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

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 elemen...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2016
Main Authors:	Koelink, E., Román, P.
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2016
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Cite this:	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.

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spelling	Koelink, E. Román, P. 2019-02-14T18:23:07Z 2019-02-14T18:23:07Z 2016 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. 1815-0659 2010 Mathematics Subject Classification: 33D45; 42C05 DOI:10.3842/SIGMA.2016.008 https://nasplib.isofts.kiev.ua/handle/123456789/147427 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. This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html. We thank I. Zurri´an for pointing out a similar example to Example 4.1 to the first author. The research of Pablo Rom´an is supported by the Radboud Excellence Fellowship. We would like to thank the anonymous referees for their comments and remarks, that have helped us to improve the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures
spellingShingle	Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures Koelink, E. Román, P.
title_short	Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures
title_full	Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures
title_fullStr	Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures
title_full_unstemmed	Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures
title_sort	orthogonal vs. non-orthogonal reducibility of matrix-valued measures
author	Koelink, E. Román, P.
author_facet	Koelink, E. Román, P.
publishDate	2016
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	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.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/147427
citation_txt	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.
work_keys_str_mv	AT koelinke orthogonalvsnonorthogonalreducibilityofmatrixvaluedmeasures AT romanp orthogonalvsnonorthogonalreducibilityofmatrixvaluedmeasures
first_indexed	2025-11-25T12:51:19Z
last_indexed	2025-11-25T12:51:19Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 008, 9 pages Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures? Erik KOELINK † and Pablo ROMÁN †‡ † IMAPP, Radboud Universiteit, Heyendaalseweg 135, 6525 GL Nijmegen, The Netherlands E-mail: e.koelink@math.ru.nl URL: http://www.math.ru.nl/~koelink/ ‡ CIEM, FaMAF, Universidad Nacional de Córdoba, Medina Allende s/n Ciudad Universitaria, Córdoba, Argentina E-mail: roman@famaf.unc.edu.ar URL: http://www.famaf.unc.edu.ar/~roman/ Received September 23, 2015, in final form January 21, 2016; Published online January 23, 2016 http://dx.doi.org/10.3842/SIGMA.2016.008 Abstract. 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. Key words: matrix-valued measures; reducibility; matrix-valued orthogonal polynomials 2010 Mathematics Subject Classification: 33D45; 42C05 1 Introduction The theory of matrix-valued orthogonal polynomials was initiated by Krein in 1949 and, since then, it was developed in different directions. From the perspective of the theory of orthogonal polynomials, one wants to study families of truly matrix-valued orthogonal polynomials. Here is where the issue of reducibility comes into play. Given a matrix-valued measure, one can construct an equivalent measure by multiplying on the left by a constant invertible matrix and on the right by its adjoint. If the equivalent measure is a block diagonal matrix, then all the objects of interest (orthogonal polynomials, three-term recurrence