New Estimates of the Spectral Dichotomy

We derive new estimates of the spectral dichotomy for matrices and matrix pencils which are based upon estimates of the restrictions of Green functions associated with the spectrum dichotomy problem onto the stable and unstable invariant subspaces and estimates of angles between these subspaces. De nouvelles estimations pour la dichotomie spectrale R esum e : Nous etablissons de nouvelles estimations pour la dichotomie spectrale des matrices et des faisceau de matrices. Elles sont bas ees d'une part sur des estimations des restrictions de fonctions de Green aux sous-espaces invariants stable et non stable (fonctions de Green associ ees au probl eme de la dichotomie spectrale) et d'autre part sur des estimations des angles entre ces sous-espaces. New estimates of the spectral dichotomy 3 In 5]-8] the author developed a theory of the spectral dichotomy for regular matrix pencils. In these papers we introduced the numerical parameter ! that serves as a condition number and enables to estimate the convergence rate of methods for solving the spectrum dichotomy problem, sensitivity of spectral characteristics with respect to perturbations of matrices and pencils and to the rounding errors on a computer. But in 7], 1] it was demonstrated that the perturbation theory also can be built in terms of norms of the resolvents on relevant contours. These latter condition numbers may be essentially better than !, in particular, when the angles between the invariant subspaces deened by the spectral dichotomy are small. Below we suggest another variant of condition numbers which simultaneously combines certain properties of ! and those of the restricted norms. In contrast with the norms of resolvents our new bounds are eeciently computed in the course of numerical solving the spectrum dichotomy problem. The present work is, in fact, a continuation of 5]-8], therefore the reader is expected to be familiar with main results stated in these papers. We highly recommend to get acquainted with 8] and, possibly, with several relevant sections of 7]. All matrix norms used in this paper are spectral. By rank we denote the minimal non-zero singular value of a matrix.