Random unconditional convergence of vector-valued Dirichlet series

Abstract

We study random unconditionality of Dirichlet series in vector-valued Hardy spaces ${\mathcal{H}}_p(X)$. It is shown that a Banach space $X$ has type 2 (respectively, cotype 2) if and only if for every choice $(x_n{)}_n\subset X$ it follows that $(x_n{n}^{-s}{)}_n$ is Random unconditionally convergent (respectively, divergent) in ${\mathcal{H}}_2(X)$. The analogous question on ${\mathcal{H}}_p(X)$ spaces for $p\ne 2$ is also explored. We also provide explicit examples exhibiting the differences between the unconditionality of $(x_n{n}^{-s}{)}_n$ in ${\mathcal{H}}_p(X)$ and that of $(x_n{z}^n{)}_n$ in $H_p(X)$.