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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Ergodic projections of continuous and discrete semigroups

Author: Sen-Yen Shaw
Journal: Proc. Amer. Math. Soc. 78 (1980), 69-76
MSC: Primary 47A35; Secondary 47D05
MathSciNet review: 548087
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Abstract: Let X be a Banach space. Let $ \{ T(t);t > 0\} $ be a uniformly bounded semigroup of operators on X, which converges strongly to P, known to be a projection, as t goes to 0. If A is its generator and $ {X_0}$ [resp., $ {X_t},t > 0$] is the set of x for which

$\displaystyle {P_0}x \equiv \mathop {\lim }\limits_{t \to \infty } {t^{ - 1}}\i... ...mits_{n \to \infty } {n^{ - 1}}\sum\limits_{i = 0}^{n - 1} {T(it)x} } \right]} $

exists, then, for each $ t \geqslant 0,{P_t}$ is a bounded projection in $ {X_t}$; when $ t = 0,{X_0} = N(A) \oplus \overline {R(A)} \oplus N(P),\;R({P_0}) = N(A)$ and $ N({P_0}) = \overline {R(A)} \oplus N(P)$; when $ t > 0$, then

$\displaystyle {X_t} = N(T(t) - I) \oplus \overline {R(T(t) - I)} ,$

$ R({P_t}) = N(T(t) - I)$ and $ N({P_t}) = \overline {R(T(t) - I)} ;\;{X_t} = X$ for all $ t \geqslant 0$ if X is reflexive. Some results on relations among the projections $ {P_t},t \geqslant 0$, are obtained. In particular, we have $ {P_t} = {P_0}$ for all sufficiently small t if A is bounded.

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Keywords: Semigroups of operators, mean ergodic theorem, ergodic projections, infinitesimal generator
Article copyright: © Copyright 1980 American Mathematical Society

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