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In this paper, we establish the quantitative mean ergodic theorems for twosubclasses of power bounded operators on a fixed noncommutative $L_p$-spacewith $1<p<\infty$, which mainly concerns power bounded invertible operators andLamperti contractions. Our approach to the quantitative ergodic theorems is thenoncommutative square function inequalities. The establishment of the latterinvolves several new ingredients such as the almost orthogonality andCalder\'on-Zygmund arguments for non-smooth kernels from semi-commutativeharmonic analysis, the extension properties of the operators underconsideration from operator theory, and a noncommutative version of theclassical transference method due to Coifman and Weiss.

Quantitative mean ergodic inequalities: power bounded operators acting on one single noncommutative $L_p$ space