Composition Operators on Spaces of Analytic Functions 9781315139920, 1315139928, 9781351459129, 1351459120, 9781351459136, 1351459139, 9781351459143, 1351459147

Table of contents :
Content: Cover
Title Page
Copyright Page
Table of Contents
Preface
1: Introduction
Exercises
Notes
2: Analysis Background
2.1 A menagerie of spaces
Spaces of functions of several variables
Exercises
Notes
2.2 Some theorems on integration
Carleson measure theorems
Exercises
Note
2.3 Geometric function theory in the disk
Exercises
Notes
2.4 Iteration of functions in the disk
Lemmas on iteration near the boundary
Exercises
Notes
2.5 The automorphisms of the ball
Exercises
Notes
2.6 Julia-Caratheodory theory in the ball
Exercises
Notes
3: Norms 3.1 Boundedness in classical spaces on the diskExercises
Notes
3.2 Compactness and essential norms in classical spaces on the disk
Exercises
Notes
3.3 Hilbert-Schmidt operators
Exercises
Notes
3.4 Composition operators with closed range
Exercises
Notes
3.5 Boundedness on Hp(BN)
Exercises
Notes
4: Small Spaces
4.1 Compactness on small spaces
Exercises
Notes
4.2 Boundedness on small spaces
Exercises
Notes
5: Large Spaces
5.1 Boundedness on large spaces
Exercises
Notes
5.2 Compactness on large spaces
Exercises
Notes
5.3 Hilbert-Schmidt operators
Exercises
Notes Notes7.8 Spectra: inner functions
Exercises
Notes
8: Normality
8.1 Normal and hyponormal composition operators
Exercises
Notes
8.2 Subnormality of adjoints
Exercises
Notes
9: Miscellanea
9.1 Adjoints of some composition operators
A norm calculation
Exercises
Notes
9.2 Equivalence of composition operators
Exercises
Notes
9.3 Topological structure
Exercises
Notes
9.4 Polynomial approximation
Exercises
Notes
Bibliography
Symbol Index
Index

Citation preview

CARL C. COWEN

BARBARA D. MACCLUER

Purdue University

University of Richmond and University of Virginia

Composition Operators on Spaces of Analytic Functions

CRC Press @) u

Taylor &Francis Group Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 ©1 1995 9 9 5by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Gro up, an In forma business

No cla im to original U.S. Government works This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher ca nnot assume responsibility for the va lidity of all materials or the co nsequences of their use. The authors and publishers have attempted to trace the copyright holders of all material repro· duced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyrig ht material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of t his book may be reprinted, reproduced, transmitted, or utili zed in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any informatio n storage or retrieval system, witho ut written permission fro m the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http: //www.copy· right.com/) or co ntact the Copyright Cleara nce Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not·for· profit organization that provides licenses and registration for a var iety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation wit hout intent to infringe. Visit the Taylor & Francis Web site at http: //www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Studies in Advanced Mathematics Series Editors STEVEN G. KRANIZ Washington University in St. Louis

Editorial Board R. Michael Beals

Gerald B . Folland

Rutgers University

University ofWashington

Dennis de Turck

William Helton

University of Pennsylvania

University of California at San Diego

Ronald DeVore

Norberto Salinas

University of South Carolina

University of Kansas

L. Craig Evans

Michael E. Taylor

University of California at Berkeley

University of North Carolina

Titles Included in the Series Real Analysis and Foundations, Steven G. Krantz CR Manifolds and the Tangential Cauchy-Riemann Complex, Albert Boggess Elementary Introduction to the Theory of Pseudodifferential Operators, Xavier Saint Raymond Fast Fourier Transforms, James S. Walker Measure Theory and Fine Properties of Functions, L.Craig Evans and Ronald Gariepy Partial Differential Equations and Complex Analysis, Steven G. Krantz The Cauchy Transform Theory, Potential Theory, and Conformal Mapping, Steven R. Bell Several Complex Variables and the Geometry of Real Hypersurfaces, John P. D'Angelo An Introduction to Operator Algebra, Kehe Zhu Modem Differential Geometry of Curves and Surfaces, Alfred Gray Wavelets: Mathematics and Applications, John Benedetto and Michael W. Frazier Vibration and Damping in Distributed Systems Volume 1: Analysis, Estimation, Attenuation, and Design, Goong Chen and Jianxin Zhou Volume II: WKB and Wave Methods, Visualization and Experimentation, Goong Chen and Jianxin Zhou A Guide to Distribution Theory and Fourier Transforms, Robert Strichartz Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Peter B. Gilkey Course in Abstract Harmonic Analysis, Gerald B . Folland Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Clark Robinson Fourier Analysis and Partial Differential Equations, Jose Garcfa-Cuerva. Eugenio Hernandez. Fernando Soria, and Jose-Luis Torrea

Contents

ix

Preface

1 6 8

1 Introduction Exercises . Notes .

2 Analysis Background .. 2.1 A menagerie of spaces Spaces of functions of several variables Exercises . .. Notes . . . .. 2.2 Some theorems on integration Carleson measure theorems Exercises . .. Notes. 2.3 Geometric function theory in the disk Exercises . . Notes .. . 2.4 Iteration of functions in the disk .. Lemmas on iteration near the boundary Exercises . Notes . . . . .. 2.5 The automorphisms of the ball Exercises . Notes. .. 2.6 Julia-Caratheodory theory in the ball

.. . .

.

. . .. . ..

Exercises . Notes .

. ..

9

9 20 26 29 29 36 44

45 46

59 62 62 81

93 95 96 101 102 102 114 ll5 v

CONTENTS

3

Norms 3.1 Boundedness in classical spaces on the disk

Exercises . . . . . . . . . . . . . . . . . . 3.2

3.3

3.4

3.5

117

117 126

Notes. . . . . . . . . . . . . . . . . . . . 128 Compactness and essential norms in classical spaces on the disk . 128 Exercises . . . . . . . . . 141 Notes. . . . . . . . . . . 143 Hilbert-Schmidt operators 143 Exercises . . . . . . . . . 148 Notes. . . . . . . . . . . 149 150 Composition operators with closed range Exercises . . . . . . . . . 159 Notes. . . . . . . . . . . 160 Boundedness on HP(BN) 160 Exercises . 172 Notes. . . . . . . . . . . 174

4 Small Spaces 4.1 Compactness on small spaces Exercises . . . . . . . . . . . Notes . . . . . . . . . . . . . 4.2 Boundedness on small spaces Exercises . Notes . . . . . . . . . . . . .

175

5

197

Large Spaces 5.1 Boundedness on large spaces . Exercises . . . . . . . . . . . Notes . . . . . . . . . . . . . 5.2 Compactness on large spaces Exercises . . . . . . . . . Notes . . . . . . . . . . . 5.3 Hilbert-Schmidt operators Exercises . Notes . . . . . . . . . . .

6 Special Results for Several Variables 6.1 Compactness revisited Exercises . . . . . Notes . . . . . . . 6.2 Wogen's theorem. Exercises . Notes . . . . . . .

175 184 185 186 193 195

197 209

211 211

214 215 215 221

221

222 222 228 228 229 242

242

CONTENTS

6.3

Examples . Exercises . Notes . . .

7 Spectral Properties 7.1 7.2

7.3

7.4

7.5

7.6

7.7 7.8

8

Introduction . . . . . . . . . . . . . . . . . . . . . . Invertible operators on the classical spaces on the disk . Exercises . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . Invertible operators on the classical spaces on the ball Exercises . . . . . . . . . . . . . . . . . Notes. . . . . . . . . . . . . . . . . . . Spectra of compact composition operators Exercises . . . . . . . . . . . . . . . . . Notes. . . . . . . . . . . . . . . . . . . Spectra: boundary fixed point. rp'(a) < 1 Exercises . . . . . . . . . . Notes. . . . . . . . . . . . Spectra: interior fixed point Exercises . .. . .. . .. . Notes . . . . . . .. . .. . Spectra: boundary fixed point. rp' (a) = 1 Exercises . . . . . . . . Notes . . . . . . . . . . Spectra: inner functions Exercises . Notes . . . . . . .. . .

Normality 8.1 Normal and hyponormal composition operators Exercises . .. . . .. . Notes . . . . . . . . . . 8.2 Subnormality of adjoints Exercises . Notes . . . . . . . .. .

9 Miscellanea 9.1

9.2

Adjoints of some composition operators A norm calculation . Exercises .. . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . Equivalence of composition operators Exercises . Notes . . . . . . . . . . . . . . . . .

243 246 247

248 248 249 255 255 256 263 263 264 276 277 277 288 289 289 298 298 299

304 304 304 307 307 308

308 312

313 313 319 320

321 321 324 326 327 327

331 331

CONTENTS

viii 9.3

Topological structure . Exercises . . . . . . .

332 344

Notes . . . . . . . . . 9.4

Polynomial approximation .. . Exercises . . ... . Notes . . . . . . . . . . . . . .

345 346 353 354

Bibliography

356

Symbol Index

377

Index

381

Preface

The study of composition operators lies at the interface of analytic function theory and operator theory. As a part of operator theory, research on composition with a fixed function acting on a space of analytic functions is of fairly recent origin, dating back to work of E. Nordgren in the mid 1960's. The first explicit reference to composition operators in the Mathematics Subject Classification Index appeared in 1990. As a glance at the bibliography will show, over the intervening years the literature has grown to a point where it would be difficult for a novice to read all of the papers in the subject. At the same time, there are themes developing so that it is possible to see important groups of papers as exploring the same theme. This book is an attempt to synthesize the achievements in the area so that those who wish to learn about it can get an overview of the field as it exists today. At the same time, we hope to bring into clearer focus the themes from the literature so it is easier to see the broad outlines of the developing theory. We have taken this opportunity to present, in addition to material that is well known to experts, some results that are appearing here for the first time. Many interesting and seemingly basic problems remain open and it is our hope that this book may point out areas in which further exploration is desirable and serve to entice others into thinking about some of these problems. One of the attractive features of this subject is that the prerequisites are minimal. This book should be suitable for second or third year graduate students who have had basic one semester courses in real analysis, complex analysis, and functional analysis. We have included a large number of exercises with the student in mind. While these exercises vary in difficulty, they are all intended to be accessible; we have not used the exercises as a place to collect major results from the literature that space did not permit a discussion of in the text. Since the exercises both illustrate and extend the theory, we urge all readers, students and non-students alike, to consider the exercises as an integral part of the book. Rather than seeking the utmost generality, the theory is developed in a context that is comfortable and illustrates the nature of the general results. In several places we consider composition operators acting on function spaces in the unit ball in eN. Typically, our study of composition operators in the several variable setting is done in separate sections (or separate chapters) from the one

ix

variable theory. Exceptions to this only occur in places where the several variable reasoning is identical to that in one variable, and where the reader interested only in the latter situation would not find it burdensome to read the arguments with N set equal to 1. However, only very little from the extensive field of complex analysis in several variables is ever needed here, and we give a complete discussion of much of the several complex variable background that is central to our subject the automorphisms of the ball and their fixed point properties, angular derivatives and the Julia-Caratheooory theory in B N, and iteration properties of self-maps of the ball. Most of the theory of several complex variables that we use without proof can be found in the first 45 pages of W. Rudin's Function Theory in the Unit Ball of [Ru80]. Statements of results used and relevant definitions are all included here, so that the reader unfamiliar with these results but willing to accept some of them without proof should find the several variable sections of the book readable. Indeed, one of our goals is to convince the reader that the unit ball of eN is an interesting place to do function-theoretic operator theory because one can quickly get to phenomena that are not seen in the disk but which can nevertheless be handled with a minimum of technical machinery. In short, we believe this is an ideal place for a first excursion into several variable function theory. This book is written from a philosophy that mathematics develops best from a base of well chosen examples and that its theorems describe and generalize what is true about the characteristic objects in a subject. This is a book about the concrete operator theory that arises when we study the operation of composition of analytic functions in the context of the classical spaces. In particular, we study the relationship between properties of Crp and properties of the symbol map ~.p: the goal is to see the norm, the spectrum, normality, etc., of Crp as consequences of particular geometric and analytic features of the function 0. Renorm the complex sequences to get a Hilbert space 1t so that S is the composition operator C.p for rp(j) = j + 1 on 'H.

8

Clulpter 1: Introduction

1.1.10 Let Y be a Banach space of functions analytic on the open unit disk that contains the polynomials and let K, be the linear functional in Y' for evaluation of the functions in Y at the point x. (a) Suppose n

K,

= LA;KIIJ j=l

where A; 1: 0 for each j and the Yi 's are distinct points. Show that n = l, At = l, and Yt = x. (b) Suppose C., and C"'i are bounded on Y such that n

c.,= LA;C.pi j=l

where the 1/J/s are distinct analytic maps of the disk into itself and A; foreachj. Sbowthatn= l,At = l,andt/Jt =!p.

1: 0

Notes The notion of functional Banach space we use in this chapter seems to have originated with P. R. Halmos [Hal82, ShW70]. Variations of many of the results and exercises in this section can be found in E. A. Nordgren's survey [No78]. Theorem 1.6 and Exercise 1.1.5 in the case of the Hardy space are results in H. J. Schwartz's thesis [Scz69] but the treatment in the generality here appears to be new. Exercises 1.1.4 and 1.1.5 arise from studying the sorts of algebraic hypotheses that should be included in an appropriate setting for the study of composition operators.

2 Analysis Background

In this chapter we collect information from geometric function theory as well as material on spaces of analytic functions that provides the necessary background for our study of composition operators. We develop this material in the form we need later, so our presentation does not constitute complete or comprehensive coverage for some topics. You may prefer to skip or skim this material on the first reading and return to it later as needed.

2.1

A menagerie of spaces

The classical Banach spaces of analytic functions are derived from various £P spaces. We begin by defining the Hardy spaces of the unit disk D = { z : Iz I < 1} in the complex plane. More complete discussions of the Hardy spaces may be found in books of Hoffman [Hof62], Duren [Dur70], Garnett [Ga81], or Koosis [Koo80]. For a function f analytic in the unit disk and 0 < r < 1, define the dilate fr by fr(ei 8 ) = f(rei 8 ). The functions fr are continuous for each r, so they are in V(8D, dO j21r).

DEFINITION 2.1 ForO< p < oo the Hardy space HP(D) is the set offunctions analytic on the unit disk for which sup

1

O(z) = diD"' k(wz) We can see why this should be true from the following calculation involving the first derivative. It can be made into a proof in many cases by worrying about convergence.

/'(w) = lim

~(/(()- /(w)) =

{-+w., - W

(/, lim -

1

{-+w ( -

_(K..z) = >..• /.(z) for all>.. inC and z in CN) and f = E /.is referred to as the homogeneous expansion of f. As in one variable, we also write /r for the function fr(z) = f(rz); it should be clear from the context whether we are speaking of this dilate of f or the rth term in the homogeneous expansion. It is clear that the monomials z 0 are orthogonal in L 2 (TN). Perhaps less obvious, but of fundamental importance is the fact that the monomials Z 0 are orthogonal in L 2(uN ), where UN is the positive Borel measure on SN that is rotation invariant (that is, invariant under the unitary group) and normalized so that u N ( S N) = 1. (In the notation u N the subscript N will occasionally be supressed when the dimension is clear from the context). The proof of this orthogonality can be found in [Ru80, § 1.4] which also includes a computation of the L 2 (uN) norms of the monomials:

liz

a 2 IILZ(crN)

(N -l)!ct!

= (N _ 1 + lal)!

(2.1.1)

DEFINITION 2.19 For 0 < p < oo the Hardy space HP(DN) is the set of functions analytic in the polydisk DN such that

sup

{

O 0. Show that

l

I d.8 "' {I - lwl2)-c II- wei'p+c 211"

z,.

0

where "' indicates that the ratio of the two expressions has positive finite limit as lwl-+ 1. Hints: Start with the binomial series formula I . ) ( I - we''

~ = ~ r(A: +~)(wei')" ~ A:!r{~) lo=ll

if ~ is not a negative integer. Apply this with 2~ = c + I to show

1

2.. I d.8 o II- we" IHe 211" -

t; A:!zrz(~)+ I

Use Stirling's formula

r(n+ I)"'

rz(A:

00

V£m

~) w

2lo

I

(;r

to show that the coefficients in this series are asymptotic to A:c-l ask-+ oo. (b) If w is in D and c > 0 show

iii_

2 2 w!ll+a+c {1 -lzl )'" dA(z) "'{1 -lwl )-c

Hint: Convert to polar coordinates and use (a). (c) Generalize (a) and (b) to BN, N > 1: if w is in BN and c > 0 then

!BBN 11- (w:()IN+c daN{()"' {I -lwl2)-c and

2.1.5 For a

lN

II - (w,

(~IN+J+a+c {1 -lzl2)a dvN(z)"' {1 -lwl2)-c

> -I, define a weighted Bergman space by

A~(D) ={!analytic in D: 11/11 2 = and a weighted Dirichlet space by

V ..

{ lf(z)I 2 {I -lzl 2)'" dA(z) < oo}

""

lv

= {/analyticinD: 11/11 2 = 1/{0)1 2 + f

lv

lf'(z)l 2 {1-lzl 2)'" dA{z) < oo}

(a) Show that the reproducing kernel functions in A~(D) are o+l Kw(z) = {I - wz)+2

1

""

Hint: Recall the formula 1

1

s'"- (1- s)

11

1 -

ds

= B(x,y) = ~~~~(=~

where B(x, y) is the Beta function and r is the Gamma function.

Clulpter 2: &ckgroullll

28

(b) Show that A~(D) and Va are weighted Hardy spaces and find estimates for the weights {J(j). (c) Show that A~(D) Va+2 and that the norms on these spaces are equivalent. are functional Hilbert spaces. (d) Show that A~(D) and (a) Suppose H 2 ({J) is a weighted Hardy space for which the generating function k is continuous on the closed disk. Show that all functions in H 2 ({J) can be extended continuously to the closed disk. (b) Use part (a) to find an example of a Hilbert space of analytic functions on the disk consisting of functions continuous on the closed disk but including at least one function that is not differentiable at 1. Prove Theorem 2.16. Suppose 'H. is a functional Hilbert space on a set X and suppose JC is a closed subspace of 'H.. (a) Given that Kz is the function in 'H. such that l(z) Kz) for all I in 'H. find the reproducing kernels for JC (note that they should be vectors in /C). (b) Is JC also a functional Hilbert space? Let A(D) be the Banach space of functions that are continuous on the closed unit disk and analytic on the open unit disk. with the supremum norm. (a) Show that A( D) is a functional Banach space on D. (b) Show that A(D) is the closure in C(D) of the analytic polynomials. < lzl < 1}. Describe the closure in C(O) of the analytic (c) Let n = {z: polynomials and describe the space of functions continuous on nand analytic

=

2.1.6

2.1. 7 2.1.8

v ..

= (/,

2.1.9

.s

inn. 2.1.10 Show that if {{J(n)} is a sequence with {J(O) lim inf {J(j) t/j ~ I, then

1t

=

I, {J(j)

>

0 for all j, and

~ { /(•) ~ ~ .,z' ' ~ I•;I'P(;)' < oo}

E.

with (1, g} = a;b;{J(j) 2 defines a weighted Hardy space. Thus, a weighted Hardy space can given as a Hilbert space of functions and the weight sequence constructed or can be given as a weight sequence and the Hilbert space constructed. 2.1 . 11 Show that if {{J( n)} is a positive, bounded sequence such that 0 is not a limit point of {{J(n)} then I is in H 2 ({J) if and only if I is in H 2 (D). 2.l.l2 Show that a weighted Hardy space H 2 ({J) contains H 00 (D} if and only if the weight sequence {{J(n)} defining the space is bounded. 2.1.13 (a) Use the polar coordinate formula

6e

1

I

dvN = 2N

CN

to calculate

roo

lo

1

l(r()r2N-l OON(() dr

SN

( lz'"l2 dvN(Z)

lsN

(b) Show that the Bergman space A 2 (BN) is equivalent to the weighted Hardy space arising from the choice {J(s) = (s + 1)- 1 • 2.1.14 Let D be the open unit disk and let 'H. denote the set of analytic functions in the unit disk such that

I>n(Z + 1)" and 11111 L 4"la..l 00

l(z) =

00

2

n=O

=

n=O

2

< 00

29

Notes

(a) Show that 1t is a Hilbert space of analytic functions on D. (b) Find the kernels Ka for lzl < l. (c) Find a linear functional k so that (pq, k) = (p, k){q, k) for all polynomials p and q but k ':/; Ka for lzl < l. This shows that 1t is not algebraically consistent. 2.1.15 Prove that the Hardy spaces HP(D) and the Bergman spaces AP(D) are algebraically consistent Banach spaces of analytic functions on the open unit disk

D.

