zbMATH — the first resource for mathematics

Ergodic properties of contraction semigroups in \(L_ p, 1<p<\infty\). (English) Zbl 0814.47010
Summary: Let \(\{T(t): t>0\}\) be a strongly continuous semigroup of linear contractions in \(L_ p\), \(1< p<\infty\), of a \(\sigma\)-finite measure space. In this paper we prove that if there corresponds to each \(t> 0\) a positive linear contraction \(P(t)\) in \(L_ p\) such that \(| T(t) f|\leq P(t)| f|\) for all \(f\in L_ p\), then there exists a strongly continuous semigroup \(\{S(t): t> 0\}\) of positive linear contractions in \(L_ p\) such that \(| T(t) f|\leq S(t)| f|\) for all \(t> 0\) and \(f\in L_ p\).
Using this and Akcoglu’s dominated ergodic theorem for positive linear contractions in \(L_ p\), we also prove multiparameter pointwise ergodic and local ergodic theorems for such semigroups.

47A35 Ergodic theory of linear operators
47D06 One-parameter semigroups and linear evolution equations
Full Text: EuDML