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We investigate the relation between the spectrum of a non-normal matrix andthe norm of its resolvent. We provide spectral estimates for the resolvent ofmatrices whose largest singular value is bounded by $1$ (so-called Hilbertspace contractions) and for power-bounded matrices. In the first case ourestimate is optimal and we present explicit matrices that achieve equality inthe bound. This result recovers and generalizes previous estimates obtained byE.B. Davies and B. Simon in the study of orthogonal polynomials on the unitcircle. In case of power-bounded matrices we achieve the strongest estimate sofar. Our result unifies previous approaches, where the resolvent was estimatedin certain restricted regions of the complex plane. To achieve our estimates werelate the problem of bounding the norm of a function of a matrix to aNevanlinna-Pick interpolation problem in a corresponding function space. Incase of Hilbert space contractions this problem is connected to the theory ofcompressed shift operators to which we contribute by providing explicit matrixrepresentations for such operators. Finally, we apply our results to study thesensitivity of the stationary states of a classical or quantum Markov chainwith respect to perturbations of the transition matrix.

Eigenvalue estimates for the resolvent of a non-normal matrix