Abstract
We obtain, by means of a classification of the eigenvalues, local estimates for holomorphic. functions of a class of linear operators on a finite dimensional linear vector space. We apply these methods to find new proofs of some theorems ofKreiss andMorton, and in addition we give a local estimate of the powers of the inverse of any nonsingular operator in this class.
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References
Kreiss, H.-O.: Über die Stabilitäts definition für Differenzengleichungen, die partielle Differentialgleichungen approximieren. B. I. T.2, 153–181 (1962).
Miller, J., andG. Strang: Matrix theorems for Partial Differential and Difference Equations. Math. Scand.18, 113–133 (1966).
Morton, K. W.: On a matrix theorem due toH.-O. Kreiss. Comm. Pure Appl. Math.17, 375–380 (1965).
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Miller, J.J.H. On power-bounded operators and operators satisfying a resolvent condition. Numer. Math. 10, 389–396 (1967). https://doi.org/10.1007/BF02162872
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DOI: https://doi.org/10.1007/BF02162872