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Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures
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- Symmetry, Integrability and Geometry: Methods and Applications, 2016, том 12
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Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures
Інший ID:
2010 Mathematics Subject Classification: 33D45; 42C05
DOI:10.3842/SIGMA.2016.008
DOI:10.3842/SIGMA.2016.008
Посилання:
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.
Підтримка:
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.
Дата:
2016
Переглядів:
591
Завантажень:
230
Анотація:
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.
