Uniformly convex operators and martingale type

The concept of uniform convexity of a Banach space was generalized to linear operators between Banach spaces and studied by Beauzamy [1976]. 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.Comment: 15 pages, to be published in Revista Matematica Iberoamerican

Ideal norms and trigonometric orthonormal systems

In this article, we characterize the $UMD$--property of a Banach space $X$ by ideal norms associated with trigonometric orthonormal systems. The asymptotic behavior of that numerical parameters can be used to decide whether or not $X$ is a $UMD$--space. Moreover, in the negative case, we obtain a measure that shows how far $X$ is from being a $UMD$--space. The main result is, that all described parameters are equivalent also in the quantitative setting

The UMD constants of the summation operators

The UMD property of a Banach space is one of the most useful properties when one thinks about possible applications. This is in particular due to the boundedness of the vector-valued Hilbert transform for functions with values in such a space. Looking at operators instead of at spaces, it is easy to check that the summation operator does not have the UMD property. The actual asymptotic behavior however of the UMD constants computed with martingales of length n is unknown. We explain, why it would be important to know this behavior, rephrase the problem of finding these UMD constants and give some evidence of how they behave asymptotically.Comment: 22 page

Superreflexivity and J-convexity of Banach spaces

A Banach space X is superreflexive if each Banach space Y that is finitely representable in X is reflexive. Superreflexivity is known to be equivalent to J-convexity and to the non-existence of uniformly bounded factorizations of the summation operators S_n through X. We give a quantitative formulation of this equivalence. This can in particular be used to find a factorization of S_n through X, given a factorization of S_N through [L_2,X], where N is `large' compared to n