Explicit bounds for the pseudospectra of various classes of matrices and operators

Abstract

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