Solid hulls of weighted Banach spaces of entire functions

TL;DR

This paper characterizes the solid hulls of weighted Banach spaces of entire functions with specific decay properties, providing explicit results for exponential-type weights and applications to multiplier spaces.

Contribution

It determines the solid hulls of weighted spaces of entire functions for a broad class of weights, including explicit formulas for exponential weights, extending previous work.

Findings

Solid hulls are explicitly characterized for weights satisfying condition (B).

Formulas are provided for weights of the form v(r)=exp(-ar^p).

Applications to the structure of multiplier spaces are demonstrated.

Abstract

Given a continuous, radial, rapidly decreasing weight $v$ on the complex plane $\mathbf{C}$, we study the solid hull of its associated weighted space $H_v^\infty(\mathbf{C})$ of all the entire functions $f$ such that $v|f|$ is bounded. The solid hull is found for a large class of weights satisfying the condition (B) of Lusky. Precise formulations are obtained for weights of the form $v(r)=\exp(-ar^p), a>0, p>0$. Applications to spaces of multipliers are included.