CONTEMPORARY MATHEMATICS

393

Recent Advances in Operator-Related

Function Theory Conference on Recent Advances

in Operator-Related Function Theory Trinity College Dublin, Ireland

August 4-6, 2004

Alec L. Matheson Michael I. Stessin

Richard M. Timoney Editors

Recent Advances in Operator-Related

Function Theory

http://dx.doi.org/10.1090/conm/393

CoNTEMPORARY MATHEMATICS

393

Recent Advances in Operator-Related

Function Theory Conference on Recent Advances

in Operator-Related Function Theory Trinity College Dublin, Ireland

August 4-6, 2004

Alec L. Matheson Michael I. Stessin

Richard M. Timoney Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeThrck, managing editor

George Andrews Carlos Berenstein Andreas Blass Abel Klein

2000 Mathematics Subject Classification. Primary 46E15, 46E20, 47B33, 47B38, 47B32, 30E20, 30E25.

Library of Congress Cataloging-in-Publication Data Conference on Recent Advances in Operator-Related Function Theory (2004 : Dublin, Ireland)

Recent advances in operator-related function theory : Conference on Recent Advances in Operator-Related Function Theory, Trinity College, Dublin, Ireland, August 4--6, 2004 / Alec L. Matheson, Michael I. Stessin, Richard M. Timoney, editors.

p. em. -(Contemporary mathematics (American Mathematical Society) ; v. 393) Includes bibliographical references. ISBN 0-8218-3925-X (alk. paper) 1. Functional analysis-Congresses. 2. Linear operators-Congresses. 3. Functions of complex

variables-Congresses. I. Matheson, Alec L., 1946- II. Stessin, Michael I., 1953- III. Timoney, Richard Martin. IV. Title. V. Series.

QA319.C66 2004 5151.7-dc22 2005053622

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10 9 8 7 6 5 4 3 2 1 11 10 09 08 07 06

This volume is graciously dedicated to Prof. Joseph A. Cima on the occasion of his seventieth birthday. Over the past 45 years Prof. Cima has dedicated himself to mathematics as a researcher, with over 100 published papers; a devoted teacher; and a sage colleague. From his very humble beginnings in a hardscrabble Pennsylvania mining town, Joseph Cima learned the value of hard work and kept a Franklinesque ethic of "early to bed, early to rise" throughout his entire career. This is evident in his research program, which is just as active now as it was when he started-with no signs of slowing down. After earning his degree from Penn State under Josephine Mitchell, Professor Cima started working in functional analysis and then moved to modern complex analysis and further to several complex variables. He keeps his hand and active mind in several other fields of analysis as well. Though he has held visiting positions and guest lectureships at various universities throughout the world, his first love is the University of North Carolina at Chapel Hill, where he has been a professor and a thesis advisor for over 45 years. He is a true believer in the egalitarian nature of public education. As a person, Professor Cima is the epitome of professionalism and integrity. His standards are high, and he demands just as much from himself as he does from the rest of us. He is always generous with his time and is known in the mathematics community as willing to seek out and develop young talent. His unending energy is an inspiration to all those who get the chance to meet him. We hope this humble offering, a collection of papers from some of Professor Cima's favorite fields of analysis, begins to show how much we appreciate all that he has done for us individually, for analysis, and for the profession in general.

Preface

List of Participants

Aleksandrov-Clark measures

Contents

IX

xi

ALEXEI POLTORATSKI and DONALD SARASON 1

Applications of spectral measures ALEC MATHESON and MICHAEL STESSIN 15

Parametrizing distinguished varieties JIM AGLER and JoHN E. McCARTHY 29

Approximation by CR functions on the unit sphere in IC2

JOHN T. ANDERSON and JOHN WERMER 35

Superposition operators and the order and type of entire functions STEPHEN M. BUCKLEY and DRAGAN VUKOTIC 51

Smooth functions in star-invariant subspaces KONSTANTIN DYAKONOV and DMITRY KHAVINSON 59

