On the generalized Fredholm spectrum in a complex semisimple banach algebra

We shall show a spectral mapping property of the generalized Fredholm spectrum in the more general context of Banach algebras. This is an extension of (Schmoeger in Demonstr Math XXXI(4):723–733, 1998, Theorem 3.3). More precisely, for a∈A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in {\mathcal{A}}$$\end{document} which is a complex semisimple Banach algebra with identity, we shall prove that if g is a analytic function on a neighbourhood of the spectrum of a and if it has no zero in the generalized Fredholm spectrum of a then g(a) is a generalized Fredholm element in A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}}$$\end{document}.