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Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures

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dc.contributor.author Koelink, E.
dc.contributor.author Román, P.
dc.date.accessioned 2019-02-14T18:23:07Z
dc.date.available 2019-02-14T18:23:07Z
dc.date.issued 2016
dc.identifier.citation 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. uk_UA
dc.identifier.issn 1815-0659
dc.identifier.other 2010 Mathematics Subject Classification: 33D45; 42C05
dc.identifier.other DOI:10.3842/SIGMA.2016.008
dc.identifier.uri http://dspace.nbuv.gov.ua/handle/123456789/147427
dc.description.abstract 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. uk_UA
dc.description.sponsorship 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. uk_UA
dc.language.iso en uk_UA
dc.publisher Інститут математики НАН України uk_UA
dc.relation.ispartof Symmetry, Integrability and Geometry: Methods and Applications
dc.title Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures uk_UA
dc.type Article uk_UA
dc.status published earlier uk_UA


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