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

Abstract

We study the $\epsilon$-pseudospectra ${\sigma }_{\epsilon }(A)$ of square matrices $A\in {\mathbb{C}}^{N\times N}$. We give a complete characterization of the $\epsilon$-pseudospectrum of any $2\times 2$ matrix and describe the asymptotic behavior (as $\epsilon \to 0$) of ${\sigma }_{\epsilon }(A)$ for any 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.

22 pages, 6 figures

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