On a class of left ideals of nest algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929651238043648 |
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| author | Costa, Pedro Ferreira, Martim Oliveira, Lina |
| author_facet | Costa, Pedro Ferreira, Martim Oliveira, Lina |
| contents | We introduce a class of left ideals (and subalgebras) of nest algebras determined by totally ordered families of partial isometries on a complex Hilbert space $H$.
Let $\mathcal{E}$ be a family of partial isometries that is totally ordered in the Halmos--McLaughlin ordering, and let $\mathcal{A}_{\mathcal{E}}$ be the subset of operators in $B(H)$ which, for all $E\in \mathcal{E}$, map the initial space of $E$ to the final space of $E$. We show that $\mathcal{A}_{\mathcal{E}}$ is a subalgebra of $B(H)$ if and only if $\mathcal{A}_{\mathcal{E}}$ is a left ideal of a certain nest algebra, and if so,
$\mathcal{E}$ consists of power partial isometries, except possibly for its supremum $\vee \mathcal{E}$, in which case the range $\operatorname{ran}(\vee \mathcal{E})$ is $H$. It is also shown that any left ideal $\mathcal{A}_{\mathcal{E}}$ is decomposable and that the subset of finite rank operators in its closed unit ball is strongly dense in the ball. Necessary and sufficient conditions to solve $Tx=y$ and $T^*x=y$ in $\mathcal{A}_{\mathcal{E}}$ are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20159 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a class of left ideals of nest algebras Costa, Pedro Ferreira, Martim Oliveira, Lina Operator Algebras Functional Analysis 47L75, 47L35, 46K50, 47A15 We introduce a class of left ideals (and subalgebras) of nest algebras determined by totally ordered families of partial isometries on a complex Hilbert space $H$. Let $\mathcal{E}$ be a family of partial isometries that is totally ordered in the Halmos--McLaughlin ordering, and let $\mathcal{A}_{\mathcal{E}}$ be the subset of operators in $B(H)$ which, for all $E\in \mathcal{E}$, map the initial space of $E$ to the final space of $E$. We show that $\mathcal{A}_{\mathcal{E}}$ is a subalgebra of $B(H)$ if and only if $\mathcal{A}_{\mathcal{E}}$ is a left ideal of a certain nest algebra, and if so, $\mathcal{E}$ consists of power partial isometries, except possibly for its supremum $\vee \mathcal{E}$, in which case the range $\operatorname{ran}(\vee \mathcal{E})$ is $H$. It is also shown that any left ideal $\mathcal{A}_{\mathcal{E}}$ is decomposable and that the subset of finite rank operators in its closed unit ball is strongly dense in the ball. Necessary and sufficient conditions to solve $Tx=y$ and $T^*x=y$ in $\mathcal{A}_{\mathcal{E}}$ are given. |
| title | On a class of left ideals of nest algebras |
| topic | Operator Algebras Functional Analysis 47L75, 47L35, 46K50, 47A15 |
| url | https://arxiv.org/abs/2412.20159 |