Ergodicity for a family of operators
DOI:
https://doi.org/10.5269/bspm.63556Resumen
The aim of this paper is to introduce the notions of power boundedness, Cesà ro boundedness, mean ergodicity, and uniform ergodicityfor a family of bounded linear operators on a Banach space. The authors present some elementary results in this setting and show that some
main results about power bounded, Cesà ro bounded, mean ergodic, and the uniform ergodic operator can be extended from the case of
a linear bounded operator to the case of a family of bounded linear operators acting on a Banach space. Also, we show that the Yosida
theorem can be extended from the case of a bounded linear operator to the case of a family of bounded linear operators acting on a Banach
space.
Referencias
1. A. Akrym, A. El Bakkali and A. Faouzi,On ergodic theorem for a family of operators, Oper. Matrices, Vol. 15, No. 3, 1161-1170, (2021).
2. Y. Derriennic, On the mean ergodic theorem for Cesaro bounded operators, Colloquium Math., 84/85 2, 443-445, (2000).
3. N. Dunford, Spectral theory I. Convergence to projection, Trans. Amer. Math. Soc. 54, 185-217, (1943).
4. R. Emilion, Mean bounded operators and mean ergodic theorems, J. Funct. Anal., 61, 1-14, (1985).
5. S. Grabiner and J. Zemanek, Ascent, descent, and ergodic properties of linear operators, J. Operator Theory 48, 69-81, (2002).
6. E. Hille, Remarks on ergodic theorems, Trans. Amer. Math. Soc., 57, 246-269, (1945).
7. U. Krengel, Ergodic Theorems, Walter de Gruyter Studies in Mathematics 6, Walter de Gruyter, Berlin - New York, (1985).
8. Z. Leka, A note on the powers of Cesaro bounded operators, Czechoslovak Math. J., 60 (135), 1091-1100, (2010).
9. M. Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc. 43, No. 2, 337-340, (1974).
10. S. Macovei, Spectrum of a Family of Operators, Surv. Math. Appl. 6, 137-159, (2011).
11. S. Macovei, Local Spectrum of a Family of Operators, Ann. Funct. Anal. 4, no. 2, 131-143, (2013).
12. M. Mbekhta and J. Zemanek, Sur le theoreme ergodique uniforme et le spectre, C. R. Acad. Sci. Paris serie I Math., 317, 1155-1158, (1993).
13. K. Yosida, Functional Analysis, Springer-Verlag, Berlin, Heidelberg, New York, (1980).
2. Y. Derriennic, On the mean ergodic theorem for Cesaro bounded operators, Colloquium Math., 84/85 2, 443-445, (2000).
3. N. Dunford, Spectral theory I. Convergence to projection, Trans. Amer. Math. Soc. 54, 185-217, (1943).
4. R. Emilion, Mean bounded operators and mean ergodic theorems, J. Funct. Anal., 61, 1-14, (1985).
5. S. Grabiner and J. Zemanek, Ascent, descent, and ergodic properties of linear operators, J. Operator Theory 48, 69-81, (2002).
6. E. Hille, Remarks on ergodic theorems, Trans. Amer. Math. Soc., 57, 246-269, (1945).
7. U. Krengel, Ergodic Theorems, Walter de Gruyter Studies in Mathematics 6, Walter de Gruyter, Berlin - New York, (1985).
8. Z. Leka, A note on the powers of Cesaro bounded operators, Czechoslovak Math. J., 60 (135), 1091-1100, (2010).
9. M. Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc. 43, No. 2, 337-340, (1974).
10. S. Macovei, Spectrum of a Family of Operators, Surv. Math. Appl. 6, 137-159, (2011).
11. S. Macovei, Local Spectrum of a Family of Operators, Ann. Funct. Anal. 4, no. 2, 131-143, (2013).
12. M. Mbekhta and J. Zemanek, Sur le theoreme ergodique uniforme et le spectre, C. R. Acad. Sci. Paris serie I Math., 317, 1155-1158, (1993).
13. K. Yosida, Functional Analysis, Springer-Verlag, Berlin, Heidelberg, New York, (1980).
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2024-05-08
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