Hardy spaces of vector-valued Dirichlet series
Volume 243 / 2018
Abstract
Given a Banach space $X$ and $1 \leq p \leq \infty $, it is well-known that the two Hardy spaces $H_p(\mathbb {T},X)$ ($\mathbb {T}$ the torus) and $H_p(\mathbb {D},X)$ ($\mathbb {D}$ the disk) have to be distinguished carefully. This motivates us to define and study two different types of Hardy spaces, $\mathcal {H}_p(X)$ and $\mathcal {H}^+_p(X)$, of Dirichlet series $\sum _n a_n n^{-s}$ with coefficients in $X$. We characterize them in terms of summing operators as well as holomorphic functions in infinitely many variables, and prove that they coincide whenever $X$ has the Analytic Radon–Nikodým Property. Consequences include a vector-valued version of the Riesz Brothers Theorem on the infinite-dimensional torus, and an answer to the question when $\mathcal {H}_1(X^{\ast })$ is a dual space.
