Vector valued multiplier spaces of $$f_\lambda $$-summability, completeness through $$c_0(X)$$-multiplier convergence and continuity and compactness of summing operators

Karakuş, Mahmut; Başar, Feyzı

doi:10.1007/s13398-020-00898-0

Cited by 4 publications

(3 citation statements)

References 18 publications

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“…Karakuş gave the vector valued multiplier spaces S Λ (T ) and S wΛ (T ) by means of Λ-convergence and a series of bounded linear operators together with the characterization the completeness of both of the normed spaces S Λ (T ) and S wΛ (T ) via c 0 (X)-multiplier convergence of an operator series, [13]. Quite recently, Karakuş and Başar [15,16] and Kama [9] introduced the vector valued multiplier spaces of almost summability, a generalized almost summability and statistical Cesàro summability, respectively.…”

Section: Discussionmentioning

confidence: 99%

“…In [20], McArthur investigated some closed linear subspaces of the space of the weakly unconditional summable sequences in a Banach space X and gave some inclusion relations between these spaces. One can refer to the Aizpuru et al [1,3], Pérez-Fernández et al [21], Kama and Altay [10], Kama et al [11], Kama [9,12], Karakuş [13], Karakuş and Başar [14,15,16] and Swartz [22,23] for recent studies on multiplier convergent series.…”

Section: Introductionmentioning

confidence: 99%

“…In this study, our main purpose is to give some results related to the some new multiplier by using Riesz summability method as a generalization of Cesàro summability, and construct some new classes of sequence spaces associated to an OVS in a Banach space having similar properties with the classes defined in [5,13,15,16] and [23]. Among other results, we give some new characterizations of c 0 (X)and ℓ ∞ (X)-multiplier convergent series through this method.…”

Section: Introductionmentioning

confidence: 99%

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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%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Karakuş¹,

Kama²

2021

Preprint

Self Cite

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Nonlinear approximation in N-dimension with the help of summability methods

Aslan

Duman

2021

RACSAM

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On some classical properties of normed spaces via generalized vector valued almost convergence

Karakuş

Başar

2022

Mathematica Slovaca

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Recently, the authors interested some new problems on multiplier spaces of Lorentz’ almost convergence and f λ-convergence as a generalization of almost convergence. f λ-convergence is firstly introduced by Karakuş and Başar, and used for some new characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in a normed space X and its continuous dual X *. In the present paper, we deal with f λ-convergence to have some inclusion relations between the vector valued spaces obtained from this type convergence and corresponding classical sequence spaces, and to give new characterizations of some classical properties like completeness, reflexivity, Schur property and Grothendieck property of normed spaces. By the way, we give a characterization of finite-dimensional normed spaces.

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Vector valued multiplier spaces of $$f_\lambda $$-summability, completeness through $$c_0(X)$$-multiplier convergence and continuity and compactness of summing operators

Cited by 4 publications

References 18 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

Nonlinear approximation in N-dimension with the help of summability methods

On some classical properties of normed spaces via generalized vector valued almost convergence

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