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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2403.05806 |
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| _version_ | 1866929270239002624 |
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| author | Barbosa, A. Raposo Jr., A. Ribeiro, G. |
| author_facet | Barbosa, A. Raposo Jr., A. Ribeiro, G. |
| contents | In this paper, we introduce the notions of $α$-quasicomplemented and totally $α$-quasicomplemented subspaces and we established some results under these contexts. We show, for example, that if $X$ is a separable or reflexive Banach space and $Y$ is a closed infinite codimensional subspace of $X$, then $Y$ is totally$\mathit{\ }α$-quasicomplemented if, and only if, $α<\aleph_{0}$ $\left( \text{this is an analogue of the theorem of Murray-Mackey and Lindenstrauss}\right) $. We also show that if $H$ is a Hilbert space and $Y,W\subset H$ are closed subspaces of $H$ such that $W$ is orthogonal to $Y$ and $\operatorname{codim}\left( Y+W\right) =\infty$, then $Y$ has a quasicomplement $Z$ containing $W$ with $\dim Z/W=\infty$. Other results in the different contexts are also included. Such results establish a connection between the theory of quasicomplemented subspaces and $\left( α,β\right) $-spaceability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05806 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a theorem due to Murray Barbosa, A. Raposo Jr., A. Ribeiro, G. Functional Analysis In this paper, we introduce the notions of $α$-quasicomplemented and totally $α$-quasicomplemented subspaces and we established some results under these contexts. We show, for example, that if $X$ is a separable or reflexive Banach space and $Y$ is a closed infinite codimensional subspace of $X$, then $Y$ is totally$\mathit{\ }α$-quasicomplemented if, and only if, $α<\aleph_{0}$ $\left( \text{this is an analogue of the theorem of Murray-Mackey and Lindenstrauss}\right) $. We also show that if $H$ is a Hilbert space and $Y,W\subset H$ are closed subspaces of $H$ such that $W$ is orthogonal to $Y$ and $\operatorname{codim}\left( Y+W\right) =\infty$, then $Y$ has a quasicomplement $Z$ containing $W$ with $\dim Z/W=\infty$. Other results in the different contexts are also included. Such results establish a connection between the theory of quasicomplemented subspaces and $\left( α,β\right) $-spaceability. |
| title | On a theorem due to Murray |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2403.05806 |