36,409 research outputs found
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
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
Ideal norms and trigonometric orthonormal systems
In this article, we characterize the --property of a Banach space by
ideal norms associated with trigonometric orthonormal systems.
The asymptotic behavior of that numerical parameters can be used to decide
whether or not is a --space. Moreover, in the negative case, we obtain
a measure that shows how far is from being a --space.
The main result is, that all described parameters are equivalent also in the
quantitative setting
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
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