On unitary equivalence to a self-adjoint or doubly–positive Hankel operator
Let A be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry, V, so that AV > 0 and A is Hankel with respect to V, i.e. V*A = AV, if and only if A is not invertible. The isometry V can be chosen to be isomorphic to N ∈...
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| Format: | Article |
| Language: | English |
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De Gruyter
2022-07-01
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| Series: | Concrete Operators |
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| Online Access: | https://doi.org/10.1515/conop-2022-0132 |
| _version_ | 1797947762849873920 |
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| author | Martin Robert T.W. |
| author_facet | Martin Robert T.W. |
| author_sort | Martin Robert T.W. |
| collection | DOAJ |
| description | Let A be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry, V, so that AV > 0 and A is Hankel with respect to V, i.e. V*A = AV, if and only if A is not invertible. The isometry V can be chosen to be isomorphic to N ∈ ℕ ∪ {+∞} copies of the unilateral shift if A has spectral multiplicity at most N. We further show that the set of all isometries, V, so that A is Hankel with respect to V, are in bijection with the set of all closed, symmetric restrictions of A−1. |
| first_indexed | 2024-04-10T21:31:51Z |
| format | Article |
| id | doaj.art-901e91f7ad4547179f49e3ba7a3b1e51 |
| institution | Directory Open Access Journal |
| issn | 2299-3282 |
| language | English |
| last_indexed | 2024-04-10T21:31:51Z |
| publishDate | 2022-07-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Concrete Operators |
| spelling | doaj.art-901e91f7ad4547179f49e3ba7a3b1e512023-01-19T13:20:29ZengDe GruyterConcrete Operators2299-32822022-07-019111412610.1515/conop-2022-0132On unitary equivalence to a self-adjoint or doubly–positive Hankel operatorMartin Robert T.W.0Department of Mathematics, University of ManitobaLet A be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry, V, so that AV > 0 and A is Hankel with respect to V, i.e. V*A = AV, if and only if A is not invertible. The isometry V can be chosen to be isomorphic to N ∈ ℕ ∪ {+∞} copies of the unilateral shift if A has spectral multiplicity at most N. We further show that the set of all isometries, V, so that A is Hankel with respect to V, are in bijection with the set of all closed, symmetric restrictions of A−1.https://doi.org/10.1515/conop-2022-0132hankel operatorslinear symmetric and self-adjoint operators (unbounded)sesquilinear forms47b3547b2547a07 |
| spellingShingle | Martin Robert T.W. On unitary equivalence to a self-adjoint or doubly–positive Hankel operator Concrete Operators hankel operators linear symmetric and self-adjoint operators (unbounded) sesquilinear forms 47b35 47b25 47a07 |
| title | On unitary equivalence to a self-adjoint or doubly–positive Hankel operator |
| title_full | On unitary equivalence to a self-adjoint or doubly–positive Hankel operator |
| title_fullStr | On unitary equivalence to a self-adjoint or doubly–positive Hankel operator |
| title_full_unstemmed | On unitary equivalence to a self-adjoint or doubly–positive Hankel operator |
| title_short | On unitary equivalence to a self-adjoint or doubly–positive Hankel operator |
| title_sort | on unitary equivalence to a self adjoint or doubly positive hankel operator |
| topic | hankel operators linear symmetric and self-adjoint operators (unbounded) sesquilinear forms 47b35 47b25 47a07 |
| url | https://doi.org/10.1515/conop-2022-0132 |
| work_keys_str_mv | AT martinroberttw onunitaryequivalencetoaselfadjointordoublypositivehankeloperator |