Mean ergodic type theorems

Year 2019, Volume: 68 Issue: 2, 2264 - 2271, 01.08.2019

Gencay Oğuz , Cihan Orhan

https://doi.org/10.31801/cfsuasmas.562214

Cited By: 1

Abstract

Let $T$ be a bounded linear operator on a Banach space $X$. Replacing the Ces\`{a}ro matrix by a regular matrix $A=(a_{nj})$ Cohen studied a mean ergodic theorem. In the present paper we extend his result by taking a sequence of infinite matrices $\mathcal{A}=(A^{(i)})$ that contains both convergence and almost convergence. This result also yields an $\mathcal{A}$-ergodic decomposition. When $T$ is power bounded we give a characterization for $T$ to be $\mathcal{A}$-ergodic.

Keywords

Infinite matrices, almost convergence, ergodic theorems

References

• Aleman, A. and Suciu, L., On ergodic operator means in Banach spaces, Integr. Equ. Oper.Theory 85, (2016), 259-287.
• Bell, H.T., Order summability and almost convergence, Proc. Amer. Math. Soc., 38 (3), (1973),
• Cohen, L.W., On the mean ergodic theorem, Ann. Math. (3), 41, (1940), 505-509.
• Krengel, U., Ergodic Theorems, de Gruyter Studies in Mathematics vol 6, Walter de Gruyter& Co., Berlin, 1985.
• Lin, M., Shoikhet, D. and Suciu L., Remaks on uniform ergodic theorems, Acta Sci. Math.(Szeged) 81, (2015), 251-283.
• Lorentz, G. G., A contribution to the theory of divergent sequences, Acta Math. 80, (1948),167-190.
• Nanda, S., Ergodic theory and almost convergence, Bull. Math, de la Soc. Sci. Math, de la R.S. de Roumanie 26, (1982), 339-343.
• Riesz, F., Some mean ergodic theorems, J. Lond. Math. Soc. 13, (1938), 274.
• Stieglitz, M., Eine verallgen meinerung des Begris Fastkonvergenz, Math. Japon. 18, (1973),53-70.
• von Neumann, J., Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. USA , 18,(1932), 70-82.
• Yoshimoto, T., Ergodic theorems and summability methods, Quart. J. Math. 38 (3), (1987),367-379.
• Yosida, K., Mean ergodic theorem in Banach space, Proc. Imp. Acad. Tokyo, 14, (1938),292-294.