relation, etc.) reduce to block diagonal matrices so that we could restrict to the study of the blocks of smaller size. An extreme situation occurs when the matrix-valued measure is equivalent to a diagonal matrix in which case we are, essentially, dealing with scalar orthogonal polynomials. Our interest in the study of the reducibility of matrix-valued measures was triggered by the families of matrix-valued orthogonal polynomials introduced in [2, 9, 10, 11]. In [10] the study of the spherical functions of the group SU(2)× SU(2) leads to a matrix-valued measure Θ and a sequence of matrix-valued orthogonal polynomials with respect to Θ. From group theoretical ?This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica- tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html mailto:e.koelink@math.ru.nl http://www.math.ru.nl/~koelink/ mailto:roman@famaf.unc.edu.ar http://www.famaf.unc.edu.ar/~roman/ http://dx.doi.org/10.3842/SIGMA.2016.008 http://www.emis.de/journals/SIGMA/OPSFA2015.html 2 E. Koelink and P. Román considerations, we were able to describe the symmetries of Θ and pinpoint two linearly inde- pendent matrices in the commutant of Θ, one being the identity. The proof that these matrices actually span the commutator required a careful computation. It then turns out that it is possi- ble to conjugate Θ with a constant unitary matrix to obtain a 2× 2 block diagonal matrix. An analogous situation holds true for a one-parameter extension of this example [9]. In [2] from the study of the quantum analogue of SU(2)×SU(2) we constructed matrix-valued orthogonal poly- nomials which are matrix analogues of a subfamily of Askey–Wilson polynomials. The weight matrix can also be unitarily reduced to a 2× 2 block diagonal matrix in this case, again arising from quantum group theoretic considerations. In [12], the authors study non-unitary reducibility for matrix-valued measures and prove that a matrix-valued measure Θ reduces into a block diagonal measure if the real vector space A of all matrices T such that TΘ(X) = Θ(X)T ∗ for any Borel set X is not trivial, in contrast to the reducibility via unitary matrices that occurs when the commutant algebra of Θ is not trivial. The aim of this paper is to develop a criterion to determine whether unitary and non-unitary reducibility of a weight matrix W coincide in terms of the ∗-invariance of A . Every reduction of Θ can be performed via a unitary matrix if and only if A is ∗-invariant, in which case A = Ah where Ah is the Hermitian part of the commutant of Θ, see Section 2. We apply our criterion to our examples [2, 9, 10] and we conclude that there is no further reduction than the one via a unitary matrix. We expect that a similar strategy can be applied to more general families of matrix-valued orthogonal polynomials as, for instance, the families related to compact Gelfand pairs given in [7]. We also discuss an example where A and Ah are not equal. It is worth noting that unitary reducibility strongly depends on the normalization of the matrix-valued measure. Indeed, if the matrix-valued measure is normalized by Θ(R) = I, then the real vector space A is ∗-invariant and by our criterion, unitary and non-unitary reduction coincide. This is discussed in detail in Remark 3.7. 