Pseudospectra of special operators and pseudospectrum preservers

Abstract

Denote by B(H) the Banach algebra of all bounded linear operators on a complex Hilbert space H. Let A∈B(H), and denote by σ(A) the spectrum of A. For ε>0, define the ε-pseudospectrum σε(A) of A asσε(A)={z∈σ(A+E):E∈B(H),‖E‖<ε}. In this paper, the pseudospectra of several special classes of operators are characterized. As an application, complete descriptions are given of the maps of B(H) leaving invariant the pseudospectra of A•B for different kinds of binary operations • on operators such as the difference A−B, the operator product AB, and the Jordan product AB+BA.