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Title: An extension of Lorentz's almost convergence and applications in Banach spaces
Authors: Mercourakis, S.
Vassiliadis, G.
Keywords: Almost Convergence
Banach Limit
Weakly Cauchy Sequence
Independent Sequence
Uniform Distribution of Sequences
Issue Date: 2006
Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation: Serdica Mathematical Journal, Vol. 32, No 1, (2006), 71p-98p
Abstract: We investigate an extension of the almost convergence of G. G. Lorentz requiring that the means of a bounded sequence converge uniformly on a subset M of N. We also present examples of sequences α∈ l∞(N) whose sequences of translates (Tn α)n≥ 0 (where T is the left-shift operator on l∞(N)) satisfy: (a) Tn α, n ≥ 0 generates a subspace E(α) of l∞(N) that is isomorphically embedded into c0 while α is not almost convergent. (b) Tn α, n ≥ 0 admits an l1-subsequence and a nontrivial weakly Cauchy subsequence while a is almost convergent. Finally we show that, in the sense of measure, for almost all real sequences taking values in a compact set K ⊆ R (with at least two points), the sequence (Tn α)n ≥ 0 is equivalent in the supremum norm to the usual l1-basis and (hence) not almost convergent.
Description: 2000 Mathematics Subject Classification: Primary 40C99, 46B99.
ISSN: 1310-6600
Appears in Collections:Volume 32, Number 1

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