Matrix moment perturbations and the inverse Szegő matrix transformation

Given a perturbation of a matrix measure supported on the unit circle, we analyze the perturbation obtained on the corresponding matrix measure on the real line, when both measures are related through the Szeg˝o matrix transformation. Moreover, using the connection formulas for the corresponding sequences of matrix orthogonal polynomials, we deduce several properties such as relations between the corresponding norms. We illustrate the obtained results with an example.


Introduction
The term moment problem was used for the first time in T. J. Stieltjes' classic memoir [32] (published posthumously between 1894 and 1895) dedicated to the study of continued fractions. The moment problem is a question in classical analysis that has produced a rich theory in applied and pure mathematics. This problem is beautifully connected to the theory of orthogonal polynomials, spectral representation of operators, matrix factorization problems, probability, statistics, prediction of stochastic processes, polynomial optimization, inverse problems in financial mathematics and function theory, among many other areas. In the matrix case, M. Krein was the first to consider this problem in [21], and later on some density questions related to the matrix moment problem were addressed in [14,15,24,25]. Recently, the theory of the matrix moment problem is used in [10] for the analysis of random matrix-valued measures. Since the matrix moment problem is closely related to the theory of matrix orthogonal polynomials, M. Krein was the first to consider these polynomials in [22]. Later, several researchers have made contributions to this theory until today. In the last 30 years, several known properties of orthogonal polynomials in the scalar case have been extended to the matrix case, such as algebraic aspects related to their zeros, recurrence relations, Favard type theorems, and Christoffel-Darboux formulas, among many others.

EDINSON FUENTES AND LUIS E. GARZA
A nice summary, as well as many references, can be found in [8]. As in the scalar case, matrix orthogonal polynomials have proved to be a useful tool in the analysis of many problems of mathematics such as differential equations [13,28], rational approximation theory [20], spectral theory of Jacobi matrices [1,29], analysis of polynomial sequences satisfying higher order recurrence relations [11,17], quantum random walks [3], and Gaussian quadrature formulas [2,12,16,31], among many others. In this contribution, we are interested in the study of some properties related with a perturbation of a sequence of matrix moments, within the framework of the theory of matrix orthogonal polynomials both on the real line and on the unit circle.
The structure of the manuscript is as follows. In Section 2, we state some properties of matrix orthogonal polynomials on the real line and on the unit circle that will be used in the sequel. Section 3 deals with the so-called Szegő matrix transformation and some of its properties. This has been previously studied in [8,9,33]. Then, we introduce a perturbation on a matrix measure supported on the unit circle that results in a perturbation of the corresponding sequence of matrix moments, and analyze the resulting perturbation on the associated matrix measure on the real line, when both measures are related through the Szegő transformation. This is the main contribution of our manuscript. Sections 4 and 5 deal with connection formulas and some other properties associated with the corresponding matrix orthogonal polynomials sequences on the real line and on the unit circle, respectively. Connection formulas for perturbations of matrix moments have been studied previously in [7,6]. Finally, we illustrate the obtained results with an example in Section 6.

