Spectral sets, extremal functions and exceptional matrices

Abstract

Let A be a square matrix and let Ω be an open set in the plane containing the spectrum of A. We consider the problem of maximizing the operator norm f (A) amongst all holomorphic functions f from Ω into the closed unit disk. If f0 is extremal for this problem and if f0(A) > 1, then it turns out that the matrix f0(A) has special properties, among them the fact that its principal left and right singular vectors are mutually orthogonal. We study this class of exceptional matrices f0(A). In particular, we are interested in the extent to which they are characterized by the aforementioned orthogonality property.