2.1.16 Prove the assertion in the text that if za precedes za' and zP precedes/ in the 01 ordering of monomials in the several variable case, then Z 0 zP precedes Z zP'. 2.1.17 Generalize Lemma 2.9 by showing that if k is the generating function for a weighted Hardy space H 2 ({3, BN) on the ball, then k is analytic on the open unit disk. 2.1.18 Prove that if the weighted Hardy space H 2 {/3, BN) contains H 00 , then IIKwll tends to infinity as w approaches the boundary and generalize Theorem 2.17 for this case. (Hint: Generalize Exercise 2.1.12 and use Equation (2.1.3).)

Notes Theorem 2.3, due to W. Rudin [Ru55], provides motivation for a particularly useful definition of the Hardy spaces on domains with nonsmooth boundaries. The property of norms of kernel functions in Exercise 2.1.18, from C. Cowen and B. MacCluer's paper [CoM94], was suggested by S. Axler; the proof outlined here is due to P. Bourdon. The fact that the kernel functions for the weighted Hardy spaces, even in several variables, are the composition of a complicated single variable function (the generating function) and a simple function of several variables ( (z, w)) seems to persist in a variety of situations; seeS. Bell's recent paper [Bel94]. The computation in Exercise 2.1.4 can be found in [Ru80, p. 17]. The results of Exercises 2.1.11 and 2.1.12 appear in the work of N. Zorboska [Zo89a].

2.2

Some theorems on integration

The Littlewood Subordination Theorem [Lit25] can be used to show that composition operators whose symbol fixes 0 are bounded in a variety of spaces. It is frequently quite a different problem to investigate the boundedness of composition operators whose symbols do not fix 0. The subordination theorem is an integral inequality for subharmonic functions. Recall that a continuous real valued function G on a plane domain is called subharmonic if for every domain U with U in and every function u harmonic on U and continuous on U for which G( z) ~ u( z) on the boundary of U, then G(z) ~ u(z) on U also. Subharmonic functions are those that satisfy the local sub-mean-value property, that is, the value ofG(z) is no more than the average value of G on small circles centered at z. The most impor-

n

n

Clulpter 2: Background

30

tant subharmonic functions for our work are the functions 1/(z)IP for f analytic andp > 0. THEOREM 2.22 (Litthwood Subordilllllion Tlleorem) Let cp be an analytic map of the unit disk into itself such that cp( 0) = 0. subharmonicfunction in D, thenforO < r < 1

2 1 ft G(cp(re19 ) dO

~

If G

is a

1

2 ft G(re") dO

Let H be the function harmonic in r D and continuous on r D that agrees with G on the circle of radius r. By Schwarz's lemma, lcp(z)l ~ r for lzl ~ r, so H(l()(z)) is well defined and harmonic in rD also. Since G(z) ~ H(z) in rD and H(cp(O)) = H(O), we have PROOF

(2ft

lo

dO G(cp(re")) 11" ~

2

= H(O) =

I

1

2ft

0

trr

lo

dO 2

H(cp(re")) 11" = H(cp(O))

dO 211"

H(re")- =

12ft 0

dO

G(re")211"

Exercise 2.2.1 gives conditions for equality to hold in this inequality. As immediate corollaries, we show that means of subharmonic functions increase as the radii of the circles increase and derive the inequality that implies a composition operator on H"(D) whose symbol fixes 0 is a contraction. COROLLARY 2.23

If G is subhannonic function in D, then for 0 < r1 < r2 < 1

1

2 ft G(rle") dO~

PROOF

I

1 2

w

G(r2e19 ) dO

Apply the theorem to the analytic function cp(z)

= r 1zfr2 with r = r 2.

COROLLARY 2.24

If cp is an analytic map of the unit disk into itself such that cp(0) H~'(D), then

Ill o cpll,

~

11/11,.

Iff is analytic, then G = Ill" is subharmonic and are the pth roots of the integrals in the theorem. I

PROOF

= 0 and f is in

II/ o cpll, and 11/11,

31

Some theorems on integration

In Section 2.1 we announced our intention to identify the space HP(D) with the closed subspace of V(8D) spanned by einiJ for n ;::: 0. The following result of J. Ryff, showing that for I in HP(D) the radial1imit of I o cp is almost everywhere equal to the composition of the radial limits of I and cp, justifies ignoring the distinction in connection with our study composition operators. (In the expression I* o cp*, we mean by I* the function on the closed disk that takes the value of I in D and the value of the radial limit of I on 8D.)

PROPOSITION 2.25

If I is in HP(D) where p > 0 and cp is an analytic map of the unit disk into itself, then(! 0 cp)* = 0 cp* almost everywhere.

r

Since every function in HP(D) is a quotient of two H 00 (D) functions ([Dur70, p. 16]) it is enough to prove the theorem for I in H 00 (D). Given such an I there is a set E c 8D of full measure so that (! o cp) and cp have radial limits at every point of E. Write E = E1 U Eh where cp has radial limits of modulus one on E 1 and radial limits of modulus less that one on Eh. For (in Eh we clearly have(! 0 cp)* = I 0 cp* = 0 cp* by the continuity of I in D. For (in E~o the definition of E guarantees that I has a limit along the arc cp(r() for 0 < r < I. Lindelof's Theorem (for example, see [Ru87, p. 259]) then applies to show that I has radial limit at cp*(() equal to (! o cp)*((), in other words, /*(cp*(()) = (! o cp)*(() as desired. I PROOF

r

The inner-outer factorization theory of HP(D) functions, which plays a major role in the study of multiplication operators and their extensions, is less important in the study of composition operators. We present only a small part of what is known and interesting from this theory.

DEFINITION 2.26 The function g is called an innerfunction ifit is a bounded analytic function on the unit disk such that limr-+ 1 lg( rei 8 ) I = 1 almost everywhere. A nonzero function Fin H 1(D) is called an outer function if log IF(O)I =

{2w dO lo log IF(ei8 )1 271"

THEOREM 2.27 (Inner-Outer FactoriZillion) If I is a nonzero function in H 1(D) there is an inner function g and an outer function Fin H 1(D) such that I = gF. Moreover; g and Fare unique up to multiplication by a constant of modulus 1. A proof of this theorem can be found in [Hof62] or [Dur70]. The outer function F is not zero in the disk, but there are also inner functions that do not vanish in the disk. We will content ourselves with proving only a more modest factorization

32

Clulpter 2: Btzclcgroutlll

into the product of an inner function carrying the information about the zeros and a nonzero function that is not necessarily outer (Theorem 2.28). The simplest inner functions are the finite Blaschke products. The function B is a finite Blaschke product if there are non-zero points of the disk a1 1 a2 1 ••• 1 an, not necessarily distinct, IAI = 1, and k a non-negative integer, so that

B(z) =

AZk

rr" Ia; I a;- z j=l

a 1· 1-a1·z

In this case, B vanishes at the points a; (and 0 if k > 0). Geometrically, a finite Blaschke product is a branched cover of the unit disk, and the finite Blaschke products are precisely the proper analytic maps of D to itself. An infinite Blaschke product is a non-zero function of the same form but with infinitely many points a; prescribed. Since the partial products are all bounded by 1 in the disk, the infinite product converges to a non-trivial analytic function if and only if the product without the factor zk converges to a non-zero value for z = 0, that is, if and only if TI Ia; 1converges, which is equivalent to E (1 - Ia; 1) < oo. A sequence satisfying this condition is called a Blaschke sequence. If B is an infinite Blaschke product. then we can see that it is an inner function. Indeed, if Bp and B 9 are partial products of B with p < q,

since these functions have modulus 1 on the unit circle. Thus, the boundary values of B 9 Bp agree with the boundary values of the function B 9 f Bp which is analytic in D. Therefore,

II Bp -

B 9 11 2= 12•1 Bp 12 - 2Re (B 9 -) Bp + IB 9 12 -d() 2 7T 0

B9 (0)) = 2 = 2Re ( 1- Bp(O)

n

= 2Re

12• 0

Bq -d() 1 - -B

p 27T

9

( 1 -,gla;l )

Since the infinite product Ia; I converges, this shows that the partial products are a Cauchy sequence in H 2(D). The limit of the sequence of partial products must be B because H 2 (D) is a functional Hilbert space and the analytic functions converge to B in the open disk. Now a norm convergent sequence in H 2 c £2 has a subsequence that converges pointwise almost everywhere. The fact that each of the partial products has modulus 1 on the circle, implies B does also, that is, B is an inner function. This observation leads to an easy factorization theorem for bounded analytic functions.

33

Some theorems on integration

THEOREM 2.28

If 1. For the Hardy spaces, the Carleson sets are certain "non-isotropic balls". Forb in eN with lbl = I, and 0 < h < I, let

S(b,h) = {z E BN:

II- (z,b}l < h}

and

S(b,h) = {z E BN: II- (z,b}l

< h}

43

Some theorems on inugnzlion

(Note that these sets specialize to those previously defined when N = I.) To visualizethesesets,considerb = e 1 = {1,0, ... ,0) sothatS(e.,h) = {z E BN: II- zd < h}. Its intersection with the complex line through 0 and e1 is the set S{l, h) in one variable which is the part of a disk of radius h. On the other hand, if z' lies in the N - 1 dimensional ball of radius ../2h - hl, which is much larger than h when his small, then {1 - h, z') will be in the closure of S(e 1, h). Since U(S(b, h)) = S(Ub, h) for any unitary map U of eN, the same comments apply for an arbitrary b. If we set Q(b, h) = S(b, h) n 8BN, then the above remarks should make it plausible that the surface measure of Q(b, h) is on the order of h( ..fh)2N-2 = hN. A precise verification of this fact can be found in Rudin's book [Ru80, p. 67]. Carleson measure theorems for Hardy spaces on the ball take the following form. THEOREM 2.37

For I' a finite, positive Borel measure on B N and 0 equivalent: (1)

There is a constant K


1 we define a nonlllngenlial approach region at(by

r((,a) ={zED: lz- (I< a(l -lzl)} Of course, the term "nontangential" refers to the fact that the boundary curves of r((,a) have a comer at(, with angle less than 1r (see Exercise 2.3.11). A function I is said to have a nonlllngentiallimit at ( if limz-+( I (z) exists in each nontangential region r{(, a). DEFINITION 2.43 We say cp has a.finite angular derivative at (on the unit circle if there is TJ on the circle so that (cp(z)- TJ)/(z- ()has a .finite nontangentiallimit as z - (. When it exists (as a .finite complex number), this limit is denoted cp'{().

Geometric funetion theory

FIGURE2.4

51

A typical nontangentlal approach region.

Our next result, the Julia-Caratheodory Theorem, is a circle of ideas which makes precise the relationship between the angular derivative rp'((), the limit of rp1(z) at (,and the quantity d(() from Julia's Lemma. THEOREM 2.44 (Julia-Ctutllhiodory Theorem) For rp : D -+ D analytic and ( in 8D, the following are equivalent: (1)

d(() = liminfz-c{I -lrp(z)I)/(I -lzl) < oo, where the limit is taken as

z approaches ( unrestrictedly in D.

rp has finite angular derivative rp' ((} at (. (3) Both rp and rp' have (finite) nontangential limits at (, with 1111 11 = li.m,._, rp(r().

(2)

Moreover; when these conditions hold, we have li.m,._, rp' (r() = rp' (() and d((} is the nontangentiallimit limz-c{I -lrp(z)i}/(I -lzl).

=

I for

= d( ()(11

The proof uses the following simple lemma about nontangential approach regions in D. LEMMA2.45 Given I < o < {3, let 6 = (/3- o)/(o +o/3}. then z +>.is in f((, /3}. PROOF

If z is in f((, o) and !>.1 :5 61(- zl,

We have

lz + >. - (I :5 lz -

(I + 1>.1

< a( l -

lzl) + 61( - zl

:5 o( I - lzl) + c5o( I - lzl)

ClulpUr 2: Bllelcgroulld

52 = (o + c5o){l -lzl)

But since 1~1 ~ 61( - zl and I( - zl

< a{l

- lzl) we get

1 -lz + ~~ ~ 1 -lzl-1~1 ~ {1 -lzl){l- 6o) Thus, lz +~-(I~ {o + 6o){l -lzl) ~

o+6o {1 -lz +~I) _ 1 60

Since /3 = (o + 6a)/{1 - c5a), the conclusion follows.

I

We will show (1) =? (2) =? (3) =? (1). For (1) =? (2) recall that by Lemma 2.41 there exists 17 on the unit circle so that for all z in D PROOF (of Theorem 2.44)

2

117- 0 }, with u(z) = 1 - z, and cfl(z) = sz. It is easily checked that cfl o u = u o 0} but u(V) is not fundamental for cfl on the whole plane. 0 The uniqueness statement of the theorem means that the only way in which n, u and cfl of the model can be varied is in the obvious way: changing variables by conjugation by an automorphism. Indeed, suppose

( a( D)) C u(D) is enough to define r.p(z) = u-• (ll>(u(z))) to give a univalent map of the disk into itself and if u(D) is a fundamental set for II> on n, then r.p has u, II>, and n as its model. The model also helps in understanding the fixed point structure of maps of the disk to itself. If b is a fixed point of r.p, the intertwining of r.p and II> implies u(b) is a fixed point of II> if u(b) exists in some sense. Since the fixed points of II> are obvious, the fixed points of r.p are the inverse images of these under u. As in the construction of Example 2.58, the model makes it possible to construct and understand analytic selfmaps of the disk with specific fixed point properties.

EXAMPLE 2.59

In this example, we will construct a map in the plane/dilation case with one fixed point on the unit circle. Let '1/J(z} = (I + z}/(I - z} be the map of the unit disk onto the right halfplane that takes -I to 0, 0 to I, and I to infinity. As suggested by the proof of Corollary 2.57, a map of the disk into itself can be constructed by letting u( z) = '1/J( z )2 - 1, multiplying by I /2, then going back by u-• (w) = '1/J- t ( v'w + 1) as is illustrated in Figure 2.9. The resulting r.p is in the plane/dilation case because r.p(O) = 0 and the corresponding automorphism of the plane to itself is II>( w) = w /2. Specifically,

r.p ( z ) = u -I

(

1 (z -u 2

>) = --:::==--~- 1 + z ~+1-z

where the branch of the square root satisfies JI = I. It can be checked directly from the formula that r.p(O} = 0 and r.p'(O) = I/2. Since 0 and oo are the only fixed points of ll>(w) = w/2, the only fixed points of r.p are 0 = u- 1 (0) and D I = u- 1 (oo). It is not difficult to check that r.p'(1} = ..fi > 1.

EXAMPLE 2.60

The model can be used to understand or construct non-univalent maps of the disk also! If the map r.p is far from univalent, the intertwining map u will of necessity be very complicated, for example, if r.p( z) = z(2z - I)/ (2 - z ), which is a Blaschke product mapping the disk two-to-one onto itself, u will be an infinite-to-one map of the disk into the plane. However, the model can be used to easily construct simple non-univalent maps of the disk with desired properties. Let

S 1 = {w

= x+iy: x > 0, y > 0, andw ::/: 2n+iyforn =

0, 1, ... andy~ 2n}

ClulpUr 2: Background

74

/

\

I

\

I

\

' ' ) ..:..... 0

, I

--+

\

I

wfl

\

I

.....

,, (/ - - ' \

---7-rl · · o _, I

\

I

'

'

\

/

1z =

(w+l) 1/2

I I

I

I I -II · · 1· T · · t ) 1

I

I \ ' \ \\

'

FIGURE2.9

,-- '

I

/

I

'

--

/

'''

/

I

'\

I/ · I....... \ /I ' I

, '

I I I

\

\

T"'

Constructing a map tp from the model.

/

'

1"'

/

-I

Iteration

Q,L-__L __ _

FIGURE2.10

75

~

_ _ _ _ _ _J __ _

The sheets S 1 and ~ of the Riemann surface.

let ~

= {w = x+iy: x > 0, IYI
l, 0 < y < I, x =F 2n} and one sheet elsewhere. Speaking loosely, if q is a point of S, so is q/2. More precisely, if q = x+iy is in St. then (x+iy)/2 is also inS1 and we write q/2 = (x+iy)/2; similarly, if q is in ~. If q is in the overlap, then so is q/2 and the equivalence class of q/2 is the same for both elements of the equivalence class of q, so the map q ....,. q/2 is well defined on S. If pis the Riemann map of D onto S that takes l to "0" and 0 to "l/2", then we can define an analytic map cp of the disk into itself by cp(z) = p- 1(p(z)/2). We see that cp is two-to-one on the subset p- 1 ( {x + iy: I < x < 2, 0 < y < l} because for such points q from St and~. the points q/2 are identified. Let 1r be the projection of S onto S1 U ~ in the right halfplane that acts by forgetting sheets and let u be the analytic function mapping the unit disk into the halfplane by u = 1r o p. Since 7r(q/2) = 1r(q)j2, the map u intertwines cp and ~(w) = w/2 on the halfplane. Thus, cp is in the halfplaneldilation case and the attractive fixed point 0 of~ corresponds to the Denjoy-Wolff point a = l of cp. Moreover, there are two other fixed points of cp: b1 that corresponds to the fixed point oo of ~ by taking w -+ oo in S 1 and ~ that corresponds to the fixed point oo of~ by taking w -+ oo in~. Notice that cp maps the part of the unit circle between a = 1 and bt that corresponds to the positive imaginary axis in S onto itself because ~maps that part of the boundary of u(D) onto itself. Moreover,

Chllpter 2: Background

76

/

to P maps the segment [0, -i) into but not onto itself and (> maps the line {x-i : x > 0 into u(D). Some points of the unit circle between b1 and b,. are mapped to the unit circle, but all such points are eventually mapped into the disk under the iteration because(> maps that part of the boundary of u(D) partly into itself and partly into

0

u(D).

Our principal interest in the model is that it enables us to solve Schroeder's functional equation. LEMMA2.61

Suppose cp, V, (>, u, and n are as in Theorem 2.53 and suppose>. "I 0 is a complex number. IfF is analytic on nand F o (> = >.F, then f o cp = >.f where f = F o u. Conversely, iff is analytic on D and f o cp = >.f, there is a function F analytic on n so that F o (> = >.F and f = F o u. PROOF

If F o (> = >.F, then f o cp

= F o u o cp = F o (> o u = >.F o u = >.f

F

Conversely, iff o cp = >.j, since u is univalent on V, we may define on u(V) by F = f o u- 1 so that F o (> = >.F on u(V). Since u(V) is a fundamental set for (>on n, we may define F on n by F(w) = >.-kp((>A:(w)) where k is an integer k +m large enough that (>A:(w) is in u(V). If k' is another such integer, say k' with m > 0, then

=

>.-k' F((>k•(w)) = >.-k-mp((>A:+m(w)) = >.-A:-mp((>m((>A:(w))) = >.-k-m (>.mF((>k(w))) = >.-kp((>k(w)) so that F is well defined. It clearly satisfies F o (> = >.F and f = F o u.

I

LEMMA2.62 (1) lj(>(z)

= szandO < Jsl < 1, thenFo(> = >.Fhasanon-zerosolution F that is analytic in the plane if and only if>. = si for j = 0, I, 2, .... Moreover, F is a solution ofF o (> = si F for some non-negative integer j if and only ifF( z) = cz; for some constant c.

