DUAL SZEGÖ PAIRS OF SEQUENCES OF RATIONAL MATRIX-VALUED FUNCTIONS

We study certain sequences of rational matrix-valued functions with poles outside the unit circle. These sequences are recursively constructed based on a sequence of complex numbers with norm less than one and a sequence of strictly contractive matrices. We present some basic facts on the rational matrix-valued functions belonging to such kind of sequences and we will see that the validity of some Christoffel-Darboux formulae is an essential property. Furthermore, we point out that the considered dual pairs consist of orthogonal systems. In fact, we get similar results as in the classical theory of Szegö’s orthogonal polynomials on the unit circle of the first and second kind.


Introduction
The theory of orthogonal polynomials is known to have numerous applications in an extensive range of engineering problems.For instance, the important role of Szegö's orthogonal polynomials on the unit circle in circuit and system theory is today well recognized (see, e.g., [1,[29][30][31][32]39] and for discussing the case of matrix polynomials [9,10,28,38,41], [11,Section 3.6]).
Starting from different points of view of applications Bultheel, González-Vera, Hendriksen, and Njåstad have formed up a fruitful collaboration and created in the 1990s a comprehensive theory of scalar orthogonal rational functions on the unit circle.In a series of research papers they worked systematically out basic parts of a concept of generalizing essential parts of the classical theory of orthogonal polynomials on the unit circle (see, e.g., [3][4][5][6][7] and probably the first work referring to the rational situation [13] by Džrbašjan).
The present paper is another contribution generalizing this topic to the case of orthogonal rational matrix-valued functions on the unit circle and continues the line of investigations stated in [25][26][27].The main objective of this paper is to discuss some dual pairs of sequences of rational matrix-valued functions which are recursively constructed based on a sequence of complex numbers with norm less than one and a sequence of strictly 2 Dual Szegö pairs of rational matrix-valued functions contractive matrices.The recurrence relations defining such pairs are natural generalizations to the situation in question of those fulfilling Szegö's orthogonal polynomials of the first and the second kind.Following the idea of Delsarte et al. [9] with respect to the case of orthogonal matrix polynomials, we only use another normalization for the orthogonal functions in the case under consideration as Szegö in his classical work [39].
Throughout the paper let n be a nonnegative integer, let q be a positive integer, let C denote the set of all complex numbers, let D := {w ∈ C : |w| < 1}, let T := {z ∈ C : |z| = 1}, and let (α j ) ∞ j=0 be a sequence of complex numbers belonging to the open unit disk D. Furthermore, I q stands for the identity matrix of size q × q and the zero matrix of size q × q is denoted by 0 q .
Similar as in [25][26][27], we consider modules q×q α,n of rational q × q matrix-valued functions with prescribed poles (using the convention 1/ 0 := ∞) at most in the set in particular, not located on the unit circle T. We will also use the notation In fact, q×q α,n denotes the set of all complex q × q matrix-valued functions X which can be represented via X =  1 π α,n P, (1.3) where P is a complex q × q matrix polynomial of degree not greater than n and where the polynomial π α,n of degree not greater than n + 1 is given by Such kind of rational matrix-valued functions are studied in [25][26][27] in a way with α 0 := 0 but for a larger set {α 1 ,α 2 ,...} of underlying complex numbers.Since the principal object of this paper is to prepare a particular approach to solve an interpolation problem for matrix-valued Carathéodory functions in D, where α 0 ,α 1 ,α 2 ,... coincide with the treated interpolation points, we make this slight modification.
In the classical case, the connection between orthogonal polynomials on T and Taylor coefficient problems is particularly given by Schur's algorithm (see [36,37]).Roughly speaking, Schur's algorithm leads to a sequence of numbers, the so-called Schur parameters, to check if the given data in the problem correspond to a holomorphic function in D which is bounded by one.As discovered later by Geronimus (see [29]), these Schur parameters are closely connected with the parameters introduced by Szegö (see, e.g., [39]) through recurrence relations for orthogonal polynomials on T. In [33], based on some results contained in [6], an analog interrelation between the parameters which appear in an algorithm of Schur-type and the parameters which appear in the recurrence relations for orthogonal rational (complex-valued) functions on T is proved and used to solve an interpolation problem of Nevanlinna-Pick type for complex-valued Carathéodory functions in D.
