# Random unconditional convergence of vector-valued Dirichlet series

## TL;DR

This paper characterizes Banach spaces with type 2 and cotype 2 via the random unconditional convergence or divergence of vector-valued Dirichlet series in Hardy spaces, extending the understanding of unconditionality in these function spaces.

## Contribution

It establishes a precise link between Banach space geometry (type and cotype) and the random unconditional behavior of vector-valued Dirichlet series in Hardy spaces, including new examples.

## Key findings

- Banach space $X$ has type 2 iff $(x_n n^{-s})_n$ is RUC in $\\mathcal H_2(X)$ for all sequences.
- Banach space $X$ has cotype 2 iff $(x_n n^{-s})_n$ is RUD in $\\mathcal H_2(X)$ for all sequences.
- Explicit examples show differences between unconditionality in $\\mathcal H_p(X)$ and $H_p(X)$.

## 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 $p\neq2$ 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 $H_p(X)$.