(2)

/f(>(z)

= z + 1 and>. "I

0, then F, analytic in the plane, the halfplane ~~:}, is a non-zero solution of

{z : Imz > 0}, or the strip {z : Jlmzl
.F if and only if

(2.4.1)

Chapter 2: Background

78

where e-r = >..and g is analytic in the punctured plane C\ {0}. the punctured disk D \ {0}, or the annulus {w : e-ln < lwl < e2""'} respectively. Toprove{l),comparetheTaylorseriesatz = Ofor F(sz} = E(a;si)zi and >..F(z) = E(>.a;)zi. Clearly functions of the form in Equation (2.4.1) satisfy the functional equation. Conversely, if F is analytic and satisfies the functional equation, define g by g( w) = e--r• F( z) where e-r = >.. and w = eZ•••. Since F satisfies the functional equation, g is well defined and analytic. I PROOF

THEOREM 2.63

Suppose cp is an analytic map of the unit disk into itself, not an automorphism. with Denjoy-Wolffpoint a.

If cp' (a) = 0, then the only non-zero solution off o cp = >.f is >.. = 1 and f constant. (2) lflal < 1 andcp'(a) :/:0, thenfocp = >.f hasanon-zerosolutionifandonly if>.. = cp'(a)i for some non-negative integer j. Moreover, f is a non-zero solution off o cp = cp' (a )if for some non-negative integer j if and only if /{z) = ro(z)i where u is the map of the model (Theorem 2.53) and cis constant. (3) lflal = 1, then the solution space of I o cp = >.f is infinite dimensional/or every>.. :/: 0. (I)

f

(1) Clearly f constant and>..= 1 satisfies the functional equation. Since

=0 is the only solution for >.. = 0, we assume >.. :/: 0.

PROOF

Suppose f o cp = >.f. Since cp'(a) = 0, the Julia-Caratbeodory Theorem implies lal < 1. Since a is a fixed point of cp, we have /(a) = f(cp(a)) = >.f(a) and either /(a) = 1 or /{a) = 0. In any case, f- /(a) satisfies the functional equation also, so we may assume /(a)= 0. Let jU>(a) = 0 for j = 0, 1, ... , k- 1. Taking kth derivatives on both sides of the equation f o cp = >.f yields (!(l:)(cp)) (cp')l:

+ terms containing JU> for j < k =

>.j(k)

Now evaluating at a gives >..j(l:)(a) = (i(k)(a}) (O)l: + 0 = 0 so j(l:)(a) = 0 also. Thus, if>..:/: 1, then f = 0 is the only solution. (2) Apply Lemma 2.61 and (1) of Lemma 2.62 recalling that cp'(a) = 4»'(u(a)). (3) If s = cp'(a) < 1 so that cp is in the halfplaneldilation case with u0 o cp = 4»o o uo for uo a map of the disk into the right halfplane and 4»o(z) = sz, then let u(z) = -(llogsl)- 1 log(uo(z)) which maps D into the strip {x + iy: IYI < 7r/(21log sl)} and let 4»(z) = z + 1. Then u o cp = 4» o u = u + 1. Thus, we have u o cp = o u where 4»(z) = z + 1 on the plane, the upper halfplane, or the strip

lteralion

79

{x + iy : IYI < 1l' /(2jlog sl)} depending on whether rp is in the plane/translation case, the halfplane/translation case, or the halfplane/dilation case. Now, we apply Lemma 2.61 and (2) of Lemma 2.62 (or in the halfplane translation case z - 1, the obvious modification of this lemma). Since the sets of analytic functions g on the punctured plane, punctured disk, and the annulus are infinite dimensional, the solution space off o rp = >.j is infinite dimensional for every >. -::/: 0. I While this theorem gives the analytic solutions of the eigenvalue equation for the composition operator C'P, it does not completely solve the problem because a function f satisfying f 0 rp = >.j is an eigenvector for c'P if and only iff is in the space of functions on which C'P is defined. As we shall see in Chapter 7, answering this question can be delicate, but the following result gives growth estimates that are helpful in determining which eigenfunctions are actually in the space. THEOREM 2.64

Suppose rp, an analytic map of the unit disk into itself, has Denjoy-Wolff point a with Ia I = 1 and rp' (a) < 1. Suppose also that b is another fixed point of rp, that rp is analytic in a neighborhood of b, and that there is 6 > 0 for which jrp( z) I < 1 whenever 0 < lz - bl < 6 and lzl ~ 1. If u is the map in the model that takes the disk into the right half plane and satisfies u o rp = rp' (a )u, then for every p > jlogrp'(a)/logrp'(b)j, there is a constant M so that

< Mlz- bj-P

ju(z)l in a neighborhood of b.

PROOF Let To= rp'(b),let s = rp'(a), and letT= exp(jlog sjfp) 0 and IYI < {Jx }. Clearly, nontangential approach for iterates of t/1 using the sectors is equivalent to nontangential approach for iterates of tp using the approach regions r( 1, a) in the disk. Given a compact set K in the halfplane, choose r < 1 so that for any point w inK, the pseudohyperbolic disk centered at w with radius r contains both K and t/J(K). Let x 0 = min{Re w : w E K}, let wo = xo + iYo be the point of K with real part x 0 and greatest imaginary part and let ~ be the pseudohyperbolic disk with center wo and radius r. (Without loss of generality, we may assume Yo ~ 0; if it is not, replace tp by ip( z) = tp(z).) Since this is a pseudohyperbolic disk with respect to the halfplane (see Exercise 2.3.6), it is the set of w for which PROOF

W-Wol

- - I, the line Re w = KXo is to the right of xo. Let x 1 = ~~:x0 and let w, = Xt + iy, be the upper point of intersection of the line Re (w) = x 1 and the boundary circle of~ (see Figure 2.12). Now w 1 has pseudohyperbolic distance r from wo, so

and

+

Construct w2 = x2 i112 by choosing x2 = KXt and letting W2 be the upper intersection point of the line Re ( w) = ~~:x 1 and the boundary circle of the pseudo-

ltel'dlion

83

,, 1,

,

, ,,

,

I . , . t " - - - - ) 1 ' - ...

,

/

"'

I

I I

I

,, '" "

" ,,"

,

, ""

1., 11

, , "" , ' , ' ', ," .... 'I

,

I

,' ,,

~-----

I

1W1

....

''

'

''

\

lji(K)

I .,. .-

r

"' " /1.

.,'

I

"0

0

FIGURE2.12 Nontangential iteration: K, ,P(K), Wo, w 1 and the disk~.

84

Clulpter 2: Background

hyperbolic disk with center w1 and radius r. It follows that

Writing d for the expression in the square root and continuing, we define Wn = Xn + iyn by Xn = ltXn-1 and Yn = Yn-1 + dxn-1· Since Xn = ltXn-1· we have

Yn = d- +1 Yn-1 -Xn

K.

K. Xn-1

By Julia's Lemma. 1/J(K) lies to the right of the line Re (w) = x 1• Since 1/J(K) is also in~. for every point w = x + iy in 1/J(K), the ratio yfx is less than or equal to the ratio ytfx 1. For win 1/J(K), if u + iv = 1/J(w), then u > K.X and the Schwarz-Pick Theorem (Exercise 2.3.6) implies that 1/J(K) and 1/J('I/J(K)) both lie in a pseudohyperbolic disk of radius r with center at w. In particular, v ~ y + xd and

v

1y

d

I Y1

d

Y2 =-

-U < -+-< -+-- K. K. X - K. K. X1

X2

In other words, for every point u + iv of 1/J('I/J(K)), the ratio vfu is no more than the ratio Y2/x2. Continuing by induction, we see the ratio of the imaginary part of a point in 1/Jn(K) to its real part is no more than Yn/Xn. Since

Yn = ~ Xn K.

+.!. Yn-1

K. Xn-1

=

dt

K.-j

+ K.-n Yo

xo

j=1

and K. is greater than 1 so that the infinite series converges, there is a ray in the right halfplane such that 1/Jn (K) is below the ray for all n. Similarly, using the lower points of intersection, we see there is a ray in the right halfplane such that 1/Jn(K) is above the ray for all n. Thus, there is a nontangential approach region that contains all the iterates 1/Jn(K). I In the case that cp' (a) = I at the Denjoy-Wolff point a on the circle, the situation is more difficult because the iterates do not have to converge nontangentially. We will state the results in terms of maps of the halfplane to itself as the geometry is clearer in this case. We begin with a lemma that gives the rate at which the real parts of the iterates of a point go to infinity. LEMMA2.67 Let 1/J be an analytic mapping of the right halfplane into itself with

"=

inf Re (1/J(x + iy))

O. < oo, so that if S is the sector {w = x + iy : x > 0 and IYI < .Bx} for some positive number .B and

U{x + iy: 6xn lw- w'l 2 + 4RcwRew' !P(w)- p(w')l 2 lw- w'l 2 so

Rep(w)Rep(w') > RewRew' lp(w)- p(w'W - lw- w'l 2 and

!P(w)- p(w')il < lw- w'l 2 Rep(w)Rep(w') - RewRew' which holds for p( w) = p( w') as well. It follows for any n that u(w) 2

2u(w)u(wn) ~ (u(w)- u(wn)) 2

-

X

X

~

I

X

X II.

I

.

X

/' //X

/ /

'' FIGURE2.13

' The region W with the points Wn•

(2.4.12)

91

Iteration

$ lp(w)- p(wn)l 2 $

lw -wnl2 XXn

u(w)u(wn)

and u(w) $ (2 + lw- Wn12) u(wn) XXn

Now if IY- Ynl $ ..Un and 6xn Xn since u/ x is decreasing as a function of x. Lemma 2.67 shows that l.itnn-oo Xn+dxn = 1 so Inequality (2.4.13) implies limw-oo u(w)fx = 0 where the limit is taken in W outside the sectorS. On the other hand, the Julia-Caratheodory Theorem implies lim10 ..... 00 p( w) / x = 0 in the sectorS. Inequality (2.4.7) implies limw-ooP'(w) = 0 for w in W. Inequality (2.4.12) implies lp{w)- p(w')l 2 < Re (p(w))Re (p(w')) lw - w'l 2 xx'

(2.4.14)

Fixing w' and letting w go to infinity in W shows lim10 ..... 00 p(w)/w = 0 also. Recalling that t/J( w) = w + p( w) gives the limits in (2). Finally, for wand w' in W from Inequality (2.4.14)

1/J(w)t/J~w') I= 11 + p(w)- p~w') I~ 1 -lp(w)- p(w') I w-w w-w w-w'

1

>1- ( -

Re (p(w))Re (p(w')))

112

xx'

Since the final term goes to zero as wand w' go to infinity in W, there is p so that

t/J is univalent in W n { w : x > p} which is (3).

I

We will put these lemmas to use in constructing the fundamental set V that used in the proof of Theorem 2.53. Recall that the goal of the proposition is to show that if rp is an analytic mapping of the disk into itself with Denjoy-Wolff point a and t.p' (a) :f: 0, then there is a fundamental set for t.p on D on which t.p is univalent. If t.p has a fixed point in the unit disk, using an automorphism of the disk, we can reduce to the case a = 0 which is covered by Exercise 2.4.3. PROOF (of Proposition 2.56)

Chapter 2: &ckground

92

Suppose a is the Denjoy-Wolff point of r.p and lal = l. From Corollary 2.47 and Lemmas 2.66 and 2.68, we see that for any compact set K in the disk. there is an integer M so that for n ~ M, the function r.p is univalent on Un>M r.pn(K). Fork= 2,3, ... ,let K~c = {z: lzl $ l - k- 1} with interior Kk ~ {z: lzl < 1 - k- 1}. Let m be the smallest integer such that r.p(O) is inK~. We will construct the fundamental set by an inductive process, starting with the integer m. Let Mm be a positive integer large enough that r.p restricted to Un~Mm r.pn(Km) is one-to-one and let Um = Un~Mm r.pn(K~). By the choice of m, Um is connected. Indeed, by continuity each r.pn(K~) is connected and r.p(O) is inK~ and r.p(K~) (since 0 is inK~). Similarly, r.p2 (0) is in r.p(K~) and r.p2 (K~). and so on. In this first step of the induction, we have found an integer Mm and a connected open set Um in D so that r.p(Um) C Um. r.p is univalent on Um. and Un~Mm r.pn(K~) CUm. Continuing with the induction, suppose integers Mm $ Mm+l $ · · · $ M1c-1 and open sets Um C Um+1 C · · · C U1c-1 have been chosen so that U; is a connected subset of D, r.p(U;) C U;, r.p is univalent on U;. and Un~MJ r.pn(KJ) C U; for j = m, m + 1, ... , k- 1. Let Mfc be an integer so that Mfc ~ M~c- 1 and r.p restricted to Un>M' r.pn(K~c) is univalent. Now the set

- .

L =closure ( U1c-1 \ ( Un~M~ r.pn(Kk))) is a compact subset of D, sor.p(L) is also. Since r.pn converges uniformly to a on K~c and a is not in r.p(L), we can find M~c ~ Mfc so that r.p(L)n{Un~M.. r.pn(Kk)) = 0. Let U1c = U1c-1 U {Un~M., r.p"(Kk)). Clearly, U~c is a connected open subset of D, r.p(U~c) C U1c and Un~M., r.p"(Kk) C U~c. We claim r.p is univalent on U~c. Suppose z and w are in U1c with z #: w. If both z and w are in U~c- 1 or both in Un~M~ r.pn(Kk), then clearly r.p(z) #: r.p(w), so we suppose z is inLand w is in Un~M.. r.p"(Kk). (Note the occurrences of M~c and Mfc!) But if z is in L, then r.p(L) is disjoint from Un~M., r.p"(Kk), so r.p(z) #: r.p(w). Thus, we construct the open sets ulc inductively for k = m, m + 1, . . .. Let V' = U1c>m U~c and let V = V'U(holes ofV'). By construction, V' is a connected open sub~t of D, r.p(V') C V', and r.p is univalent on V'. By the maximum principle and the argument principle, r.p(V) C V and r.p is univalent on V as well [Boa87, p. 115] (or see Exercise 2.4.4). It follows that Vis a fundamental set for r.p on D as we were required to show. I It can be shown that in case the Denjoy-Wolff point a is on the unit circle, if the iterates of one point of the disk converge to a nontangentially, then the iterates of every point of the disk do. Moreover, in this case, V can be chosen to contain small sectors with vertex at a and opening 0 for all 0 < 1r. We omit the proof since we will not need this result in our subsequent work.

Exercises

93

Exercises 2.4.1 In the case of an interior fixed point, the function a of Theorem 2.53 is sometimes called the Koenigs' function after G. Koenigs who proved [Koe84] that if tp is an analytic map of the disk into itself, not an automorphism, with tp{O) = 0, then tp·(Z) u(z) = lim -'-.;-oa sJ

exists and satisfies tp{a(z)) = sa(z) in the disk for s = tp'(O). Prove Koenigs' theorem. 2.4.2 We say the functions tp and 1/J commute with each other if tp o 1/J = 1/J o tp. Suppose tp is an analytic map of the unitdiskinto itself with tp{O) = 0 andO < ltp'(O)I < 1. (a) Show that if 1/J is an analytic map of the disk to itself and 1/J commutes with tp, then ,P(O) = 0 also. (b) Show that if 1/J is an analytic map of the disk to itself and 1/J commutes with tp, then I(J and 1/J have the same Koenigs' function. (c) Show, conversely, that if 1/J is an analytic map of the disk to itself and tp and 1/J have the same Koenigs' function, then 1/J commutes with tp. (d) Show that if a denotes the Koenigs' function of tp, then {,P: 1/Jotp

= tpo,P} = {,P: ao,P = ,P'(O)a}

(e) Show that the operators Cv> and C.,p commute if and only if the functions lfJ and 1/J commute. 2.4.3 Suppose that tp is an analytic map of the disk into itself with tp{O) = 0 and 0 < llfJ'(O)I < 1. Show that there is 6 > 0 so that if V = {z : lzl < 6}, then tp(V) C V, tp is univalent on V, and if K is a compact subset of the disk, there is a positive integer n so that if j ~ n then I(J;(K) C V. 2.4.4 Suppose V' is a connected open subset of D and V = V' U (holes ofV'). (a) Show that if I(J is an analytic map of the disk into itself and tp(V') c V', then

tp(V) 2.4.5

2.4.6

c V.

(b) Show that if tp is an analytic map of the disk into itself and tp is univalent on V', then it is univalent on V also. a; zi is an analytic map of the disk into itself, not an Show that if tp{ z) = automorphism, with tp{O) = 0, then the Taylor coefficients of the Koenigs function a can be determined recursively from those of I(J. Show that if tp is an analytic mapofthedisk into itselfwithtp{O) = tp'(O) = 0, then there is no non-constant analytic function a and linear fractional transformation ~ so that ~ o a a o tp. Prove the assertion of Example 2.58 that if tp is a linear fractional map of the disk into itself that has Denjoy-Wolff point 1 with I(J1 ( 1) = 1 then I(J maps the disk onto itself. Show that if tp has Denjoy-Wolffpoint 1 with....i.{l) = 1 and its modelis f! = {z: lrnz > O},and~(z) = z+ l, then~(z) = tp{z) has model f! = {z: lrnz > 0}, and ~(z) = z- 1. (a) In the halfplaneltranslation case, show that the choices ~(z) = z + 1 and ~(z) = z- 1 are actually different. (b) Show that any automorphism ~ of a halfplane or the whole plane is conjugate to one of the four cases or else is an elliptic automorphism.

E

=

2.4. 7

2.4.8

2.4.9

94

2.4.10

Clulpt~r 2:

&d:grollntl

(a) Suppose tp has Denjoy-Wolff point 1 and is analytic in a neighborhood of I, tp1 ( 1) = 1 and Re tp11 ( 1) i 0. Show that the domain 0 of the model is the plane. (Caution: the second derivative at a fixed point is not invariant under conjugation by an automorphism, so a = 1 is important here (see Exercise 2.3.4).) (b) Give examples to show that both the halfplanel and the plane/translation cases can occur when tp has Denjoy-Wolff point I and tp' (I) = I and Re tp" (I) =

0. 2.4.11 Prove: lftpisananalyticmapofthediskintoitselfwithtp(x)realfor-1 < x 8D. (b) Construct a univalent map tp of the disk into itself with Denjoy-Wolff point 1 and tp1 (1) = 1 such that tp(lD) n 8D consists of four fixed points. Suppose tp, an analytic map of the unit disk into itself, has Denjoy-Wo1ffpoint a with lal = 1, tp1 (a) = 1, and the model for tp is halfplane/translation. Show that

Notes

95

if { z;} is a sequence in the disk with rp( z;) sequence.

= Z;+lo then { z;} is an interpolating

Notes This circle of ideas has a very long history. In his paper of 1871 [Scr71], E. Schroeder considers the problem of fractional iteration and relates it to the solution of various functional equations including the functional equation f o rp = sf that bears his name. In 1884, G. Koenigs [Koe84] constructed a solution of Schroeder's equation near a fixed point a when rp is analytic at a and rp'(a) = s,lsl :f: 0, 1. In the years since these classic works, much has been written; the interested reader might consult the bibliography of [Co81] or that of M. Kuczma [Kuc63]. For the most part, the literature has concentrated on cases when

.1 2 ~ 1 - lzd 2 - 31>.1 so that

1 -lzt + >.12 -lz'l2 ~~IIzd- 31>.1 Q and hence if 1>.1 $

611 - ztl for some positive oless than 1/2 1 -lzt + >.1 2 -lz'l 2 ~ ( ~ - 36)11 - ztl Q

But

and (zl

2 /ill- ZJ- >.1

$

2 2 /i(ll- Ztl + 1>.1) $ /i(l +6)11- zd

+ >., z') will lie in r(e,, {3) provided 2

~ - 3o ~

since ( Zt

2

:B (1 + o)

+ ..\, z') is in f( e1, {3) if

~II - Zt - >.I < 1 - lz• + >.1 2 - lz'l 2 {3

(2.6.1)

CluJpur 2: &ckgrourul

106

Inequality (2.6.1) holds provided 6 is less than the positive number

2/a- 2//J 3 + 2/{j

the minimum of this value and I /2 is the 6 we seek.

