Variational Analysis of Pseudospectra

The $\epsilon$-pseudospectrum of a square matrix $A$ is the set of eigenvalues attainable when $A$ is perturbed by matrices of spectral norm not greater than $\epsilon$. The pseudospectral abscissa is the largest real part of such an eigenvalue, and the pseudospectral radius is the largest absolute value of such an eigenvalue. We find conditions for the pseudospectrum to be Lipschitz continuous in the set-valued sense and hence find conditions for the pseudospectral abscissa and the pseudospectral radius to be Lipschitz continuous in the single-valued sense. Our approach illustrates diverse techniques of variational analysis. The points at which the pseudospectrum is not Lipschitz (or more properly, does not have the Aubin property) are exactly the critical points of the resolvent norm, which in turn are related to the coalescence points of pseudospectral components.