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

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]**Heinz-Otto Kreiss,*Über Matrizen die beschränkte Halbgruppen erzeugen*, Math. Scand.**7**(1959), 71–80 (German). MR**0110952****[2]**John Miller and Gilbert Strang,*Matrix theorems for partial differential and difference equations*, Math. Scand.**18**(1966), 113–133. MR**0209308****[3]**John J. H. Miller,*On power-bounded operators and operators satisfying a resolvent condition*, Numer. Math.**10**(1967), 389–396. MR**0220080****[4]**K. W. Morton,*On a matrix theorem due to H. O. Kreiss*, Comm. Pure Appl. Math.**17**(1964), 375–379. MR**0170460**

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

Article copyright:
© Copyright 1968
American Mathematical Society