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Oscillation and the mean ergodic theorem for uniformly convex Banach spaces. (English) Zbl 1355.37009
Summary: Let \(\mathbb{B}\) 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 \((1/ n)\sum_{i< n} {T}^{n} x\). We prove the following variational inequality in the case where \(T\) is power bounded from above and below: for any increasing sequence \(({t}_{k})_{k\in \mathbb{N}}\) of natural numbers we have \(\sum_{k} \| {A}_{{t}_{k+ 1}} x-{A}_{{t}_{k}} x\|^{p} \leq C\| x\|^{p}\), 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.

MSC:
37A30 Ergodic theorems, spectral theory, Markov operators
47A35 Ergodic theory of linear operators
47L10 Algebras of operators on Banach spaces and other topological linear spaces
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