Noncommutative Functional Calculus Theory and Applications of Slice
This book presents a functional calculus for n-tuples of not necessarily commuting linear operators. In particular, a functional calculus for quaternionic linear operators is developed. These calculi are based on a new theory o
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Series Editors H. Bass J. Oesterl´e A. Weinstein
Fabrizio Colombo Irene Sabadini Daniele C. Struppa
Noncommutative Functional Calculus Theory and Applications of Slice Hyperholomorphic Functions
Fabrizio Colombo Dipartimento di Matematica Politecnico di Milano Via Bonardi 9 20133 Milano Italy [email protected]
Irene Sabadini Dipartimento di Matematica Politecnico di Milano Via Bonardi 9 20133 Milano Italy [email protected]
Daniele C. Struppa Schmid College of Science Chapman University Orange, CA 92866 USA [email protected]
2010 Mathematical Subject Classification: 30G35, 47A10, 47A60 ISBN 978-3-0348-0109-6 e-ISBN 978-3-0348-0110-2 DOI 10.1007/978-3-0348-0110-2 Library of Congress Control Number: 2011924879
© Springer Basel AG 2011 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. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com
Contents 1 Introduction 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Plan of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Slice monogenic functions 2.1 Clifford algebras . . . . . . . . . . . . . . . . . . . . 2.2 Slice monogenic functions: definition and properties . 2.3 Power series . . . . . . . . . . . . . . . . . . . . . . . 2.4 Cauchy integral formula, I . . . . . . . . . . . . . . . 2.5 Zeros of slice monogenic functions . . . . . . . . . . 2.6 The slice monogenic product . . . . . . . . . . . . . 2.7 Slice monogenic Cauchy kernel . . . . . . . . . . . . 2.8 Cauchy integral formula, II . . . . . . . . . . . . . . 2.9 Duality Theorems . . . . . . . . . . . . . . . . . . . 2.10 Topological Duality Theorems . . . . . . . . . . . . . 2.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Functional calculus for n-tuples of operators 81 3.1 The S-resolvent operator and the S-spectrum . . . . . . . . . . . . 82 3.2 Properties of the S-spectrum . . . . . . . . . . . . . . . . . . . . . 86 3.3 The functional calculus . . . . . . . . . . . . . . . . . . . . . . . . 88 3.4 Algebraic rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5 The spectral mapping and the S-spectral radius theorems . . . . . 93 3.6 Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.7 Functional calculus for unbounded operators and algebraic properties101 3.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4 Quaternionic Functi
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