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Biconcave-function characterisations of UMD and Hilbert spaces

Published online by Cambridge University Press:  17 April 2009

Jinsik Mok Lee
Jinsik Mok Lee
Affiliation:
Department of Mathematics, Sungwha University, 381-7 Samyong-Dong Chonahn 330-150, Republic of Korea
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Abstract

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Suppose that X is a real or complex Banach space with norm |·|. Then X is a Hilbert space if and only if

for all x in X and all X-valued Bochner integrable functions Y on the Lebesgue unit interval satisfying EY = 0 and |x − Y| ≤ 2 almost everywhere. This leads to the following biconcave-function characterisation: A Banach space X is a Hilbert space if and only if there is a biconcave function η: {(x, y) ∈ X × X: |x − y| ≤ 2} → R such that η(0, 0) = 2 and

If the condition η(0, 0) = 2 is eliminated, then the existence of such a function η characterises the class UMD (Banach spaces with the unconditionally property for martingale differences).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 47 , Issue 2 , April 1993 , pp. 297 - 306
Copyright
Copyright © Australian Mathematical Society 1993

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