Maps preserving Fredholm or semi-Fredholm elements relative to some ideal

Abstract

We consider the Calkin algebra CR(A) and the Fredholm theory in a Banach algebra A, relative to some fixed ideal F of A. Our aim is to study linear maps between unital Banach algebras A and B which are surjective up to the inessential elements relative to F, and preserve Fredholm or semi-Fredholm elements in both directions or equivalently different relatively essential spectral sets such as essential spectrum, left or right essential spectrum, the boundary of essential spectrum or the full essential spectrum. We characterize such mappings when one of CR(A) or CR(B) is commutative and also investigate similar problems when A is assumed to be a unital C^∗-algebra of real rank zero and B is an arbitrary Banach algebra.