Geodesic
Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {Erik Koelink and Pablo Rom\'an},
     title = {Orthogonal vs. {Non-Orthogonal} {Reducibility} of {Matrix-Valued} {Measures}},
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}
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Erik Koelink; Pablo Román. Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a7/

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