Means of Hilbert Space Operators
The monograph is devoted to a systematic study of means of Hilbert space operators by a unified method based on the theory of double integral transformations and Peller's characterization of Schur multipliers. General properties on means of operators such
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Fumio Hiai Hideki Kosaki
Means of Hilbert Space Operators
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Authors Fumio Hiai Graduate School of Information Sciences Tohoku University Aoba-ku, Sendai 980-8579 Japan e-mail: [email protected] Hideki Kosaki Graduate School of Mathematics Kyushu University Higashi-ku, Fukuoka 812-8581 Japan e-mail: [email protected]
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Mathematics Subject Classification (2000): 47A30, 47A64, 15A60 ISSN 0075-8434 ISBN 3-540-40680-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10949634
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Preface
Roughly speaking two kinds of operator and/or matrix inequalities are known, of course with many important exceptions. Operators admit several natural notions of orders (such as positive semidefiniteness order, some majorization orders and so on) due to their non-commutativity, and some operator inequalities clarify these order relations. There is also another kind of operator inequalities comparing or estimating various quantities (such as norms, traces, determinants and so on) naturally attached to operators. Both kinds are of fundamental importance in many branches of mathematical analysis, but are also sometimes highly non-trivial because of the non-commutativity of the operators involved. This monograph is mainly devoted to means of Hilbert space operators and their general properties with the main emphasis on their norm comparison results. Therefore, our operator inequalities here are basically of the second kind. However, they are not free from the first in the sense that our general theory on means relies heavily on a certain order for operato
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