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Proceedings of the American Mathematical Society

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



Weak convergence of semigroups implies strong convergence of weighted averages

Author: Humphrey Fong
Journal: Proc. Amer. Math. Soc. 56 (1976), 157-161
MSC: Primary 47A35; Secondary 28A65
Erratum: Proc. Amer. Math. Soc. 61 (1976), 186.
MathSciNet review: 0405133
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Abstract: For a fixed $p,1 \leqslant p < \infty$, let $\{ {T_t}:t > 0\}$ be a strongly continuous semigroup of positive contractions on ${L_p}$ of a $\sigma$-finite measure space. We show that weak convergence of $\{ {T_t}:t > 0\}$ in ${L_p}$ is equivalent with the strong convergence of the weighted averages $\int _0^\infty {{T_t}f{\mu _n}(dt)(n \to \infty )}$ for every $f \in {L_p}$ and every sequence $({\mu _n})$ of signed measures on $(0,\infty )$, satisfying ${\sup _n}||{\mu _n}|| < \infty ;{\lim _n}{\mu _n}(0,\infty ) = 1$; and for each $d > 0,{\lim _n}{\sup _{c \geqslant 0}}|{\mu _n}|(c,c + d] = 0$. The positivity assumption is not needed if $p = 1$ or 2. We show that such a result can be deduced-not only in ${L_p}$, but in general Banach spaces-from the corresponding discrete parameter version of the theorem.

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Keywords: Banach space, <IMG WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${L_p}$">-space, semigroup of operators, weak convergence, ergodic theorem
Article copyright: © Copyright 1976 American Mathematical Society