Pseudospectra of Special Operators and Pseudosectrum 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 σε(A) of A as σε(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 kind of binary operations • on operators such as the difference A−B, the operator product AB, and the Jordan product AB +BA.