Vector-valued Hausdorff summability methods and ergodic theorems
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- by Takeshi Yoshimoto PDF
- Proc. Amer. Math. Soc. 107 (1989), 915-926 Request permission
Abstract:
Suppose $X$ and $Y$ are two general Banach spaces. Let $H = ({\Lambda _{n,k}})$ be a general ${\mathbf {B}}[X,Y]$-operator valued Hausdorff summability method: ${\Lambda _{n,k}} = (_k^n){\Delta ^{n - k}}{U_k}$ for $k \leq n$ and ${\Lambda _{n,k}} = {\theta _{X,Y}}$ for $k > n$, where $\{ {U_k}\} _{k = 0}^\infty$ is a sequence of operators in ${\mathbf {B}}[X,Y]$ and $\Delta$ denotes the backward difference (operator) and ${\theta _{X,Y}}(x) = {0_Y}$ (the zero element in $Y$) for all $x \in$. Then some necessary and sufficient conditions are given for the mean and uniform convergence of the averages \[ \sum \limits _{k = 0}^n {(_k^n){\Delta ^{n - k}}{U_k}({T^k}x)} \quad (x \in X,T \in {\mathbf {B}}[X]).\]References
- S. D. Chatterji, Les martingales et leurs applications analytiques, École d’Été de Probabilités: Processus Stochastiques (Saint Flour, 1971) Lecture Notes in Math., Vol. 307, Springer, Berlin, 1973, pp. 27–164 (French). MR 0448536
- L. W. Cohen, On the mean ergodic theorem, Ann. of Math. (2) 41 (1940), 505–509. MR 2027, DOI 10.2307/1968732
- Nelson Dunford, Spectral theory. I. Convergence to projections, Trans. Amer. Math. Soc. 54 (1943), 185–217. MR 8642, DOI 10.1090/S0002-9947-1943-0008642-1 N. Dunford and J. T. Schwartz, Linear operators, part 1, Interscience, New York, 1958.
- G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
- L. C. Kurtz and D. H. Tucker, Vector-valued summability methods on a linear normed space, Proc. Amer. Math. Soc. 16 (1965), 419–428. MR 199592, DOI 10.1090/S0002-9939-1965-0199592-9
- Lynn C. Kurtz and Don H. Tucker, An extended form of the mean-ergodic theorem, Pacific J. Math. 27 (1968), 539–545. MR 236561
- Michael Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc. 43 (1974), 337–340. MR 417821, DOI 10.1090/S0002-9939-1974-0417821-6
- Robert Sine, A mean ergodic theorem, Proc. Amer. Math. Soc. 24 (1970), 438–439. MR 252605, DOI 10.1090/S0002-9939-1970-0252605-X
- Don H. Tucker, A representation theorem for a continuous linear transformation on a space of continuous functions, Proc. Amer. Math. Soc. 16 (1965), 946–953. MR 199722, DOI 10.1090/S0002-9939-1965-0199722-9
- Takeshi Yoshimoto, Ergodic theorems and summability methods, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 151, 367–379. MR 907244, DOI 10.1093/qmath/38.3.367 —, General ergodic theorems for linear operators, Publ. I.R.M.A. Strasbourg, 376/P-208, 1988.
- Kôsaku Yosida and Shizuo Kakutani, Operator-theoretical treatment of Markoff’s process and mean ergodic theorem, Ann. of Math. (2) 42 (1941), 188–228. MR 3512, DOI 10.2307/1968993
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 107 (1989), 915-926
- MSC: Primary 47A35
- DOI: https://doi.org/10.1090/S0002-9939-1989-0984826-7
- MathSciNet review: 984826