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| Format: | Preprint |
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2007
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| Online Access: | https://arxiv.org/abs/0705.2576 |
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| _version_ | 1866917999917662208 |
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| author | Frank, Michael Sharifi, Kamran |
| author_facet | Frank, Michael Sharifi, Kamran |
| contents | In this notes unbounded regular operators on Hilbert $C^*$-modules over arbitrary $C^*$-algebras are discussed. A densely defined operator $t$ possesses an adjoint operator if the graph of $t$ is an orthogonal summand. Moreover, for a densely defined operator $t$ the graph of $t$ is orthogonally complemented and the range of $P_FP_{G(t)^\bot}$ is dense in its biorthogonal complement if and only if $t$ is regular. For a given $C^*$-algebra $\mathcal A$ any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules is regular, if and only if any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules admits a densely defined adjoint operator, if and only if $\mathcal A$ is a $C^*$-algebra of compact operators. Some further characterizations of closed and regular modular operators are obtained.
Changes 1: Improved results, corrected misprints, added references. Accepted by J. Operator Theory, August 2007 / Changes 2: Filled gap in the proof of Thm. 3.1, changes in the formulations of Cor. 3.2 and Thm. 3.4, updated references and address of the second author. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_0705_2576 |
| institution | arXiv |
| publishDate | 2007 |
| record_format | arxiv |
| spellingShingle | Adjointability of densely defined closed operators and the Magajna-Schweizer Theorem Frank, Michael Sharifi, Kamran Operator Algebras Mathematical Physics Functional Analysis 46L08, 47L60 In this notes unbounded regular operators on Hilbert $C^*$-modules over arbitrary $C^*$-algebras are discussed. A densely defined operator $t$ possesses an adjoint operator if the graph of $t$ is an orthogonal summand. Moreover, for a densely defined operator $t$ the graph of $t$ is orthogonally complemented and the range of $P_FP_{G(t)^\bot}$ is dense in its biorthogonal complement if and only if $t$ is regular. For a given $C^*$-algebra $\mathcal A$ any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules is regular, if and only if any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules admits a densely defined adjoint operator, if and only if $\mathcal A$ is a $C^*$-algebra of compact operators. Some further characterizations of closed and regular modular operators are obtained. Changes 1: Improved results, corrected misprints, added references. Accepted by J. Operator Theory, August 2007 / Changes 2: Filled gap in the proof of Thm. 3.1, changes in the formulations of Cor. 3.2 and Thm. 3.4, updated references and address of the second author. |
| title | Adjointability of densely defined closed operators and the Magajna-Schweizer Theorem |
| topic | Operator Algebras Mathematical Physics Functional Analysis 46L08, 47L60 |
| url | https://arxiv.org/abs/0705.2576 |