2 Reducibility of matrix-valued measures Let MN (C) be the algebra of N × N complex matrices. Let µ be a σ-finite positive measure on the real line and let the weight function W : R→MN (C) be strictly positive definite almost everywhere with respect to µ. Then Θ(X) = ∫ X W (x)dµ(x), (2.1) is a MN (C)-valued measure on R, i.e., a function from the σ-algebra B of Borel subsets of R into the positive semi-definite matrices in MN (C) which is countably additive. Note that any positive matrix measure can be obtained as in (2.1), see for instance [4, Theorem 1.12] and [5]. More precisely, if Θ̃ is a MN (C)-valued measure, and Θ̃tr denotes the scalar measure defined by Θ̃tr(X) = Tr(Θ̃(X)), then the matrix elements Θ̃ij of Θ̃ are absolutely continuous with respect to Θ̃tr so that, by the Radon–Nikodym theorem, there exists a positive definite function V such that dΘ̃i,j(x) = V (x)i,j dΘ̃tr(x). Note that we do not require the normalization Θ(R) = I as in [5]. A detailed discussion about the role of the normalization in the reducibility of the measure is given at the end of Section 3. Going back to the measure (2.1), we have dΘtr(x) = Tr(W (x)) dµ(x) so that Θtr is absolutely continuous with respect to µ. Note that Tr(W (x)) > 0 a.e. with respect to µ so that µ is absolutely continuous with respect to Θtr. The unicity of the Radon–Nikodym theorem implies W (x) = V (x)Tr(W (x)), i.e., W is a positive scalar multiple of V . Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures 3 We say that two MN (C)-valued measures Θ1 and Θ2 are equivalent if there exists a constant nonsingular matrix M such that Θ1(X) = MΘ2(X)M∗ for all X ∈ B, where ∗ denotes the adjoint. A MN (C)-valued measure matrix Θ reduces to matrix-valued measures of smaller size if there exist positive matrix-valued measures Θ1, . . . ,Θm such that Θ is equivalent to the block diagonal matrix diag(Θ1(x),Θ2(x), . . . ,Θm(x)). If Θ is equivalent to a diagonal matrix, we say that Θ reduces to scalar measures. In [12, Theorem 2.8], the authors prove that a matrix-valued measure Θ reduces to matrix-valued measures of smaller size if and only if the real vector space A = A (Θ) = { T ∈MN (C) \|TΘ(X) = Θ(X)T ∗ ∀X ∈ B } , contains, at least, one element which is not a multiple of the identity, i.e., RI ( A , where I is the identity. Note that our definition of A differs slightly from the one considered in [12]. If W is a weight matrix for Θ, then we require that T ∈ A satisfies TW (x) = W (x)T ∗ almost everywhere with respect to µ. If there exists a subspace V ⊂ CN such that Θ(X)V ⊂ V for all X ∈ B, since Θ(X) is self-adjoint for all X ∈ B, it follows that Θ(X)V ⊥ ⊂ V ⊥ for all X ∈ B. If ιV : V → CN is the embedding of V into CN , then PV = ιV ι ∗ V ∈MN (C) is the orthogonal projection on V and satisfies PV Θ(X) = Θ(X)PV , for all X ∈ B. Hence, the projections on invariant subspaces belong to the commutant algebra A = A(Θ) = { T ∈MN (C) \|TΘ(X) = Θ(X)T ∀X ∈ B } . Since Θ(X) is self-adjoint for all X ∈ B, A is a unital ∗-algebra over C. We denote by Ah the real subspace of A consisting of all Hermitian matrices. Then it follows that A = Ah ⊕ iAh. If CI ( A, then there exists T ∈ Ah such that T /∈ CI. The eigenspaces of T for different eigenvalues are orthogonal invariant subspaces for Θ. Therefore Θ is equivalent via a unitary matrix to matrix-valued measures of smaller size. Remark 2.1. Let S ∈ A and T ∈ A . Then we observe that S∗ ∈ A and therefore STS∗Θ(x) = Θ(x)ST ∗S∗ = Θ(x)(STS∗)∗ for all X ∈ B. Hence there is an action from A into A which is given by S · T = STS∗. Lemma 2.2. A does not contain non-zero skew-Hermitian elements. Proof. Suppose that S ∈ A is skew-Hermitian. Then S is normal and thus unitarily diago- nalizable, i.e., there exists a unitary matrix U and a diagonal matrix D = diag(λ1, . . . , λN ), λi ∈ iR, such that S = UDU∗, see for instance [8, Chapter 4]. Since S ∈ A , we get DU∗Θ(X)U = U∗Θ(X)UD∗ = −U∗Θ(X)UD, for all X ∈ B. The (i, i)-th entry of the previous equation is given by λi(U ∗Θ(X)U)i,i = −(U∗Θ(X)U)i,iλi. Take any X0 ∈ B such that Θ(X0) is