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 Tn is equivalent to the strong convergence of Cesaro averages 1m+1∑mn=0Tkn for any strictly increasing sequence {kn}. A similar property is true for linear contractions on lp-spaces (1≤p1, or for positive linear contractions on Lp-spaces. We prove that this property holds for any linear contractions on a separable p-convex Banach lattices of sequences.