Operator valued series, almost summability of vector valued multipliers and (weak) compactness of summing operator

Karakuş, Mahmut; Başar, Feyzı

doi:10.1016/j.jmaa.2019.123651

Cited by 7 publications

(7 citation statements)

References 10 publications

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“…By bearing in mind Theorem 2.4 together with previous results given in [5] and [15], we have the following:…”

Section: Vector Valued Multiplier Spaces Through Riesz Summability An...mentioning

confidence: 62%

“…From Theorem 2.21 and the conclusions due to Altay and Kama [5] and Karakuş and Başar [15], we have the following: Corollary 2.22. Let X and Y be two normed spaces, and T k ∈ B(X : Y ) for all k ∈ N. Then, the following statements are equivalent: 23.…”

Section: Since the Inclusion Mmentioning

confidence: 82%

“…In the following, we introduce the spaces of R-convergence and weakly R-convergence of multipliers with association of an OVS, and obtain some new characterizations of c 0 (X)-and ℓ ∞ (X)-multiplier convergent (Cauchy) series. We also present the necessary and sufficient conditions for the continuity and (weak) compactness of the summing operator S defined from these multiplier spaces to another normed space Y , and derive some results by the previous works [5,15] and [23].…”

Section: Vector Valued Multiplier Spaces Through Riesz Summability An...mentioning

confidence: 95%

“…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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“…By bearing in mind Theorem 2.4 together with previous results given in [5] and [15], we have the following:…”

Section: Vector Valued Multiplier Spaces Through Riesz Summability An...mentioning

confidence: 62%

Section: Since the Inclusion Mmentioning

confidence: 82%

Section: Vector Valued Multiplier Spaces Through Riesz Summability An...mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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show abstract

Characterization of summing operators in multiplier spaces of deferred Nörlund summability

2022

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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

Karakuş

Başar

2020

RACSAM

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Operator valued series, almost summability of vector valued multipliers and (weak) compactness of summing operator

Cited by 7 publications

References 10 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

Characterization of summing operators in multiplier spaces of deferred Nörlund summability

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

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