Solid hulls and cores of weighted $H^\infty$-spaces

TL;DR

This paper characterizes the solid hull and core of weighted $H^$ spaces of analytic functions on the disc and entire plane, providing new precise results for various weights and exploring their basis properties.

Contribution

It determines the solid hull and core for a broad class of weighted $H^$ spaces, including concrete weights, and analyzes the basis structure of these spaces.

Findings

Solid hull and core are characterized for various weights.

Monomials form an unconditional basis in the closure of polynomials.

Weighted spaces of entire functions can differ from their solid hulls and cores.

Abstract

We determine the solid hull and solid core of weighted Banach spaces $H_v^\infty$ of analytic functions $f$ such that $v|f|$ is bounded, both in the case of the holomorphic functions on the disc and on the whole complex plane, for a very general class of radial weights $v$. Precise results are presented for concrete weights on the disc that could not be treated before. It is also shown that if $H_v^\infty$ is solid, then the monomials are an (unconditional) basis of the closure of the polynomials in $H_v^\infty$. As a consequence $H_v^\infty$ does not coincide with its solid hull and core in the case of the disc. An example shows that this does not hold for weighted spaces of entire functions.