Comparison of K-spectral set bounds on norms of functions of a matrix or operator

Abstract

We use results in [M. Crouzeix and A. Greenbaum, Spectral sets: numerical range and beyond, SIAM Jour. Matrix Anal. Appl., 40 (2019), pp. 1087-1101] to derive upper bounds on the norm of a function f of a matrix or other bounded linear operator A based on the infinity-norm of f on various regions in the complex plane. We compare these results to those that can be derived from a straightforward application of the Cauchy integral formula by replacing the norm of the integral by the integral of the resolvent norm. While, in some cases, the new upper bounds on ‖f(A)‖ are much tighter than those from the Cauchy integral formula, we show that in many cases of interest, the two bounds are of the same order of magnitude, with that from the Cauchy integral formula actually being slightly smaller. We give a partial explanation of this in terms of the numerical range of the resolvent at points near an ill-conditioned eigenvalue. At such points we show that the resolvent is close to a rank one matrix whose numerical range is a disk about a point near the origin, and we argue that in this case the two bounds are almost the same.