Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^*$-modules
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arXiv
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| Format: | Preprint |
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2008
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| _version_ | 1866917999938633728 |
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| author | Frank, Michael Sharifi, Kamran |
| author_facet | Frank, Michael Sharifi, Kamran |
| contents | In this note we show that an unbounded regular operator $t$ on Hilbert $C^*$-modules over an arbitrary $C^*$ algebra $ \mathcal{A}$ has polar decomposition if and only if the closures of the ranges of $t$ and $|t|$ are orthogonally complemented, if and only if the operators $t$ and $t^*$ have unbounded regular generalized inverses. For a given $C^*$-algebra $ \mathcal{A}$ any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules has polar decomposition, if and only if any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules has generalized inverse, if and only if $\mathcal A$ is a $C^*$-algebra of compact operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_0806_0162 |
| institution | arXiv |
| publishDate | 2008 |
| record_format | arxiv |
| spellingShingle | Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^*$-modules Frank, Michael Sharifi, Kamran Operator Algebras Functional Analysis 46L08, 47L60, 46C05 In this note we show that an unbounded regular operator $t$ on Hilbert $C^*$-modules over an arbitrary $C^*$ algebra $ \mathcal{A}$ has polar decomposition if and only if the closures of the ranges of $t$ and $|t|$ are orthogonally complemented, if and only if the operators $t$ and $t^*$ have unbounded regular generalized inverses. For a given $C^*$-algebra $ \mathcal{A}$ any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules has polar decomposition, if and only if any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules has generalized inverse, if and only if $\mathcal A$ is a $C^*$-algebra of compact operators. |
| title | Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^*$-modules |
| topic | Operator Algebras Functional Analysis 46L08, 47L60, 46C05 |
| url | https://arxiv.org/abs/0806.0162 |