Oscillation and the mean ergodic theorem for uniformly convex Banach spaces
نویسندگان
چکیده
Let B be a p-uniformly convex Banach space, with p ≥ 2. Let T be a linear operator on B, and let Anx denote the ergodic average 1 n 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∈N of natural numbers we have k At k+1 x − At k x p ≤ Cx 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 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.
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