There is a similar connection between orthogonal rational matrix-valued functions and solving certain interpolation problems of Nevanlinna-Pick type for matrix-valued Carathéodory functions (i.e., matrix interpolation problems which are studied with other methods, e.g., in [2,8,15]).But it takes more technical effort to verify such a connection in that case.The main task of this paper is to go some steps towards generalizing the results presented in [33] to the matrix case.In fact, we provide particular formulae starting from the recurrence relations for orthogonal rational matrix-valued functions stated in [26].In a forthcoming work, these formulae will finally play a key role by solving interpolation problems of Nevanlinna-Pick type for matrix-valued Carathéodory functions in D via orthogonal rational matrix-valued functions including an interrelation between the parameters which appear in the recurrence relations studied in the present paper and the parameters which appear in the algorithm discussed in [24,Section 5].
Similar as in [25,Definition 3.3], here a sequence (X j ) τ j=0 of matrix-valued functions is called a left (resp., right) orthonormal system corresponding to (α j ) τ j=0 and a nonnegative Hermitian q × q matrix-valued Borel measure F on T if the following two conditions are satisfied.
(i) For each integer j ∈ {0, 1,...,τ}, the function X j belongs to q×q α, j .(ii) For all integers j,k ∈ {0, 1,...,τ}, where δ jk := 1 if j = k and δ jk := 0 if j = k.Recall that a nonnegative Hermitian q × q Borel measure on T is a countably additive mapping from the σ-algebra B T of all Borel subsets of T into the set of nonnegative Hermitian q × q matrices.For basic facts on the integration theory with respect to nonnegative Hermitian Borel measures we refer to [35] (see also [23] concerning the special situation of rational matrix-valued functions).Note that a measure F has to fulfill some additional conditions if orthonormal systems of rational matrix-valued functions as above do exist (see, e.g., [25,Corollary 4.4]).
In [27] it is shown that a pair of orthonormal systems corresponding to (α j ) τ j=0 and F, that is, a pair [(X j ) τ j=0 ,(Y j ) τ j=0 ] consisting of a left (resp., right) orthonormal system (X j ) τ j=0 (resp., (Y j ) τ j=0 ) corresponding to (α j ) τ j=0 and some nonnegative Hermitian q × q Borel measure F on T, meets some specific recurrence relations.An essential characteristic of these recurrence relations is marked by an intensive interplay between the elements of the left and the right orthonormal systems although the left and the right versions come in without connection to each other per definition.This phenomenon already occurred in the case of matrix polynomials on T by finding the analogon of Szegö's recursions for that situation (see [9]).
Using a special normalization for the orthonormal systems of rational matrix-valued functions, the recurrence relations stated in [27] gain a simpler structure (see [26]).In fact, [26,Theorems 2.11,3.5,and 3.7] imply a parametrization of these particular pairs [(X j ) τ j=0 ,(Y j ) τ j=0 ] of orthogonal rational matrix-valued functions in terms of an initial condition and a sequence (E ) τ =1 of strictly contractive q × q matrices.These considerations are the starting point for the present paper.The crucial idea here is that we associate to such a pair [(X j ) τ j=0 ,(Y j ) τ j=0 ] a dual pair [(X # j ) τ j=0 ,(Y # j ) τ j=0 ] which satisfies analog recurrence relations depending on (−E ) τ =1 instead of (E ) τ =1 .Since this duality concept given by recurrence relations forms the main part in the proofs of the results below (not directly the orthogonality of the underlying systems), we center such dual pairs of sequences of rational matrix-valued functions and we return to some questions concerning the orthogonality only in the last section of the paper.
