Oscillation and the mean ergodic theorem for uniformly convex Banach spaces
Abstract
Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear operator on 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 N} of natural numbers we have sum_k || A_{t_{k+1}} x - A_{t_k} x ||^p <= C || x ||^p, where the constant C depends only on p and the modulus of uniform convexity. For T a nonexpansive operator, we obtain a weaker bound on the number of epsilon-fluctuations in the sequence. We clarify the relationship between bounds on the number of epsilon-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.
Keywords
Cite
@article{arxiv.1203.4124,
title = {Oscillation and the mean ergodic theorem for uniformly convex Banach spaces},
author = {Jeremy Avigad and Jason Rute},
journal= {arXiv preprint arXiv:1203.4124},
year = {2015}
}