I

PROOF (of Tbeorem 2.81)

We leave it as an exercise to check that it is sufficient to prove the theorem in the case ( = 71 = e~o which will simplify notation. For the implication (I) => (2), assumed d(et) < oo; our goal is to show (1 - cp 1 (z))/(I - z 1) has restricted limit at et. By Julia's Lemma for the ball (Lemma 2.77)

=

For0 lw'l 2 = 1Wll2 + .. ·lwNI 2 } The Cayley transform ~(z) = i(e, + z)/(1 - z,), defined for z, =F 1, is a biholomorphic map of BN onto n that extends (with ~(e 1 ) = oo) to a homeomorphism of B N onto 0 U 80 U { oo }, the one point compactification of the closure of 0. The automorphisms of B N that fix onJy e 1 correspond, under conjugation by ~, to automorphisms of n fixing onJy { oo}. An example of a class of such automorphisms are the Heisenberg translations: DEFINITION 2.89

For each (bt, b') in 8n set

These "translations" are called the Heisenberg translations of 0; they form a subgroup of Aut(O) and for each b # 0, hbfixes oo only. The corresponding map in the disk. ~-I o hb o cJ? will be called a Heisenberg translation of BN.

Julia-Cartllhlodory in the baU

111

Unlike the situation in one variable, these "translations" are not the complete set of all automorphisms fixing only e 1• For example, if >.; is a complex number of modulus 1 for j = 21 31 ••• 1 N and b I 0 is real, the map (wl! w') -+ (wJ + bl A2t.IJ21 ••• I ANWN) is an automorphism of n fixing 00 only but is not a Heisenberg translation. Notice that this mapping fixes as a set the image under 4> of the complex line through 0 and e 1; that is the set {(w 11 w') En: w' = 0}. That this is no accident is the content of the next theorem. We say r.p fixes A as a set if r.p(A) C A. THEOREM 2.90

An automorphism of BN which fixes e1 only is either a Heisenberg translation of B N or fixes as a set some non-empty proper affine subset of B N . Before we can prove this theorem we need to establish some other results about the automorphisms of B N fixing e 1, or equivalently, the automorphisms of n fixing oo. We will find it convenient to transfer back and forth between the ball and the upper half space by means of the Cayley transform 4> and temporarily adopt the convention of using lower case letters for automorphisms of B N and the corresponding upper case letters for the associated automorphisms in n obtained by composition on the right and left by 4> and 4>- 1 respectively. LEMMA2.91

Given any (a. I a') inn with Im OJ -la'l 2 hb so that hb(aJ 1 a1 ) = (i 1 0').

= 1 there exists a Heisenberg translation

Write a1 = c + i(l + la'l 2) with creal. Set b = (-c + ila'l 2 1 -a') so I that b is in on. The Heisenberg translation hb has the desired property. PROOF

LEMMA2.92

Suppose g inAut(O)fixes oo only. Thenforevery w = (wJ 1 W 1 ) inn Img1(w) -lg'(w)l 2

where g

=

(91.!121 ... 1YN)

= Imw1

-lw'l 2

= (g.,g').

Set g( i 1 0') = ( a 11 a') and lett = Im a 1 -la'l 2, a positive quantity since a') is in n. For s > 0 define the non-isotropic dilDtion in Aut(n) by

PROOF (OJ I

(2.6.3)

If s I 1 the fixedpointsetofc5 8 is {01 oo }. Withs = 1/Vtwe mayuseLemma2.91 to find a Heisenberg translation hb so that hb o 6, o g fixes {i,O') and oo. The corresponding automorphism of BN fixes e 1 and 0, hence is unitary. Moreover, it must fix as a set the orthogonal complement of the complex line through 0 and eJ.

Cluzpter 2: &ckground

112

This implies

hb o 6. o 9(w11 w') = (wt. Uw') for some unitary U on eN -J. Equivalently

9(w"w') = 6..;e o hb" 1(wt,Uw') = (t(wJ- bt- 2i(Uw',b')), v'i(Uw'- b')) We claim that the hypothesis on 9 shows that t must be 1. If not, we may solve v'i(Uw' - b') = w' since (U - t- 112I) is non-singular. Denote its solution by v1• Set VJ = o: + ilv'1 2 for real o: to be specified. We have (vt. v') is in and therefore 9( VJ' v') is in since 9 is an automorphism. By our choice of v'

an

an

9( v" v') = (t( o: + ilvl 2 - bt - 2i(Uv', b'} ), v') Since this is a point of the boundary of n we must have

1m (t(o: + ilv'l 2

-

bt - 2i(Uv', b'} )) = lv'l 2 = Im (o: + ilv'l 2 )

Thus (v1, v') will be a fixed point of 9 if o: real is chosen to satisfy Re(t(o: + ilv'l 2

-

bt - 2i(Uv', b'})) = o:

or to:+ tRe( -b1 - 2i(Uv', b'})

= o:

If t '::/: 1 this may be solved for real o: and then the fixed point set of 9 will contain more than one point, a contradiction. Thus t must be 1 and therefore

9(w 1,w') = (wJ -b.-2i(Uw',b'},Uw' -b') from which it follows by direct calculation that Im91(w) -l9'(w)l 2 = Imw1 -lw'l 2 for all (w., w') inn.

I

Note that the proof of Lemma 2.92 shows that any automorphism 9 of n fixing oo only can be written as 9( w 1, w') = he( w 1, Uw') for some unitary U on eN -J and Heisenberg translation he. If ~(z) = w, where~ is the Cayley transform, then Imw1 -lw'l 2 = (l-lzl 2 )/(l1- ztl 2 ), so the geometric meaningofLemma2.92 is as follows: if an automorphism 9 of n fixes oo only, the corresponding automorphism G = ~-· o 9 o ~in Aut(BN) maps the boundary of each ellipsoid E(k, e 1) into (and therefore onto) itself. This should be compared with the result of Exercise 2.3. 7 for parabolic automorphisms in the disk. PROOF (of Theorem 2.90) Let G be the given automorphism of B N and let 9 be the corresponding automorphism of!l. Set 9(i,O') = (a 1,a'). By Lemma 2.92 we know Ima 1 -la'l 2 = 1 so we may apply Lemma 2.91 to find a Heisenberg

113

Julia-Cturllhlodory in the ball

translation hb so that hbog fixes both oo and (i, 0'). As in the proofofLemma2.92 the corresponding ball automorphism F = Hb o G is unitary and has the form F(z1, z') = (z1, Uz') for some unitary U. Since

F o ~- 1 (w.,w')

= (w 1 - ~. -

2 -.Uw')

WJ +l WJ +l

= ~-l o F(w.,w')

on all points of eN with first coordinate not equal to -i. we have /( w 1, w') = (w., Uw') where I=~ 0 F 0 ~-I and (w., w') is inn. At this point we distinguish two cases. If every eigenvalue of U is I then U, and hence F. is the identity and our original automorphism G is the Heisenberg translation H;; 1. On the other hand, ifUhasaneigenvalue ei 8 -=F l findO -=F A = (>.2 , >.3 , ••• , >.N) so that A(U) = ei8 A where (U) is the matrix of the operator U with respect to the standardbasisoncN-l. Recallthatg = ~oGo~-l = ~oH;; 1 oFo~-l = h!; 1of I where h!; = ft;;, b = ( -b1, -b'). Let V be the column vector ( -~, ... , -bN )t so that AV = 2 -b;>.;. Consider the set

E,;': A

=

{

( WJ' ... 'w N) E

N AV } n:L >.;w; = --..8

1- e

j=l

which is a non-empty proper affine subset of n. We claim that g fixes A as a set. To see this, let (w1, Wl, ... , WN) be in A. Now g(w1, w') = h!; 1 o f(w., w') = h!; 1(w., Uw'). Writing W' = (w2 , ••• , WN )t we see that the last N- 1 coordinates of h!; 1(w., Uw') are ((U)W' + V)t. Since ,

)

iB

,

A (( U ) W + V = e AW + AV = (

·e

e' AV . ) 1- e'8

+

AV AV = - - .8 1- e'

and g( w 1, w') is in A as desired. Since ~ preserves affine sets (Exercise 2.6.4) the corresponding map G on B N fixes as a set some non-empty, proper, affine subset of BN. I We can now prove Theorem 2.83 in the case that cp is an automorphism fixing one point of 8B N. We may normalize so that the fixed point is e 1. THEOREM 2.93

Suppose cp is an automorphism of BN fixing e1 only. Then 'Pn on compact subsets of B N.

eJ, uniformly

PROOF If cp is a Heisenberg translation ~- 1 o hb o ~ then the result is immediate from the easy observation that (hb)n - oo. The remainder of the proof is an inductive argument. By Theorem 2.51 the result holds for N = 1. We assume it holds for k < N and that cp is not a Heisenberg translation. Then by Theorem 2. 90 cp fixes as a set

114

C#uzpkr 2: Backgroutul

some affine subset A of B N with dimension k, 1 ~ k < N. The restriction cp of cp to A~ Bk may be considered as an automorphism of Bk fixing e 1 only (it is

easy to check from the description of A in the proof of Theorem 2.90 that e 1 is in aA). By induction, IPn - et and then by Lemma 2.84, 'Pn - eJ. I A final observation before we prove Theorem 2.83: LEMMA2.94

If some subsequence of {cpn} converges to the identity map, then cp must be an automorphism. PROOF Suppose 'Pn, - I. By passing to a subsequence if necessary we may assume 'Pn,-t - T. 'lben 'Pn,-t o cp converges toTo cp and also to I, so we must haverocp =I. Inparticular,r{BN) C BNandthereforecpn, = cpocpn,-t- cpor which implies that cp o .,. = I as well. I PROOF (of Theorem 2.83.) We have already established the result for an automorphism with no fixed point in B N. Suppose now that cp is an arbitrary fixed point free self map of BN. If every subsequential limit of { 'Pn} is constant, then 'Pn - ({cp) by Corollary 2.86, and we are done. If there is a non-constant subsequential limit, then there is a non-constant idempotent t/J so that for some { n,}, cp"' - t/J. Let A be the fixed point set of t/J, an affine set (by Theorem 2.76) of dimension at least I. We claim that cp maps A into A. To see this choose zo in A. Since 'Pn,(ZO)t/J(zo) = Zo we have cp( cp"' (.zo)) - cp(.zo). But

cp(cp"'(Zo)

= 'Pn.(cp(.zo))- t/J(cp(.zo))

so t/J(cp(.zo)) = cp(.zo) and cp(.zo) is in A Moreover, since 'Pn, restricted to A converges to the identity on A, Lemma 2.94 (with B replaces by A) implies that cp restricted to A is an automorphism of A, and is clearly a fixed point free automorphism. Thus by the result already established for automorphisms, the sequence of restrictions of 'Pn, to A converges to a constant, contradicting 'Pn, t/J. We conclude that all subsequential limits of {'Pn} must be constant and we are I done.

Exercises 2.6.1 Verify that the normalizations '1 = ( = e 1 cause no loss of generality in the proof of Theorem 2.81 2.6.2 Suppose that rp is an analytic map of B N into B N with finite angular derivative at et and rp(et) = e1. Show that for j = 2, 3, ... , N

Nous

115

(a)

'P;(rel) . I1m = 0

r-1

(l - r)l/2

(b)

is bounded in every region r(el, a). Hence

'P;(z) (l- %1)1/2

has restricted limit 0 at e1. 2.6.3 If

0 and 0 < s < 1 so that IG n ~(b, s)l ~ 6"(1 -lbl)

2

for all binD, where ~(b, s) is the disk {z: lz- bl < s(I -lbl)}. 3.4.4 Show that (l) => (2) in Theorem 3.33. Hints: By the previous exercise it is enough to show that 2 IG n D(b, t:)l ~ 6(1 -lbl) for some t:, 6

> 0 and all bin D.

Use the test functions (1 -lbl2)(a+2)/2

(1- bz)a+2

3.4.5 Suppose 1/J = 'Pu o cp where 'Pu = (u- z)/(1 - uz) and cp(O) = u. Complete the proof of Theorem 3.32 by showing that there exist c1, 61 > 0 such that IGt. n S((,h)l ~ 61h2 foralll(l = 1 andO < h < 1 ifandonlyifthereexistsC2,~ > 0 such that IG~ n S((, h)l ~ ~h2 for alll(l = 1 and 0 < h < l. Hint: By the Exercise 3.4.3 it is enough to verify this equivalence with the Carleson sets S((, h), for ICI = 1 and 0 < h < I replaced by pseudohyperbolic disks D(b, t:) for b in D and some fixed 0 < t: < l. Automorphisms of the disk permute the pseudohyperbolic disks, and for any measurable subset E of D we have I'Pu(E)I "' lEI where"' indicates that the ratio of the two quantities is bounded above and below by constants depending only on u. 3.4.6 If cp is inner, must Cv> be bounded below on the Bergman spaces A~(D) for 0

> -1?

Cluzpter 3: Norms

16()

Notes The Fredholm composition operators on H 2 (D) were first identified by J. Cima, J. Thomson and W. Wogen ([CiTW74]) and later by a different and more general method by P. Bourdon in [Bou90] where Lemma 3.26 appears. Fredholm composition operators on the Dirichlet space V in the disk were characterized by J. Cima ([Cim77]) and the proof given here of Theorem 3.29 uses some ideas from that paper. In [CiTW74] the closed range composition operators on H 2 (D) are characterized in terms of boundary behavior of cp : C., has closed range if and only if there exists M < oo suchthatcr(E) ~ Mcr{cp- 1{E)nW)foreveryBorelmeasurablesetEinaD;i.e. ~ is essentially bounded away from 0. Theorem 3.32 can be extended to the Bergman spaces A!(D), for a > -1. The result in this context is: C., has closed range on A!(D) if and only if there exist positive numbers c and 6 such that lc:''a n S{(, h}l ~ 6h2 for

1(1 = 1 and 0 < h < 1 where c:'' 0

= {z: T.,,a+2(z) > c}

and T fP,a+2 is defined in terms of the generalized Nevanlinna counting function: T.

'~'• 11

and

NfP,fl(w) =

(w)

=

-

NfP,fl(w)

(-log lwl)ll

L

(-loglzJ(w}l) 11

1 and

1, the spaces HP( B N) can still be described [Ru80] in terms of harmonic or M-harmonic majorants where a function is called harmonic it is annihilated by the ordinary Laplacian~ where ~g :: 4 L~LD,D,g and M-harmonic if it is annihilated by the invariant Laplacian~ where ~g(a) =~(go r.p4 ){0) and 'Pa is the automorphism defined in Section 2.5. However, when N > I a harmonic

Boundedness in HP(BN)

161

or M-harmonic function need not be the real part of an analytic function. The reader is invited to also look again at the proof of Theorem 3.1 with an eye to finding a similar problem in attempting to generalize it to the case N > 1. In spite of these difficulties in extending the proof of boundedness, it is also not obvious that there exist unbounded composition operators when N > 1. For N = 2 the first examples to be discovered included the following maps:

cp(zhz2) = (2z1z2,0) (ii) cp(z1 1 Z2) = (zl + zi 1 0) (iii) cp(z,, z2) = (Acp 1, Bcp2 ) where A, B ~ 0, A 2 + B 2 = 1 and cpi are inner functions on B2. (i)

There are analogous examples for N > 2. The class of examples in (iii) requires some comment. An inner function on B N is a function fin H 00 ( B N) whose radial limits have modulus 1 almost everywhere with respect to normalized surface area measure UN. There are no higher dimensional analogues of the finite Blaschke products and the existence of non-constant inner functions when N > 1 was a long standing open problem; the first proofs of their existence were given by A. Aleksandrov [Alv82] and E. L0w [L0w82]. While the original arguments to show that the maps in examples (i)-(iii) gave unbounded operators used ad hoc methods, we will treat these examples in a unified way shortly. Notice however that all of these maps exhibit a certain degeneracy. In (i) and (ii) the image of B2 under cp is contained in the complex line [e 1] = {.Xe, : ,\ E C}, and certain arcs in 8B2 are collapsed to single points in 8B2 • The maps in (iii) collapse a set of full measure in 8B2 into a set of measure 0 (either a torus if A, B > 0 or a circle if either A or B =0). The first attempt at a generalized understanding of boundedness of composition operators when N > 1 involved a Carleson measure criterion. Recall from Section 2.2 our notation for Carleson sets in B N. For ( in 8BN and h > 0 we have

S((, h)= {z E BN: II- (z,(}l Q((, h) = S((, h) n 8BN

< h}

In the next theorem cp* denotes the radial limit of the mapping cp considered as a mapof8BN-+ BN. THEOREM 3.35

Let p < oo and suppose f : B N -+ B N is analytic. Define a Borel measure J1. on BN by f.J.(A) = uN(cp*- (A)). Then ( 1)

c'P is bounded on HP( B N) if and only if there exists c < 00 so that J.l.(S((, h)) $ ChN for ( E 8BN, h

(2)

CVJ is compact on HP(BN)

> 0.

if and only if

J.l.(S((, h)) = o(hN) ash-+ 0, uniformly in(.

Chapter 3: Norms

162

As we observed in Section 2.2, the quantity hN appearing on the right hand side is best interpreted as comparable to uN(Q((, h)). Part (1) of this theorem is a direct consequence of Theorem 2.37. In the first part of the proof of Theorem 3.35 below, we will, as promised in Section 2.2, give the argument for the easier direction of Theorem 2.37. Note that an immediate and useful consequence ofTheorem 3.35 is the following corollary. COROllARY 3.36

/fC'P is bounded (respectively compact) on HP(BN) for some finite value ofp, then c'P is bounded (compact) on HP(BN )for all p, 0 < p < 00. First suppose that C'P is bounded on HP(BN) and fix

PROOF (of Theorem 3.35)

any .,., in aBN and 0

< h < 1. Consider the test functions fw(z) = (1 - {z, w))- 4 Nfp

where w

= (1 - h )TJ. llfw

0

For these functions we have

cpll~ = {

laaN

lfw

0

cp*IP duN

= { lfwlp df.l. ~

1

where f.1. = CTNcp*- and where we have used the relationship Uw o cp)*(() = fw o cp*((} a.e.[uN ], which is obvious since fw extends continuously to BN. Our assumption that C'P is bounded guarantees that there exists C depending only on cp such that

Thus

{

ls(,,h)

ifwiPdf.l.$ {

~

ifwiPdJ1.$C {

laaN

ifwiPduN

A computation (Exercise 3.5.2) shows that 1/wiP ~ (2h)- 4 N on S(TJ, h). As outlined in Exercise 2.1.4, we may use the binomial series expansion, the orthogonality of (w, (}k and (w, (}m fork -::/= m, and Equation (2.1.1) to see that ll!wll~"' h- 3N; see [Ru80, p. 18] for the details of this calculation. Thus

f.l.(S(TJ, h})(2h}-4N $ Cth- 3 N and therefore f.l.(S(TJ, h))$ C2 hN. This gives the "only if' direction of (1). Conversely, if f.J.(S(TJ, h)} $ KhN for all 11 in 8BN and h > 0 we apply the harder direction ((1) implies (2)) of Theorem 2.37 to see that

f

~

IJIP df.l. $ c

f

laaN

1/*IP duN

Boundedness in HP(B N)

163

for some C < oo and every fin HP(BN ). In particular iff is in the ball algebra A(BN) of functions analytic in BN and continuous in BN

II! 0 ~PII: =

r

JMN

r

=

}1fN

l(f 0 cp)*l" duN= lfi'P dJ.I. $ C

r

JMN

r

JMN

If 0 cp*I'P duN=

lfi'P dUN

Since A(BN) is dense in HP(BN) this completes the proof of (1). For (2) first assume J.L(S((, h)) :f o(hN), and choose sequences (n in 8BNand hn decreasing to zero and /3 > 0 so that J.L(S((n, hn)) ~ f3h;{. For Wn = (1- hnKn we use again as test functions fn(z) = (1- (z, w71 ) )-•NIP, for which llfnll~ "' h;;, 3N. Let 9n = /n/11/nllp· Note that 9n tends to 0 uniformly on compact subsets of B N and llYn ° ~PII~ =

f l(9n ° cp)*IP duN= f l9n ° cp*IP duN JMN JMN

=f

.Ia;

1

,. f 11:1'P dp. . 111n 11 'P Js((n,hn)

IYni'P dp. 2:

Again we use the estimates IfniP 2: (2hn)- 4 N and 11/nll~"' h;;, 3N to see that this last integral is bounded away from zero as n tends to oo. Conversely, assume J.L(S((, h)) = o(hN), uniformly in (in 8BN . It is easier, and equivalent, to replace the sets S((, h) by the windows

-

W((, h)= {z E BN: 1 -lzl

z

< h, TZf

E Q((, h)}.

