Precompactness in the uniform ergodic theory

Yu. Lyubich; J. Zemánek

Studia Mathematica (1994)

  • Volume: 112, Issue: 1, page 89-97
  • ISSN: 0039-3223

Abstract

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

How to cite

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Lyubich, Yu., and Zemánek, J.. "Precompactness in the uniform ergodic theory." Studia Mathematica 112.1 (1994): 89-97. <http://eudml.org/doc/216140>.

@article{Lyubich1994,
abstract = {We characterize the Banach space operators T whose arithmetic means $\{n^\{-1\}(I + T + ... + T^\{n-1\})\}_\{n ≥ 1\}$ form a precompact set in the operator norm topology. This occurs if and only if the sequence $\{n^\{-1\} T^n\}_\{n ≥ 1\}$ is precompact and the point 1 is at most a simple pole of the resolvent of T. Equivalent geometric conditions are also obtained.},
author = {Lyubich, Yu., Zemánek, J.},
journal = {Studia Mathematica},
keywords = {Banach space operators; arithmetic means; precompact set in the operator norm topology},
language = {eng},
number = {1},
pages = {89-97},
title = {Precompactness in the uniform ergodic theory},
url = {http://eudml.org/doc/216140},
volume = {112},
year = {1994},
}

TY - JOUR
AU - Lyubich, Yu.
AU - Zemánek, J.
TI - Precompactness in the uniform ergodic theory
JO - Studia Mathematica
PY - 1994
VL - 112
IS - 1
SP - 89
EP - 97
AB - We characterize the Banach space operators T whose arithmetic means ${n^{-1}(I + T + ... + T^{n-1})}_{n ≥ 1}$ form a precompact set in the operator norm topology. This occurs if and only if the sequence ${n^{-1} T^n}_{n ≥ 1}$ is precompact and the point 1 is at most a simple pole of the resolvent of T. Equivalent geometric conditions are also obtained.
LA - eng
KW - Banach space operators; arithmetic means; precompact set in the operator norm topology
UR - http://eudml.org/doc/216140
ER -

References

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  12. [12] M. Mbekhta et J. Zemánek, Sur le théorème ergodique uniforme et le spectre, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 1155-1158. 
  13. [13] H. C. Rönnefarth, Charakterisierung des Verhaltens der Potenzen eines Elementes einer Banach-Algebra durch Spektraleigenschaften, Diplomarbeit, Technische Universität Berlin, Berlin, 1993. 
  14. [14] A. Świech, Spectral characterization of operators with precompact orbit, Studia Math. 96 (1990), 277-282; 97 (1991), 266. Zbl0725.47003
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  16. [16] J. Zemánek, On the Gelfand-Hille theorems, in: Functional Analysis and Operator Theory, J. Zemánek (ed.), Banach Center Publ. 30, Warszawa, 1994, 369-385. Zbl0822.47005

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