Orlicz–Pettis Theorem for λ-multiplier convergent operator series

Tao, Yuanhong; Li, Ronglu

doi:10.1017/s0004972700039186

Cited by 4 publications

(2 citation statements)

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“…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]. In [27], Yuanghong and Ronglu proved that multiplier convergence of OVS is closely related to the AK property of the sequence spaces, and obtained corresponding versions of Orlicz-Pettis theorems. Kama and Altay [10] obtained some new multiplier space by using Fibonacci sequence spaces, and Kama et al [11] gave similar results for multiplier spaces which were obtained from backward difference matrix.…”

Section: Discussionmentioning

confidence: 99%

“…The Orlicz-Pettis Theorem is one of the important results in the Theory of Functional Analysis. Many generalizations and applications of this theorem can be found in [5,17,18,22,24,25,27]. Before stating and proving the Orlicz-Pettis Theorem by means of Riesz convergence, we will give the following definition (see [22]):…”

Section: A Version Of Orlicz-pettis Theorem For Riesz Summabilitymentioning

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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