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

• Дата: 2016
• Автори: Koelink, E., Román, P.
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут математики НАН України 2016
• Назва видання: Symmetry, Integrability and Geometry: Methods and Applications
• Онлайн доступ: http://dspace.nbuv.gov.ua/handle/123456789/147427
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: 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.

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.