On the Complex Symmetric and Skew-Symmetric Operators with a Simple Spectrum
In this paper we obtain necessary and sufficient conditions for a linear bounded operator in a Hilbert space H to have a three-diagonal complex symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in H. It is shown that a set of all such operators is a proper sub...
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2011-02-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Online Access: | http://dx.doi.org/10.3842/SIGMA.2011.016 |
| _version_ | 1811214001300832256 |
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| author | Sergey M. Zagorodnyuk |
| author_facet | Sergey M. Zagorodnyuk |
| author_sort | Sergey M. Zagorodnyuk |
| collection | DOAJ |
| description | In this paper we obtain necessary and sufficient conditions for a linear bounded operator in a Hilbert space H to have a three-diagonal complex symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in H. It is shown that a set of all such operators is a proper subset of a set of all complex symmetric operators with a simple spectrum. Similar necessary and sufficient conditions are obtained for a linear bounded operator in H to have a three-diagonal complex skew-symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in H. |
| first_indexed | 2024-04-12T05:55:59Z |
| format | Article |
| id | doaj.art-5b824b9e2481438ca120e528f16c23c3 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-04-12T05:55:59Z |
| publishDate | 2011-02-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-5b824b9e2481438ca120e528f16c23c32022-12-22T03:45:09ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592011-02-017016On the Complex Symmetric and Skew-Symmetric Operators with a Simple SpectrumSergey M. ZagorodnyukIn this paper we obtain necessary and sufficient conditions for a linear bounded operator in a Hilbert space H to have a three-diagonal complex symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in H. It is shown that a set of all such operators is a proper subset of a set of all complex symmetric operators with a simple spectrum. Similar necessary and sufficient conditions are obtained for a linear bounded operator in H to have a three-diagonal complex skew-symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in H.http://dx.doi.org/10.3842/SIGMA.2011.016complex symmetric operatorcomplex skew-symmetric operatorcyclic operatorsimple spectrum |
| spellingShingle | Sergey M. Zagorodnyuk On the Complex Symmetric and Skew-Symmetric Operators with a Simple Spectrum Symmetry, Integrability and Geometry: Methods and Applications complex symmetric operator complex skew-symmetric operator cyclic operator simple spectrum |
| title | On the Complex Symmetric and Skew-Symmetric Operators with a Simple Spectrum |
| title_full | On the Complex Symmetric and Skew-Symmetric Operators with a Simple Spectrum |
| title_fullStr | On the Complex Symmetric and Skew-Symmetric Operators with a Simple Spectrum |
| title_full_unstemmed | On the Complex Symmetric and Skew-Symmetric Operators with a Simple Spectrum |
| title_short | On the Complex Symmetric and Skew-Symmetric Operators with a Simple Spectrum |
| title_sort | on the complex symmetric and skew symmetric operators with a simple spectrum |
| topic | complex symmetric operator complex skew-symmetric operator cyclic operator simple spectrum |
| url | http://dx.doi.org/10.3842/SIGMA.2011.016 |
| work_keys_str_mv | AT sergeymzagorodnyuk onthecomplexsymmetricandskewsymmetricoperatorswithasimplespectrum |