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 A(n)x denote the ergodic average (1/n)Sigma(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 is an element of N) of natural numbers we have Sigma(k) parallel to A(tk+1) x - A(tk) x parallel to(p) <= C parallel to x parallel to(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 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.