R-bounded Representations of L1 (G)

Abstract

We investigate R-bounded representations $$\Psi: L^{1}\left( G\right) \rightarrow {\mathcal{L}}\left( X\right) $$ , where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism $$U:G\rightarrow {\mathcal{L}}\left( X\right) $$ , we are then able to analyze certain classical homomorphisms U (e.g. translations in L p (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators.