Oscillation and the mean ergodic theorem for uniformly convex Banach spaces

Let B be a p -uniformly convex Banach space, with p≥ 2 . Let T be a linear operator on B , and let Aₙ x denote the ergodic average (1/ n)∑ i< n Tⁿ x . We prove the following variational inequality in the case where T is power bounded from above and below: for any increasing sequence (tₖ )k∈ N of natural numbers we have ∑ ₖ A_tk+ 1 x- A_tₖ x ᵖ ≤ 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 ε -fluctuations in the sequence. We clarify the relationship between bounds on the number of ε -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.