GivenEchoosehosmallenoughsothatJ.L(W((,h)) $ EhN forh :S hoandall(in anN. Let J.1.1 be the restriction of J.l. to BN \ (1- ho)BN. We claim J.1.1 is aCarleson measure, with J.L'(W((, h)) :S CEhN for some constant C depending only on N. This is clearly true, with C = 1, for h :S ho. Suppose h > ho. Now Q( (, h) can be covered by a finite collection {Q(w;, ho/3)} where thew; are in Q((, h). Since Q( w;, ho/3) and Q( (,h) intersect, and h > ho/3, the absorption property described in Exercise 3.5.7 guarantees that there is a constants > 0 independent of h so that Q((, sh) :::> Q(w;, ho/3) for each j. Furthermore, there is (see [Ru80, p. 68]) a disjoint subcollection r of {Q(w;, ho/3)} so that Q((,h) C UQ(w;,ho) r

We can obtain an upper estimate on card r: UN(UQ(w; , ho/3)) ~(card r)cJ(ho/3)N r

CluzpUr 3: Norms

164

since uN(Q(wj, ho/3)) 2: c1(ho/3)N for some constant Ct depending only on N. Also

O"N(UQ(w;,~/3)) $ UN(Q((,sh)) $ c2sNhN r

for some 02 depending only on N, so

C2sNhN card r $ Ct(ho/3)N

( h )N

= c ho

From this it follows that

J£'(W((,h)) $ I>'(W(wi,ho)) $ C r

(~) N E~ = CEhN

as desired. Finally, suppose {/n} is a bounded sequence in HP(BN) with fn converging to 0 uniformly on compact subsets of BN. By the first part of the theorem, is bounded on HP(BN) and by Exercise 3.5.1, this implies Uno It')* = o It'* almost everywhere on 8BN. Thus

f!

{

18EJN

I(In 0 ct')*IP duN= =

{

.h;\(1-lto)Jfii

{

18EJN

If! 0 lt'*IP duN=

IJ:IPdJ.L+j

{

hi;

c'P

IJ:IP dJ.L

_IJ:IPdJ.L =I +II .

(1-ho)BN

Integral (II) can be made as small as desired by choosing n large. By Theorem 2.37 and the remark following it, integral (I) is less than KE(sup llfnll~) for some finite constant K. Since E is arbitrary we are done. I As expected, there is a completely analogous version of Theorem 3.35 for the weighted Bergman spaces A~(BN ): THEOREM 3.37

Let p < oo and suppose It': BN- BN is analytic. Define a Borel measure J.L on BN by J.L(A) = Vact'- 1(A) where dva = (1 -lzl 2 ) 0 dvN. Then ( 1)

C'P is bounded on A~ ( B N) if and only if there exists C

< oo so that

J.L(S((,h)) $ ChN+a+l for( E 8BN, h > 0. (2)

C'P is compact on A~(BN) if and only if

J.L(S((, h))= o(hN+a+l) ash- 0, uniformly in(. The proof is left to the reader in Exercise 3.5.8. As a corollary, we see that boundedness and compactness of C'P on A~ ( B N) is independent of p, p < oo.

Boundedness in HP(BN)

165

We will finish this section by examining some other easy consequences of Theorem 3.35, while Chapter 6 will explore some of its deeper consequences. We begin with a result that prevents the symbol map of a bounded operator from having a particularly strong kind of degeneracy. COROIL4.RY 3.38 If c"' is bounded on HP(BN) then cp* cannot carry a set of positive Lebesgue measure in 8BN into a set of Lebesgue measure 0. PROOF Suppose cp*(A) C E C 8BN where aN(E) = 0. For any t exists Carleson sets Q((~c, h~c) with

>

0 there

CXl

and CXl

L O'N(Q((Ic, h~c))
a > 1/2 where [x] denotes the greatest integer less than or equal to x. Some remarks may help put Condition (4.1.1) in perspective. If ,B( n) = R" where R > 1 then Condition (4.1.1) is satisfied and all functions in the corresponding space Bl(,B) are actually analytic in the disk of radius R. However if ,B(n} = exp(n 4 ) for any 0 < a < 1 then Condition (4.1.1) is still satisfied (see Exercise 4.1.4) but it follows from a basic result about random series [Kah85, p. 40] that the space H 2 (,B) contains functions which are not continuable across any point of 8D. We will see later(Corollary 7.15), however, that for any sequence ,B(n} satisfying Condition (4.1.1), all functions in H 2 (,B) are in C 00 (D). Finally, the weight sequences ,B( n) = ( n + 1) 4 for a > I /2 give rise to boundary-regular small spaces which fail to satisfy Condition (4.1.1 ). Our interest in Condition (4.1.1) is in the following example. EXAMPLE4.6

Whenever an increasing sequence ,B( n) satisfies Condition (4.1.1) then the function + 1}/2 gives rise to a compact operator on H 2 (,B), and consequently

rp(z) = (z

181

ContpGehless

H 2 ({J) can not be automorphism invariant. We will verify that C"' is compact by doing a calculation which actually shows that it is trace class. To this end, begin with the orthonormal basis {en} where en(z) = zn/{J(n). We have

where C{n, k) are the appropriate binomial coefficients. Our goal is to show that I

oo

I

n

L {J(n) 2n L C(n, k){J(k)

n=O

(4.1.2}

k=O

converges so C"' satisfies the hypotheses of Theorem 3.24. We concentrate first on the inner sum, remembering that {J( n) is increasing so we may write, for our fixed a > I /2. n

~~

n

k=O

k=O

k=(an)+l

L C(n, k){J(k) = L C(n, k){J(k) + L

C(n, k){J(k)

(4.1.3}

n

L

~ {J([an]}2n +

C(n, k){J(k)

k=[an)+l

Since oo

L

n=O

I I {J(n) 2n

[an]

oo

L C(n, k){J(k) ~ n=O L k=O

{J{[an]} {J(n)
I /2. Thus we can make the simple estimate

L n

k=(an]+l

C(n, k){J(k) ~ nC(n, [an]+ I}{J{n} ~ nC(n, [an]}{J{n)

Clulpt~r 4:

182

For n sufficiently large, say n

~

Snulll SJNIC~s

No, use Stirling's formula

lim .../iWi(nfe)n

n-oo

n! and the bounds [an] > an - 1 and n - [an]

~

=1 n - an to obtain the estimate

c(a)yn C(n, [an]) ~ (ao 1/2 yields a 0 (1- a)l-a > 1/2. Thus 00

L

11

n=No

{J(n) 2n

L n

k=(an)+l

00

C(n, k){J(k) ~

L

n=No

11

{J(n) 2n nC(n, [an]){J(n)

So for boundary-regular spaces that are so small as to fail to be automorphism invariant there can exist compact Cv> with II~PIIoo = 1 and finite angular derivative at some point of 8D. We conclude this section by looking at the question of whether rp in the small spaceY and II~PIIoo < 1 implies Cv> compact on Y. We will confine our attention to the case of weighted Hardy spaces H 2 ({J). For the choice {J(n) = JlR where R > 1 it is an easy computation to show that acting on H 2 ({J) is trace class. In particular then. Cv> is compact. PROOF

'f/J(z} =

Set lllloo = rand pick r with r < r < 1. It is easy to check that for rz, ct/J is trace class- simply compute E IICt~J(en)ll for the orthonormal

Compactness

183

basis en= zn f[j(n). Now r.p = 1/J o (/;where (j;(z) = r.p(z)fr, so we will be done if we can show C-is bounded on H 2 ([3). Using II({JIIoo < 1 find a disk D :J D 'P_

-

with(/; analytic on D and (/;(D) c D. Then for any I in H 2 ([3), I o (/;is analytic in the neighborhood D of the closed disk and so is contained in H 2 ([3) as desired.

I

An informal statement of Theorem 4. 7 is "for good spaces and good symbols, II'PIIoo < 1 implies Ccp is compact." An informal contrapositive of this is "if r.p is a good symbol with II'PIIoo < 1 but Ccp is not compact, then H 2([3) is a bad space." This contrapositive can be made precise for a large class of weighted spaces. Consider an arbi~ace H 2 ([3) where the weight sequence is regular in the sense that limn-+oo \I[3( n) exists, either finite or infinite. Examples of sequences with such regularity include [3( n) = na, [3( n) = exp( na ), [3( n) = exp( {log n )9), as well as products and quotients (defined in the obvious way) of regular sequences. By comparison, if [3( n) = 22• for 2k ~ n < 2k+1 then [3( n) is a monotone sequence that is not regular in this sense. If the weight sequence {3 is regular and H 2 {/3) supports a non-compact composition operator Ccp with r.p analytic in a neighborhood of the closed unit disk and II'PIIoo < 1 then all functions in H 2 ([3) are, in fact, analytic in RD for some fixed Rwith R > 1. Indeed, if H 2 ([3) does not include every function analytic in a neighborhood of the disk, find I = 2: a,.zn that is analytic in a neighborhood of RD for some R > 1 but which is not in H 2 {[3), so that 2: lanl 2 [3(n) 2 = oo. Now if lanlf3{n) ~ 1 for infinitely many n, then 2: lanlf3{n) = oo. On the other hand, if lanlf3{n) < 1 for all but finitely many n, then since lanl 2 [3{n) 2 < lanlf3{n) for all but finitely many n, the divergence of 2: lanl 2 f3(n) 2 implies the divergence of 2: lanlf3{n) and the Cauchy-Schwarz inequality implies

Since I is analytic in a disk larger than RD, the series 2: lani.R'l converges. In particular, for all but finitely many terms, lani.R'l < 1 which implies the series 2: Ian 12 R2n converges and we must have

~ [j(n)2 = oo L- R2n

n=O

The divergence of this sum implies n

f"i!ii"::Vi

lim sup V P\"r n-+oo R2

>1 -

{4.1.4)

Clulpter 4: SmaU SptKeJ

184

Our regularity hypothesis lets us replace the limit supremum by the limit in Inequality (4.1.4). Fix any 8 with 1 < 8 .I where >. :F 0, 1 or a product of the F.(z). eigenvalues {D~r'P~~:(O)} and where I has homogeneous expansion Show inductively that F. 0 for s = 0, I, 2, .... 4.1.2 Suppose Y is small but not necessarily boundary-regular. Assume 'P maps D into D with 'P(O) = 0. Show that if 'P has finite angular derivative at any point of 8D then C., is not compact on )1. Hint: If 'P has finite angular derivative at 1 with nontangentiallimit 1 at 1, then {'P( r) : 0 < r < I} is a nontangential curve at 1. Apply Lindelof's Theorem. 4.1 .3 Re-do the previous exercise for Y a small space on the ball B N , for N > 1. Hints: The following consequence of Theorem 2.79 is relevant.

=

Theorem ([Ru80, p. 171])

Suppose I is in H 00 (BN ), limit L at(.

I< I = 1 and limr-1 l(r() = L. Then I

E::O

has restricted

(The definition of restricted limit is Definition 2.78.) The exercise can then be done for N > 1 by showing • If 'P has finite angular derivative at (in aBN. then A(r) = 'P(r() satisfies properties (1) and (2) in Definition 2.78. • If 'P has finite angular derivative at e1 with 'P* ( e1) = e1, then 'Po 'P also has finite angular derivative ate,, with radial limit e 1 there. Exercise 2.6.2 will be helpful here.

185

Notes 4.1.4 Let,B(n) = exp(na) forO
1/2 and automorphism invariant for all a. Conclude that for all a > 1/2, if C"' is compact then II'PIIoo < l. If H 2 (,B) supports a non-compact C"' with II'PIIoo < 1, then H 2 (,B) supports an unbounded c"' with 'P E H 2 (,8) and II'PIIoo < 1. (a) Note that Theorem 4.4 does not apply to the small space H 00 (D), but recall from Exercise 3.2.2 that if c"' is compact on H 00 (D) then II'PIIoo < 1. Generalize this to any small space in which the functions zn are a bounded set. (b) Show that if c"' is compact on H 00 (BN}, N > 1 then cp(BN) c BN. Show that if there are functions analytic in a neighborhood of the closed unit disk not in H 2 (,B}, then H 2 (,B) is not automorphism invariant. Let cp{z) = z2• Show that C"' is unbounded on H 2 (,B) for (a) ,B(n) = exp(na), a > 0 (b) ,B(n) = exp((logn) 9 ),q > 1 Give an example of a weighted Hardy space H 2 (,B) and an analytic map cp of the disk into itself such that

w.

sup wED 2

IIC:(Kw)ll IIKwll

< oo

yet C"' is not bounded on H (,B).

Notes R. Roan made the first study of composition operators on analytic Lipschitz spaces in the disk in (Roa80a] and on sP(D) in (Roa78b]. An error in Proposition 7 of (Roa78b] is corrected by B. D. MacCiuer in (Mc87], where a Carleson criterion (Theorems 4.11 and 4.12 of the next section) is given for boundedness and compactness of C"' on sP(D). This Carleson criterion is used in [Mc87] to prove Theorem 4.5 for the particular choices .Y = sP(D), p ~ 1. The general version of Theorem 4.5 presented here is due to J. H. Shapiro (Sho87b] who also first noted that some boundary-regular non-automorphism invariant small spaces could support compact C"' with II'PIIoo = 1. Our examples are a simplification of Theorem 3.4 in that paper. P. Thran [TuS8] investigated the question of automorphism invariance of convergence of power series on the unit circle and found that, surprisingly, if the series for I converges at ( 0 and cp is an automorphism of the disk, the series for I o cp need not converge at f( a with ' (1 - r!)(aH)/2 /n (z) = (1 - rn'7z)(a+2)

=

where r., ll and /n(O) = 0. Show that C"' is not bounded below on{!.. }. (b) Using Theorem 4.8 show that cp is a one-to-one map of{}[) onto{}[). Hint: By [CoP82, Lemma 8.2] if cp: D-+ Dis univalent and cp((I) = cp((2) = A where IC1I = IC2I = IAI = I then cp cannot have finite angular derivative at both (I and (2. (c) Show that C"' is Fredholm on Va(D) for some -I < a < 0 if and only if cp is in Aut( D). 4.2.19 Modify the argument in the previous problem to identify the Fredholm composition operators on S 2 (D).

Notes Theorem 4.9 is due to K. Madigan [Mad93b]. The proof given here, using explicit test functions, is a modification of Madigan's original argument. Theorem 4.8 was essentially noted by J. H. Shapiro in [Sho87b]; the analogous results in Theorem 4.10 and Theorem 4.I3 appear to be new. Theorems 4.II and 4.12 are due to B. D. MacCluer [Mc87], as is Exercise 4.2.11. The result of Exercise 4.2.12 was shown to us by T. Carroll. Exercises 4.2.I3, 4.2.14, 4.2.15, and 4.2.16 are all taken from K. M. Madigan and A. Matheson's study of compact composition operator on the Bloch and little Bloch spaces in the disk [MaM93].

196

Cluzpter 4: Snudl Spt~£es

The result of Exercise 4.2.2 appears in the work of R. Roan [Roa80a], where the main result is the converse statement to this exercise. However, the proof given there appears to be in error (the estimates in (8) on p. 37.5 are not correct). We believe it is an open question as to whether or not the result still holds. The results of Exercise 4.2.18 are in the paper of M. Jovovic and B. MacCiuer [JoM9.5]. The conclusion of part (a) of Exercise 4.2.18 on closed range operators on the small spaces 'Da(D) as well as the analogous result on S2 (D) (Exercise 4.2.19) can be obtained by a quite different argument that makes use of the fact that these spaces are, under equivalent norms, Banach algebras; seeR. Roan [Roa78b] for this point of view.

5 Large Spaces

5.1

Boundedness on large spaces

In contrast to the function spaces considered in the last chapter, here we consider weighted Bergman or Hardy spaces that include functions which grow much more rapidly than functions in H 2 (D) or the standard weighted Bergman spaces A! (D). We will confine our attention to function spaces on the disk, and concentrate on spaces AMD) defined for weights G positive, continuous, and non-increasing 1 on (0, 1) with J0 G(r)rdr < oo so that G{lzl)dA{z) is a positive, circularly symmetric finite measure on D. Set

A~(D) = {!analytic: f

jD

lf(zWG(Izl) dA(z) 7r

< oo}

with

11/11~ = { lf(zWG(Izl) dA(z) jD 7r The space A~ (D) is equivalent to the weighted Hardy space H 2 (/3) where /3( n )2 = 2pn/c where {Pn} are the moments

Pn

=

1 1

r 2n+ 1G(r) dr

1

and c = 2J0 G(r)rdr, chosen so that /3(0) = l. If G(r) = (1- r) 0 a > -1) or (- log r ) 0 , we have the standard weighted Bergman spaces, equivalent norms. The large spaces of principal interest here will arise from weights"; functions G(r) that decrease more rapidly than any power of (I DEFINITION 5.1

(for with "fast r).

We say G(r) is afast weight if

lim r--+1

G(r)

{1- r) 0

= 0

(5.1.1)

197

Clulpter 5: lArge SpGCes

198

for all a > 0. If, in addition, the quantity in Equation (5.1.1) is decreasing for r near 1, for all a > 0, we say that G is /liSt and regular.

J;

Throughout this first section our standing hypothesis on G is that it is positive, conG(r)rdr < oo. Specific examples tinuous, and non-increasing on (0, 1) with of such G that are, in addition, both fast and regular include

for c

> 0 and P > 1 and

for c > 0 and p > 0. The spaces Ab{D) all have reproducing kernel functions that are obtained from the orthonormal basis { zn / ~} in the usual way:

:=0

Kw(z) =

2L 1

00

1

-wnzn n=OPn

Since the moments Pn are bounded, we see that IIKwll tends to infinity as w tends to 1. By multiplying the integral inequality of Littlewood's Subordination Principle (Theorem 2.22) by rG(r) and integrating with respect tor from 0 to 1 we see that whenever 0.

Denoting the kernel function for this space

K~

we have

00

IIK~II2 =

2)n + l)a+llwl2n n=O

which is comparable to (1 - lwl)-(a+2) on D. For any a > 0 the hypothesis guarantees that (n+ 1)a+ 1/3( n )2 is bounded and hence that there is a finite constant C so that for w in the disk (5.1.2)

Suppose (is a point with I I for all (in an then inf{I~P'(()I : ( e aD} > I since, by Proposition 2.46, this infimum is actually attained. By Exercise 2.3.17 lim r-+1

logM(r) logr

>1

Thus, ignoring the trivial case limr-1 M(r) < I from which it easily follows that C"' is compact on Ab{D), for r sufficiently close to l, E(-logM(r)) > E( -log r) and Theorem 5.5 guarantees that C"' is bounded on Ab{D): COROUARY 5.7

If I~P'(()I > I for every ( in aD. then c"' is bounded on Ab{D), where G is any positive, continuous, non-increasing weight function. Much more is actually true, as we shall see in the next section. A theorem analogous to Theorem 5.5, but with a hypothesis involving the function G(r) directly, gives a sufficient condition for C"' to be bounded. THEOREM5.8

If G is a non-increasing weight function and .

G(r)

h~_!~P G(M(r)) < oo then C"' is bounded on Ab{D). The proof of this theorem, which we will not give here, requires only a technical lemma that shows that this assumption on G implies that either the hypothesis of Theorem 5.5 holds or that limr-+l M(r) < I (so that C"' is in fact compact on

Ab{D)).

The next theorem provides almost the converse to Theorem 5.8. THEOREM5.9

Suppose G is a non-increasing weight function and . inf 1

•:,n-1

then

Our desired corollary is immediate:

G(r) G(M(r)) =

00

Clulpter 5: Large Spt~Ces

206

COROLLARY 5.10

If G is a non-increasing weight function and . inf 11

G{r) G(M{r)) =

:,rt-1

00

then Crp is unbounded on Ab{D). PROOF (or1beorem 5.9) If liP' ((}I > 1 for all ( on the unit circle, then for r sufficiently close to 1, M(T) ~ r. Since G is non-increasing this says

.

G(r)

h~~~P G(M(r)) ~

1

Thus for the hypothesis to hold we must have a point (o in By Julia's Lemma

M(T)

~ jcp(r(o}l ~

T forO< T < 1

Let .

B = h~~~p

av with lcp'((o}l

IIKM(r)ll IIK,.II2

~

1.

