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Banach Spaces and Inequalities Associated with New Generalization of Cesàro Matrix

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Abstract

Let the triangle matrix Aru be a generalization of the Cesàro matrix and U ∈ {c0, c, ℓ∞}. In this study, we essentially deal with the space U(Aru) defined by the domain of Aru in the space U and give the bases, and determine the Köthe-Toeplitz, generalized Köthe-Toeplitz and bounded-duals of the space U (Aru). We characterize the classes (ℓ∞ (Aru):ℓ∞), (ℓ∞(Aru): c), (c(Aru): c), and (U: V(Aru)) of infinite matrices, where V denotes any given sequence space. Additionally, we also present a Steinhaus type theorem. As an another result of this study, we investigate the ℓp-norm of the matrix Aru and as a result obtaining a generalized version of Hardy’s inequality, and some inclusion relations. Moreover, we compute the norm of well-known operators on the matrix domain ℓp (Aru).

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Acknowledgements

The authors would like to express their sincere thanks to Professor Bilâl Altay, Department of Mathematical Education, İnönü University, 44280–Malatya, Türkiye, who carefully revised Sections 3, 4 and 5 in the original version of the article and made necessary corrections.

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Authors and Affiliations

  1. Department of Primary Mathematics Teacher Education, İnönü University, 44280, Malatya, Türkiye

    Feyzi Başar

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, Canada

    Hadi Roopaei

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  1. Feyzi Başar
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  2. Hadi Roopaei
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Correspondence to Hadi Roopaei.

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The authors declare no conflict of interest.

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Başar, F., Roopaei, H. Banach Spaces and Inequalities Associated with New Generalization of Cesàro Matrix. Acta Math Sci 43, 1518–1536 (2023). https://doi.org/10.1007/s10473-023-0404-0

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  • DOI: https://doi.org/10.1007/s10473-023-0404-0

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