Explicit Bounds for the Pseudospectra of Various Classes of Matrices and Operators

Abstract

We study the $ε$-pseudospectra $σ_ε(A)$ of square matrices $A \in \mathbb{C}^{N \times N}$. We give a complete characterization of the $ε$-pseudospectrum of any $2 \times 2$ matrix and describe the asymptotic behavior (as $ε\to 0$) of $σ_ε(A)$ for any square matrix $A$. We also present explicit upper and lower bounds for the $ε$-pseudospectra of bidiagonal matrices, as well as for finite rank operators.