Holomorphically Decomposable Fredholm Elements in Semi-Simple Banach Algebras

Abstract

The notion of holomorphically decomposable Fredholm operators is extended in case of complex semi-simple Banach algebra A. Indeed, let x ∈A , we studies σhF (x) the holomorphically decomposable Fredholm spectrum of x. We prove that σhF (x) is countable if and only if the usual spectrum σ(x) is. Also we discuss the spectral mapping theorem for σhF (x). On the other hand we prove that the class HΦ(A) for the holomorphically decomposable Fredholm elements in A is an upper