Lecture Notes in Mathematics 2209

Transcription

1 Lecture Notes in Mathematics 2209 Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Board: Michel Brion, Grenoble Camillo De Lellis, Zurich Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gábor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York Anna Wienhard, Heidelberg

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3 Piotr Budzyński Zenon Jabłoński Il Bong Jung Jan Stochel Unbounded Weighted Composition Operators in L 2 -Spaces 123

4 Piotr Budzyński Katedra Zastosowań Matematyki Uniwersytet Rolniczy w Krakowie Il Bong Jung Department of Mathematics Kyungpook National University Daegu, Republic of Korea Zenon Jabłoński Instytut Matematyki Uniwersytet Jagielloński Jan Stochel Instytut Matematyki Uniwersytet Jagielloński ISSN ISSN (electronic) Lecture Notes in Mathematics ISBN ISBN (ebook) Library of Congress Control Number: Mathematics Subject Classification (2010): 47B38, 47B37, 47B33, 47B20, 47B25, 44A60, 47A80 Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

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6 Preface Since the celebrated Banach theorem on surjective linear isometries between spaces of continuous functions appeared, there has been a growing interest in studying weighted composition operators between various function spaces including spaces of continuous, holomorphic and p-summable functions. These operators have been investigated from different points of view. Ergodic theory utilizes isometric composition operators known as Koopman operators, while operator theory deals mostly with non-isometric weighted composition operators. Weighted composition operators are still candidates for the negative solution to the invariant subspace problem (Bishop operators are among them). A recently introduced and intensively studied class of operators called weighted shifts on directed trees overlaps with that of weighted composition operators and connects operator and graph theories. In this book we establish the foundations of the theory of bounded and unbounded weighted composition operators in L 2 -spaces over -finite measure spaces. We develop the theory in full generality, meaning that the corresponding composition operators are not assumed to be well defined. We investigate basic properties of weighted composition operators including, inter alia, welldefiniteness, dense definiteness, closedness and boundedness. We study their powers and answer the question of when their C 1 -vectors are dense in the underlying L 2 -space. We describe explicitly their polar decompositions as well as polar decompositions of their adjoints. We characterize their hyponormality, cohyponormality, quasinormality, normality, selfadjointness and positive selfadjointness. We provide the first-ever criteria for the subnormality of unbounded weighted composition operators. They require the existence of measurable families of probability measures on the closed half line satisfying a kind of consistency condition. In the bounded case, these criteria are in fact characterizations. They also generalize the previously known criteria for the subnormality of weighted shifts on countable directed trees. We adapt our general results to the context of weighted composition operators over discrete measure spaces. We indicate the subtle interplay between the classical moment problem, the graph theory and the injectivity problem. We investigate the relationships between weighted composition operators and the corresponding multiplication and composition operators. We study the subnormality of weighted vii

7 viii Preface composition operators with matrix symbols via the Berg-Durán transformation of Hausdorff moment sequences. We address the question of when the tensor product of finitely many weighted composition operators can be regarded as a weighted composition operator and prove that if the factors are densely defined, then the closure of the tensor product coincide with an appropriate weighted composition operator. We illustrate the optimality of obtained results by a variety of examples including those of discrete and continuous types. The majority of results included in this book are new and they have not been published elsewhere. Daegu, Republic of Korea December 9, 2017 Piotr Budzyński Zenon Jabłoński Il Bong Jung Jan Stochel

8 Acknowledgements A substantial part of this book was written while the first, the second and the fourth author visited Kyungpook National University in 2015 and 2016, and the third author visited Jagiellonian University in They wish to thank the faculties and the administrations of these units for their warm hospitality. The research of the first author was supported by the Ministry of Science and Higher Education of the Republic of Poland. The research of the second and fourth authors was supported by the NCN (National Science Center), decision No. DEC-2013/11/B/ST1/ The research of the third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2015R1A2A2A ). ix

9 Contents 1 Preliminaries Introduction Notations andprerequisites Preparatory Concepts Measure-Theory Background Invitation to Weighted Composition Operators Assorted Classes of Weighted Composition Operators ConditionalExpectation Adjoint and Polar Decomposition A Basic Characterization of Quasinormality Subnormality: General Criteria General Scheme Injectivity Versus.CC/ The Condition.CC 1 / Subnormality via.cc 1 / C 1 -Vectors Powers of C ;w Generating Stieltjes Moment Sequences Subnormality in the Bounded Case Seminormality Hyponormality Cohyponormality Normality and Formal Normality Selfadjointness Discrete Measure Spaces Background Seminormality Subnormality xi

10 xii Contents 6.4 Moments andinjectivity Examples Relationships Between C ;w and C M w C Versus C ;w Radon-Nikodym Derivative and Conditional Expectation Application to Subnormality Subnormality in the Matrix Symbol Case Examples Miscellanea Tensor Products Modifying the Symbol Quasinormality Revisited A Non-probabilistic Expectation B Powers of Operators References Symbol Index Author Index Subject Index