Spectral properties of continuous representations of topological groups
The investigations are motivated by investigations of spectral properties and characterizations of uniform continuity of one-parameter \(C_0\)-operator groups \(\{T_t\}\) -- hence representations of the group \(\mathbb{R}\) -- e.g., by the angular distribution of the spectrum \(\sigma^1(T_t) :=\{\lambda/|\lambda|: \lambda\in \mathrm{Spec}(T_t)\}\), cf. [\textit{K. Latrach, J. M. Paoli} and \textit{P. Simonnet}, ``A spectral characterization of the uniform continuity of strongly continuous groups'', Arch. Math. 90, No.~5, 420--428 (2008; Zbl 1154.47033)]. As shown by counterexamples, these results are not valid for locally bounded representations of arbitrary locally compact Abelian groups. This motivates investigations of strongly, but not uniformly continuous representations \(\theta\) of (second countable) locally compact Abelian groups \(G\) and of related characterizations of uniform continuity. A representation \(\theta\) of \(G\) by linear operators of a Banach space generates a commutative Banach algebra (of operators), this leads to representations of \(G\) on a commutative Banach algebra, and hence Gelfand-theoretical methods and tools are available for analyzing spectral properties of \(\theta\), cf., e.g., [\textit{J. Esterle}, ``The zero-\(\sqrt{3}\) and zero-two laws for representations of locally compact abelian groups'', J. Contemp. Math. Anal., Armen. Acad. Sci. 38, No.~5, 9--19 (2003); translation from Izv. Nats. Akad. Nauk Armen., Mat. 38, No.~5, 11--22 (2003; Zbl 1160.22301)]. A key result for characterizations of uniform continuity is Proposition 2.4, saying that strongly continuous representations \(\theta\) of \(G\) on a commutative Banach algebra \(\mathcal{A}\) are uniformly continuous iff, for all \(\chi\in\widehat{\mathcal{A}}\), \(\chi\circ\theta\) is continuous. The main result (Theorem 4.6) characterizes uniform continuity of \(\theta\) by spectral properties, similar (but not equal) to the aforementioned characterizations of uniform continuity of \(C_0\)-semigroups.
strongly continuous linear representations
scientific article; zbMATH DE number 5886758