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

Mercourakis, S.; Vassiliadis, G.

Serdica Mathematical Journal (2006)

- Volume: 32, Issue: 1, page 71-98
- ISSN: 1310-6600

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topMercourakis, 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 -

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