Almost summability and unconditionally Cauchy series

Aizpuru, A.; Armario, R.; Pérez-Fernández, Francisco

doi:10.36045/bbms/1225893944

Cited by 14 publications

(6 citation statements)

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“…in the norm topology, uniformly in m ∈ N ∀m, n, k ∈ N. We can write wG hk z k = z 0 if the series is weakly G h -almost convergent to z 0 in the weak topology. To obtain the definition given in [3], we will take h n = n + 1, p n+m = 1, γ = 0 such that q k = q m+n z k , where q n = n, ∀n ∈ N.…”

Section: Definition 24mentioning

confidence: 99%

“…For a detailed study, one can refer to [10]. Using the completeness property of a subspace of ∞ obtained by almost convergence, a depiction of wuC and uc series along with a new form of the Orlicz-Pettis theorem was presented by Aizpuru et al [3]. Recently, a vector valued multiplier space through Cesàro convergence was introduced by Altay and Kama [5].…”

Section: Introductionmentioning

confidence: 99%

“…wG h ( k z k ) of almost summability related with k z k was studied by Aizpuru et al[3] which is given asS wAC k z k = t = (t k ) ∈ ∞ : wAC k t k z k exists .…”

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confidence: 99%

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Applications of Orlicz–Pettis theorem in vector valued multiplier spaces of generalized weighted mean fractional difference operators

2021

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In this study, we deal with some new vector valued multiplier spaces $S_{G_{h}}(\sum_{k}z_{k})$ S G h ( ∑ k z k ) and $S_{wG_{h}}(\sum_{k}z_{k})$ S w G h ( ∑ k z k ) related with $\sum_{k}z_{k}$ ∑ k z k in a normed space Y. Further, we obtain the completeness of these spaces via weakly unconditionally Cauchy series in Y and $Y^{*}$ Y ∗ . Moreover, we show that if $\sum_{k}z_{k}$ ∑ k z k is unconditionally Cauchy in Y, then the multiplier spaces of $G_{h}$ G h -almost convergence and weakly $G_{h}-$ G h − almost convergence are identical. Finally, some applications of the Orlicz–Pettis theorem with the newly formed sequence spaces and unconditionally Cauchy series $\sum_{k}z_{k}$ ∑ k z k in Y are given.

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Section: Definition 24mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Applications of Orlicz–Pettis theorem in vector valued multiplier spaces of generalized weighted mean fractional difference operators

2021

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“…Notice that these sequence spaces associated to a weakly unconditionally Cauchy series (in brief wuc series) were defined originally [6] in terms of the norm topology and the usual weak topology of the space. Since then, these kinds of results have been investigated in several convergence spaces associated with a weakly unconditionally Cauchy series using different types of convergence [7][8][9][10][11][12]. In fact, several questions remain unsolved for different kinds of convergence.…”

Section: Introductionmentioning

confidence: 99%

Ideal Convergence and Completeness of a Normed Space

et al. 2019

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We aim to unify several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated to the wuc series. If, additionally, the space is completed for each wuc series, then the underlying space is complete. In the process the existing proofs are simplified and some unanswered questions are solved. This research line was originated in the PhD thesis of the second author. Since then, it has been possible to characterize the completeness of a normed spaces through different convergence subspaces (which are be defined using different kinds of convergence) associated to an unconditionally Cauchy sequence.

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“…In [2,3,5,17,23,27,28], there are important results that relate the case of a convergence method to classical properties of scalar and vector-valued multiplier spaces. However, in the case of the statistical Cesàro convergence, such spaces have not yet been introduced.…”

Section: Introductionmentioning

confidence: 99%