Conjugation and Clark operators STEPHAN RAMON GARCIA 67

A class of conformal mappings with applications to function spaces DANIEL GIRELA 113

Composition operators between Bergman spaces of functions of several variables

HYUNGWOON Koo and WAYNE SMITH 123

Isometries of some classical function spaces among the composition operators

MARIA J. MARTIN and DRAGAN VUKOTIC

New results on a classical operator ALFONSO MONTES-RODRIGUEZ and STANISLAV A. SHKARIN

Size conditions to be in a finitely generated ideal of H 00

JORDI PAU

The classical Dirichlet space WILLIAM T. Ross

vii

133

139

159

171

viii CONTENTS

Comparing topologies on the space of composition operators EERO SAKSMAN and CARL SUNDBERG

Brennan's conjecture for weighted composition operators WAYNE SMITH

199

209

Preface

This volume comprises the conference proceedings of a conference entitled "Re-cent advances in operator-related function theory", held at 'Itinity College, Dublin, August 4-6, 2004. The main theme of the conference was the role of Aleksandrov-Clark measures in modern operator theory and its attendant function theory. An introduction to these measures and their context for operator theory and function theory can be found in the paper of Poltoratski and Sarason. Further applications are found in too paper of Matheson and Stessin.

Garcia's paper provides a survey of recent results related to conjugation oper-ators, Clark's operators and related topics. Ross's paper provides an introduction to the classical Dirichlet space, with an exposition of known results and a list of open problems.

The remaining papers deal with recent results in specific directions related to the theme of the conference. We believe they serve to illustrate the richness of the subject and the wealth of recent progress. We hope also that this collection will help inspire further work in the near future.

