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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: Let $ A$ be a densely defined closed (linear) operator, and $ \{ {A_\alpha }\} $, $ \{ {B_\alpha }\} $ be two nets of bounded operators on a Banach space $ X$ such that $ \vert\vert{A_\alpha }\vert\vert = O(1),{A_\alpha }A \subset A{A_\alpha },\vert\vert A{A_\alpha }\vert\vert = o(1)$, and $ {B_\alpha }A \subset A{B_\alpha } = I - {A_\alpha }$. Denote the domain, range, and null space of an operator $ T$ by $ D(T)$, $ R(T)$, and $ N(T)$, respectively, and let $ P(\operatorname{resp} .B)$ be the operator defined by $ Px = {\lim _\alpha }{A_\alpha }x(resp. By = {\lim _\alpha }{B_\alpha }y)$ for all those $ x \in X(\operatorname{resp} .y \in \overline {R(A)} )$ for which the limit exists. It is shown in a previous paper that $ D(P) = N(A) \oplus \overline {R(A)} ,R(P) = N(A),D(B) = A(D(A) \cap \overline {R(A)} ),R(B) = D(A) \cap \overline {R(A)} $, and that $ B$ sends each $ y \in D(B)$ to the unique solution of $ Ax = y{\text{ in }}\overline {R(A)} $. In this paper, we prove that $ D(P) = X$ and $ \vert\vert{A_\alpha } - P\vert\vert \to 0$ if and only if $ \vert\vert{B_\alpha }\vert D(B) - B\vert\vert \to 0$, if and only if $ \vert\vert{B_\alpha }\vert D(B)\vert\vert = O(1)$, if and only if $ R(A)$ is closed. Moreover, when $ X$ is a Grothendieck space with the Dunford-Pettis property, all these conditions are equivalent to the mere condition that $ D(P) = X$. The general result is then used to deduce uniform ergodic theorems for $ n$-times integrated semigroups, $ (Y)$-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, $ n$-times integrated semigroup, cosine operator function
Article copyright: © Copyright 1992 American Mathematical Society

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