{5.1.10)

2

Assume, for a contradiction, that B is finite. There exists T 1 < 1 so that IIKM(r) 11

2

IIK,.II 2

if r 1

~ T

< 1. Choose A

< 28 -

> 2B and use the hypothesis on G to find T2 G(r)

G(M(r))

< 1 with

~A

if r2 ~ T < 1. Set To= max{rt. r2}. Let h(T) = IIK,.II 2G(T). Since the identity map does not induce a HilbertSchmidt composition operator we have, by Exercise 3.3.1 of Section 3.3,

L11Kzii

2

G(Izl) dA(z) = oo

so that h(T) must be unbounded as r approaches 1. Now for To ~ r < 1

A

>

28

> -

2 IIKM(r)ll = h(M(r)) G(r) > h(M(r)) A IIK,.II 2 h(r) G(M(r)) h(r)

so that

h(M(r)) h(r}

sup{h(T) : 0 ~ T ~ M(ro)} and let t1 = inf{t : h(t) ~ N}. Then

for To

207

Boundedness

h(tt) = Nand h(s) < N for all s < t 1. Let r* be such that M(r*) = t1; by the definition oft1 it follows that r* > ro. Also, by Inequality (5.1.10) we have r* $ M(r*). Ifr* = M(r*) then h(M(r*)) h(r*)

=

1

contradicting Inequality (5.1.11). If r* < M(r*) then h(r*) < N by the definition of t1 while h(M(r*)) = h(tt) = N, so again Inequality (5.1.11) is I contradicted. Thus B is infinite and we are done. Although the main interest in Theorems 5.5, 5.8, and 5.9 and Lemma 5.6 is for fast weights G, the only requirement on G in these results is that it is a positive, continuous, non-increasing weight function. The next result, another application of Lemma 5.6, also holds whenever G is positive, continuous, and non-increasing, and is of interest even in the case of the standard weights. The functions H, F, and E continue to be defined from G via Equations (5.1.4), (5.1.5), and (5.1.6). THEOREM 5.11

Let cp : D - D be analytic and suppose there exist ro < I and A > 0 with jcp(z)l $ lziA for ro $ lzl < 1. Then for any to > 0, there exists c independent off so that

II/ 0

cpll~ $ c ( O 0 where I < (J < 3 and 0 < t < 21-IJ. (a) ShowthatrpmapsDintoD. Notethatrp(l) =I andrphasangularderivative I at I. Show that lrp( ei') I < I if ei' =F I so that rp has finite angular derivative onlyatl. (b) If {zn} is a sequence in D with z..- I tangentially then lrp(z..)l < lz..l for n large. (c) Use Theorem 5.8 and Corollary 5.10 to show that Cop is bounded on Ab(D) if and only if (J ~ a + I. 5.1.9 Set G( r) = (I - r )0 where a ~ 0 and let rp be an analytic map of D into D with rp(O) = 0. Let M = {/ E Ab{D) : /(0) = 0}. (a) Given ro with 0 < ro < I, show that there exists c < oo so that IICopiMII 2 ~ cA-(a+l) provided lrp(z)l ~ lziA on ro < lzl < I. (b) Let rp(z) = zn. Part (a) shows that n'"+ 1 IIC... IMII 2 is bounded as n goes to infinity. Is it also bounded away from zero? (c) Suppose {w;} are the zeros of rp repeated with multiplicities. Show that there exists b < oo so that IICcplMII2

~ b (~)1 ~ lw;l))

a+l

where b depends only on a. 5.1.1 0 Suppose {V'n} is a sequence of analytic self-maps of D such that Cop,. is bounded on Ab(D) for some positive, continuous decreasing weight function G. (a) Let P be the orthogonal projection of A~(D) onto the constants defined by P(f) = /(0). Prove the converse of Theorem 5.12; that is, show that if II Cop,. - Pll- 0 then V'n(z)- 0 for each z in D. (b) Define a not necessarily orthogonal projection of Ab(D) onto the constants by Po(/)= f(a) for a in D. Show that if V'n is a sequence of maps of the disk into itself such that IICop,.- Poll converges to 0, then IPn(z)- a for every z in D. (c) Suppose further that t/Ja(z) = (z - a)/(1 -liz) induces a bounded composition operator on Ab(D). Show that if V'n(z) - a pointwise in D then II Cop,. - Po II converges to 0. Notice that the conclusion of (c) holds whenever Ab(D) is automorphism invariant and 'l'n is the nt" iterate of rp provided rp, not an elliptic automorphism of D, has fixed point a in D.

Notes

211

Notes The study of composition operators on large spaces was begun by T. L. Kriete and B. D. MacCluer in [KrM92] and most of the results of this section, as well as most of the exercises, are contained in, or essentially contained in, that paper. The proof of Theorem 5.8 can be found there as well. Exceptions are Theorem 5.9 which was recently obtained by T. L. Kriete ([Kri94 ]), and Theorem 5.12 and the related Exercise 5.1.1 0, also due toT. L. Kriete (private communication). Exercise 5.1.8 is due toT. L. Kriete and B. D. MacCluer [KrM94] and [KrM92]. Note that a consequence of this exercise is that liP' ((}I ~ 1 does not characterize boundedness on large spaces. The results of this exercise remain true for (J = 3 as well. In particular !p(z} z + t(l- z} 3 , with 0 < t < 1/4 give bounded operators on A~(D) for G(r) exp(-B( 1 ~,_}2]. As. we will see in the next section, C.p is not compact since IP'(I} 1. The main theorem of [KrM94] shows that for weights that decay more rapidly than G( r) = exp(- B ( 1~r there are no bounded, non-compact operators, other than those induced by rotations. The (J = 3 example shows that this result is sharp.

=

=

=

?J

5.2

Compactness on large spaces

Recall that by Proposition 3.13, in any weighted Hardy space H 2 ({3) where E f3(n)- 2 = oo the essential nonn satisfies

IIK 1 at all ( in the unit circle. A result which lies deeper than this is that, for fast weights, this condition is sufficient for compactness. Before proceeding to a proof of this, we need to assemble several relevant ideas. In Section 3.2, we described a procedure for estimating essential norms from above which worked in a fairly general setting including that of weighted Bergman spaces Ab(D) with G fast and regular. If Ccp is bounded on Ab{D) and H(r) = G(s) ds dx we have by Inequality (3.2.4)

I: I:

IICcpll~ ~ C rlim 'Yr -t where

'Yr

=

sup E H(lz;(w)l) r:Siwl e 1/(2c) for all(. 5.3.3 Suppose that G(r) = exp( -clogl(I - r)) for some c > 0. Show that if C., is Hilbert-Schmidt on A~(D) then I~P'(()I > e 1/(2c) almost everywhere.

Notes The main results of this section, Theorems 5.16 and 5.18, have analogues for the Schatten p-classes 8 11 , p > 0 given by T. L. Kriete and B. D. MacCluer in [KrM92]. There it is shown that for fast weights G which decay sufficiently rapidly every compact C., on A~(D) will be in 8 11 for every p > 0, while for slower fast weights a necessary condition for C., to lie in any S, class is I~P'(()I = oo almost everywhere. The kernel function approach given here for the Hilbert-Schmidt class is due toT. L. Kriete [Kri94], as are the results of Exercises 5.3.2 and 5.3.3.

6 Special Results for Several Variables

6.1

Compactness revisited

Our goal in this chapter is to continue the study begun in Section 3.5 ofboundedness on function spaces in BN when N > 1. The Carleson and compactness of measure criterion (Theorem 3.35) continues to underlie the results in this chapter, but our focus wiU be to obtain deeper consequences of this characterization than those in Section 3.5. We begin with a compactness question which is motivated by one of the earliest results on compactness in one variable: if 1p(D) is contained is compact on H"(D) for all in a nontangential approach region in D, then p < oo (Proposition 3.25). We will consider C., when 1p(BN) is contained in a Koranyi approach region

c.,

c.,

r((,a)

= {z E BN: 11- (z,(}l < ~(1 -lzl 2)}

for some a > I and ( in aD. Our first lemma exploits the compatibility of the Carleson sets S('7, t) with the Koranyi approach regions where, as in Section 2.2,

S('7,t) = {z E BN: 11- (z,'7}1 < t} LEMMA6.1

There exists C are such that

< oo, depending only on a, so that whenever 17 in 8BN and t > 0 S(TJ,t) nr((,a)

then

11 - ((, TJ}I

::/:0

$ Ct.

PROOF For z, w in the closed ball, let d(z, w) = 11 - (z, w} j. Recall that d(z, w) is a quasi-metric on BN (Exercise 3.5.7). If w is in S('7, t) nr((, a) then d(w, () $ (a/2)(1 -lwl 2 ) and d(w, '1) $ t. So we have

d{(, 17) $ 3 (d((, w) + d(w, 77)) $ 3 ((a/2)(1 222

-lwl 2 ) + t)

Compoctness

223

But since w is in S(TJ, t), 1 -lwl

:5 t. Thus

d((, 71) :5 3{ot + t)

= 3{o + l)t

which gives the conclusion with C = 3{o + 1).

I

LEMMA6.2

Qr.p(BN)

c r{(, a) then on HP(BN) we have

Crp is bounded if and only if UN 0. If S(TJ, t) and r{(, o) are disjoint. then r.p- 1S(TJ, t) is empty and uNr.p- 1S(TJ, t) = 0. On the other hand, if S (TJ, t) n r( (, a) is non-empty then Lemma 6.1 and the absorption property of the Carleson balls (Exercise 3.5.7) show that there exists K < oo, independent of TJ, t so that

S{(, Kt) :J S(TJ, t) and hence

UN(BN) contained in r(e~oa). Since inner functions mapping 0 to 0 are measure preserving maps of 8BN into 8D ([Ru80, p. 405]) O'NIP-l (E)

=

u, (E)

for every Borel set E in 8D. Roughly speaking, the map 1/J must behave like 1 - ( 1 - z )11 near 1, where

b = 2cos- 1{1/a) 7r

To make this precise, note that the boundary curves of the region parametrized near 1 by

w+(O) = 1 - {2 cos 0-

r

in D may be

~ )ei8 a

1

and w_(O) = w+(O) forcos- {1/a)- 6:5 0 :5 cos- 1{1/a) with 6 small. Since w~ and w~ are continuous, a theorem of Warschawski [Pom92, p. 52) shows that if b is as defined above, then

1- .,P(z) _ {1- z)b = g(z) is continuous and non-zero in D 1, and

nV

where Vis a small closed disk centered at

,p- 1S{l,t) c DnV for t sufficiently small. For such t

{z:

~(z)

E S{elt t)} = {z: II- 1/J o cp(z)l < t} = {z: lg(- 1S(et, t) ~ uNrp- 1S(I, (tfM) 1111 ) = O"Nrp- 1Q(I, (tfM) 1111 ) = u 1Q(1, (t/M)tfb) "'cttfb

Choosing a= aN = &«(7r/(2N)) gives 1/b = Nand thus Cto is not compact

onHP(BN)· If we choose a> aN then 1/b - 1S(et, t) =F O(tN) so that Cto is not bounded on HP(BN ).

I

The same slice integration technique which is at the heart of the proof of Theorem 6.4 will be used in the next theorem, which gives a boundedness result not by restricting rp but rather by enlarging the range space for the operator C.p. THEOREM6.5

If IP is any analytic map of B N

into itself. then the weighted Bergman space A~_ 2 (BN ).

c

-1. By Theorem 2.36 and Exercise 2.2.7 this translates into the estimate H

T.-1 S(ei8 't)< cto+2

,-a,

where 1-'o is the weighted area measure on D given by dl-'a = (1 - lzl 2 ) 0 dA where C does not depend on 1J or (since each map rf'/ fixes 0. Write A for rp- 1S((, t), convert to polar coordinates (Exercise 2.1.13) and use slice integration to obtain

/

JBN

XA(z)(1-lzi 2 )N- 2 dvN(z)

Chopur 6: Several Variables

228

=

t r2N-1 (1 - ,_l)N-2 JBBN f {21f XA(re"11) dfJ da(11) dr lo 211"

lo

~c{

{

J8BN JD

XA(Z11)(1

-lzi 2)N-2 dA(z) 00(71)

Now XA(Z11) = 1 if and only if r,o(z71) is inS((, t) which occurs if and only if 11- (r,o(z71),()l < t if and only if r,(z) is in 8(1, t). so the inner integral is bounded by some absolute constant times tN and therefore

1

op-IS((,t)

(1

-lzi 2)N- 2 dvN(z) ~ CtN

where C does not depend on either ( or t.

I

Exercises 6.1.1 Let~: BN-+ BN be an arbitrary analytic map. Show CT~- 1 8((, t) ~ Ctfor all (in E1BN and t > 0, where C depends only on ~(0). 6.1.2 If~ : BN -+ BN has no finite angular derivative at any point of E1BN then C, mapping H 11 (BN) into A~_ 2 (BN) is compact. 6.1.3 Consider ~(z1, z2, ••• , ZN) = (2z1Z2,0'). Show that C, does not map H 2(BN) into A~(BN) for any a < (N - 3)/2. Compare this with the statement of Theorem 6.5. Do you get a better result by considering ~(zt, z2, ••. , ZN) = (c(a)z 0 ,0') where a is a multi-index other than (1, I, 0') with at least two nonzero entries, and

Notes Theorem 6.4 is due to B. D. MacCluer [Mc85]. Theorem 6.5 is a special case of a result due to B. D. MacCluer and P. R. Mercer [McM93]. Both Theorem 6.4 and Theorem 6.5 have generalizations that hold for composition operators on smoothly bounded. sttongly convex domains in CN; see [McM93] for these results. Theorem 6.5 has also been generalized by J. A. Cima and P.R. Mercer [CiMe94] to composition operators between two weighted Bergman spaces in the ball (or more generally in smoothly bounded sttongly convex domains in CN). Their result shows, in particular, that C, will always map A~(BN) boundedly into A:+N-l(BN)· The result of Exercise 6.1.3 also appears in [McM93].

229

6.2

Wogen's theorem

In this section we look at a remarkable result that gives a necessary and sufficient condition for a smooth cp to induce a bounded composition operator on HP( B N) for 0 < p < oo. By smooth we will mean that cp is of class C 3 on B N; this will be a standing hypothesis on cp throughout this section, and we will regard cp as being defined on B N. Our condition for boundedness of Crp will involve a relationship between certain directional derivatives of the coordinate functions of cp at points of 8B N which map to 8B N. We begin with a few facts about some directional derivatives at a fixed point of 8B N. LEMMA6.6

Forcp: BN-+ BN analytic and ofclass C3 on BN withfixedpointatet = (1,0')

>0

(2)

Dt'Pt(et) ~ (1 -lrpt(O)I)/(1 + I'Pt(O)I) DA:rp1(et) = Ofor k = 2, 3, ... , N

(3)

Dt'Pt(et) ~ IDA:k'Pl (et)lfor k = 2, 3, ... , N

(1)

where cp = (rp 1, rp2 , ••• , 'PN ). PROOF If rp 1(0} = 0 then (1) follows immediately from the Schwarz Lemma (applied to the restriction of rp 1 to the complex line through e 1) and the JuliaCaratheodory Theorem (Theorem 2.81). If rp 1(0} = a :f: 0 consider 1/J : D-+ D defined by 1/J(z) ='Po o [rp 1]e1 where [rp 1]e 1 (w) = rp 1(wet) and rp0 (w) =(aw)/(1- aw). Since 1/J(O) = 0 and 11/J(l}l = 1 the Julia-Caratheodory Theorem implies that 11/J'(l}l ~ 1. This translates into lrp~(l)Dtrp 1 (et)l ~ 1 or

IDt'Pt(et)l

~

II -

al 2 1 -lal2

~

11+

lal lal

as desired. For (2) and (3) fix k, 2 :5 k :5 N and consider cp on the circle r(t) = (cost, 0, ... , Asin t, ... , 0} where A is any constant of modulus 1 and A sin t appears as the kth component of r(t). Set h~(t) = Rerp 1(r(t)); this function has a maximum at t = 0, since rp 1(e 1) = 1. A calculation shows that h~(O) = ReADA:rp 1(et) for all A with !AI = 1. Since h~(O) = 0 we have DA:'Pt (e1) = 0, giving (2). Similarly, one computes h1(0) = -Dtrp 1(et) +ReA2 DA:A:'P1(et) and this must be non-positive. So 2

ReA DA:A:'P1(et) for all A with IAI = 1 and (3) follows.

I

:5 Dt'Pt(et)

230

Chllpter 6: Se'l'eral Variables

From this point on, we confine our attention to the case N = 2 and write B for B2 and u for u 2. This allows certain notational simplifications while still capturing all the essential ingredients of the arguments. At the end of the section, we will indicate how the results generalize for N > 2. PROPOSITION 6.7

lfv>(eJ) = e1 and DJv> 1(eJ) = IDnv> 1(eJ)I then Crp is unbounded on HP(B)for 0

1(e1). Forl~l = 1definetheunitarymapU(zltz2) = (zlt~z2). Choose~ so that ~2 D22rp 1(eJ) = ID22v> 1(eJ)j. If 1/J = rp o U then 1/J(eJ) = e~o D11/JJ(eJ) = D1rp1(eJ), and D22¢1(e1) = ID221PJ(e1)l. Thus 1/J fixes e1 and satisfies D1 1/J1 (e1) = D221/J1 (e1 ). But, since Cu is an isometric isomorphism of HP(B), C"' is unbounded if and only if Ct/J = CuCrp is unbounded, so we may assume DIIPJ(eJ) = D22v> 1(e1). By Theorem 3.35 it will suffice to show that u!p- 1 S(e~,6) -::/: 0(62 ). To do this we will produce sets A( 6) C aB and a positive constant C so that for all sufficiently small 6 > 0, PROOF

(i) (ii)

v>(A(6))

c

S(e~, C6)

u(A{6)) "'62 log ~

Parametrize a neighborhood of e1 in aB by the one-to-one map A : {x E R3 : lxl < 1/2}- aB where A(x1,x2,x3)

= (v't-lxl 2 + ix~,x2 + ix3)

Note that for a set A in the image of A, u(A) "' m3(A- 1{A)) where dm3 is Lebesgue volume measure on R3, since

du =

dm3 Jt-lxl 2

We will describe our desired sets A(6) as A{S1(6)) where (see Figure 6.1) 0(6) = {(X1,X2 1 X3): 0

< XJ < 6,62/ 3 < X3 < 6 1/ 2,0 < X2 < 6jx3}

Calculate m3(S1(6)) =

r

lo

1 6

=

dxl

rl/l!.... dx3

}62/3 x3

6{log 6 112 - log 6213) dx 1

I

= d 2 log-

6

Wogen \' theonm

231

l

Thus u(A(6)) "'62 log which is (ii). To verify (i) setg(x) = cp 1(A(x)) -1 so thatg is C 3 on {lxl < 1/2} c R 3 with g(O) = 0. From the definition of S(e1!6) we will have cp(A(0(6)) C S(ell C6) if lg(x)l < C6for all x in 0(6). Using the chain rule and the fact that D2cp1(ei) = 0 we compute that {}g

8x2

(0) = 0 and

{}g

8x3

(0)

=0

while the hypothesis that D1cp 1(ei) = D22cp1(e1) gives that

&g (0) = 0

{}x~

We use these calculations to write the Taylor expansion of g about 0:

{}g g(x) = B(O)x1 X!

