Essentially spectrally bounded linear maps

Abstract

Let L(H) be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space H. We characterize essentially spectrally bounded linear maps from L(H) onto L(H) itself. As a consequence, we characterize linear maps from L(H) onto L(H) itself that compress different essential spectral sets such as the the essential spectrum, the (left, right) essential spectrum, and the semi-Fredholm spectrum.