strictly positive definite. Since U is unitary, U∗Θ(X0)U is strictly positive definite and therefore (U∗Θ(X0)U)i,i > 0 for all i = 1, . . . , N , which implies that λi = 0. � Theorem 2.3. A ∩A ∗ = Ah. 4 E. Koelink and P. Román Proof. Observe that if T ∈ Ah, then TΘ(X) = Θ(X)T = Θ(X)T ∗ for all X ∈ B, and thus Ah ⊂ A . Since T is self-adjoint, we also have T = T ∗ ∈ A ∗. On the other hand, let T ∈ A ∩A ∗. Then T ∗ ∈ A ∗ ∩A ⊂ A , and since A is a real vector space, (T − T ∗) ∈ A . The matrix (T − T ∗) is skew-Hermitian and therefore by Lemma 2.2 we have (T − T ∗) = 0. Hence T is self-adjoint and T ∈ Ah. � Corollary 2.4. If T ∈ A ∩A ∗, then T = T ∗. Proof. The corollary follows directly from the proof of Theorem 2.3. � Corollary 2.5. A is ∗-invariant if and only if A = Ah. Proof. If A = Ah then A is trivially ∗-invariant. On the other hand, if we assume that A is ∗-invariant then the corollary follows directly from Theorem 2.3. � Remark 2.6. Corollary 2.4 says that if A is ∗-invariant, then it is pointwise ∗-invariant, i.e., T = T ∗ for all T ∈ A . Remark 2.7. Suppose that there exists X ∈ B such that Θ(X) ∈ R>0I, then every T ∈ A is self-adjoint and Corollary 2.5 holds true trivially. Since TΘ(X) = Θ(X)T ∗ for all X ∈ B, if there is a point x0 ∈ supp(µ) such that lim δ→0 ∥∥∥∥ 1 µ((x0 − δ, x0 + δ)) Θ((x0 − δ, x0 + δ))− I ∥∥∥∥ = 0, then it follows that T = T ∗ and so Corollary 2.5 holds true. This is the case, for instance, for the examples given in [3, 1], where W (x0) = I for some x0 ∈ supp(µ). For Examples 4.2 and 4.3, in general, there is no x0 ∈ [−1, 1] for which W (x0) = I. 3 Reducibility of matrix-valued orthogonal polynomials Let MN (C)[x] denote the set of MN (C)-valued polynomials in one variable x. Let µ be a finite measure and W be a weight matrix as in Section 2. In this section we assume that all the moments Mn = ∫ xnW (x) dµ(x), n ∈ N, exist and are finite. Therefore we have a matrix-valued inner product on MN (C) 〈P,Q〉 = ∫ P (x)W (x)Q(x)∗ dµ(x), P,Q ∈MN (C)[x], where ∗ denotes the adjoint. By general considerations, e.g., [5, 6], it follows that there exists a unique sequence of monic matrix-valued orthogonal polynomials (Pn)n∈N, where Pn(x) = n∑ k=0 xkPnk with Pnk ∈ MN (C) and Pnn = I, the N × N identity matrix. The polynomials Pn satisfy the orthogonality relations 〈Pn, Pm〉 = δnmHn, Hn ∈MN (C), where Hn > 0 is the corresponding squared norm. Any other family (Qn)n∈N of matrix-valued orthogonal polynomials with respect to W is of the form Qn(x) = EnPn(x) for invertible matri- ces En. The monic orthogonal polynomials satisfy a three-term recurrence relation of the form xPn(x) = Pn+1(x) +BnPn(x) + CnPn−1(x), n ≥ 0, (3.1) where P−1 = 0 and Bn, Cn are matrices depending on n and not on x. Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures 5 Lemma 3.1. Let T ∈ A . Then we have (1) The operator T : MN (C)[x]→MN (C)[x] given by P 7→ PT is symmetric with respect to Θ. (2) TPn = PnT for all n ∈ N. (3) THn = HnT ∗ for all n ∈ N. (4) TMn = MnT ∗ for all n ∈ N. (5) TBn = BnT and TCn = CnT for all n ∈ N. Proof. Let P,Q ∈MN (C)[x]. Then 〈PT,Q〉 = ∫ P (x)TW (x)Q(x)∗ dµ(x) = ∫ P (x)W (x)T ∗Q(x)∗ dµ(x) = ∫ P (x)W (x)(Q(x)T )∗ dµ(x) = 〈P,QT 〉, so that T is a symmetric operator. This proves (1). It follows directly from (1) that the monic matrix-valued orthogonal polynomials are eigenfunctions for the operator T , see, e.g., [6, Proposition 2.10]. Thus, for every n ∈ N there exists a constant matrix Λn(T ) such that PnT = Λn(T )Pn. Equating the leading coefficients of both sides of the last equation, and using that Pn is monic, yields T = Λn(T ). This proves (2). The proof of (3) follows directly from (2) and the