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
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
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 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.