An extension of Lorentz's almost convergence and applications in Banach spaces

Mercourakis, S., and Vassiliadis, G.. "An extension of Lorentz's almost convergence and applications in Banach spaces." Serdica Mathematical Journal 32.1 (2006): 71-98. <http://eudml.org/doc/281361>.

@article{Mercourakis2006,
abstract = {2000 Mathematics Subject Classification: Primary 40C99, 46B99.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.},
author = {Mercourakis, S., Vassiliadis, G.},
journal = {Serdica Mathematical Journal},
keywords = {Almost Convergence; Banach Limit; Weakly Cauchy Sequence; Independent Sequence; Uniform Distribution of Sequences; almost convergence; Banach limit; weakly Cauchy sequence; independent sequence; uniform distribution of sequences},
language = {eng},
number = {1},
pages = {71-98},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {An extension of Lorentz's almost convergence and applications in Banach spaces},
url = {http://eudml.org/doc/281361},
volume = {32},
year = {2006},
}

TY - JOUR
AU - Mercourakis, S.
AU - Vassiliadis, G.
TI - An extension of Lorentz's almost convergence and applications in Banach spaces
JO - Serdica Mathematical Journal
PY - 2006
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 32
IS - 1
SP - 71
EP - 98
AB - 2000 Mathematics Subject Classification: Primary 40C99, 46B99.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.
LA - eng
KW - Almost Convergence; Banach Limit; Weakly Cauchy Sequence; Independent Sequence; Uniform Distribution of Sequences; almost convergence; Banach limit; weakly Cauchy sequence; independent sequence; uniform distribution of sequences
UR - http://eudml.org/doc/281361
ER -