Linear structures in the set of non-norm-attaining operators on Banach spaces
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917353738993664 |
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| author | Dantas, Sheldon Falcó, Javier Jung, Mingu Rodríguez-Vidanes, Daniel L. |
| author_facet | Dantas, Sheldon Falcó, Javier Jung, Mingu Rodríguez-Vidanes, Daniel L. |
| contents | We study large linear structures inside sets arising in the theory of norm-attaining operators. We provide several results in the context of lineability, spaceability, maximal-spaceability, and $(α, β)$-spaceability for sets of non-norm-attaining bounded linear operators whenever such sets are nonempty. To be more specific, we show that if $Y$ is a strictly convex renorming of $c_0 (Γ)$, then the set
$$
\mathcal{L}(c_0 (Γ),Y)\setminus \overline{\text{NA} (c_0 (Γ),Y)} $$
is $2^{|Γ|}$-spaceable. We also prove that
$$
\mathcal{L}(d_* (w,1) ,\ell_p )\setminus \overline{\text{NA} (d_* (w,1),\ell_p )} $$
is maximal-spaceable. Finally, we establish that whenever the set of non-norm-attaining operators from a Banach space $X$ into $\ell_p (Γ)$ (respectively, $c_0 (Γ)$) is nonempty, it contains a subspace linearly isometric to $\ell_p(Γ)$ (respectively, $c_0 (Γ)$). These results extend and complement several known results in the literature concerning large linear structures in sets of non-norm-attaining operators. Our results are obtained in a more general framework involving group-invariant operators, which allows us to treat classical spaces of operators as special cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_17426 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Linear structures in the set of non-norm-attaining operators on Banach spaces Dantas, Sheldon Falcó, Javier Jung, Mingu Rodríguez-Vidanes, Daniel L. Functional Analysis We study large linear structures inside sets arising in the theory of norm-attaining operators. We provide several results in the context of lineability, spaceability, maximal-spaceability, and $(α, β)$-spaceability for sets of non-norm-attaining bounded linear operators whenever such sets are nonempty. To be more specific, we show that if $Y$ is a strictly convex renorming of $c_0 (Γ)$, then the set $$ \mathcal{L}(c_0 (Γ),Y)\setminus \overline{\text{NA} (c_0 (Γ),Y)} $$ is $2^{|Γ|}$-spaceable. We also prove that $$ \mathcal{L}(d_* (w,1) ,\ell_p )\setminus \overline{\text{NA} (d_* (w,1),\ell_p )} $$ is maximal-spaceable. Finally, we establish that whenever the set of non-norm-attaining operators from a Banach space $X$ into $\ell_p (Γ)$ (respectively, $c_0 (Γ)$) is nonempty, it contains a subspace linearly isometric to $\ell_p(Γ)$ (respectively, $c_0 (Γ)$). These results extend and complement several known results in the literature concerning large linear structures in sets of non-norm-attaining operators. Our results are obtained in a more general framework involving group-invariant operators, which allows us to treat classical spaces of operators as special cases. |
| title | Linear structures in the set of non-norm-attaining operators on Banach spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2311.17426 |