Lyubich, Yu.; Zemánek, Jaroslav Precompactness in the uniform ergodic theory. (English) Zbl 0817.47014 Stud. Math. 112, No. 1, 89-97 (1994). 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. Cited in 3 Documents MSC: 47A35 Ergodic theory of linear operators 47A10 Spectrum, resolvent 47D03 Groups and semigroups of linear operators Keywords:Banach space operators; arithmetic means; precompact set in the operator norm topology PDFBibTeX XMLCite \textit{Yu. Lyubich} and \textit{J. Zemánek}, Stud. Math. 112, No. 1, 89--97 (1994; Zbl 0817.47014) Full Text: EuDML