Abstract
For the Cesàro matrix \(C=(c_{nm})_{n,m\in \mathbb {N}}\), where \(c_{nm}=\frac{1}{n}\), if \(n\ge m\) and \(c_{nm}=0\) otherwise, the Cesàro sequence spaces \(ces_0,\, ces_p\) (for \(1<p< \infty \)) and \(ces_\infty \) are defined. These spaces turn out to be real vector lattices and with respect to a corresponding (naturally introduced) norm they are all Banach lattices, and so possess (or not possess) some interesting properties. In particular, the relations to their generating ideals \(c_0, \, \ell _p\) and \(\ell _\infty \) are investigated. Finally the ideals of all finite, totally finite and selfmajorizing elements in \(ces_0, ces_p\) (for \(1<p< \infty \)) and \(ces_\infty \) are described in detail.
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Notes
The converse is not true, in general.
A matrix A will be called non-negative, if \(A\ge 0\).
One has \(\left\| e_n\right\| ^p_{ces_p}=\left\| Ce_n\right\| _p^p=\sum \nolimits _{k=n}^\infty \frac{1}{k^p}\le \sum \nolimits _{k=1}^\infty \frac{1}{k^p}<\infty \).
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Funding was provided by Türkiye Bilimsel ve Teknolojik Araştirma Kurumu (Grant No. 1059B191900662).
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F. Polat was supported for a one-year stay in 2022 at Technische Universität Dresden by the Scientific and Technological Research Council of Turkey (TUBITAK) in the context of the 2219-Post Doctoral Fellowship Program.
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Gönüllü, U., Polat, F. & Weber, M.R. Cesàro vector lattices and their ideals of finite elements. Positivity 27, 27 (2023). https://doi.org/10.1007/s11117-023-00977-7
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DOI: https://doi.org/10.1007/s11117-023-00977-7