Uniformly convex operators and martingale type

The concept of uniform convexity of a Banach space was generalized to linearoperators between Banach spaces and studied by Beauzamy [1976]. Under thisgeneralization, a Banach space X is uniformly convex if and only if itsidentity map I_X is. Pisier showed that uniformly convex Banach spaces havemartingale type p for some p>1. We show that this fact is in general not truefor linear operators. To remedy the situation, we introduce the new concept ofmartingale 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 ofidentity 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 ofall linear operators and to study the factorization of the finite summationoperators. There is a certain analogy with the theory of Rademacher type.