Geodesic
An extension of Lorentz's almost convergence and applications in Banach spaces
Serdica Mathematical Journal, Tome 32 (2006) no. 1, pp. 71-98

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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.
Keywords: Almost Convergence, Banach Limit, Weakly Cauchy Sequence, Independent Sequence, Uniform Distribution of Sequences 
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     title = {An extension of {Lorentz's} almost convergence and applications in {Banach} spaces},
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Mercourakis, S.; Vassiliadis, G. An extension of Lorentz's almost convergence and applications in Banach spaces. Serdica Mathematical Journal, Tome 32 (2006) no. 1, pp. 71-98. http://geodesic.mathdoc.fr/item/SMJ2_2006_32_1_a6/