A brief synopsis is as follows.In Section 2 we introduce the central notations of this paper and explain basics on the recurrence relations defining these dual pairs of sequences of rational matrix-valued functions.By using certain well-known results on Potapov's Jtheory (see, e.g., [11,12,14,16,34]) we get in Section 3 some important properties of the rational matrix-valued functions belonging to such special pairs.In fact, the considerations there are motivated by the studies in [17-19, 21, 22] (see [9] and [11,Section 3.6]) on particular matrix polynomials solving Taylor coefficient problems.In Section 4 we will see that the pairs in question fulfill so-called Christoffel-Darboux formulae.As the treatments in Section 5 imply, the realization of such kind of Christoffel-Darboux formulae is in a way also a sufficient condition for rational matrix-valued functions to be dual Szegö pairs of sequences of rational matrix-valued functions.Finally, we extend in Section 6 the investigations stated in [26,Section 3] on the connection between recurrence relations and orthogonality of rational matrix-valued functions including an alternative proof of [26,Theorem 3.5].The essential new information in Section 6 is that, based on the duality concept introduced here, one has more insight into the structure of the nonnegative Hermitian q × q Borel measure occurring already in [26,Theorem 3.5].Following this train of thoughts, we will obtain two particular choices of measures, where the one corresponds to the pair [(X j ) τ j=0 ,(Y j ) τ j=0 ], the other corresponds to the dual pair [(X # j ) τ j=0 ,(Y # j ) τ j=0 ], and both can be recovered from each other similar as in the special case of orthogonal matrix polynomials on T (see, e.g., [11,Definition 3.6.10,Proposition 3.6.9,and Lemma 3.6.28]).In particular, the dual pairs of rational matrix-valued functions are modelled on Szegö's classical orthogonal polynomials of the first and the second kind.

Some basic facts
As the studies in [25][26][27] (see also [6]) suggest, the following transform of a rational function into another is an essential tool for the consideration on orthonormal systems of rational matrix-valued functions.If X ∈ q×q α,n , then the adjoint rational matrix-valued function X [α,n] of X (with respect to the underlying points α 0 ,α 1 ,...,α n ∈ D) is the rational matrix-valued function (belonging to q×q α,n as well) which is uniquely determined by the formula and where b αj denotes the elementary Blaschke factor corresponding to α j , that is, (2.3) Some information on further interrelations between X [α,n] and the underlying function X can be found in [25, Section 2].Note that the results on adjoint rational matrix-valued functions in [25] are explained relating to the special case α 0 = 0.But it is not hard to restate these with their proofs to the present situation.For instance, if X,Y ∈ q×q α,n , then also in that case the following properties are fulfilled. (I) We study in the following certain sequences of rational matrix-valued functions formed by given sequences of points belonging to D and of parameters which are strictly contractive matrices.Recall that a complex q × q matrix A is said to be contractive (resp., strictly contractive) if I q −A * A is a nonnegative (resp., positive) Hermitian matrix, where A * denotes the adjoint matrix of A. For instance, the zero matrix 0 q of size q × q is a strictly contractive matrix.
If τ is a nonnegative integer or ∞, if (E ) τ =1 is a sequence of strictly contractive q × q matrices, and if X 0 and Y 0 are nonsingular complex q × q matrices fulfilling the condition X * 0 X 0 = Y 0 Y * 0 , then we define sequences of rational matrix-valued functions (X j ) τ j=0 and (Y j ) τ j=0 by the relations and, for all integers ∈ {1, 2,...,τ} and points u ∈ C \ P α, , recursively, (2.5) Here and in the sequel A 1/2 stands for the (unique) nonnegative Hermitian square root of a nonnegative Hermitian q × q matrix A, the notation A −1 stands for the inverse of a 6 Dual Szegö pairs of rational matrix-valued functions nonsingular q × q matrix A, and hence A −1/2 denotes the inverse matrix of the nonnegative Hermitian square root of a positive Hermitian q × q matrix A tantamount to the nonnegative Hermitian square root of A −1 .Similar as in [26], we call [(X j ) τ j=0 ,(Y j ) τ j=0 ] the Szegö pair of rational matrix-valued functions generated by [(α j ) τ j=0 ;(E ) τ =1 ;X 0 ,Y 0 ].In addition, we consider simultaneously the Szegö pair [(X # j ) τ j=0 ,(Y # j ) τ j=0 ] of rational matrix-valued functions generated by the special choice [(α j ) τ j=0 ;(−E ) τ =1 ;(X −1 0 ) * ,(Y −1 0 ) * ] and call this the dual Szegö pair of [(X j ) τ j=0 ,(Y j ) τ j=0 ] in the following.In fact, we have and, for all integers ∈ {1, 2,...,τ} and points u ∈ C \ P α, , the recurrence relations (2.7) ].The definition of a Szegö pair of rational matrix-valued functions is inspired by the recurrence relations presented in [26,Section 2].This will be emphasized by the following theorem on particular orthogonal systems of rational matrix-valued functions.A left (resp., right) orthonormal system (X j ) τ j=0 corresponding to (α j ) τ j=0 and some nonnegative Hermitian q × q matrix-valued Borel measure F on T is said to be of left (resp., right) Szegö-type if in addition the matrices are positive Hermitian, where the numbers η j , j ∈ {0, 1,...,τ}, are defined by (2.9) Note that if there exists a left (resp., right) orthonormal system (Y j ) τ j=0 corresponding to (α j ) τ j=0 and F, then one can always choose such a special sequence (X j ) τ j=0 of orthonormal systems (cf.[25,Corollary 4.4] and [26,Remark 2.2]).
