Uniform convergence of ergodic limits and approximate solutions

Author:
Sen-Yen Shaw

Journal:
Proc. Amer. Math. Soc. **114** (1992), 405-411

MSC:
Primary 47A35; Secondary 47D03

DOI:
https://doi.org/10.1090/S0002-9939-1992-1089413-0

MathSciNet review:
1089413

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a densely defined closed (linear) operator, and , be two nets of bounded operators on a Banach space such that , and . Denote the domain, range, and null space of an operator by , , and , respectively, and let be the operator defined by for all those for which the limit exists. It is shown in a previous paper that , and that sends each to the unique solution of . In this paper, we prove that and if and only if , if and only if , if and only if is closed. Moreover, when is a Grothendieck space with the Dunford-Pettis property, all these conditions are equivalent to the mere condition that . The general result is then used to deduce uniform ergodic theorems for -times integrated semigroups, -semigroups, and cosine operator functions.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1992-1089413-0

Keywords:
Uniform ergodicity,
linear functional equation,
Grothendieck space with the Dunford-Pettis property,
-times integrated semigroup,
cosine operator function

Article copyright:
© Copyright 1992
American Mathematical Society