Ergodic properties of contraction semigroups in $L_p$, $1$

Sato, Ryotaro

Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) no. 2, pp. 337-346.

Résumé

Let $\{T(t):t>0\}$ be a strongly continuous semigroup of linear contractions in $L_p$, $1$, 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.

Détail

Classification : 47A35, 47B38, 47D03, 47D06
Keywords: contraction semigroup; semigroup modulus; majorant; pointwise ergodic \newline theorem; pointwise local ergodic theorem