Spectral theory of group representations and their nonstandard hull
Let G denote a locally compact abelian group and let U be a bounded strongly continuous representation of G on the Banach space E. We introduce the notion of the Riesz part \(R\sigma\) (U) of the Arveson spectrum \(\sigma\) (U) of U. The representation U is called R-compact if every bounded subset \(C\subset E\) satisfying \(\lim_{t\to e}\sup \{\| U_ tx-x\|:x\in C\}=0\) is relatively compact. Then the following assertions are equivalent (i) \(\sigma (U)=R\sigma (U);\) (ii) U is R- compact; (iii) For all \(f\in L^ 1(G)\) the associated operator \(U_ f\) is compact. This theorem is applied to the complete characterization of R-compact representations on Banach lattices, which generalizes results of \textit{G. Greiner} [Über das Spektrum stark stetiger Halbgruppen positiver Operatoren, Diss. Tübingen (1980; Zbl 0475.47025)] as well as of \textit{H. Uhlig} [Derivationen und Verbandshalbgruppen, Diss. Tübingen (1979; Zbl 0453.47019)] even in the case of \(G={\mathbb{R}}\). As an important tool for the study of the spectrum we introduce the nonstandard hull of an arbitrary group representation which facilitates all proofs of known results as well as the new ones represented here.
bounded strongly continuous representation
scientific article; zbMATH DE number 3906765
scientific article