\(C_{0}\)-continuity of the Fröbenius-Perron semigroup
Summary: We consider the Fröbenius-Perron semigroup of linear operators associated to a semidynamical system defined in a topological space \(X\) endowed with a finite or a \(\sigma\)-finite regular measure. We prove that if there exists a faithful invariant measure for the semidynamical system, then the Fröbenius-Perron semigroup of linear operators is \(C_{0}\)-continuous in the space \(L_\mu^1(X)\). We also give a geometrical condition which ensures \(C_{0}\)-continuity of the Fröbenius-Perron semigroup of linear operators in the space \(L_\mu^p(X)\) for \(1\leq p<\infty\), as well as in the space \(L_{\text{loc}}^1\).
scientific article; zbMATH DE number 1780426