Semigroups of multipliers associated with semigroups of operators

Abstract

Let G G be an infinite compact group with dual object Σ \Sigma . Corresponding to each semigroup T = { T ( ξ ) ; ξ ≥ 0 } \mathcal {T} = \{ T(\xi );\xi \geq 0\} of operators on L p ( G ) , 1 ≤ p > ∞ {L_p}(G),1 \leq p > \infty , which commutes with right translations, there is a semigroup E = { E ξ ( σ ) ; ξ ≥ 0 , σ ∈ Σ } \mathcal {E} = \{ {E_\xi }(\sigma );\xi \geq 0,\sigma \in \Sigma \} of L p ( G ) {L_p}(G) multipliers. If T \mathcal {T} is strongly continuous, then { E ξ ( σ ) ; ξ ≥ 0 } \{ {E_\xi }(\sigma );\xi \geq 0\} is uniformly continuous for each σ \sigma . Conversely a semigroup E \mathcal {E} of L p ( G ) {L_p}(G) -multipliers determines a semigroup T \mathcal {T} of operators on L p ( G ) {L_p}(G) , is strongly continuous if each E ξ ( σ ) {E_\xi }(\sigma ) is uniformly continuous; and then there exist a function A A on Σ \Sigma and Σ 0 ⊂ Σ {\Sigma _0} \subset \Sigma such that E ξ ( σ ) = E 0 ( σ ) exp ⁡ ( ξ A σ ) {E_\xi }(\sigma ) = {E_0}(\sigma )\exp (\xi {A_\sigma }) if σ ∈ Σ 0 \sigma \in {\Sigma _0} and E ξ ( σ ) = 0 {E_\xi }(\sigma ) = 0 if σ ∉ Σ 0 \sigma \notin {\Sigma _0} .