1 ~ ~

+2

j,k=l

&g

{} {} Xj

(O)x;xk

Xk

+ O(ixi

3

)

(6.2.1)

where fzi(O) = 0. If x = (xllx2,x3) is in 0(6) we certainly have lx1l < 6, lx2l < 6ll3, ixJI < 6 112, and lx2x3 i < 6 and all terms on the right side of Equation (6.2.1) are 0(6) for x in 0(6) and 6 < 1, giving (i). I

FIGURE6.1

The sets 0(6) in R3 •

Clulpur 6:

232

s~~~~rul

Variables

Lemma 6.6 and Proposition 6.7 have non-normalized versions that do not assume rp(e 1) = e 1• To state these, we first set some notation. For 'I in OBN let r,o,(z) = (rp(z), 'I) be the coordinate of rp in the ~tion. In the special case 'I= e 1 = (1,0) or e2 = (0, 1) we will continue to write rp 1 and r,o2 for these coordinate functions of rp. For (in 8BN with ( = ((t,(z), let (l. denote (-(2,(1 ) so that ((, (l.) = 0. Note that ef = ez. The directional derivative De in the ( direction satisfies Dcr,o,(z) = (rp'(z)(, 'I)· With this notation we have the following generalization of Lemma 6.6. LEMMA6.8 /frp(() ='I where

1 2. If(, '1· and rare in anN with ~P(() = 17 and ((, r) = 0 then we always have Dc~P.,(() > 0 and Dc~P.,(() ~ ID.,..,.~P.,(()I. The following two results characterize the bounded composition operators on HP(nN) with smooth symbol. THEOREM 6.14

If !p : n N - n N is analytic and !p is of class C 3 on B N and if there exist (, '1· and Tin aBN With fP(() = '7 and((, r) = 0 SO that

Dc~P.,(() = IDn~P.,(()I

then C'P is unbounded on HP(BN ).

THEOREM 6.15

If !p : n N - B N is analytic and !p is of class C 3 on n N and if Dc~P.,(()

for all (in 8BN with ~P(() bounded on HP(nN ).

>

IDn~P.,(()I

= 17 in 8BN andr in 8BN with {(,r) = 0, then C'P is

Chapter 6: Several Variables

242

Exercises 6.2.1 Show that there is an absolute constant c so that lz 8B and z in B. 6.2.2 If cp is of class C 3 on B with sup { ucp-•;("1, t) : 'I E 8B, t

(1 2 ~ cd(z, () for all (

> 0}

in

= oo

then there is an 110 in 8B such that sup {

CT 0} = oo

that is, if the Carleson criterion fails to hold, then it fails at a single point. 6.2.3 If 1/J =

11"/3. 6.2.6 Assume that cp : B -+ B is analytic and cp' is uniformly bounded. Note that by Lemma 3.40 there exist positive constants c and to so that if cp(() = 'I where where 1 - cp11 (et) - R.,(z, et)

I'll=

1 then

cp(Q(C, t))

c s(.,, ct)

for all t < to. The constant c depends on cp(O) but not on t fort < to. Use this to show that there exists a finite constant B so that if d(z, () < to where (is in So then d(cp(z), cp(()) $ Bd(z, (). 6.2.7 Suppose A is anN x N matrix with IIAII $ I and set cp(z) Az. Show that C"' is bounded on H'P(BN ). Indeed, show that C"' is bounded on any weighted Hardy space H 2 ({J, BN ).

=

Notes We have followed W. Wogen's [Wog88] original argument very closely; we have however corrected a missing hypothesis in Lemmas 6 and 7 of that paper.

243

Exampks

6.3

Examples

Propositions 6.7 and 6.9 of the previous section do not, by themselves, make it any easier to construct examples of smooth r.p: BN-+ BN for which will be unbounded on HP(BN ). In this section we develop some tools which will help construct some unbounded composition operators whose symbols, in contrast to the examples in Chapter 3, are highly non-degenerate maps of B N into B N. We will concentrate on examples in B2 and we write B for B 2 throughout this section. We begin with the disk automorphism r.p( z) = ( z + a) f (1 + az) for 0 < a < 1 and its second order Taylor polynomial P2 ( z) at z = 1. A computation shows that

c.,

p2 ( z )

=

a+ 3a2 1 + 2a- 3a2 a(1 -a)~ (l+a)2 + {l+a)2 z- {l+a)2

(6.3.1)

Notice that P2(1} = 1 and P2(8D) has second order contact with 8D at 1. Set Ct = a+ 3a2, c2 = 1 + 2a - 3a2, and c3 = -a(1 - a) so that P2(z) = (ct + c2z + CJzl)/(1 + a) 2. The next lemma will allow us to estimate IP2(z)l on circles of radius r, 0 < r < 1". LEMMA6.16

If f(z) = Ct +c2z +c3z2 with c,, c2, and C3 as above, then for any r in (0, 1} and z with lzl = r we have 1/(z)l :::; f(r); moreover the inequality is strict if z f. r. PROOF

= x + iy where lzl = r.

Write z

lf(z)l 2 = =

Cf + Cf +

Then

~r2 + ~r4 + 2CtC2X + 2 (ctC3.z2- CtC3Y2 + C2CJX(z2 + yl)) ~r2 + ~r4 + 2CtC2X + 2ctCJ(z2- yl) + 2c2C3X~

From the definitions of c,,c2 andc3 we have -4ctc3 = CtC2 +c2c3. Some algebra shows that lf(z)il is equal to

Cf +c1c2 +c2c3 + (~- 2ctc3)~+~r4 - (c1c2 +c2c3)(1-x} 2 -2c2c3x(1-r2) Since -c2c3 and Ct c2 + c2c3 are positive this is bounded above by

Cf + CtC2 + C2C3 +

(~- 2CtC3)r2 +~r4 - {ctC2 + C2CJ}(1- r) 2 - 2c2CJT(1- r 2)

which is f(r) 2 and equality holds only if x = r.

I

As a consequence of this lemma notice that if z is in D with lzl = r then

1 1 IP2(z)l = ( 1 + a) 2 1/(z)l :::; (I+ a) 2 f(r):::; 1 where the last inequality uses the fact that 0 < a < 1. Thus P2 (z) maps D into D. Moreover P2 is one-to-one on D since if Ct + c2z + c3~ = c, + c2w + c3wl we

Cluzpt~r 6: Sev~ral

244

Variables

have c2lz- wl = lclllz- wllz + wl so that c2 = IC3IIz + wl when z :F w. This implies 1 + 2a - 3a2 = a( 1 - a) lz + wl $ 2a - 2a2 which contradicts a < 1.

Next we will construct a mapping of the form

~(Zt1z2) = (P2(z1) + kt~1k2~)

(6.3.2)

with k2 :F 0 and real so that ~(B) c Band yet C"' is unbounded on H 11 (B). For such a map ~( Zt 1z2) = ~( Wt 1toz) if and only if Zt = Wt and ~ = since P 2 is one-to-one on 15, so that~ will be at worst two-to-one on B. Notice that rp(e 1) = et and Dt~t (et) = (1-a)/(1 +a) while D22~ 1 (et) = 2k~o so provided ~(B) c B, C"' will be unbounded (by Proposition 6.7) if 2 k 1 = ~ = (-a-) c2 +2c3 2+2a 1 +a 2a2

wi,

Thus it remains only to see whether, with this choice of k 1, there is a choice of k2 so that ~(B) C B. To this end we write l~(z~tz2)12 = l~ 1 (z~tz2)1 2 + l~2 (zt1z2)l 2 and let z = (zt I Z2) be in aB with lztl = r so that lz21 2 = 1 - rl. We have

= P2(zt) + kt~

~ 1 (z)

Using Lemma 6.16 we have 2

l~t(z)l $

a ) ( 1 +a

2 $ (-a-) 1+ a Since ez + 2c3

l~.(z)l $

c

(~ + a2

c2 r + c3 r2 +I c2 + 2c31 (l _ rl)) a2 a2 2a2

~ay (c• +ezr+C3rl + c2 ~2c3- (ez +~3)rl) 1

(

= (1 ~a)2

> 0.

(I a2Ct + a2zt c2 c3 21 lc2 +2c3 Jl) + a2Zt + 2a2 ~

> 0 we have

= ( 1 + a ) 2 c,

since ez

J)

2 a ( Ct c2 c3 2 c2 + 2c3 =(l+a)2 a2+a2zt+a2zt+ 2a2 ~

C2 + Cz + C) - 2 (1 - r) 2)

((1 +a)2- ~(1-r)2) $1

Thus

l~(z)l 2 Sl~,(z)l 2 + 1~2(z)l 2 Sl~t(z)l + 1~2(z)l 2 $ (1

~ a)2

((1 + a)2- ~ (1 -

= 1 - ( 1 - r )2 ( 2( 1

~ a)2 -

r)2) +4,q(l - r)2

4,q)

245

Exampks

Any choice of k2 satisfying C2 2{1

2

+ a)2 > 4 ~

will guarantee that cp(B) C B; for each a in {0, 1) some such choice of k2 will be possible since c2 = 1 + 2a - 3a2 > 0. As a specific example let a = 1/2 so that Ct = 5/4, c2 = 5/4, c3 = -1/4 and P2{z) = {5 + 5z- z 2)/9. Then cp{zt, Z2) =

5

5

12

(9 + 9z1- 9ZJ +

1...2 2 6~' k2Z2)

where k2 is chosen to satisfy 0 < k2 < ../5f(6.fi) will be an at worst two-to-one map of B into B, analytic in B, for which Crp is unbounded on HP(B). Our next goal is to construct a biholomorphic map of B into B which gives rise to an unbounded composition operator. Again the Taylor polynomial a + 3a2 1 + 2a - 3a2 a( 1 - a) z2 p2 (z ) = {1

+ a)2

+

{1 + a)2

z- {1 + a)2

will form the basis of our example. Let l(zt, z2) be the first coordinate function of cp as defined in Equation (6.3.2), that is let

l(zt. z2) = P2(zt) + ktzi where kt = {1-a)/(2+2a) with a in (0, 1), so that as before we have Dtf(et) = Dnl(et)· From I we construct a map 1/J by 1 1/J(Zt.Z2) = 2{1 + l(zt.Z2),z2(1- l(zt,Z2))

(6.3.3)

Note that the components of 1/J are polynomials of degree at most 3. Several observations are essentially immediate: • 1/J(et) = et • Dt'I/Jt(et) = Dn'I/Jt(et) • 1/J is one-to-one on B For the third of these observations note that 1/J( Zt, z2) = 1/J(Wt, W2) for points (zt,Z2) and {Wt.U12) in B implies that l(zt,Z2) = I(Wt.W2) and z2 = U12. since l(z) = 1 for z in B only if z = et. The assertion then follows since l(zt. z2) = P2(zt) + kt~ and P2 is one-to-one on D. We claim that 1/J maps B into B. To verify this we estimate 11/J( Zt, z2) 12 when lztl =rand lz2l 2 = 1- r2 forO~ r ~ 1. Then 2 411/J(zt. z2)l = =

II + l(zt. z2)l 2 + lz2l 2ll - l(zt. z2)l 2 1 + 2 ReI+ 111 2 + {1 - rl){l - 2 ReI+ 111 2)

1/1 2 -

2 2 2 r + 2r Ref+ (1 - rl)l/1 ~ 2 + 111 2 - r2 + 2r2111 + {1 - rl)lll 2

= 2+

246

Chllpter 6: Several Variables

Recall our previous estimate on 1/(zh z2)l when lztl = rand lz2l 2 = I -

Ill $; where 0

l is biholomorphic to the Siegel upper half space n = {(WJ I • • • I w N) = (W1 I w') : Im WJ > lw'l 2 } via the Cayley transform

;,..( ) .e1 + z 'A' z = 1 - l - ZJ

and for each b = (b 11 b') in 00, we have a Heisenberg translation hb in Aut(O) defined by hb(w., w') = (w 1 + b1 + 2i(w11b'} 1 w' + b') which fixes oo only; the corresponding map ~-I o hb o ~is an automorphism of BN fixing e1 only, and is called a Heisenberg translation of BN. By Theorem 2.90 the automorphisms of BN that fix e 1 only are either Heisenberg translations or they fix, as a set, some non-empty, proper, affine subset of B N. This fact allows us to treat the nonHeisenberg translation automorphisms in an inductive fashion. We first dispense with those t.p which are Heisenberg translations. LEMMA7.9

Suppose (b., b') is in 80 with associated Heisenberg translation h(~,1t) and let H(b.,b') = ~-I oh(b.,b') o~ be the corresponding automorphism ofB N. Then every ~ in the unit cirde is an eigenvalue of infinite multiplicity for C., on H 00 ( B N ).

We first suppose that b' :f: 0 and claim that the composition operator with symbol H(b.,b') is similar to the operator with symbol H(tReba+i,r') where lb;l 2 = 1/t and r' = (r2 1 . . . 1 TN) with r; = v'tlb;l 1 j = 2 1 . . . 1 N. To see this it suffices to find an automorphism T ofn with roh r} into itself, then the first r coordinate functions for cp depend only on the variables Zt, • ••

,z...

7.3.3 Complete the proof of Lemma 7.9 for the case h(w 1 , w') = (w 1 + b1, w') where b, is real and non-zero. 7.3.4 What is the spectrum of C., on H"(BN) for p ~ 1, p =f. 2 in the case that cp is an automorphism fixing two points of aB N?

Notes The spectrum of C., acting on H"(BN ), N > 1 and p ~ 1, when cp is an automorphism of BN was identified by B. D. MacCluer in [Mc84b]. The results given here are minor modifications of the arguments in that paper.

264

7.4

Cluzpter 7: Spectra

Spectra of compact composition operators

The general principle that the structure of the composition operator Crp is closely related to the fixed point behavior of the function r.p is well illustrated by compact composition operators. Determining the spectrum of a compact operator is equivalent to finding the eigenvalues of the operator. We will prove here a theorem of considerable generality, which will show that in essentially all of the spaces of interest to us these eigenvalues are determined by the derivative of r.p at the Denjoy-Wolff fixed point of r.p. This statement will need some clarification in the case of spaces defined on BN. N > I as then ''the" Denjoy-Wolffpoint may not be well-defined. Our first result generalizes Theorem 2.17 on the behavior of the normalized kernel functions in weighted Hardy spaces in the disk. PROPOSITION 7.13

Suppose H 2({3) is a weighted Hardy space in the disk.

If

2k

00

L {3~n)2 = oo

n=O

for some non-negative integer k, then lening K~k) denote the kernel function for evaluation of the kth derivative at w, we have

K( ) - ""' . w z - ~ (n- k)! n=k

=:nw -lczn

{3(n)2

so

Since n" /2" ~ n!/( n- k)! ~ n" for n ~ 2k, the last sum tends to oo as

lwl -+ 1

265

CorrqH~£1

precisely when 21c

00

L

n=O

.B7n)2 = oo

which is our hypothesis. Now consider any weak limit point f of K~k)

IIK~Ic)ll for a sequence of w's tending to the boundary of the disk. If pis any polynomial we have

l(p, ,)1

-

lim l(p

- w-8D

K~k)

, IIK~>n

)I -

-

lim IP(k)(w)l -0

w-8D IIK~Ic)ll

Since the polynomials are dense in H 2 (,B) we must have f = 0.

-

I

The same expression for IlK~) II used in the proof of this result shows that if

for some j, then IlK~) II is bounded as w approaches the unit circle, a fact that we will need shortly. The next result will be key to describing the spectrum of when is compact.

c'P

c'P

THEOREM 7.14 Let H 2 (,B) be a weighted Hardy space on the disk. Suppose 00

21c

L .B7n)2 = oo n=O for some non-negative integer k. /fC.p is compact on H 2 (,B), then r.p has a unique fixed point in the open unit disk. PROOF Let k be the least non-negative integer for which the sum in the statement of the lemma diverges. If k = 0 the argument is familiar: Assume r.p has no fixed point in D and let ( denote its Denjoy-Wolff point in 8D, so that the angular derivative at (is not greater than 1. By Julia's Lemma (Theorem 2.41), r.p maps the disks

Clulpt~r

266

into themselves, for each c have, by Proposition 3.13

> 0.

7:

S~ctra

In particular,jrp(r()l ~ r for all r in (0, 1). We

so Crp cannot be compact, which is our desired contradiction. Uniqueness of the interior fixed point is obvious, since rp is not the identity. For the case k ~ I we must work a little harder. We begin by identifying c:(K~>). To do this notice that for J in H 2 ({3) (J,c;K~Ic)) = f'(rp(w))rp'(w) when k =I

and fork> 1

(!, c;Kf:>) =Die(! o rp)(w) = J(lc)(rp(w))(rp'(w))lc + J'(rp(w))rp(lc)(w) +lower order terms

where j(lc) and rp(lc) indicate the kth. derivative and "lower order terms" means terms involving products of derivatives of f at rp( w) of order less than k and derivatives of rp at w of order less than k. Thus if k = I 1>) = ·"'(w)K(t) C*(K< rp w ~ rp(w)

and if k >I

c:(K~Ic)) = (rp'(w)]lc K~~~> + rp(lc)(w)K~1~w) + lower order terms where the "lower order terms" involve kernels for derivatives of order less than k at rp(w) with coefficients involving products of derivatives of rp at w of order less than k. If rp has no fixed point in D and therefore has Denjoy-Wolff point at some ( in an consider --(1)

llI

K;~>

rp(r()

IlK;~> II

=

(rp,

IlK;~> II) -

0

as r tends to 1 by Proposition 7 .13, since rp is in H 2 ({3). As I K~ >II is bounded as

267

Compact

w tends to 8D, we have for the case k

>

1

IP(k) (r{)K(t) II is bounded as w tends to aD, as is IIPU>(w)l = I(~,D,KM>}I :5 II1PIIIIKM>11 while IlK~) II- oo as lwl- 1, so that, in the case k > 1, the terms in Equation (7 .4.1) that are not explicitly shown all tend to 0 as r tends to 1. Putting it all together we see that if is compact and k > 0 is the least integer for which E n 2k / (3( n )2 = oo we must have

c;

IIP'(r{)lkiiK~~~c>ll _ IIK$~>11 as r 1,0

1

( {)

1. Since { is the Denjoy-Wolff point of

'# 0 and therefore

0 !p,

we must have

IP'(r{) -

IIK~~~c>ll- 0 IIK~~>n

which is a contradiction since IIP(r{)l ;::: r. Thus 1,0 must have fixed point in D; and the uniqueness of this fixed point is obvious. I The proof shows that the important features of the weighted Hardy spaces that were used were that the norms of the kernel functions only depend on the modulus of the point and that the norms for the kernel functions for the appropriate order derivatives tend to infinity as one approaches the boundary. The more difficult part of the proof of this last result is only needed in the case that E (3( n) - 2 < oo and H 2 ((3) is not automorphism invariant, since otherwise Theorem 4.5 would apply to show that compact implies ll1Plloo < 1 and 1,0 then clearly has a unique fixed point in D. This theorem can be used to show that some weighted Hardy spaces contain only smooth functions.

c.,

COROLLARY 7.15

Let {(3(n)} be an increasing sequence satisfying

~ (3((an})

~ (3(n) < oo for some a, 1/2 < a < 1, where [x) denotes the greatest integer less than or equal to x. If H 2((3) is the associated weighted Hardy space on the disk, then eachfunction in H 2 ((3) is in C 00 (D).

268

Clulpter 7: Speclnl

PROOF In Example 4.6, we showed that under the hypothesis above, the composition operator C"' with symbol r.p(z) = (z + 1)/2 is compact on l{l([J). Since a = I is the only fixed point of r.p in the closed disk, Theorem 7.14 implies 00

2k

~{J7n)2 < oo for every positive integer k. Now suppose /(z) = E11nzn is a function in l{l([J). Then by the Cauchy-Scbwarz inequality

(Lnklllnlf ~

(L P7:;2) (LIIlni2P(n)2)

2 = ( L P7:;2) 11111 < oo Since 1/(k)(z)l ~ in C 00 (D). I

E nklllnl for all points of the closed unit disk, we see that f

is

The part of the proof of Theorem 7.14 that works with kernel functions for evaluation of derivatives does not generalize to several variables, chiefly because the expression for the adjoint of c"' applied to kernel functions for derivatives of order at least one involve, as coefficients, derivatives of all the components of 0, but this is incompatible with the assumption that a subsequence { z~~:1 } of the iteration sequence converges to b with k; tending to -oo. On the other hand, if b '::/: a, then there is f > 0 so that Dt = { z : lz - bl ~ f} is contained in D and does not contain a. In this case, the iterates of Dt converge to a and there is n so that 'Pn(Dt) n Dt = 0, which also contradicts the assumption that a subsequence { z~~:1 } of the iteration sequence converges to b with k; tending to -oo. If cp is not an elliptic automorphism of the disk onto itself and Zo is a point of the disk, then = cp,.(zo) defines an iteration sequence of distinct points for k ~ 0 or else there is a least M so that cp M zo = a, the Denjoy-Wolff point of cp in D, and there is an iteration sequence of distinct points defined for 0 ~ k ~ M. Moreover, either there is no point w of D with cp( w) = Zo or we can find z_ 1 so that cp( z_ 1) = Zo· So every point of the disk is in at least one iteration sequence {z,. }~K of distinct points for which either M = oo and Iim,._. 00 z,. = a or M is finite and 'PM(ZO) =a and either K = -oo and Iim~~: .... -oo lz~~:l = 1 or K is finite and there is no point win the disk with cp(w) = ZK· The case {z,.n=-oo in which IimA:-+-oo ZA: = b where b is a fixed point of cp on the circle is especially interesting. If b is not the Denjoy-Wolff point, then we

c;

z,.

