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Precompactness in the uniform ergodic theory. (English) Zbl 0817.47014

Summary: We characterize the Banach space operators \(T\) whose arithmetic means \(\{n^{-1} (I+T+ \dots +T^{n-1} )\}_{n\geq 1}\) form a precompact set in the operator norm topology. This occurs if and only if the sequence \(\{n^{-1} T^ n \}_{n\geq 1}\) is precompact and the point 1 is at most a simple pole of the resolvent of \(T\). Equivalent geometric conditions are also obtained.

MSC:

47A35 Ergodic theory of linear operators
47A10 Spectrum, resolvent
47D03 Groups and semigroups of linear operators
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