Linear maps preserving essential spectral functions and closeness of operator ranges

Abstract

Let H be an infinite-dimensional complex Hilbert space and ℬ(H) the algebra of all bounded linear operators on H. Some characterizations are obtained of linear maps Φ on ℬ(H) preserving essential spectral functions such as the intersection of left essential spectrum and right essential spectrum of operators, preserving Fredholm operators, and preserving semi-Fredholm operators, and preserving closeness of operator ranges. A result of Mbekhta, Rodman and Šemrl (Integral Equations Operator Theory 55 (2006) 93-109) on generalized invertibility preserving linear maps is improved.