Characterization of multipliers on vector-valued Hardy spaces

TL;DR

This paper characterizes the multipliers on vector-valued Hardy spaces over infinite polydisks, polytori, and Dirichlet series, highlighting differences from scalar cases and identifying conditions for their equivalence.

Contribution

It provides a detailed characterization of multiplier spaces on vector-valued Hardy spaces in various complex domains, revealing key differences from scalar-valued cases and extending to Dirichlet series.

Findings

Multiplier space on infinite polydisk is $H__^^_2, B(X))

On the infinite polytorus, multiplier space is $H_^{sot}_^^, B(X)) for separable X

Spaces coincide when X has the analytic Radon-Nikodym property

Abstract

This work characterizes the multipliers on vector-valued Hardy spaces over the infinite polydisk and the infinite polytorus, as well as in the context of Dirichlet series. Unlike the scalar-valued setting, where these frameworks are completely analogous reformulations of one another, there are significant differences in the vector-valued context. We prove that while the space of multipliers on the infinite polydisk is $H_\infty(\mathbb{D}^\infty_2, B(X))$, the situation on the infinite polytorus is distinct; assuming $X$ is separable, the multiplier space can be identified as $H_\infty^{sot}(\mathbb{T}^\infty, B(X))$, consisting of essentially bounded SOT-measurable functions. These spaces coincide when $X$ possesses the analytic Radon-Nikodym property. Finally, we extend these results to the associated Hardy spaces of Dirichlet series, $\mathcal{H}_p^+(X)$ and $\mathcal{H}_p(X)$, providing characterizations for their respective multiplier spaces.