Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures
نویسنده
چکیده
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 nontrivial. If the subspace Ah of self-adjoints elements in the commutant algebra A of Θ is nontrivial, 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.
منابع مشابه
The Analytic Theory of Matrix Orthogonal Polynomials
We survey the analytic theory of matrix orthogonal polynomials. MSC: 42C05, 47B36, 30C10 keywords: orthogonal polynomials, matrix-valued measures, block Jacobi matrices, block CMV matrices
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