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March, 2002 Uniformly convex operators and martingale type
Jörg Wenzel
Rev. Mat. Iberoamericana 18(1): 211-230 (March, 2002).

Abstract

The concept of uniform convexity of a Banach space was generalized to linear operators between Banach spaces and studied by Beauzamy. Under this generalization, a Banach space $X$ is uniformly convex if and only if its identity map $I_X$ is. Pisier showed that uniformly convex Banach spaces have martingale type $p$ for some $p>1$. We show that this fact is in general not true for linear operators. To remedy the situation, we introduce the new concept of martingale subtype and show, that it is equivalent, also in the operator case, to the existence of an equivalent uniformly convex norm on $X$. In the case of identity maps it is also equivalent to having martingale type $p$ for some $p>1$. Our main method is to use sequences of ideal norms defined on the class of all linear operators and to study the factorization of the finite summation operators. There is a certain analogy with the theory of Rademacher type.

Citation

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Jörg Wenzel. "Uniformly convex operators and martingale type." Rev. Mat. Iberoamericana 18 (1) 211 - 230, March, 2002.

Information

Published: March, 2002
First available in Project Euclid: 18 February 2003

zbMATH: 1021.46006
MathSciNet: MR1924692

Subjects:
Primary: 46B03
Secondary: 46B07 , 47A30

Keywords: ‎Banach spaces , linear operators , martingale subtype , martingale type , summation operator , superreflexivity , uniform convexity‎

Rights: Copyright © 2002 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.18 • No. 1 • March, 2002
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