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 map...

In this paper we describe the form of those continuous multiplicative maps on B(H) (H being a separable complex Hilbert space of dimension not less than 3) which preserve the rank, or the corank. Furthermore, we characterize those continuous *-semigroup endomorphisms of B(H) which are spectrum non-increasing.