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 Vladikavkaz. Mat. Zh., 2017, Volume 19, Number 3, Pages 3–10 (Mi vmj619)

Blum–Hanson ergodic theorem in a Banach lattices of sequences

A. N. Azizov, V. I. Chilin

National University of Uzbekistan named after M. Ulugbek, Tashkent

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 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.

Key words: Banach solid lattice, $p$-convexity, linear contraction, ergodic theorem.

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UDC: 517.98

Citation: A. N. Azizov, V. I. Chilin, “Blum–Hanson ergodic theorem in a Banach lattices of sequences”, Vladikavkaz. Mat. Zh., 19:3 (2017), 3–10

Citation in format AMSBIB
\Bibitem{AziChi17} \by A.~N.~Azizov, V.~I.~Chilin \paper Blum--Hanson ergodic theorem in a Banach lattices of sequences \jour Vladikavkaz. Mat. Zh. \yr 2017 \vol 19 \issue 3 \pages 3--10 \mathnet{http://mi.mathnet.ru/vmj619}