Boutulary fixed point, cp' (a)

pogsl log rp'(b;)

~~:-:'-:­

so by Theorem 2.64, there are M; and t:; so that lu(z)l ~

M;lz- b;l-flog•l/l21og~l

for lz- b;l < t:;. Putting these inequalities together, we see that there isM so that for z in D,

II lz- b;l-flos•I/I2Jog~l k

lu(z)l ~ M

j=l

and for r"0

112

< ..\o < ,\
I so that 1 -I~P(z)l >A

1-lzl

for all z with

lzl 2:: r.

Chapter 7: Spectra

292

PROOF The Schwarz lemma and the fact that

.25}

Since {lz~cl} is a decreasing sequence converging to 0, it follows that n ;::: 0 and lz~:l < .25 fork> nand lz~cl ;::: .25 fork~ n.) If IZnl > .5, then IZn+tl < .25 implies IZn+tl ~ .51Znl· Lemma 7.35, together with this observation, shows there is a number c with .5 ~ c < 1 so that lzlc+tl ~ clz~:l whenever k ;::: n. Applying this inequality repeatedly, we find, for k ;::: n,

(7.6.2) Let M be the interpolation constant for Lemma 7.34 with r = .25. (Of course, M ;::: 1.) Choose a positive integer m large enough that 1 w.I is not bounded below on 'H.m. If {zk}~K is an iteration sequence for cp with IZ .S, then the series

L 00

>,-kK:;

k=-K

converges absolutely. Indeed, for k

> n, since lz~; I < .25, we see that

and it follows that

We also need a lower bound for II E:.-K >.-kK:;II· By Lemma 7.34, we can find a function fin H 00 (D) so that 11/lloo ~ M and fork ~ n,

z;:' f(z~;)

1/(z~;)l = I and ).k(l _ Z()Z~;)

>0

(7.6.4)

Now by Eq. (7.6.4)

=

4MI.znlm fiT < 4MI.znlm rll < 4Miznlm _1_ 3l>.ln 1 - fiT - 3l>.ln 1 - ;k - 3l>.ln 6M

This means

=

2l.znlm 9l>.ln

Interior fixed point

295

We have used II - ZoZn I $ 2 and the final inequality is obtained by ignoring terms with k 1= 0. Since

zmf

Ill- zozll

M

$

y'l-l.zol2

the Cauchy-Schwarz inequality now yields the desired lower bound: oo

II ~

>.- k Km II > Zk

1("

00

LJ-K

-

>.-kKm zk• ~}~ 1-zoz >

l"~lm -v

- 2M y't -lzol2

llt%11

1 Km -2M II Zo II

--

Note that since 'H.m is invariant for C"', for any g in 'H.m,

It follows that

(C*m

>.I)(~ >.-kKm) = ~

~

->.K+1Km ~K

-K

and

Finally we make our choice of iteration sequence { zk}~K· Since

and we are assuming p

> 0 we must have lim sup IIK'Pk(w) II lwl-1

= oo

296

Chapter 7:

S~ctrtz

for every k. Since m is fixed and IlK~ II :5: IIKull :5: IlK~ II+ m for all u in D, for the w near 8D important in the computation of p

so the essential spectral radius may be computed .

.

p = k-ooo bm ( hmsup lwl-ol

II) 1/k II Km .Sand

llx;'x(w) II ( ')K IIK:all > p Let Z-K = wand Zk+l = tp(z~c) fork ~ -K. Thus the iteration sequence { z~c}~K has lzol > .Sand our calculations give, for this iteration sequence

Since 1.>.1 < p' we may choose K sufficiently large so that this is as small as desired. Thus C!- >.I is not bounded below and Xis in u(C.p) as desired. I Kamowitz [Kam75] was the first to investigate spectra of composition operators whose symbol is not an inner function and has a fixed point in the disk. The hypothesis of his theorem is analyticity in a neighborhood of the closed disk. From the perspective of the proof of Theorem 7 .30, this hypothesis allows him to calculate the essential spectral radius in terms of the derivative of the function tp at its fixed points on the unit circle and to find iteration sequences on which the estimates in the proof are still valid. Specifically, if tp is analytic in a neighborhood of the closed disk and is not an inner function, Exercise 7.5.2 guarantees that there is an integer n so that Sn = { w : lwl = 1 and I~Pn(w)l = 1} is either empty or consists only of the (finitely many) fixed points of ~Pn on the circle. If tp has a fixed point a in the open disk, then, for such an n, the derivatives of ~Pn at each of the fixed points on the circle are bigger than I and the essential spectral radius of is

C;

max{tp~(w)- 1 12 : wE Sn}

Applying the spectral mapping theorem for the Calkin algebra, we see the essential spectral radius of C

, then ei9 A is also an eigenvalue

ofGrpfor each positive number 0. PROOF

If g is an eigenvector for Crp with eigenvalue A, since

/9

is in H""(D),

3()()

ClulpUr 7: Spectra

/99 is in JI2(D) and Ccp(/99){z) = /9(cp(z))9(cp(z)) = ei8 /9(z)>..9(z) so /99 is an eigenvector for C., with eigenvalue e19 >... Since the constant function 1 is an eigenvector for C., with eigenvalue 1, each point of the unit circle is a eigenvalue of infinite multiplicity for C'P and the unit circle is contained in the spectrum and the essential spectrum of C'P. I

c.,

However, the functions f 9 do not induce a similarity between and eiiC'P as in the case cp' (a) < 1 because f 9 is never invertible in H 00 (D). Indeed, since the set u(V) of Theorem 2.53 is a fundamental set for translation in the upper halfplane, for each positive number M, there is z in D so that Im u( z) > M, so for each t: > 0, there is z in D so that lh(z)l < t:. Results pertaining to the fixed points of cp besides the Denjoy-Wolff point, such as Theorems 7.23 and 7.29, apply to c'(J for cp in the halfplane/translation case as well. For cp not an automorphism but in the halfplaneltranslation case, the picture that emerges is that the spectrum of C., is a subset of the closed unit disk that contains the unit circle and a disk (perhaps degenerate) centered at the origin. However, no examples are known in which the spectrum is a proper subset of the unit disk for this case and the spectrum of C'P for lal = 1 and cp'(a) < 1 (given by Theorem 7 .26) suggests that the spectrum should be the unit disk. However, the situation is radically different for cp in the plane/translation case. The most obvious reason this should be so is that the weighted shift analogy that works in the halfplane/dilation case depends on the near orthogonality of the kernel functions for the iterates of a point but in the plane/translation case, these kernels are not close to orthogonal. The evidence that this is more than just a technical inconvenience is that the spectra for c'(J when cp is in the plane/translation case need not show any circular symmeby. This case is poorly understood; we present a class of examples that permit calculation of spectra but do not suggest plausible general techniques. DEFINITION 7.40 A set { ll't : t ~ 0} will be called a one-parameter semigroup of anlllytic functions on D if ll't is an analytic map of the disk into itselffor each t with cp0 (z) z, ll't o cp. = ~~'•+t and (z, t) 1-+ ll't(z) is jointly continuous. A semigroup indexed by a complex parameter t will be called a holomorphic semigroup of aMiytic functions if there is T > 0 such that the semigroup properties hold for Iargtl < T and on this sector the map t 1-+ ll't(z) is analytic for each z in D.

=

An easy example of a one-parameter semigroup of analytic functions is {rt z : t ~ 0} where lrl ~ 1 and other simple examples can be constructed from other linear fractional maps of the disk into itself (see Exercise 7. 7.1 ). For 0 < r < 1,

the one-parameter semigroup above can be extended to a holomorphic semigroup {rtz : Ret > 0}. The model of Theorem 2.53 provides a good way to under-

Boundary fixed point, rp'(a)

=I

301

stand semigroups of analytic functions. Since automorphisms of the plane or the halfplane are always part of a semigroup, the intertwining 4> o u = u o t.p, which implies 4>n o u = u o 'Pn for positive integers n, can be formally extended to get 4>t o u = u o 'Pt· It is not difficult to show that every function in a semigroup of analytic functions is univalent (Exercise 7.7.2), so the intertwining map u is univalent and we see that we should have 'Pt = u-• o 4>t o u. This relation can be used to construct a semigroup 'Pt as long as the set u(D) is invariant under 4>t for t ~ 0.

If 'Pt is a semigroup of analytic functions, {C'Pe} is a strongly continuous semigroup of operators which can be studied with the tools developed in that theory [HiP57]. (A set of bounded operators {Tt} is called a strongly continuous semigroup of operators if T.Ti. = T•+t• To = I, and t ~--+ Texis continuous for each x in the space.) The examples of spectra of composition operators that we will exhibit arise as part of a semigroup in this way. Given 0 with 0 < 0 $ 1r,let G be the sector G = {(: I arg(l < 9}, and let u be the map 1 +z)29/w

u(z)= ( 1-z

which is a conformal map of D onto G. Let r( G) = G if 0 $ 1r /2 and r( G) = {t : Iarg tl < 1r -0} if 0 > 11' /2 so that, in either case, (+tis in G whenever (is in G and tis in r(G). Now, fort in r(G), define 'Pt by 'Pe(z) = u-• (u(z)+t) so that 'Pt is analytic in D with 'Pt(D) CD. Replacing u by uft, we see the model for 'Pt is also halfplaneltranslation and the function 'Pt also has Denjoy-Wolff point 1 with 'PH I) = 1. We will write Ct for the composition operator CV'e. In the case 0 = 1r /2 (so G is a halfplane) t + (2- t)z 'Pt(z) = (2 + t)- tz

for Ret

> 0.

In the case 0 =

11' /4,

we find

t 2 + 2tJI=ZI + (2 - t 2 )z 'Pt z = (2 + t 2 ) + 2tJI=ZI- t 2 z ( )

for I argtl < 71'/4. We will prove a spectral containment theorem using the theories of holomorphic semigroups and Banach algebras that will allow us to identify the spectrum of Ct in some cases. THEOREM 7.41

Let 0, G, r(G), and Ct be as above. Then on H 2 (D)

a(Ct) for all t in T( G).

c {e-.Bt: larg,81 ~ 111'/2-0I}U{O}

Clulpter 7: SpectrG

302

PROOF The set {Ct : t E r{ G)} is a holomorphic semi group of operators. Indeed, since operator valued functions are analytic in the norm topology if and only if they are analytic in the weak-operator topology (Theorem 3.1 0.1 of [HiP57, p. 93]), it is sufficient to check that the map t 1-+ (Ce{/), K,.) is holomorphic in r(G) for each fin H 2(D) and each z in D. But. (Ct{f), K,.) = f(u- 1(u(z)+t)) which is holomorphic in t because u-• and f are holomorphic. In particular, t 1-+ Ct is a continuous and holomorphic function of t in the norm topology for t in r(G). Let A be the norm closed algebra of operators generated by {f} U {Ct : t E r( G)}. Thus A is a commutative Banach algebra with identity and the Gelfand theory applies: the spectrum of Ct as an element of A. denoted u A ( Ce). is the set

{A(Ct) :A is a multiplicative linear functional on A} For A a multiplicative linear functional on A. let .X(t) = A(Ct) fort in r(G). Since A is multiplicative, the Gelfand theory implies IIAII = 1 and since Ct is a norm-holomorphic semigroup, we see that .X(t) is a holomorphic function on r{G) such that

+ t2) =

.X(tt

A(Ct,+t2 ) = A(Ct.)A(Ce2 ) = .X(tt).X(t2)

=

This means, either .X(t) 0 or .X(t) = e-flt for some complex number {3. In addition, we see that for every t in r(G), using Theorem 3.9 and the fact that IIAII = l, le-Ptl

= n~oo lim le-/Jntll/n = lim IA(G:')II/n n-+oo t ~ lim llc;'lll/n = tp~(I)-1/2 = l. n-oo

The definition of r(G) and this inequality imply I arg ,BI ~ l1r/2- 91. Putting all our information together, we have

u( Ct) C u A ( Ct) = {A( Ct) : A is a multiplicative linear functional on A} C {e-/Jt: I arg,BI ~ l1r/2-

as was to be proved.

91} U {0}

I

COROLLARY 7.42

Suppose 0 < 9 ~ 1r /2 and G, r(G), and Ct are as above. Then on H 2 (D)

u(Ct) = {e-/Jt: I arg,BI ~ 1rj2- 9} U {0} for all t in r( G). PROOF When I arg/31 ~ 1rj2- 9, the real part of {3u(z) is positive, so f(z) = exp( -{3u(z)) is in H 00 (D). Since f( ) with IAol = r. But this would imply, as A approaches Ao in the essential resolvent of Cv> with I.XI = r that the norm of the essential inverse of Cv> -.XI approaches infinity. For the appropriate(}, since Ls is a left inverse of Cv> - AI, it must be the essential inverse of Cv> - AI but

IlLs lie$ IlLs II$ IIAII(l- rniiAII)- 1(11Cv>ll

+ r)n-J

for all real(}. That is, the boundary of u e ( Cv>) does not intersect the circle of radius r but ue(Cv>) does intersect the circle, so we conclude that ue(Cv>) includes this circle. We have shown that Cv> - AI is left invertible but not Fredholm for all A with I.XI = r for r < rp'(a) 112 , as desired. I THEOREM 7.45

Let rp be an inner function, not an automorphism, with Denjoy-Wolff point a on the unit circle. Then on H 2 (D)

u(CV') = ue(CV') ={.X: I-XI$ rp'(a)- 112 }

c;

Theorem 7.44 shows {A : IAI < rp' (a) 112 } is a set of eigenvalues of of infinite multiplicity and Lemma 7.24 shows {.X: rp'(a) 112 n or, writing it as an integral,

c.,

Since C., is an isometry, its adjoint is the projection onto the range followed by

321

322

Clrllpter 9: Miscelllmea

its inverse, that is,

c:l = 'L.(f, ~n) zn or (C* f)(z) = •

J

l(eiB~

dO

1- ~(e'8 )z 211"

If cp is a general analytic function mapping D into itself, no satisfactory formula on H 2 (D) is known. Here we obtain a simple formula for when cp is for a linear fractional transformation. The adjoint is a product of Toeplitz operators and a composition operator.

c:

c:

LEMMA9.1

Jfcp(z) = (az + b)(cz + d}- 1 is a linear fractional transfonnation mapping D into itself, where ad- be= 1, then u(z) = (az- c)( -bz + d)- 1 maps D into itself. Linear fractional transformations may be regarded as one-to-one mappings of the Riemann sphere onto itself. Let iJ denote the open set

PROOF

iJ =

{z: lzl

> 1} U {oo}

Now, cp maps D into itself, so -y(z) = cp(z) also maps D into itself. It follows that -y- 1( z) maps iJ into itself. An easy calculation shows that 1

u(z) = which implies u(z) maps D into D.

-y-l(~)

I

Recall that for gin L 00 (8D), the Toeplitz operator T9 is the operator on H 2 given by T9 (f) = PIg for I in H 2 and P the orthogonal projection of L 2 onto H 2 • (For general properties ofToeplitz operators, see [Do72, Chapter 7].) THEOREM9.2

Let cp( z) = ( az + b) (cz + d) -I be a linear fractional transfonnation mapping D into itself, where ad - be = 1. Then u( z) = (az - c) ( -bz + d) -I maps D into itself, g(z) = (-bz + d)- 1 and h(z) = cz +dare in H 00 , and c: = T 9 CtrTt The function his clearly in H 00 • By Lemma 9.1, u maps D into itself and since the denominators of u and g are the same, g is in H 00 • This means the formula makes sense. Now, for winD, let Kw(z) = (1 - iiiz}- 1• This function is the reproducing kernel at w, that is, (f,Kw) = l(w) for I in H 2 (D). It is easily proved that Tt Kw = h(w)Kw and c:Kw = Ktp(w)· Calculation gives PROOF

T 9 CtrTt(Kw)(z) = h(w)T9 Ctr(Kw)(z)

Adjoints

323

-bz + d - waz +we

=

1

1- cp(w)z

= K.p(w)(z) = c:(Kw)(z)

Since the Kw span a dense set of H 2, the desired equality holds.

I

EXAMPLE 9.3 A Unitary Equivalence. Fort> 1, let cp(z) = z(t2 - (t2 - 1)z)- 1• We want to understand the operator C'P on H 2 (D). Easy calculations show that cp(O) = 0, cp(1) = 1, and cp(D) is the disk of radius t 2 (2t 2 - 1)- 1 inside the unit circle and tangent to it at 1, so c

0 and 6 > 0 so that for each z E D

or equivalently log( I - lcp(z)l) ~log 6 +log I(- cp(z)lk

Chapter 9: Miscellanea

340

This inequality persists at almost all points of the unit circle. Since (( - cp(z) )A: is in H 00 (D), log Ia). a ~ 1. Recall that C'P is compact on I>a, a > l if and only if

is not isolated. 9.3.2 (This exercise extends Exercise 1.1.10 on the linear independence of composition operators to their independence in the Calkin algebra.) Suppose cp1, cp2, ... , IPn are distinct analytic maps of D into D and a1, az, .. ., and an are complex constants. Recall that E(1p) = {( e 8D: IIP(()I =I}. (a) Show that on H 2(D)

9.3.1

II~·,C·,I[ "~I•;I'IE(,;ll (b) If ( in 8D is such that cp; (() exists and has modulus I for each j = 1, 2, •.. , nand if no pair { 'Pi• 'P; }, i #= j, has the same data at (then on 'Da for a ~ I

~~~.,c., I[ ;, 1. (a) As a substitute for Lemma 9.I2 first show that for fin 'Da and winD

11/11 .. 1/'( w )I -< M a (1 - lwl2)(o+4)/2 where M 0 is a constant depending only on a. From this show

IIC.,/-C,p/11~ ~ M~ll/11~ {

JD

lfP(z)- 1/J(zW(I -lzlz)o-:+4 dA(z) (min(1 -lcp(z)l, 1 -11/J(z)l))

e

L 1(aD) let w be an outer function with lwl = I - IIPI almost everywhere on the unit circle. Set v = w

.

1-lzl

inf{ I 0.

= L2 (E) $ Ab(D) if

350

Clulpter 9: Miscellanea

THEOREM 9.22

SupposetheintegralinExpression(9.4.2)converges. /fP2 (~) = L 2(E)eAb(D) then II\ El > 0 for every open arc I in aD. We will prove a result due to Hru~ev which gives a converse to Theorem 9.22 when the weight function G satisfies a certain growth condition. THEOREM 9.23

Suppose that

1 C:r) 1

exp

for every a > 0. L 2 (E) EB Ab{D).

G(r)dr

< oo

If II \ E! > 0 for every open

arc

I in aD then P2 (~) =

The proof of this theorem requires two lemmas, the first of which gives a simple condition equivalent to splitting. LEMMA9.24 For~= XE d8 is in P2 (~).

+ G dAj1r we have P

2

(~) = L2 (E) EB Ab{D) if and only ifXD

The "only if" direction is obvious. For the converse suppose P2 (~) contains XD· Since the polynomials are dense in P2 (~). P2 (~) is invariant under multiplication by z. Thus if pis a polynomial, PXD is in P2 (~) and the map p ....... PXD extends to a bounded operator M on P2 (~). Since M 2 = M and M* = M, M is an orthogonal projection. Density of the polynomials also shows that range M = Ab(D) and range (I- M) = L 2 (E). I PROOF

Thus, to prove Theorem 9.23 we will be interested in the distance from XD to P 2 (~); our goal is to show

inf

{

/Ep2(p).m

lxn- /l 2 d~ = o

when the hypotheses of Theorem 9.23 hold. It is in estimating this distance that the composition operators will appear. We will also have need of the function

cb

8(E) = defined for 0 8(E) is finite.