Classical Summation in Commutative and Noncommutative L<sub>p</sub>-Spaces
The aim of this research is to develop a systematic scheme that makes it possible to transform important parts of the by now classical theory of summation of general orthonormal series into a similar theory for series in noncommutative $L_p$-spaces constr
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For further volumes: http://www.springer.com/series/304
2021
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Andreas Defant
Classical Summation in Commutative and Noncommutative Lp-Spaces
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Andreas Defant Carl von Ossietzky University Oldenburg Department of Mathematics Carl von Ossietzky Strasse 7-9 26111 Oldenburg Germany [email protected]
ISBN 978-3-642-20437-1 e-ISBN 978-3-642-20438-8 DOI 10.1007/978-3-642-20438-8 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011931784 Mathematics Subject Classification (2011): 46-XX; 47-XX c Springer-Verlag Berlin Heidelberg 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
In the theory of orthogonal series the most important coefficient test for almost everywhere convergence of orthonormal series is the fundamental theorem of Menchoff and Rademacher. It states that whenever a sequence (αk ) of coefficients satisfies the “test” ∑k |αk log k|2 < ∞, then for every orthonormal series ∑k αk xk in L2 (μ ) we have that ∑k αk xk converges μ -almost everywhere. The aim of this research is to develop a systematic scheme which allows us to transform important parts of the now classical theory of almost everywhere summation of general orthonormal series in L2 (μ ) into a similar theory for series in noncommutative L p -spaces L p (M , ϕ ) or even symmetric spaces E(M , ϕ ) constructed over a noncommutative measure space (M , ϕ ), a von Neumann algebra M of operators acting on a Hilbert space H together with a faithful normal state ϕ on this algebra. In Chap. 2 we present a new and modern understanding of the classical theory on pointwise convergence of orthonormal series in the Hilbert spaces L2 (μ ), and show that large parts of the classical theory transfer to a theory on pointwise convergence of unconditionally convergent series in spaces L p (μ , X ) of μ -integrable functions with values in Banach spaces X, or more generally Banach function spaces E(μ , X ) of X -valued μ -integrable functions. Here our tools are strongly based on Grothendie
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