On the resolvent of a linear operator associated with a well-posed Cauchy problem

Author:
John Miller

Journal:
Math. Comp. **22** (1968), 541-548

MSC:
Primary 47.30; Secondary 35.00

DOI:
https://doi.org/10.1090/S0025-5718-1968-0233220-X

MathSciNet review:
0233220

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Abstract: We show how local estimates may be obtained for holomorphic functions of a class of linear operators on a finite-dimensional linear vector space. This is accomplished by classifying the spectrum of each operator and then estimating its resolvent on certain contours in the left half-plane. We apply these methods to prove some known theorems, and in addition we obtain new estimates for the inverse of these operators. Analogous results for power-bounded operators are given in [3].

**[1]**H.-O. Kreiss, ``Über Matrizen die beschränkte Halbgruppen erzeugen,''*Math. Scand.*, v. 7, 1959, pp. 71-80. MR**0110952 (22:1820)****[2]**John Miller & Gilbert Strang, ``Matrix theorems for partial differential and difference equations,''*Math. Scand.*, v. 18, 1966, pp. 113-133. MR**35**#206. MR**0209308 (35:206)****[3]**John Miller, ``On power-bounded operators and operators satisfying a resolvent condition,''*Numer. Math.*, v. 10, 1967, pp. 389-396. MR**0220080 (36:3147)****[4]**K. W. Morton, ``On a matrix theorem due to H.-O. Kreiss,''*Comm. Pure Appl. Math.*, v. 17, 1965, pp. 375--380. MR**30**#698. MR**0170460 (30:698)**

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DOI:
https://doi.org/10.1090/S0025-5718-1968-0233220-X

Article copyright:
© Copyright 1968
American Mathematical Society