The Hardy Space of a Slit Domain

If H is a Hilbert space and T : H ? H is a continous linear operator, a natural question to ask is: What are the closed subspaces M of H for which T M ? M? Of course the famous invariant subspace problem asks whether or not T has any non-trivial invariant

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Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta) Benoît Perthame (Université Pierre et Marie Curie, Paris) Laurent Saloff-Coste (Cornell University, Ithaca) Igor Shparlinski (Macquarie University, New South Wales) Wolfgang Sprössig (TU Bergakademie Freiberg) Cédric Vilani (Ecole Normale Supérieure, Lyon)

Alexandru Aleman Nathan S. Feldman William T. Ross

The

Hardy Space of a

Slit

Domain

Birkhäuser Basel . Boston . Berlin

Author: Alexandru Aleman Centre for Mathematical Sciences Lund University 221 00 Lund Sweden e-mail: [email protected]

Nathan S. Feldman Department of Mathematics Washington & Lee University Lexington, VA 24450 USA e-mail: [email protected]

William T. Ross Department of Mathematics and Computer Science University of Richmond Richmond, VA 23173 USA e-mail: [email protected]

2000 Mathematical Subject Classification: 30D55, 47A15, 47A16

Library of Congress Control Number: 2009931266

Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

ISBN 978-3-0346-0097-2 Birkhäuser Verlag AG, Basel · Boston · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Cover design: Birgit Blohmann, Zürich, Switzerland Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-0346-0097-2

e-ISBN 978-3-0346-0098-9

987654321

www.birkhauser.ch

Preface If H is a Hilbert space and T : H → H is a continous linear operator, a natural question to ask is: What are the closed subspaces M of H for which T M ⊂ M? Of course the famous invariant subspace problem asks whether or not T has any non-trivial invariant subspaces. This monograph is part of a long line of study of the invariant subspaces of the operator T = Mz (multiplication by the independent variable z, i.e., M z f = z f ) on a Hilbert space of analytic functions on a bounded domain G in C. The characterization of these Mz -invariant subspaces is particularly interesting since it entails both the properties of the functions inside the domain G, their zero sets for example, as well as the behavior of the functions near the boundary of G. The operator M z is not only interesting in its own right but often serves as a model operator for certain classes of linear operators. By this we mean that given an operator T on H with certain properties (certain subnormal operators or two-isometric operators with the right spectral properties, etc.), there is a Hilbert space of analytic function