Power growth of mean-L-stable operators on Banach spaces

Abstract

We study the growth of powers of mean-L-stable operators on Banach spaces. We show that, for linear operators, mean-L-stability is equivalent to uniform boundedness in density; this yields $\|T^n\|=O(n)$ on every Banach space. On Hilbert spaces we prove that mean-L-stability is equivalent to absolute Cesàro boundedness and obtain $\|T^n\|=O(n^{1/2-c_T})$ for some $c_T>0$. For positive mean-L-stable operators on abstract $L^p$-spaces, $1\le p<\infty$, we similarly obtain $\|T^n\|=O(n^{1/p-c_T})$, where in both cases the positive constant $c_T$ cannot be chosen uniformly over all such operators. For positive mean-L-stable operators on $p$-convex Banach lattices, we prove the bound $O(n^{1/p})$ and construct positive topologically mixing operators $T_p$ for which $\|T_p^n\|\asymp n^{1/p}$, where $1\leq p<\infty$. These operators satisfy a uniform weak $(p,p)$ orbit estimate, while the averages of $\|T_p^nx\|^s$ are bounded for $s<p$, of order $\log N$ for $s=p$, and of order $N^{s/p-1}$ for $s>p$. The operator $T_1$ is uniformly Kreiss bounded and has linear power growth, answering a question of Montes-Rodríguez, Sánchez-Álvarez and Zemánek (2005). Moreover, $T_1$ is mean-L-stable and mean Li--Yorke chaotic, while it is not distributionally chaotic. This answers a question of Bernardes, Bonilla and Peris (2020).