Reduction of a mean ergodic Markov semigroup on \(C(X)\) into its irreducible components
Let \(T=(T(t))_{t\geq 0}\) be an ergodic \(C_ 0\)-semigroup of Markov operators on \(C(K)\), where \(K\) is compact. The author shows that there exists a mutually disjoint family \((S_ \lambda)_{\lambda\in\Lambda}\) of closed subsets of \(K\) such that \(T\) naturally induces irreducible Markov-semigroups on \(C(S_ \lambda)\). If \(T\) is uniformly ergodic then the peripheral spectrum of the generator \(A\) of \(T\) is determined by that of the components. The results are analogous to those of Niiro-Sawashima in the discrete case. Their proofs depend on Greiner's Perron-Frobenius theory [see \textit{R. Nagel} (ed.), One parameter semigroups of positive operators. Lect. Notes Math. 1184 (1986; Zbl 0585.47030)].
ergodic \(C_ 0\)-semigroup of Markov operators
scientific article; zbMATH DE number 64385
scientific article