The spectra of algebras of Lorch analytic mappings

Abstract

For a complex Banach algebra $E$, let $H_L\left(E\right)$ be the space of the mappings from $E$ into $E$ that are analytic in the sense of Lorch. We will show that $H_L\left(E\right)$ is a closed subalgebra of $H_b(E,E)$ and we will give a description of its spectrum. As an application we will show that $E$ is semi-simple if and only if $H_L\left(E\right)$ is semi-simple. We will also give descriptions of the spectra of other algebras of analytic mappings in the sense of Lorch. In particular we will study the spectrum of the Banach algebra $H_L^{\infty }\left(\mathrm{int}{B}_E\right)$ of the bounded mappings from $\mathrm{int}{B}_E$ into $E$ that are analytic in the sense of Lorch.