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 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re- printing, re-use of illustrations, broadcasting, reproduction yb photocopying machine or similar means, dna storage ni data banks. Under § 54 of the German Copyright Law where copies are fmoard e other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. Berlin Heidelberg © Springer-Verlag by 7791 Printed ni Germany Printing dna binding: Beltz Offsetdruck, Hemsbach/Bergstr. 012345-0413/1412 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. û in, >^viû< ~ = JT )8(^v)~(û 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! =

= 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 -- 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 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(~). =

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