Mark Embree, Rice, Definitions of the Pseudospectrum for Analyzing Behavior of Dynamical Systems

Abstract: For an operator on a Banach space, the usual approach to the pseudospectrum leads to two equivalent definitions, one in terms of the norm of the resolvent, the other involving eigenvalues of perturbed operators. The equivalence of these definitions depends on the use of the induced operator norm applied to a standard eigenvalue problem; for different norms (e.g., Hilbert-Schmidt) or more exotic eigenvalue problems (e.g., matrix pencils), the equivalence fails, and the definition one prefers depends largely on the motivation application. While many favor generalizations based upon eigenvalue perturbations, we argue that in a variety of settings one learns more from definitions based on the resolvent and fundamental solutions to the motivating dynamical system. We shall illustrate the merits of this perspective through examples involving the Hilbert-Schmidt norm, differential-algebraic equations, and polynomial eigenvalue problems.