JournalsrmiVol. 18, No. 1pp. 211–230

Uniformly convex operators and martingale type

  • Jörg Wenzel

    Friedrich-Schiller-Universität Jena, Germany
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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 XX is uniformly convex if and only if its identity map IXI_X is. Pisier showed that uniformly convex Banach spaces have martingale type pp for some p>1p>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 XX. In the case of identity maps it is also equivalent to having martingale type pp for some p>1p>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.

Cite this article

Jörg Wenzel, Uniformly convex operators and martingale type. Rev. Mat. Iberoam. 18 (2002), no. 1, pp. 211–230

DOI 10.4171/RMI/316