Uniformly convex operators and martingale type

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.