Similar operator topologies on the space of positive contractions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912726772613120 |
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| author | Gillet, Valentin |
| author_facet | Gillet, Valentin |
| contents | In this article, we study the similarity of the Polish operator topologies $\texttt{WOT}$, $\texttt{SOT}$, $\texttt{SOT}\mbox{$_{*}$}$ and $\texttt{SOT}\mbox{$^{*}$}$ on the set of the positive contractions on $\ell_p$ with $p > 1$. Using the notion of norming vector for a positive operator, we prove that these topologies are similar on $\mathcal{P}_1(\ell_2)$, that is, they have the same dense sets in $\mathcal{P}_1(\ell_2)$. In particular, these topologies will share the same comeager sets in $\mathcal{P}_1(\ell_2)$. We then apply these results to the study of typical properties of positive contractions on $\ell_p$-spaces in the Baire category sense. In particular, we prove that a typical positive contraction $T \in (\mathcal{P}_1(\ell_2), \texttt{SOT})$ has no eigenvalue. This stands in strong contrast to a result of Eisner and Mátrai, stating that the point spectrum of a typical contraction $T \in (\mathcal{B}_1(\ell_2), \texttt{SOT})$ contains the whole unit disk. As a consequence of our results, we obtain that a typical positive contraction $T \in (\mathcal{P}_1(\ell_2), \texttt{WOT})$ (resp. $T \in (\mathcal{P}_1(\ell_2), \texttt{SOT}\mbox{$_{*}$})$) has no eigenvalue. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_11587 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Similar operator topologies on the space of positive contractions Gillet, Valentin Functional Analysis 46A45, 47A15, 54E52, 47B65 In this article, we study the similarity of the Polish operator topologies $\texttt{WOT}$, $\texttt{SOT}$, $\texttt{SOT}\mbox{$_{*}$}$ and $\texttt{SOT}\mbox{$^{*}$}$ on the set of the positive contractions on $\ell_p$ with $p > 1$. Using the notion of norming vector for a positive operator, we prove that these topologies are similar on $\mathcal{P}_1(\ell_2)$, that is, they have the same dense sets in $\mathcal{P}_1(\ell_2)$. In particular, these topologies will share the same comeager sets in $\mathcal{P}_1(\ell_2)$. We then apply these results to the study of typical properties of positive contractions on $\ell_p$-spaces in the Baire category sense. In particular, we prove that a typical positive contraction $T \in (\mathcal{P}_1(\ell_2), \texttt{SOT})$ has no eigenvalue. This stands in strong contrast to a result of Eisner and Mátrai, stating that the point spectrum of a typical contraction $T \in (\mathcal{B}_1(\ell_2), \texttt{SOT})$ contains the whole unit disk. As a consequence of our results, we obtain that a typical positive contraction $T \in (\mathcal{P}_1(\ell_2), \texttt{WOT})$ (resp. $T \in (\mathcal{P}_1(\ell_2), \texttt{SOT}\mbox{$_{*}$})$) has no eigenvalue. |
| title | Similar operator topologies on the space of positive contractions |
| topic | Functional Analysis 46A45, 47A15, 54E52, 47B65 |
| url | https://arxiv.org/abs/2412.11587 |