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.

References in corpus (2)

• Variational bounds for a dyadic model of the bilinear Hilbert transform
• Uncomputably noisy ergodic limits

Cited by in corpus (6)

• Norm-variation of ergodic averages with respect to two commuting transformations
• Norm convergence of multiple ergodic averages on amenable groups
• Fluctuation bounds for ergodic averages of amenable groups
• On the speed of convergence in the ergodic theorem for shift operators
• A quantitative multiparameter mean ergodic theorem
• Norm-variation of triple ergodic averages for commuting transformations