Geodesic
Blum–Hanson ergodic theorem in a Banach lattices of sequences
Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 3, pp. 3-10 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is well known that a linear contraction $T$ on a Hilbert space has the so called Blum–Hanson property, i. e., that the weak convergence of the powers $T^n$ is equivalent to the strong convergence of Ĉesaro averages $\frac1{m+1}\sum_{n=0}^m T^{k_n}$ for any strictly increasing sequence $\{k_n\}$. A similar property is true for linear contractions on $l_p$-spaces ($1\le p<\infty$), for linear contractions on $L^1$, or for positive linear contractions on $L^p$-spaces ($1< p<\infty$). We prove that this property holds for any linear contractions on a separable $p$-convex Banach lattices of sequences. 
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A. N. Azizov; V. I. Chilin. Blum–Hanson ergodic theorem in a Banach lattices of sequences. Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 3, pp. 3-10. http://geodesic.mathdoc.fr/item/VMJ_2017_19_3_a0/

[1] Akcoglu M., Sucheston L., “On operator convergence in Hilbert space and in Lebesgue space”, Period. Math. Hungar., 2 (1972), 235–244 | DOI | MR | Zbl

[2] Akcoglu M. A., Huneke J. P., Rost H., “A couterexample to Blum-Hanson theorem in general spaces”, Pacific J. of Math., 50 (1974), 305–308 | DOI | MR | Zbl

[3] Akcoglu M. A., Sucheston L., “Weak convergence of positive contractions implies strong convergence of averages”, Zeitschrift feur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 32 (1975), 139–145 | DOI | MR | Zbl

[4] Bellow A., “An $L_p$-inequality with application to ergodic theory”, Hous. J. Math., 1:1 (1975), 153–159 | MR | Zbl

[5] Bennet C., Sharpley R., Interpolation of Operators, Acad. Press, Inc., N. Y., 1988 | MR | Zbl

[6] Blum J. R., Hanson D. L., “On the mean ergodic theorem for subsequences”, Bull. Amer. Math. Soc., 66 (1960), 308–311 | DOI | MR | Zbl

[7] Creekmore J., “Type and cotype in Lorentz of $L_{p,q}$ spaces”, Indag. Math., 43 (1981), 145–152 | DOI | MR | Zbl

[8] Dunford N., Schwartz J. T., Linear Operators, v. I, General Theory, Wiley, 1988 | MR

[9] Kantorovich L. V., Akilov G. P., Functional Analysis, Pergamon Press, Oxford–N. Y. etc, 1982 | MR | Zbl

[10] Krengel U., Ergodic Theorems, De Gruyter Stud. Math., 6, Walter de Gruyter, Berlin–N. Y., 1985 | MR | Zbl

[11] Lefevre P., Matheron E., Primot A., “Smoothness, asymptotic smoothness and the Blum–Hanson property”, Israel J. Math., 211 (2016), 271–309 | DOI | MR | Zbl

[12] Lin M., “Mixing for Markov operators”, Z. Wahrsch. Verw. Geb., 19 (1971), 231–242 | DOI | MR

[13] Lindenstrauss J., Tzafriri L., Classical Banach Spaces, Springer-Verlag, Berlin–N. Y., 1996 | MR | Zbl

[14] Jones L. K., Kuftinec V., “A note on the Blum–Hanson theorem”, Proc. Amer. Math. Soc., 30 (1970), 202–203 | MR

[15] Hao M. C., Kami'nska A., Tomczak-Jaegermann N., “Orlicz spaces with convexity or concavity constant one”, J. Math. Anal. Appl., 320 (2006), 303–321 | DOI | MR | Zbl

[16] Millet A., “Sur le théorème en moyenne d'Akcoglu–Sucheston”, Mathematische Zeitschrift, 172 (1980), 213–237 | DOI | MR | Zbl

[17] Muller V., Tomilov Y., “Quasisimilarity of power bounded operators and Blum–Hanson property”, J. Funct. Anal., 246 (2007), 385–399 | DOI | MR | Zbl