Modulated (C,α)-ergodic theorems in noncommutative Lp-spaces

Abstract

In this paper, we first establish the weighted $(C,\alpha )$-maximal ergodic inequalities for positive Dunford-Schwartz operators acting on noncommutative $L_p$ spaces for $\alpha >0$ and $1\le p\le \infty$. Utilizing these results, we then present the noncommutative Besicovitch weighted $(C,\alpha )$-individual ergodic theorems. Additionally, we also investigate the norm convergence of the Besicovitch weighted $(C,\alpha )$-ergodic averages. Some of them appear to be new even in the commutative setting.