New Results in Analysis of Orlicz-Lorentz spaces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913363399802880 |
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| author | Bernal-González, Luis Rodríguez-Vidanes, Daniel L. Seoane-Sepúlveda, Juan B. Tag, Hyung-Joon |
| author_facet | Bernal-González, Luis Rodríguez-Vidanes, Daniel L. Seoane-Sepúlveda, Juan B. Tag, Hyung-Joon |
| contents | In this article, we investigate the existence of closed vector subspaces (i.e.spaceability) in various nonlinear subsets of Orlicz-Lorentz spaces $Λ_{φ,w}$, equipped with the Luxemburg norm. If a family of Orlicz functions $(φ_n)_{n=1}^{\infty}$ satisfies certain order relations with respect to a given Orlicz function $φ$, the subset of the order-continuous subspace $(Λ_{φ,w})_a$ whose elements do not belong to $\bigcup_{n=1}^{\infty}Λ_{φ_n,w}$ is spaceable, and even maximal-spaceable when $φ$ satisfies the $Δ_2$-condition. We also show that this subset is either residual or empty. In addition, sufficient conditions for this subset not being $(α, β)$-spaceable are provided. A similar analysis is also performed on the subset $Λ_{φ,w} \setminus (Λ_{φ,w})_a$ when $φ$ does not satisfy the $Δ_2$-condition.
The comparison between different Orlicz-Lorentz spaces is characterized via the generating pairs $(φ,w)$. For a fixed Orlicz function that satisfies the $Δ_2^{\infty}$-condition, we provide a characterization of disjointly strictly singular inclusion operators between Orlicz-Lorentz spaces with different weights. As a consequence, there are certain subsets of Orlicz-Lorentz spaces on $[0,1]$ for which lineability problem is not valid. Moreover, various types of $(α,β)$-lineability and pointwise lineability properties on other nonlinear subsets of Orlicz-Lorentz spaces are examined. These results extend a number of previously known results in Orlicz and Lorentz spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_13903 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | New Results in Analysis of Orlicz-Lorentz spaces Bernal-González, Luis Rodríguez-Vidanes, Daniel L. Seoane-Sepúlveda, Juan B. Tag, Hyung-Joon Functional Analysis In this article, we investigate the existence of closed vector subspaces (i.e.spaceability) in various nonlinear subsets of Orlicz-Lorentz spaces $Λ_{φ,w}$, equipped with the Luxemburg norm. If a family of Orlicz functions $(φ_n)_{n=1}^{\infty}$ satisfies certain order relations with respect to a given Orlicz function $φ$, the subset of the order-continuous subspace $(Λ_{φ,w})_a$ whose elements do not belong to $\bigcup_{n=1}^{\infty}Λ_{φ_n,w}$ is spaceable, and even maximal-spaceable when $φ$ satisfies the $Δ_2$-condition. We also show that this subset is either residual or empty. In addition, sufficient conditions for this subset not being $(α, β)$-spaceable are provided. A similar analysis is also performed on the subset $Λ_{φ,w} \setminus (Λ_{φ,w})_a$ when $φ$ does not satisfy the $Δ_2$-condition. The comparison between different Orlicz-Lorentz spaces is characterized via the generating pairs $(φ,w)$. For a fixed Orlicz function that satisfies the $Δ_2^{\infty}$-condition, we provide a characterization of disjointly strictly singular inclusion operators between Orlicz-Lorentz spaces with different weights. As a consequence, there are certain subsets of Orlicz-Lorentz spaces on $[0,1]$ for which lineability problem is not valid. Moreover, various types of $(α,β)$-lineability and pointwise lineability properties on other nonlinear subsets of Orlicz-Lorentz spaces are examined. These results extend a number of previously known results in Orlicz and Lorentz spaces. |
| title | New Results in Analysis of Orlicz-Lorentz spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2312.13903 |