Semigroups of Multipliers Associated with Semigroups of Operators

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 $\sigma \notin {\Sigma _0}$.