A characterization of S-pseudospectra of linear operators in a Hilbert space

Abstract

In this work, we introduce and study the S-pseudospectra of linear operators defined by non-strict inequality in a Hilbert space. Inspired by A. Böttcher’s result [3], we prove that the S-resolvent norm of bounded linear operators is not constant in any open set of the S-resolvent set. Beside, we find a characterization of the S-pseudospectrum of bounded linear operator by means the S-spectra of all perturbed operators with perturbations that have norms strictly less than ε.