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Operator-valued multiplier theorems characterizing Hilbert spaces

Part of: Nontrigonometric harmonic analysis Harmonic analysis in one variable Normed linear spaces and Banach spaces; Banach lattices General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

Wolfgang Arendt  and
Shangquan Bu
Wolfgang Arendt
Affiliation:
Abteilung Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany e-mail: arendt@mathematik. uni-ulm.de
Shangquan Bu
Affiliation:
Department of Mathematical Science, University of Tsinghua, Beijing 100084, China e-mail: sbu@math.tsinghua.edu.cn
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Abstract

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We show that the operator-valued Marcinkiewicz and Mikhlin Fourier multiplier theorem are valid if and only if the underlying Banach space is isomorphic to a Hilbert space.

Keywords

Operator-valued Fourier multiplierHilbert space and Rademacher boundedness

MSC classification

Secondary: 42A45: Multipliers 42C99: None of the above, but in this section 46C15: Characterizations of Hilbert spaces 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 2 , October 2004 , pp. 175 - 184
Copyright
Copyright © Australian Mathematical Society 2004

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