Matrix orthogonal polynomials
We will use the following notation. Let M l = M l (C) be the ring of l ×l matrices with complex entries. The matrices I and 0 will denote the l × l identity matrix and zero matrix, respectively. An Hermitian matrix A ∈ M l satisfies A = A H (its conjugate transpose), and the set of Hermitian matrices is a real vector space. We say A is positive definite if u H Au > 0 for every nonzero vector u ∈ C l , and positive semi-definite if the inequality is not strict. For 0 ≤ k ≤ j − 1, let where I is in the k-th position, and define χ (j) supported on E ⊆ R is said to be Hermitian (resp. positive semi-definite, positive definite), if for every Borel subset B on E, the matrix σ(B) is a Hermitian (resp. positive semi-definite, positive definite) matrix. Notice that the assumption of positive definiteness implies the Hermitian character of the measure. For more information about matrix measures, see [8] 2.1. Matrix orthogonal polynomials on the real line. Let µ = (µ i,j ) l i,j=1 be a Hermitian matrix measure supported on E ⊆ R and denote by P l [x] the linear space of polynomials in x ∈ R with coefficients in M l . If the scalar measures µ i,j , i, j = 1, . . . , l, are absolutely continuous with respect to the scalar Lebesgue measure dx, then according to the Radon-Nikodym theorem, it can always be expressed as is also a Hermitian matrix, and thus Notice that µ induces right and left sesquilinear forms, defined respectively by On the other hand, the j-th moment associated to the matrix measure µ is defined by Notice that every moment is a Hermitian matrix, i.e., µ H j = µ j for j ≥ 0, since µ is Hermitian. We denote by H j ∈ M lj and F j ∈ M l(j+1) the block matrices , with x ∈ R and j ≥ 1, and the block vector v j,2j−1 = [µ j µ j+1 · · · µ 2j−1 ] H ∈ M lj×l . Notice that H j is a Hankel invertible and Hermitian matrix for j ≥ 1. With these matrices, it is shown in [27] that the sequence of (right) matrix polynomials (P R j (x)) j≥0 defined by (i.e., each P R j is defined as the Schur complement of x j I in the matrix F j ), is orthogonal with respect to the sesquilinear form (2.2), that is, Conversely, any sequence of matrix moments (µ j ) j≥0 satisfying that the corresponding Hankel matrices are invertible and Hermitian defines a Hermitian matrix measure. In this situation, we will say (µ j ) j≥0 is a Hermitian sequence. The positiveness of µ can also be characterized in terms of the norms s j , j ≥ 0, as follows. In a similar way, a sequence of left matrix polynomials (P L n (x)) n≥0 can be defined such that it is orthogonal with respect to the sesquilinear form (2.3).
The kernel matrix polynomial of degree j for x, y ∈ R is defined by which can be expressed also in terms of the moments by (see [27]) Denoting K (i,r) j (y, x) the i-th (resp. r-th) derivative of K j (y, x) with respect to the variable y (resp. x), we have from (2.7), for 0 ≤ i ≤ j and 0 ≤ r ≤ j,

Matrix orthogonal polynomials on the unit circle.
Let T := {z ∈ C : |z| = 1} and D := {z ∈ C : |z| < 1} be the unit circle and unit disc, respectively, where z ∈ T is parametrized as z = e iθ with θ ∈ (−π, π]. In what follows, σ = (σ i,j ) l i,j=1 will be a Hermitian matrix measure supported on T (on all or part of T) and P l [z] will denote the linear space of polynomials in z ∈ C with coefficients in M l . A Hermitian matrix measure σ can be decomposed as where the Hermitian matrix W(θ) denotes the Radon-Nikodym derivative of the absolutely continuous part of σ with respect to the scalar Lebesgue measure, and σ s represents the singular part.
As in the real line case, a matrix measure σ induces right and left sesquilinear forms, defined respectively as On the other hand, the j-th (matrix) moment associated with the matrix measure σ is Notice that T j is a Toeplitz invertible and Hermitian matrix for j ≥ 1. As in the real line case, any sequence (c j ) j∈Z such that T j is invertible and Hermitian for j ≥ 1 will be called a Hermitian sequence, and has an associated Hermitian matrix measure σ supported on T. The sequence of monic matrix polynomials (Φ R j (z)) j≥0 defined by (see [26]) is orthogonal with respect to the sesquilinear form (2.8) and satisfies where the Hermitian matrix S R j is the Schur complement of T j+1 with respect to In a similar way, a sequence (Φ L n (z)) n≥0 can be constructed such that it is orthogonal with respect to the sesquilinear form (2.9). Unlike the real line case, where the right-orthogonal matrix polynomial is just the conjugate transpose of the left-orthogonal matrix polynomial, on the unit circle there is not a direct relation between the right and left-orthogonal polynomials. This is a consequence of the fact that zf, g R,σ = f,zg R,σ . Nevertheless, we will consider only the right matrix polynomials, denoted by Φ j = Φ R j and the corresponding norms S j = S R j for j ≥ 0, since the results for the left polynomials are analogous.

