A generalization of almost convergence, completeness of some normed spaces with wuC series and a version of Orlicz-Pettis theorem

Karakuş, Mahmut; Başar, Feyzı

doi:10.1007/s13398-019-00701-9

Cited by 8 publications

(2 citation statements)

References 14 publications

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“…In [3], they also generalized the results given in [1] to any regular matrix. In recent times, Karakuş and Başar [14] introduced a slight generalization of almost convergence and gave some new multiplier spaces associated to the series k x k in a normed space by means of this new summability method. Swartz constructed some new versions of the Orlicz-Pettis theorem for multiplier convergent series under continuity assumptions on some linear operators, and gave applications to spaces of continuous linear operators, [24,25].…”

Section: Discussionmentioning

confidence: 99%

Riesz multiplier convergent spaces of operator valued series and a version of Orlicz- Pettis theorem

Karakuş¹,

Kama²

2021

Preprint

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It is not usual to characterize an operator valued series via completeness of multiplier spaces. In this study, by using a series of bounded linear operators, we introduce the space M ∞ R k T k of Riesz summability which is a generalization of the Cesàro summability. Therefore, we give the completeness criteria of these spaces with c0(X)-multiplier convergent operator series. It is a natural consequence that one can characterize the completeness of a normed space through M ∞ R k T k which will be assumed that is complete for every c0(X)-multiplier Cauchy operator series. Then, we characterize the continuity and the (weakly) compactness of the summing operator S from the multiplier space M ∞ R k T k to an arbitrary normed space Y through c0(X)-multiplier Cauchy and ℓ∞(X)-multiplier convergent series, respectively. We also prove that if k T k is ℓ∞(X)-multiplier Cauchy, then the multiplier space of weakly Riesz-convergence associated to the operator valued seriesAmong other results, finally, we obtain a new version of the well-known Orlicz-Pettis theorem by using Riesz-summability.

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Section: Discussionmentioning

confidence: 99%

Riesz multiplier convergent spaces of operator valued series and a version of Orlicz- Pettis theorem

Karakuş¹,

Kama²

2021

Preprint

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“…Tripathy and Mahanta [31] also studied vector valued sequences associated with multiplier sequences. Furthermore, Karakus and Basar introduced the multiplier spaces S (T), S w (T) and studied some new multiplier spaces by using generalization of almost summability in [18,19]. To know more about multiplier spaces, one may refer to [13,14,16,29].…”

Section: Introductionmentioning

confidence: 99%

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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Spaces of vector sequences defined by the f-statistical convergence and some characterizations of normed spaces

Kama

2020

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A generalization of almost convergence, completeness of some normed spaces with wuC series and a version of Orlicz-Pettis theorem

Cited by 8 publications

References 14 publications

Riesz multiplier convergent spaces of operator valued series and a version of Orlicz- Pettis theorem

Riesz multiplier convergent spaces of operator valued series and a version of Orlicz- Pettis theorem

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

Spaces of vector sequences defined by the f-statistical convergence and some characterizations of normed spaces

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