Pseudospectra of special operators and pseudospectrum preservers

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 σ ε ( A ) of A as σ ε ( 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 A B + B A .