EDINSON FUENTES AND LUIS E. GARZA
Theorem 2.1 is valid also for the matrix measure σ with support on T. The kernel matrix polynomial of degree j for z, w ∈ C is defined in this case by and its corresponding expression in terms of the matrix moments is (see [26]) As in the real case, K

Matrix moments perturbation and its connection through the inverse Szegő transformation
In this section, we define the Szegő matrix transformation of a matrix measure supported on the real line, as well as some of its properties. Later on, we will define a perturbation of a matrix measure supported on T that consists in the perturbation of a single matrix moment of such measure, and analyze the resulting perturbation on the real line when the measures are related through the Szegő matrix transformation.
is symmetric and positive semi-definite, then its associated matrix moments are Hermitian matrices, since On the other hand, if µ(x) is a positive semi-definite matrix measure supported on [−1, 1], it is possible to define a symmetric positive semi-definite matrix measure σ(θ) supported on (−π, π] by This is the so-called Szegő matrix transformation (see [8,9,33]). The scalar version of the Szegő transformation can be found in [30]. We will write dσ = Sz(dµ) if the matrix measures µ and σ satisfy for any integrable function f on [−1, 1]. In a similar way, the inverse Szegő matrix transformation can be defined by the relation for any integrable function g on T such that g(θ) = g(−θ). We will denote this relation by dµ = Sz −1 (dσ). Notice that if µ has the form (2.1), then (3.1) becomes which can be written as If dµ(x) = ω(x) dx, and dσ(θ) = W(θ) dθ 2π , then they are related through the Szegő matrix transformation when In particular, for W(θ) = cos k θ, with k ≥ 0, we have The proof is analogous to the one given in [ Given a positive semi-definite matrix measure σ supported on T, we can consider the perturbation for a fixed j and m j ∈ M l such that σ j is positive semi-definite. In particular, if σ is symmetric and m j is a Hermitian matrix, then (3.5) becomes and thus σ j is a symmetric, positive semi-definite matrix measure supported on the unit circle. Notice that the right sesquilinear form associated with (3.5) is From (3.7), one easily sees that the corresponding matrix momentc k satisfies In other words, σ j represents an additive perturbation of the matrix moments c j and c −j of σ.
The following result shows the matrix measure supported on [−1, 1] that is related with (3.5) through the inverse Szegő matrix transformation. This result constitutes a generalization of the corresponding result on the scalar case studied in [18,19]. Proposition 3.1. Let σ be a positive semi-definite matrix measure supported on T, and let m j ∈ M l be a Hermitian matrix such that σ j defined as in (3.5) is positive semi-definite. Assume dµ = Sz −1 (dσ). Then, dµ j = Sz −1 (dσ j ) is given by where T j (x) := cos(jθ) is the j-th degree Chebyshev polynomial of the first kind on R (see [5]).

MATRIX MOMENT PERTURBATIONS, INVERSE SZEGŐ TRANSFORMATION 581
Since σ j (θ) is a symmetric matrix measure, by applying the inverse Szegő matrix transformation and taking into account (3.4), we have which is (3.8).
Recall that the double factorial or semifactorial of a number n (denoted by n!!) is the product of all the integers from 1 up to n that have the same parity (odd or even) as n. That is, if n is even, The next proposition expresses the considered perturbation in terms of the moments associated with the matrix measure (3.8).

EDINSON FUENTES AND LUIS E. GARZA
where [j/2] = j/2 if j is even and [j/2] = (j − 1)/2 if j is odd. As a consequence, if j ≤ n and n + j is even, and, since for n even, and equals zero if n is odd, (3.10) becomes (3.9).
This means that the perturbation of the matrix moments c j and c −j of a matrix measure σ supported on the unit circle results in a perturbation, defined by (3.9), of the moments µ n associated with a measure µ supported in [−1, 1], when both measures are related through the inverse matrix Szegő transformation.

