Oscillation and the mean ergodic theorem for uniformly convex Banach spaces

Abstract

AbstractLet $ \mathbb{B} $ be a $p$-uniformly convex Banach space, with $p\geq 2$. Let $T$ be a linear operator on $ \mathbb{B} $, and let ${A}_{n} x$ denote the ergodic average $(1/ n){\mathop{\sum }\nolimits}_{i\lt n} {T}^{n} x$. We prove the following variational inequality in the case where $T$ is power bounded from above and below: for any increasing sequence $\mathop{({t}_{k} )}\nolimits_{k\in \mathbb{N} } $ of natural numbers we have ${\mathop{\sum }\nolimits}_{k} \mathop{\Vert {A}_{{t}_{k+ 1} } x- {A}_{{t}_{k} } x\Vert }\nolimits ^{p} \leq C\mathop{\Vert x\Vert }\nolimits ^{p} $, where the constant $C$ depends only on $p$ and the modulus of uniform convexity. For $T$ a non-expansive operator, we obtain a weaker bound on the number of $\varepsilon $-fluctuations in the sequence. We clarify the relationship between bounds on the number of $\varepsilon $-fluctuations in a sequence and bounds on the rate of metastability, and provide lower bounds on the rate of metastability that show that our main result is sharp.