Random unconditional convergence of vector-valued Dirichlet series

Abstract

We study random unconditionality of Dirichlet series in vector-valued Hardy spaces Hp(X). It is shown that a Banach space X has type 2 (respectively, cotype 2) if and only if for every choice (xn)n⊂X it follows that (xnn−s)n is random unconditionally convergent (respectively, divergent) in H2(X). The analogous question on Hp(X) spaces for p≠2 is also explored. We also provide explicit examples exhibiting the differences between the unconditionality of (xnn−s)n in Hp(X) and that of (xnzn)n in Hp(X).