Oscillation and the mean ergodic theorem for uniformly convex Banach spaces

TL;DR

This paper establishes new variational inequalities and fluctuation bounds for ergodic averages in uniformly convex Banach spaces, advancing understanding of convergence behaviors for power bounded and nonexpansive operators.

Contribution

It introduces novel bounds on ergodic averages and fluctuation counts in uniformly convex Banach spaces, linking these to metastability and demonstrating sharpness of results.

Findings

Proves a variational inequality for power bounded operators in p-uniformly convex Banach spaces.

Provides bounds on epsilon-fluctuations for nonexpansive operators.

Shows the main results are sharp through lower bounds on metastability.

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.