fact that T ∈ A . We have THn = ∫ Pn(x)TW (x)Pn(x)∗ dµ(x) = ∫ Pn(x)W (x)T ∗Pn(x)∗ dµ(x) = ∫ Pn(x)W (x)(Pn(x)T )∗dµ(x) = ∫ Pn(x)W (x)(TPn(x))∗ dµ(x) = HnT ∗. The proof of (4) is analogous to that of (3), replacing the polynomials Pn by xn. Finally, we multiply the three-term recurrence relation (3.1) by T on the left and on the right and we subtract both equations. Using that TPn = PnT and TPn+1 = Pn+1T we get TBnPn + TCnPn−1 = BnTPn + CnTPn−1. The coefficient of xn is TBn = BnT and therefore we also have TCn = CnT . � Corollary 2.5 provides a criterion to determine whether the set of Hermitian elements of the commutant algebra A is equal to A . However, for explicit examples, it might be cumbersome to verify the ∗-invariance of A from the expression of the weight. Our strategy now is to view A as a subset of a, in general, larger set whose ∗-invariance can be established more easily and that implies the ∗-invariance of A . Motivated by Lemma 3.1 we consider a sequence (Γn)n of strictly positive definite matrices such that if T ∈ A , then TΓn = ΓnT ∗ for all n. Then for each n ∈ N and I ⊂ N we introduce the ∗-algebras AΓ n = A(Γn) = {T ∈MN (C) \|TΓn = ΓnT}, AΓ I = ⋂ n∈I AΓ n, and the real vector spaces A Γ n = A (Γn) = {T ∈MN (C) \|TΓn = ΓnT ∗}, A Γ I = ⋂ n∈I A Γ n . (3.2) It is clear from the definition that A ⊂ A Γ n for all n ∈ N. 6 E. Koelink and P. Román Remark 3.2. For any subset I ⊂ N, the sequence (Γn)n induces a discrete MN (C)-valued measure supported on I dΓI(x) = ∑ n∈I Γnδn,x. Theorem 2.3 applied to the measure dΓI yields that A Γ I ∩ (A Γ I )∗ is the subset of Hermitian matrices in AΓ I . Theorem 3.3. If A Γ I is ∗-invariant for some non-empty subset I ⊂ N, then A = Ah. In particular, the statement holds true if A Γ n is ∗-invariant for some n ∈ N. Proof. If T ∈ A , then T ∈ A Γ n for all n ∈ I. Since A Γ I is ∗-invariant, then T ∗ ∈ A Γ n for all n ∈ I. If we apply Corollary 2.4 to the measure in Remark 3.2, we obtain T = T ∗. Therefore T ∈ Ah ⊂ A and thus A is ∗-invariant. Hence the theorem follows from Corollary 2.5. � Remark 3.4. Two obvious candidates for sequences (Γn)n are given in Lemma 3.1, namely the sequence of squared norms (Hn)n and the sequence of even moments (M2n)n. Remark 3.5. Let Θ be a MN (C)-valued measure, not necessarily with finite moments and take a positive definite matrix Γ such that TΓ = ΓT ∗ for all T ∈ A (Θ). Let S be a positive definite matrix such that Γ = S2. We can now consider the MN (C)-valued measure S−1ΘS−1. By a simple computation, we check that T ∈ A (Θ) if and only if S−1TS ∈ A (S−1ΘS−1). This gives A ( S−1ΘS−1 ) = S−1A (Θ)S. Moreover, if T ∈ A (Θ), then TΓ = ΓT ∗ implies that S−1TS = ST ∗S−1 = (S−1TS)∗. Hence S−1TS is self-adjoint for all T ∈ A (Θ). Then we have by Corollary 2.5 A ( S−1ΘS−1 ) h = A ( S−1ΘS−1 ) . On the other hand, if U ∈ Ah(Θ), then S−1USS−1Θ(X)S−1 = S−1UΘ(X)S−1 = S−1Θ(X)S−1SUS−1 = S−1Θ(X)S−1 ( S−1US )∗ , for all X ∈ B. Therefore S−1A(Θ)hS ⊂ A (S−1ΘS−1) = A(S−1ΘS−1)h. In general this is an inclusion, see Example 4.1. Remark 3.6. Suppose that Θ is a MN (C)-valued measure with finite moments and that M2n ∈ R>0I, respectively Hn ∈ R>0I, for some n ∈ N. Then it follows from Lemma 3.1 and (3.2) that T = T ∗ for all T ∈ A M 2n , respectively T ∈ A N n . Then Theorem 3.3 says that A = Ah. Remark 3.7. Let Θ be a MN (C)-valued measure such that the first moment M0 is finite. Then there exists a positive definite matrix S such that M0 = S2. The measure Θ̃ = S−1ΘS−1 satisfies Θ̃(R) = S−1Θ(R)S−1 = S−1M0S −1 = I. Therefore by Remark 3.6 we have that A (S−1ΘS−1) = A(S−1ΘS−1)h. Observe that the nor- malization Θ̃(R) = I is assumed in [5] so that in the setting of that paper the real subspace of Hermitian elements in the commutant coincides with the real vector space A (Θ). Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures 7 4 Examples In this section we discuss three examples of matrix-valued weights that exhibit different features. The first example is a slight variation of [12, Example 2.6]. Example 4.1. Let µ be the Lebesgue measure on the interval [0, 1], and let W be the weight W (x) = ( x2 + x x x x ) = ( 1 √ 6 3 0 1 )( x2 0 0 x )( 1 0√ 6 3 1 ) . A simple computation gives that A and A are given by A = CI, A = R ( 1 0 0 1 ) + R ( 1 − √ 6 3 0 0 ) . Observe that A is clearly not ∗-invariant since ( 1 − √ 6 3 0 0 )∗ /∈ A . Now we consider the sequen- ce (M2n)n of even moments. The first moment is given by M0 = ( 2 3 √ 6 6√ 6 6 1 2 ) and the algebras AM0 and A M 0 are AM0 = C ( 1 0 0 1 ) + C (√ 6 6 1 1 0 ) , A M 0 = R ( 1 0 0 1 ) + R (√ 6 6 1 1 0 ) + R ( 1 − √ 3 6 0 0 ) + iR ( 1 −2 √ 6 3√ 6 2 −1 ) . (4.1) This gives the inclusions RI ( (AM0 )h ( A M 0 . It is also clear from (4.1) that A M 0 ∩ (A M 0 )∗ = (AM0 )h. Now we proceed as in Remark 3.7, we take the positive definite matrix S such that M0 = S2. Here S = 1 15 ( √ 6+9 3 √ 6−3 3 √ 6−3 3 2 √ 6+6 ) . Then S−1W (x)S−1 = 1 25 ( (33 + 12 √ 6)x2 + (28− 8 √ 6)x −(6 + 9 √ 6)x2 + (4 + 6 √ 6)x −(6 + 9 √ 6)x2 + (4 + 6 √ 6)x (42− 12 √ 6)x2 + (22 + 8 √ 6)x ) . We finally have that A (S−1ΘS−1) = RI + RE + RF, A (S−1M0S −1) = RI + RE + RF + RG, where E = S−1 (√ 6 6 1 1 0 ) S = (√ 6 6 1 1 0 ) , G = iS−1 ( 1 −2 √ 6 3√ 6 2 −1 ) S = ( 0 −i i 0 ) , F = S−1 ( 1 − √ 3 6 0 0 ) S = 1 25 ( 11 + 4 √ 6 −(2 + 3 √ 6) −(2 + 3 √ 6) 14− 4 √ 6 ) . Then we have the following inclusions: RI = S−1A(Θ)hS ( A ( S−1ΘS−1 ) h = A (S−1ΘS−1), and S−1 ( AM0 ) h S ( A ( S−1M0S −1 ) h = A ( S−1M0S −1 ) . 8 E. Koelink and P. Román Example 4.2. Our second example is a family of matrix-valued Gegenbauer polynomials in- troduced in [9]. For ` ∈ 1 2N and ν > 0, let dµ(x) = (1 − x2)ν−1/2dx where dx is the Lebesgue measure on [−1, 1] and let W (ν) be the (2`+ 1)× (2`+ 1) matrix ( W (ν)(x) ) m,n = m∑ t=max(0,n+m−2`) α (ν) t (m,n)C (ν) m+n−2t(x), α (ν) t (m,n) = (−1)m n!m!(m+ n− 2t)! t!(2ν)m+n−2t(ν)n+m−t (ν)n−t(ν)m−t (n− t)!(m− t)! (n+m− 2t+ ν) (n+m− t+ ν) × (2`−m)!(n− 2`)m−t(−2`− ν)t (2`+ ν) (2`)! , where n,m ∈ {0, 1, . . . , 2`} and n ≥ m. The matrix W (ν) is extended to a symmetric matrix,( W (ν)(x) ) m,n = ( W (ν)(x) ) n,m . In [9, Proposition 2.6] we proved that A is generated by the identity matrix I and the involution J ∈M2`+1(C) defined by ej 7→ e2`−j Now we will use Theorem 3.3 to prove that Ah = A . This says that there is no further non-unitary reduction of the weight W . As a sequence of positive definite matrices we take the squared norms of the monic polynomials, (Γn)n = (Hn)n, that were explicitly calculated in [9, Theorem 3.7] and are given by the following diagonal matrices( H(ν) n ) i,k = δi,k √ π Γ(ν + 1 2) Γ(ν + 1) ν(2`+ ν + n) ν + n n!(`+ 1 2 + ν)n(2`+ ν)n (ν + k)n(2`+ 2ν + n)n(2`+ ν − k)n × k!(`+ ν)n(2`− k)!(n+ ν + 1)2` (2`+ ν + 1)n(2`)!(n+ ν + 1)k(n+ ν + 1)2`−k . For any n ∈ N we choose I = {n, n + 1}. If we take T ∈ A Γ I , i.e., TH (ν) n = H (ν) n T ∗ and TH (ν) n+1 = H (ν) n+1T ∗, it follows that Ti,j = ( H (ν) n ) j,j (H (ν) n )i,i T j,i = j!(2`− j)!(ν + i)n(2`+ ν − i)n(n+ ν + 1)i(n+ ν + 1)2`−i i!(2`− i)!