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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Weak convergence of semigroups implies strong convergence of weighted averages
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by Humphrey Fong PDF
Proc. Amer. Math. Soc. 56 (1976), 157-161 Request permission

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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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 56 (1976), 157-161
  • MSC: Primary 47A35; Secondary 28A65
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0405133-8
  • MathSciNet review: 0405133