Spectra of weighted algebras of holomorphic functions

TL;DR

This paper investigates the spectral properties of weighted algebras of holomorphic functions on Banach spaces, establishing conditions for algebraic structure, analyzing spectra with analytic structures, and exploring composition operators and algebra homomorphisms.

Contribution

It provides new criteria for when weighted holomorphic function spaces form algebras and introduces an analytic framework for their spectra, including the study of composition operators.

Findings

Conditions for weighted spaces to be algebras or have polynomial Schauder decompositions

Spectra endowed with an analytic structure

Behavior of composition operators and algebra homomorphisms on spectra

Abstract

We consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the spectra of weighted algebras and endow them with an analytic structure. We also deal with composition operators and algebra homomorphisms, in particular to investigate how their induced mappings act on the analytic structure of the spectrum. Moreover, a Banach-Stone type question is addressed.