ix

List of Participants

Alexandru Aleman University of Lund

John T. Anderson College of the Holy Cross

Catherine Beneteau Seton Hall University

Christopher Boyd University College Dublin

Patrick A. Brown SUNY Albany

Stephen Buckley National University of Ireland Maynooth

Tom Carroll University College Cork

Joseph Cima University of North Carolina

Olivia Constantin University of Lund

Sean Dineen University College Dublin

Ron Douglas Texas A&M University

Konstantin Dyakonov Universitat de Barcelona

Eva A. Gallardo-Gutierrez Universidad de Zaragoza

Stephan R. Garcia University of California, Santa Barbara

Daniel Girela U niversidad de Malaga

xi

Mark Halinan Trinity College Dublin

Mary Hanley University College Dublin

Remo Hugli Trinity College Dublin

James Jamison University of Memphis

Hans Jarchow U niversitiit Zurich

Siarhei Katsialouski Belarus

Angeliki Kazas SUNY Oneonta

Dmitry Khavinson University of Arkansas

Greg Knese Washington University, St. Louis

Urs Kollbrunner Universitiit Zurich

Jussi Laitila University of Helsinki

Wayne Lawton National University of Singapore

David Malone Communications Network Research Institute

Maria Jose Martin Gomez Universidad Autonoma de Madrid

Alec Matheson Lamar University

xii PARTICIPANTS

John McCarthy Washington University, St. Louis

Alfonso Montes-Rodriguez Universidad de Sevilla

Jennifer Moorhouse Colgate University

Raymond Mortini Universite de Metz

Don Moynihan Trinity College Dublin

Artur Nicolau Universitat Autonoma de Barcelona

Pekka Nieminen University of Helsinki

Maria Nowak Lublin, Poland

Donal O'Donovan Trinity College Dublin

Jordi Pau Universitat Autonoma de Barcelona

Serpil Pehlivan Suleyman Demirel University, Turkey

Alexei Poltoratski Texas A&M University

Manuel Ponce Escudero Universidad de Sevilla

Arthur Prendergast Trinity College Dublin

Stefan Richter University of Tennessee

Alejandro Rodriguez Martinez Universidad de Sevilla

William Ross University of Richmond, Virginia

Ray Ryan National University of Ireland, Galway

Eero Saksman University of Jyvaskyla

Rebecca Schmitz University of Virginia

Antonio Serra Instituto Superior Tecnico, Lisbon

Aristomenis G. Siskakis Aristotle University of Thessaloniki

Wayne Smith University of Hawaii

Michael Stessin SUNY Albany

Karel Stroethoff University of Montana

Carl Sundberg University of Tennessee

Richard M. Timoney Trinity College Dublin

Hans-Olav Tylli University of Helsinki

Mitsuru Uchiyama Fukuoka University of Education

Prasada R. Vegulla Washington University, St. Louis

Dragan Vukotic Universidad Aut6noma de Madrid

David Walsh National University of Ireland, Maynooth

Warren R. Wogen University of North Carolina

Lawrence Zalcman Bar-Ilan University

Titles in This Series

393 Alec L. Matheson, Michael I. Stessin, and Richard M. Timoney, Editors, Recent advances in operator-related function theory, 2006

392 Stephen Berman, Brian Parshall, Leonard Scott, and Weiqiang Wang, Editors, Infinite-dimensional aspects of representation theory and applications, 2005

391 Jiirgen Fuchs, Jouko Mickelsson, Grigori Rozenblioum, Alexander Stolin, and Anders Westerberg, Editors, Noncommutative geometry and representation theory in mathematical physics, 2005

390 Sudhir Ghorpade, Hema Srinivasan, and Jugal Verma, Editors, Commutative algebra and algebraic geometry, 2005

389 James Eells, Etienne Ghys, Mikhail Lyubich, Jacob Palis, and Jose Seade, Editors, Geometry and dynamics, 2005

388 Ravi Vakil, Editor, Snowbird lectures in algebraic geometry, 2005 387 Michael Entov, Yehuda Pinchover, and Michah Sageev, Editors, Geometry,

spectral theory, groups, and dynamics, 2005 386 Yasuyuki Kachi, S. B. Mulay, and Pavlos Tzermias, Editors, Recent progress in

arithmetic and algebraic geometry, 2005 385 Sergiy Kolyada, Yuri Manin, and Thomas Ward, Editors, Algebraic and topological

dynamics, 2005 384 B. Diarra, A. Escassut, A. K. Katsaras, and L. Narici, Editors, Ultrametric

functional analysis, 2005 383 Z.-C. Shi, Z. Chen, T. Tang, and D. Yu, Editors, Recent advances in adaptive

computation, 2005 382 Mark Agranovsky, Lavi Karp, and David Shoikhet, Editors, Complex analysis and

dynamical systems II, 2005 381 David Evans, Jeffrey J. Holt, Chris Jones, Karen Klintworth, Brian Parshall,

Olivier Pfister, and Harold N. Ward, Editors, Coding theory and quantum computing, 2005

380 Andreas Blass and Yi Zhang, Editors, Logic and its applications, 2005 379 Dominic P. Clemence and Guoqing Tang, Editors, Mathematical studies in

nonlinear wave propagation, 2005 378 Alexandre V. Borovik, Editor, Groups, languages, algorithms, 2005 377 G. L. Litvinov and V. P. Maslov, Editors, Idempotent mathematics and mathematical

physics, 2005 376 Jose A. de Ia Peiia, Ernesto Vallejo, and Natig Atakishiyev, Editors, Algebraic

structures and their representations, 2005 375 Joseph Lipman, Suresh Nayak, and Pramathanath Sastry, Variance and duality for

cousin complexes on formal schemes, 2005 37 4 Alexander Barvinok, Matthias Beck, Christian Haase, Bruce Reznick,

and Volkmar Welker, Editors, Integer points in polyhedra-geometry, number theory, algebra, optimization, 2005

373 0. Costin, M. D. Kruskal, and A. Macintyre, Editors, Analyzable functions and applications, 2005

