Eigenvalue estimates for the resolvent of a non-normal matrix

TL;DR

This paper derives optimal and strong spectral estimates for the resolvent of non-normal matrices, unifying previous methods and applying to Markov chain stability analysis.

Contribution

It provides new optimal bounds for resolvent norms of Hilbert space contractions and power-bounded matrices, connecting matrix analysis with Nevanlinna-Pick interpolation.

Findings

Optimal resolvent bounds for Hilbert space contractions.

Strongest estimates for power-bounded matrices.

Application to stability of Markov chains.

Abstract

We investigate the relation between the spectrum of a non-normal matrix and the norm of its resolvent. We provide spectral estimates for the resolvent of matrices whose largest singular value is bounded by $1$ (so-called Hilbert space contractions) and for power-bounded matrices. In the first case our estimate is optimal and we present explicit matrices that achieve equality in the bound. This result recovers and generalizes previous estimates obtained by E.B. Davies and B. Simon in the study of orthogonal polynomials on the unit circle. In case of power-bounded matrices we achieve the strongest estimate so far. Our result unifies previous approaches, where the resolvent was estimated in certain restricted regions of the complex plane. To achieve our estimates we relate the problem of bounding the norm of a function of a matrix to a Nevanlinna-Pick interpolation problem in a corresponding function space. In case of Hilbert space contractions this problem is connected to the theory of compressed shift operators to which we contribute by providing explicit matrix representations for such operators. Finally, we apply our results to study the sensitivity of the stationary states of a classical or quantum Markov chain with respect to perturbations of the transition matrix.