Connection formulas on the real line and further results
Let µ be a Hermitian matrix measure supported on E ⊆ R with an associated sequence of monic matrix orthogonal polynomials (P n (x)) n≥0 , and a Hermitian sequence of matrix moments (µ n ) n≥0 . Define a new matrix moments sequence (μ n ) n≥0 byμ n = µ n + M n ∈ M l with M n Hermitian, in such a way that (μ n ) n≥0 is also a Hermitian sequence. Then, there exists an associated measureμ that is Hermitian, with a corresponding sequence of matrix orthogonal polynomials (P n (x)) n≥0 . Also, define s j = P j , P j R,µ ands j = P j ,P j R,μ .
In this section, we will show the relation between the matrix polynomialsP j and P j , as well as the relation betweens j and s j , in terms of the sequence of matrix moments (µ n ) n≥0 . Notice that the perturbation of a matrix measure µ supported on E = [−1, 1] defined by (3.8) constitutes a particular case with M n = m j B(n, j), if j ≤ n and n + j is even, 0, otherwise.

MATRIX MOMENT PERTURBATIONS, INVERSE SZEGŐ TRANSFORMATION 583
We will use the following notation: For a fixed k, with 0 ≤ k ≤ 2j − 1, it has been proved in [7] that As a consequence we have the next result.
LetH j ∈ M lj be the block Hankel matrix associated with the perturbed matrix moments sequence, i.e.

Theorem 4.3.
Let µ be a Hermitian matrix measure with matrix moments (µ k ) 2n−1 k=0 . Define a new Hermitian sequence of matrix moments (μ k ) 2n−1 k=0 byμ k = µ k + M k with 0 ≤ k ≤ 2n − 1, and denote byμ its corresponding Hermitian matrix measure. Then, for 1 ≤ j ≤ n, we havẽ Proof. Let us assume first that only the k-th moment is perturbed. From (2.5) we have that if 2j < k, thens j = s j . We will show that if k ≤ 2j, we havẽ and thus from (2.5) we get From (4.5) and using (2.6), we deduce (4.10). On the other hand, for j ≤ k ≤ 2j −1 we haveμ From (2.5) and (2.6), proceeding as above, we get and from (4.6) and using (2.6), Since

EDINSON FUENTES AND LUIS E. GARZA
we get (4.10). Finally, for k = 2j (notice that if j = n the result is also valid with M 2j = 0), On the other hand, using the notation given in (4.1), (4.10) can be written as , which, using (4.2) and (4.3), becomes The general case, i.e. when all moments are perturbed (k = 0, . . . , 2n − 1) is easy to deduce, since we have (4.8) and so (4.9) follows.

Connection formulas for the unit circle and further results
Let σ be a Hermitian matrix measure supported on T, with associated monic matrix orthogonal polynomials (Φ n (z)) n≥0 and matrix moments (c n ) n≥0 . We define a new Hermitian sequence of matrix moments by (c n ) n≥0 , wherec n = c n + m n ∈ M l for n ≥ 0, and denote its corresponding Hermitian matrix measure and monic matrix orthogonal polynomials byσ and (Φ n (z)) n≥0 , respectively. Furthermore, we define S j = Φ j , Φ j R,σ andS j = Φ j ,Φ j R,σ . In this section, in a similar way as in the real case, we will show the relation between the matrix polynomialsΦ j and Φ j , as well as the relation betweenS j and S j , in terms of the matrix moments (c n ) n≥0 . Notice that (3.5) constitutes a particular case, when only one of the moments is perturbed. Let us define the following notation: D j+1 m k := diag{m k · · · m k } ∈ M (j+1)l , D k,j+k := diag{k!I · · · (j + k)!I} ∈ M (j+1)l .
As in the real line case, for a fixed k, with 0 ≤ k ≤ j − 1, it has been proved in [6] that As a consequence, we have the following result. The above theorem is a generalization to the matrix case of what was studied in [4,19]. The particular case of a perturbation of a single matrix moment given by (3.5) is shown in the following corollary.

EDINSON FUENTES AND LUIS E. GARZA
On the other hand, for 1 ≤ j ≤ n, letÑ j ∈ M jl be defined as  Notice that and thus substituting N j,k in (5.6), (5.5) follows.