372 Jose Burillo, Sean Cleary, Murray Elder, Jennifer Taback, and Enric Ventura, Editors, Geometric methods in group theory, 2005

371 Gui-Qiang Chen, George Gasper, and Joseph Jerome, Editors, Nonlinear partial differential equations and related analysis, 2005

370 Pietro Poggi-Corradini, Editor, The p-harmonic equation and recent advances in analysis, 2005

369 Jaime Gutierrez, Vladimir Shpilrain, and Jie-Tai Yu, Editors, Affine algebraic geometry, 2005

TITLES IN THIS SERIES

368 Sagun Chanillo, Paulo D. Cordaro, Nicholas Hanges, Jorge Hounie, and Abdelhamid Meziani, Editors, Geometric analysis of PDE and several complex variables, 2005

367 Shu-Cheng Chang, Bennett Chow, Sun-Chin Chu, and Chang-Shou Lin, Editors, Geometric evolution equations, 2005

366 Bernheim Booi3-Bavnbek, Gerd Grubb, and Krzysztof P. Wojciechowski, Editors, Spectral geometry of manifolds with boundary and decompositon of manifolds, 2005

365 Robert S. Doran and Richard V. Kadison, Editors, Operator algebras, quantization, and non-commutative geometry, 2004

364 Mark Agranovsky, Lavi Karp, David Shoikhet, and Lawrence Zalcman, Editors, Complex analysis and dynamical systems, 2004

363 Anthony To-Ming Lau and Volker Runde, Editors, Banach algebras and their applications, 2004

362 Carlos Concha, Raul Manasevich, Gunther Uhlmann, and MichaelS. Vogelius, Editors, Partial differential equations and inverse problems, 2004

361 Ali Enayat and Roman Kossak, Editors, Nonstandard models of arithmetic and set theory, 2004

360 Alexei G. Myasnikov and Vladimir Shpilrain, Editors, Group theory, statistics, and cryptography, 2004

359 S. Dostoglou and P. Ehrlich, Editors, Advances in differential geometry and general relativity, 2004

358 David Burns, Christian Popescu, Jonathan Sands, and David Solomon, Editors, Stark's Conjectures: Recent work and new directions, 2004

357 John Neuberger, Editor, Variational methods: open problems, recent progress, and numerical algorithms, 2004

356 Idris Assani, Editor, Chapel Hill ergodic theory workshops, 2004 355 William Abikoff and Andrew Haas, Editors, In the tradition of Ahlfors and Bers, III,

2004 354 Terence Gaffney and Maria Aparecida Soares Ruas, Editors, Real and complex

singularities, 2004 353 M. C. Carvalho and J. F. Rodrigues, Editors, Recent advances in the theory and

applications of mass transport, 2004 352 Marek Kubale, Editor, Graph colorings, 2004 351 George Yin and Qing Zhang, Editors, Mathematics of finance, 2004 350 Abbas Bahri, Sergiu Klainerman, and Michael Vogelius, Editors, Noncompact

problems at the intersection of geometry, analysis, and topology, 2004 349 Alexandre V. Borovik and Alexei G. Myasnikov, Editors, Computational and

experimental group theory, 2004 348 Hiroshi Isozaki, Editor, Inverse problems and spectral theory, 2004 347 Motoko Kotani, Tomoyuki Shirai, and Toshikazu Sunada, Editors, Discrete

geometric analysis, 2004 346 Paul Goerss and Stewart Priddy, Editors, Homotopy theory: Relations with algebraic

geometry, group cohomology, and algebraic K-theory, 2004

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstoref.

The articles in this book are based on talks at a conference devoted to interrelations between function theory and the theory of operators. The main theme of the book is the role of Alexandrov-Clark measures. Two of the articles provide the introduction to the theory of Alexandrov-Clark measures and to its applications in the spectral theory of linear operators. The remaining articles deal with recent results in specific directions related to the theme of the book.

ISBN 0- 8218- 3925- X

9 780821 839256

CONM/393