determinanten einer reellen positiven Funktion",
Hath. Ann. 76, 490-503 (1915). In the last fifteen years interest has
been focused on the asymptotic behavlour as the section parameter increases
to = of the very large and the very small e£gen 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
Page
CRAFTER ,I Introduction
.i The first eigen value problem ............. I
2. The second eigen value problem ............. I0
C RAFIER .II Hilbert Space Background-Small Eigen Values
.i A perturbation problem . ................ 61
.2 The resolvent equation ................. 20
.3 Spectral resolutions .................. 24
.4 Convergence in dimension ................ 28
CHAPTER llI. The Fourier Transform Theorem
.i Spaces and operators .................. 31
.2 The application of the perturbation theory ....... 39
.3 Convergence of S^(t) ~ to ~^)~ ............ 44
4. Convergence of F ^(t) to ^ F . . . . . . . . . . . . 49
.5 Convergence of --ot--)t(^F@)t('S_ ~)~" on M~, Part I • • • 54
6. Convergence of S^ (t)½F^ (t) to (S^) ~ one, Part II . . 57
7. The asymptotic formula, I . . . . . . . . . . . . . . . 65
8. The asymptotic formula, II . . . . . . . . . . . . . . 70
CHAPTER .VI The Fourier Series Theorem
.1 Spaces and operators . . . . . . . . . . . . . . . . . 74
.2 Application of the perturbation theory . . . . . . . . 78
.3 Convergence of S_ ^(t) ½ to ~^)½ . . . . . . . . . . . 83
.4 Convergence of F_ "(t) to ^ F . . . . . . . . . . . . . 83
.5 Convergence of ! ^(t}~F'(t)to ~^~½F', Part I ..... 92
.6 Convergence of _S _F½)t( (t)to ~ (f) one, Part II . . 93
.7 The asympotic formula, 1 . . . . . . . . . . . . . . . 93
.8 The asympotic formula, II . . . . . . . . . . . . . . . 96
TABLE OF CONTENTS
Pa~e
CHAPTER V. Hilbert Space Theory - Large Eigen Values
.I A perturbation theorem . . . . . . . . . . . . . . . . . 97
.2 Convergence in dimension . . . . . . . . . . . . . . . . 102
CHAPTER VI. The Fourier Series and Fourier Transform Theorems
.I Spaces and operators . . . . . . . . . . . . . . . . . . 105
2. Further operators . . . . . . . . . . . . . . . . . . . II0
3. "_F (t) and ^ F (t)* 113
4. <t)f (t) and fF:, V: (t)F: (t) and V:f ...... 118
5. "v_ (t) and __V (t) ~ and A. V . . . . . . . . . . . . . . 121
.6 ._N ~ (t) . . . . . . . . . . . . . . . . . . . . . . . . . 122
7. The asympotic formula . . . . . . . . . . . . . . . . . . 125
8. The Fourier transform . . . . . . . . . . . . . . . . . . 134
9. Asymptotic formulas for the large eigen value
problem continued . . . . . . . . . . . . . . . . . . . 137
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . 139
INDEX OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . 142
Chapter I
INTRODUCTION
§i. The First Ei~en Value Problem
This paper is devoted to the study of two problems concerning extreme
eigen values of Toeplitz operators. It si partly a very systematic
exposition (self-contained apart from a knowledge of basic harmonic
analysis and Hilbert space theory) of results previously known in a
slightly less general form and partly a presentation of new results.
In order to explain the first of these problems we recall Szego's
fundamental theorem on the global distribution of the eigen values of
Toeplitz operators. Let T be the real numbers R modulo 2~ and
let L2(T) be the Hilbert space of those complex Lebesgue measurable
functions u^(@) on T for which l!^u!! <~ where
u]l ^ lt)e( = <u'l >^u in,
>^vi^u< ~ = JT )8(^v)~(^u de.
Given a real function f^(8) in LI(T) let M be the operator
Mu ^'(o) = f^(O)u ^(o)
where the domain of M consists of these functions u^(@) E L2(T) for
which I ^f (@)u^ l)e( 2 6 LI(T). It is easy to see that M si self-adjoint
and that its spectrum si equal to the essential range of f(e).
