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Semigroups of multipliers associated with semigroups of operators


Author: A. Olubummo
Journal: Proc. Amer. Math. Soc. 49 (1975), 161-168
MSC: Primary 43A22
DOI: https://doi.org/10.1090/S0002-9939-1975-0621871-5
MathSciNet review: 0621871
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Abstract: Let $ G$ be an infinite compact group with dual object $ \Sigma $. Corresponding to each semigroup $ \mathcal{T} = \{ T(\xi );\xi \geq 0\} $ of operators on $ {L_p}(G),1 \leq p < \infty $, which commutes with right translations, there is a semigroup $ \mathcal{E} = \{ {E_\xi }(\sigma );\xi \geq 0,\sigma \in \Sigma \}$ of $ {L_p}(G)$ multipliers. If $ \mathcal{T}$ is strongly continuous, then $ \{ {E_\xi }(\sigma );\xi \geq 0\} $ is uniformly continuous for each $ \sigma $. Conversely a semigroup $ \mathcal{E}$ of $ {L_p}(G)$-multipliers determines a semigroup $ \mathcal{T}$ of operators on $ {L_p}(G)$, is strongly continuous if each $ {E_\xi }(\sigma )$ is uniformly continuous; and then there exist a function $ A$ on $ \Sigma $ and $ {\Sigma _0} \subset \Sigma $ such that $ {E_\xi }(\sigma ) = {E_0}(\sigma )\exp (\xi {A_\sigma })$ if $ \sigma \in {\Sigma _0}$ and $ {E_\xi }(\sigma ) = 0$ if $ \sigma \notin {\Sigma _0}$.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0621871-5
Article copyright: © Copyright 1975 American Mathematical Society

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