Blum–Hanson ergodic theorem in a Banach lattices of sequences

It is well known that a linear contraction $T$ on a Hilbert space has the so called Blum–Hanson property, i. e., that the weak convergence of the powers $T^n$ is equivalent to the strong convergence of Ĉesaro averages $\frac1{m+1}\sum_{n=0}^m T^{k_n}$ for any strictly increasing sequence $\{k_n\}$. A similar property is true for linear contractions on $l_p$-spaces ($1\le p<\infty$), for linear contractions on $L^1$, or for positive linear contractions on $L^p$-spaces ($1< p<\infty$). We prove that this property holds for any linear contractions on a separable $p$-convex Banach lattices of sequences.