(ν + j)n(2`+ ν − j)n(n+ ν + 1)j(n+ ν + 1)2`−j T j,i. It follows directly from this equation that Ti,i ∈ R and Ti,2`−i = T 2`−i,i. Now we observe that Ti,j = ( H (ν) n ) j,j( H (ν) n ) i,i T j,i = ( H (ν) n ) j,j (H (ν) n )i,i ( H (ν) n+1 ) i,i( H (ν) n+1 ) j,j Ti,j = (ν + j + n)(ν + j + n+ 1)(2`+ n+ ν − j)(2`+ n+ ν + 1− j) (ν + i+ n)(ν + i+ n+ 1)(2`+ n+ ν − i)(2`+ n+ ν + 1− i) Ti,j . (4.2) Equation (4.2) implies that Ti,j = 0 unless j = i or j = 2` − i. Hence T is self-adjoint and thus A Γ I is ∗-invariant and from Theorem 3.3 we have A = Ah. We conclude that A is the real span of {I, J}, and so there is no further non-unitary reduction. Example 4.3. Our last example is a q-analogue of the previous example for ν = 1. This sequence of matrix-valued orthogonal polynomials matrix analogues of a subfamily of Askey– Wilson polynomials and were obtained by studying matrix-valued spherical functions related to the quantum analogue of SU(2)× SU(2). For any ` ∈ 1 2N and 0 < q < 1, we have the measure dµ(x) = √ 1− x2 dx supported on [−1, 1] and a (2` + 1) × (2` + 1) weight matrix W which is given in [2, Theorem 4.8], we omit the explicit expression here and we give, instead, the explicit expression for the squared norms Hn of the monic orthogonal polynomials (Hn)i,j = δi,j q−2`2−2n(q2, q4`+4; q2)2 n(1− q4`+2)2 (q2i+2, q4`−2i+2; q2)2 n(1− q2n+2i+2)(1− q4`−2i+2n+2) . Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures 9 The expression for Hn is obtained combining Theorem 4.8 and Corollary 4.9 of [2]. The com- mutant algebra is generated by {I, J} as in the previous example, see [2, Proposition 4.10]. We take (Γn)n = (Hn)n, I = {n, n + 1} for any n ∈ N and observe that THn = HnT ∗ and THn+1 = Hn+1T ∗ implies Ti,j = ( H (ν) n ) j,j( H (ν) n ) i,i T j,i = ( H (ν) n ) j,j( H (ν) n ) i,i ( H (ν) n+1 ) i,i( H (ν) n+1 ) j,j Ti,j = (1− q2n+2j+2)(1− q2n+2j+4)(1− q4`+2n−2i+2)(1− q4`+2n−2i+4) (1− q2n+2i+2)(1− q2n+2i+4)(1− q4`+2n−2j+2)(1− q4`+2n−2j+4) Ti,j . As in the previous example, it follows that T is self-adjoint and therefore, A = Ah. Hence there is no further non-unitary reduction for W . Remark 4.4. Theorem 3.3 can be used to determine the irreducibility of a weight matrix. In fact, with the commutant algebras already determined in [9] and [2], Theorem 3.3 implies that the restrictions of the weight matrices of Examples 4.2 and 4.3 to the eigenspaces of the matrix J are irreducible. For some explicit cases see [10, Section 8]. Acknowledgements We thank I. Zurrián for pointing out a similar example to Example 4.1 to the first author. The research of Pablo Román is supported by the Radboud Excellence Fellowship. We would like to thank the anonymous referees for their comments and remarks, that have helped us to improve the paper. References [1] Aldenhoven N., Koelink E., de los Ŕıos 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 (SU(2) × SU(2), diag), arXiv:1507.03426. [3] Álvarez-Nodarse R., Durán A.J., de los Ŕıos 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 Ŕıos 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 (SU(2) × SU(2), 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 (SU(2) × SU(2), 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. http://dx.doi.org/10.1016/j.jat.2014.10.007 http://arxiv.org/abs/1308.2540 http://arxiv.org/abs/1507.03426 http://dx.doi.org/10.1016/j.jat.2013.02.003 http://arxiv.org/abs/0711.2703 http://dx.doi.org/10.1007/s00020-007-1517-x http://dx.doi.org/10.1007/s00020-007-1517-x http://arxiv.org/abs/1310.5134 http://dx.doi.org/10.1017/CBO9780511810817 http://arxiv.org/abs/1403.2938 http://dx.doi.org/10.1093/imrn/rnr236 http://arxiv.org/abs/1012.2719 http://dx.doi.org/10.4171/PRIMS/106 http://arxiv.org/abs/1203.0041 http://arxiv.org/abs/1501.04059 1 Introduction 2 Reducibility of matrix-valued measures 3 Reducibility of matrix-valued orthogonal polynomials 4 Examples References

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

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