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This article is cited in 10 scientific papers (total in 10 papers)
Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures
Erik Koelinka, Pablo Románb a IMAPP, Radboud Universiteit, Heyendaalseweg 135, 6525 GL Nijmegen, The Netherlands
b CIEM, FaMAF, Universidad Nacional de Córdoba, Medina Allende s/n Ciudad Universitaria, Córdoba, Argentina
Abstract:
A matrix-valued measure $\Theta$ reduces to measures of smaller size if there exists a constant invertible matrix $M$ such that $M\Theta M^*$ is block diagonal. Equivalently, the real vector space $\mathcal{A}$ of all matrices $T$ such that $T\Theta(X)=\Theta(X) T^*$ for any Borel set $X$ is non-trivial. If the subspace $A_h$ of self-adjoints elements in the commutant algebra $A$ of $\Theta$ is non-trivial, then $\Theta$ is reducible via a unitary matrix. In this paper we prove that $\mathcal{A}$ is $*$-invariant if and only if $A_h=\mathcal{A}$, i.e., every reduction of $\Theta$ can be performed via a unitary matrix. The motivation for this paper comes from families of matrix-valued polynomials related to the group $\mathrm{SU(2)}\times \mathrm{SU(2)}$ and its quantum analogue. In both cases the commutant algebra $A=A_h\oplus iA_h$ is of dimension two and the matrix-valued measures reduce unitarily into a $2\times 2$ block diagonal matrix. Here we show that there is no further non-unitary reduction.
Keywords:
matrix-valued measures; reducibility; matrix-valued orthogonal polynomials.
Received: September 23, 2015; in final form January 21, 2016; Published online January 23, 2016
Citation:
Erik Koelink, Pablo Román, “Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures”, SIGMA, 12 (2016), 008, 9 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1090 https://www.mathnet.ru/eng/sigma/v12/p8
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Abstract page: | 208 | Full-text PDF : | 69 | References: | 72 |
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