Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures
A matrix-valued measure Θ reduces to measures of smaller size if there exists a constant invertible matrix M such that MΘM∗ is block diagonal. Equivalently, the real vector space A of all matrices T such that TΘ(X)=Θ(X)T∗ for any Borel set X is non-trivial. If the subspace Ah of self-adjoints elemen...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Zitieren: | Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures / E. Koelink, P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862546773585690624 |
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| author | Koelink, E. Román, P. |
| author_facet | Koelink, E. Román, P. |
| citation_txt | Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures / E. Koelink, P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A matrix-valued measure Θ reduces to measures of smaller size if there exists a constant invertible matrix M such that MΘM∗ is block diagonal. Equivalently, the real vector space A of all matrices T such that TΘ(X)=Θ(X)T∗ for any Borel set X is non-trivial. If the subspace Ah of self-adjoints elements in the commutant algebra A of Θ is non-trivial, then Θ is reducible via a unitary matrix. In this paper we prove that A is ∗-invariant if and only if Ah=A, i.e., every reduction of Θ can be performed via a unitary matrix. The motivation for this paper comes from families of matrix-valued polynomials related to the group SU(2)×SU(2) and its quantum analogue. In both cases the commutant algebra A=Ah⊕iAh is of dimension two and the matrix-valued measures reduce unitarily into a 2×2 block diagonal matrix. Here we show that there is no further non-unitary reduction.
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| id | nasplib_isofts_kiev_ua-123456789-147427 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-25T12:51:19Z |
| publishDate | 2016 |
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| spelling | Koelink, E. Román, P. 2019-02-14T18:23:07Z 2019-02-14T18:23:07Z 2016 Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures / E. Koelink, P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 12 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D45; 42C05 DOI:10.3842/SIGMA.2016.008 https://nasplib.isofts.kiev.ua/handle/123456789/147427 A matrix-valued measure Θ reduces to measures of smaller size if there exists a constant invertible matrix M such that MΘM∗ is block diagonal. Equivalently, the real vector space A of all matrices T such that TΘ(X)=Θ(X)T∗ for any Borel set X is non-trivial. If the subspace Ah of self-adjoints elements in the commutant algebra A of Θ is non-trivial, then Θ is reducible via a unitary matrix. In this paper we prove that A is ∗-invariant if and only if Ah=A, i.e., every reduction of Θ can be performed via a unitary matrix. The motivation for this paper comes from families of matrix-valued polynomials related to the group SU(2)×SU(2) and its quantum analogue. In both cases the commutant algebra A=Ah⊕iAh is of dimension two and the matrix-valued measures reduce unitarily into a 2×2 block diagonal matrix. Here we show that there is no further non-unitary reduction. This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications.
 The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html.
 We thank I. Zurri´an for pointing out a similar example to Example 4.1 to the first author. The
 research of Pablo Rom´an is supported by the Radboud Excellence Fellowship. We would like to
 thank the anonymous referees for their comments and remarks, that have helped us to improve
 the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures Article published earlier |
| spellingShingle | Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures Koelink, E. Román, P. |
| title | Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures |
| title_full | Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures |
| title_fullStr | Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures |
| title_full_unstemmed | Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures |
| title_short | Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures |
| title_sort | orthogonal vs. non-orthogonal reducibility of matrix-valued measures |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147427 |
| work_keys_str_mv | AT koelinke orthogonalvsnonorthogonalreducibilityofmatrixvaluedmeasures AT romanp orthogonalvsnonorthogonalreducibilityofmatrixvaluedmeasures |