The pseudospectra of linear operator pencils in a Hilbert space

The purpose of this paper is to introduce and study some basic proprieties of the pseudospectra of linear operator pencils (or S-pseudospectra of linear operators) defined by non-strict inequality in a Hilbert space. Inspired by Böttcher’s result (J Integral Equ Appl 6(3):267–301, 1994), we show that the S-resolvent of a bounded operator acting in Hilbert space cannot have constant norm on any open set where S is not invertible. After that, we characterize 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}.