Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 008, 9 pages      arXiv:1509.06143      https://doi.org/10.3842/SIGMA.2016.008
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

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

Erik Koelink ^a and Pablo Román ^ab
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

Received September 23, 2015, in final form January 21, 2016; Published online January 23, 2016

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 ${\mathscr A}$ 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 ${\mathscr A}$ is $*$-invariant if and only if $A_h={\mathscr A}$, 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 ${\rm SU}(2)\times {\rm SU}(2)$ 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.

Key words: matrix-valued measures; reducibility; matrix-valued orthogonal polynomials.

pdf (358 kb)   tex (15 kb)

References

1. Aldenhoven N., Koelink E., de los Ríos A.M., Matrix-valued little $q$-Jacobi polynomials, J. Approx. Theory 193 (2015), 164-183, arXiv:1308.2540.
2. Aldenhoven N., Koelink E., Román P., Matrix-valued orthogonal polynomials for the quantum analogue of $({\rm SU}(2)\times{\rm SU}(2), \mathrm{diag})$, arXiv:1507.03426.
3. Álvarez-Nodarse R., Durán A.J., de los Ríos A.M., Orthogonal matrix polynomials satisfying second order difference equations, J. Approx. Theory 169 (2013), 40-55.
4. Berg C., The matrix moment problem, in Coimbra Lecture Notes on Orthogonal Polynomials, Editors A. Branquinho, A. Foulquié Moreno, Nova Sci. Publ., New York, 2008, 1-57.
5. Damanik D., Pushnitski A., Simon B., The analytic theory of matrix orthogonal polynomials, Surv. Approx. Theory 4 (2008), 1-85, arXiv:0711.2703.
6. Grünbaum F.A., Tirao J., The algebra of differential operators associated to a weight matrix, Integral Equations Operator Theory 58 (2007), 449-475.
7. Heckman G., van Pruijssen M., Matrix-valued orthogonal polynomials for Gelfand pairs of rank one, Tohoku Math. J., to appear, arXiv:1310.5134.
8. Horn R.A., Johnson C.R., Matrix analysis, Cambridge University Press, Cambridge, 1985.
9. Koelink E., de los Ríos A.M., Román P., Matrix-valued Gegenbauer polynomials, arXiv:1403.2938.
10. Koelink E., van Pruijssen M., Román P., Matrix-valued orthogonal polynomials related to $({\rm SU}(2)\times{\rm SU}(2),{\rm diag})$, Int. Math. Res. Not. 2012 (2012), 5673-5730, arXiv:1012.2719.
11. Koelink E., van Pruijssen M., Román P., Matrix-valued orthogonal polynomials related to $({\rm SU}(2)\times{\rm SU}(2),{\rm diag})$, II, Publ. Res. Inst. Math. Sci. 49 (2013), 271-312, arXiv:1203.0041.
12. Tirao J., Zurrián I., Reducibility of matrix weights, arXiv:1501.04059.