A spectral criterion of asymptotic almost periodicity for uniformly continuous representations of Abelian semigroups
Let S be a topological abelian semigroup with unit and T a representation of S as linear operations in a Banach space B. T is supposed continuous with respect to the strong topology of linear operations in B and is bounded (i.e. \(\sup_{s}\| T(s)\| \leq \infty)\). We make S a directed space by assuming \(s\geq t\) iff \(s=tu\) for some \(u\in S\). This induces a direction in \(\{\) T(s)\(\}\). We call T asymptotically almost periodic (a.a.p.) iff every directed subset of \(\{\) T(s)\(\}\) contains a strongly convergent directed sequence. This is a weakened form of almost periodicity of T - a condition which means that for every \(x\in B\) the orbit \(O(x)=\{T(s)x\}\) is relatively compact. According to a result of \textit{M. Yu. Lyubich} and \textit{Yu. I. Lyubich} [Teor. Funkts., Funkts. Anal. Prilozh. 45, 69-84 (1986; Zbl 0629.47033)] the a.a.p. property of T is equivalent to the possibility of representing B as the direct sum \(B_ o+B_ 1\) of the ``interior spectrum'' \(B_ 0=\{x:\) \(\lim_{s} T(s)x=0\}\) and the ``boundary spectrum'' \(B_ 1\) which is a topological direct sum of all \(V_{\chi}=\{x:\) \(x\in B\), \(T(s)x=\chi (s)x\) (s\(\in S)\}\) where \(\chi\) denotes a unitary character of S such that \(V_{\chi}\neq \{0\}\). (The set of all such characters is called the unitary asymptotic spectrum or unitary a-spectrum of T). The author infers therefrom that, if T is a.a.p., and \(\chi\) a unitary character of S then the spaces \(V_{\chi}\) and \(W_{\chi}=\{f:\) \(f\in B^*\), \(T^*(s)f=\chi (s)f\) (s\(\in S)\}\) are total sets of linear functionals for each other (in other words: T satisfies the spectral duality condition). The main result of the paper consists in some kind of an inverse theorem: suppose that 1) T is uniformly continuous (i.e. continuous with respect to the operator norm), 2) its unitary a-spectrum is countable and 3) T satisfies spectral duality. Then t is a.a.p.. Instead of assuming spectral duality for T one can assume in this theorem that T is weakly almost periodic (i.e. the orbits O(x) are weakly relatively compact) since spectral duality then follows. Especially, if a representation T in a reflexive space is uniformly continuous and its unitary a-spectrum is countable then T is a.a.p.
scientific article; zbMATH DE number 4146282