Let ~ be the integers and let L2(Z) be the Hilbert space of those
complex functions u(k) on Z for which !!ul! < m where
llul! = <ul u> if2 < -,
<u Iv> = I u(k)v(k).
kEZ
We denote by ~ the Fourier transform
:p~ u >-- u^(e) --~ u(k)e ik@
Z
i-
which maps L2(Z)- unitarily onto 2(T), L and we denote by ~ the
inverse Fourier transform
u^(e)e "ike de k q .Z
^ : U >-- u.(k) = ~ T J
It is not difficult to see that if
-I
T=~O QKM
then (formally)
Tu-(k) = f(k-j)u(j) u E L2(~),
jE_z
where
f(k) =I__ 2~ j e~ f^(e)e -ike .ed
In a variety of applications ti si necessary to consider the truncations of T
n
T(n)u-(k) = i f(k-j)u(j) k = 0~l~.--~n
j=0
and to study in detail the behaviour of the spectrum of T )n( as n >-- m.
The first and one of the most important results in the area is due to
.G Szego. Suppose that [kk(n)}~=O are the (necessarily real) eigen values
of T (n) and that for -~ < a < b < ~
J
N[(a,b); T ])n( = [kk(n) E (a,b)]#~
where [ }# is the number of elements in the set [ }. If
:eil
l[e: f^(e) = a]ly = f^(e) = b})l T = o,
where [I ]IT is the Lebesgue measure of the set { } in T, then Szego
showed that
)I( lim N[(a,b); r(n)]/(n + )i = 2~ I{~: f~(6) E (a,b)}IT -
n ~
We will have occasion to make use of (I). This striking formula has been
generalized in many directions.
We now describe in a special case the first of the problems whose
solution si the object of this paper. If T(n)u =~ u and lul = ~i then
k = (2~) -I ~ ^f l)@( u^ (@)12d@
~T
_n
where u^(@) ~\= u(k)e ikS. Let {)~k(n)}k=l be the eigen values of .T )n(
written in non-decreasing order. It follows that if 1 = ess M inf f^(8)~
2 = M ess sup f^(O), (M I may be -m ~ and 2 M may be +~), and if
I M <M 2 then
I < M Xn, "'<-2,n~<--l n,n~--<" < ~"
It is not difficult to see that
limkn,k = MI k = 1,2~...,
n~ ~
)2(
lim kn~n+l_k = M2 k = 1,2~...
We are interested in "how" this convergence takes place -- under suitable
assumptions on f^(O). In this section we suppose that 1 = M 0, P~ < +~
and that there exists 81E T such that:
]"0 ~-(~l]tO el,
f^(@) ~ as @ >--
where g > 0~ ~ > 0, and where if U is any open set in T containing
1 O
then
g.l.b. I f^(a)[ > o.
T\U
If in (i) we set a = ~njk and b = m then for n large and k fixed
we find that "essentially"
k i
n~ ~ l{o: °le-°l ~°[ < ~n,k}[T ~
which implies that
(3) ~n,k ~ a(~k) ~ n ~0-
for k fixed as n -> ~. Unrigorous as this argument is the result it
suggests is almost correct since~ as we shall show~ there exists a
sequence of constants
(4) 0 <~i <-~2 ~--''" lim ~k = ~
k-~a
depending upon ~ such that
)5( A n,k ~ a~k n-w
for k fixed as n--> ~.
In order to give an indication of how )5( is proved it will be
convenient to replace T by ~2 (2-dimensional Euclidean space)~ a
setting which has the advantage that in it our problems take on a more
typical and general form.
Let ~ be a locally star-shaped set in ~2 which si of finite
measure.
Let L~ be the Hilbert space of complex measurable functions on ~2
determined by the inner product
<u Iv> L = ~ u(~)v(~)d~
2~-
where ~ = (~i,~2). For each t > 0 we define a projection E(t) on L
by the formula
E(t)u" )~( Xtf~(~)u(~). =