Blum-Hanson Ergodic Theorem in a Banach Lattices of Sequences

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2017.3.7107 Blum-Hanson Ergodic Theorem in a Banach Lattices of Sequences Azizov A. N. , Chilin V. I. Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 3. Abstract: 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 Cesaro 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. Keywords: Banach solid lattice, \(p\)-convexity, linear contraction, ergodic theorem Language: Russian Download the full text For citation: Azizov A. N., Chilin V. I. Blum-Hanson ergodic theorem in a Banach lattices of sequences // Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp. 3-10. DOI 10.23671/VNC.2017.3.7107 + References 1. Akcoglu M., Sucheston L. On operator convergence in Hilbert space and in Lebesgue space. Period. Math. Hungar. 1972. Vol. 2. P. 235-244. 2. Akcoglu M. A., Huneke J. P. and Rost H. A couterexample to Blum–Hanson theorem in general spaces. Pacific J. of Math. 1974. Vol. 50. P. 305-308. 3. Akcoglu M. A., Sucheston L. Weak convergence of positive contractions implies strong convergence of averages. Zeitschrift feur Wahrscheinlichkeitstheorie und Verwandte Gebiete. 1975. Vol. 32. P. 139-145. 4. Bellow A. An \(L_p\)-inequality with application to ergodic theory. Hous. J. Math. 1975. Vol. 1, № 1. P. 153-159. 5. Bennet C., Sharpley R. Interpolation of Operators. N.Y.: Acad. Press, Inc., 1988. 6. Blum J. R., Hanson D. L. On the mean ergodic theorem for subsequences. Bull. Amer. Math. Soc. 1960. Vol. 66. P. 308-311. 7. Creekmore J. Type and cotype in Lorentz of \(L_{p,q}\) spaces. Indag. Math. 1981. Vol. 43. P. 145-152. 8. Dunford N., Schwartz J. T. Linear Operators. Part I: General Theory. Wiley, 1988. 9. Kantorovich L. V., Akilov G. P. Functional Analysis. Oxford-N.Y. etc: Pergamon Press, 1982. 10. Krengel U. Ergodic Theorems. De Gruyter Stud. Math. Vol. 6. Walter de Gruyter. Berlin-N.Y., 1985. 11. Lefevre P., Matheron E. and Primot A. Smoothness, asymptotic smoothness and the Blum-Hanson property. Israel J. Math. 2016. Vol. 211. P. 271-309. 12. Lin M. Mixing for Markov operators. Z. Wahrsch. Verw. Geb. 1971. Vol. 19. P. 231-242. 13. Lindenstrauss J., Tzafriri L. Classical Banach Spaces. Berlin-N.Y.: Springer-Verlag, 1996. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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