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

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.