The spectra of algebras of Lorch analytic mappings

For a complex Banach algebra E, let H-L(E) be the space of the mappings from E into E that are analytic in the sense of Lorch. We will show that H-L(E) 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(E) 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(infinity)(int B-E) of the bounded mappings from int B-E into E that are analytic in the sense of Lorch. (C) 2009 Elsevier Ltd. All rights reserved.