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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].


References [Enhancements On Off] (What's this?)

  • [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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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1968-0233220-X
Article copyright: © Copyright 1968 American Mathematical Society

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