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	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. 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. ← Contents of issue


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