Observe that the difference between the backward recurrence relations stated in Proposition 2.4 for a Szegö pair of rational matrix-valued functions and for its dual Szegö pair consists in the different signs in front of the parameters E , ∈ {1, 2,...,τ}, similar to the case of the forward recursions defining such pairs of rational matrix-valued functions.

Connection to Potapov's J-theory
We will show in this section that one can use Potapov's J-theory (see, e.g., [11,12,14,34]) to obtain some information on the rational functions belonging to dual Szegö pairs.In fact, we get certain formulae which can be considered as a generalization of results on matrix polynomials in [21] (with respect to an approach solving Taylor coefficient problems for matrix-valued Carathéodory functions via orthogonal matrix polynomials) to the rational case.
Recall that if p is a positive integer and if J 1 and J 2 are complex p × p signature matrices (i.e., unitary and Hermitian) respectively, then a complex p × p matrix A is called J 2 -J 1 -contractive (resp., J 2 -J 1 -unitary) when J 2 −A * J 1 A is a nonnegative Hermitian matrix (resp., the zero matrix 0 p ).In the particular case J 1 = J 2 we write shortly J 1 -contractive (resp., J 1 -unitary) instead of J 1 -J 1 -contractive (resp., J 1 -J 1 -unitary).The special choice of the 2q × 2q signature matrices j qq := I q 0 q 0 q −I q , J q := 0 q −I q −I q 0 q (3.1) will be essential in the considerations below.
Theorem 3.1 yields in view of some well-known results on j qq -J q -contractive matrices (see, e.g., [16,Lemma 8] and use in addition [11, Lemma 2.1.5])particularly the following result.
Proof.Let ∈ {1, 2,...,τ} and u ∈ (D ∪ T) \ Z α, .We prove only the assertion with respect to Θ (u).Similarly, one can verify the others by using the same arguments.From Corollary 3.6 we know that the matrix Θ −1 (u) is nonsingular.Consequently, follows from (2.13).Hence (cf. the proof of Theorem 3.1), the matrix and j qq -unitary if u ∈ T. Therefore, the identity implies immediately the assertion referring to Θ (u).

Andreas Lasarow 19
Observe that the first and second Christoffel-Darboux formulae in Theorem 4.2 coincide with the identities for orthogonal rational matrix-valued functions proved in [25, Section 5] (see also [27,Theorem 3.10]).In view of Remark 2.1, these relations yield for each integer j ∈ {0, 1,...,τ} and points u,v ∈ C \ P α, j directly (4.8) The essential new information concerns the third and fourth Christoffel-Darboux formulae in Theorem 4.2 which can be regarded as a matricial version of [6, the first identity in Corollary 4. 3.4].Furthermore, the Christoffel-Darboux formulae in Theorem 4.2 can be obviously restated as follows.

A characterization of Szegö pairs
In the previous section (see, e.g., Theorem 4.2), we have explained that a Szegö pair of rational matrix-valued functions along with its dual Szegö pair fulfills some Christoffel-Darboux formulae.Referring to this, we study now an inverse problem.Roughly speaking, we will see that the realization of Christoffel-Darboux formulae is in a way also a sufficient condition for rational matrix-valued functions to be dual Szegö pairs of rational matrix-valued functions (cf.[27, Theorem 3.10]).
(i) The first (resp., second, third, or fourth) formula of Theorem 4.2 is satisfied.
(ii) The first (resp., second, third, or fourth) formula of Corollary 4.3 is satisfied.
Lemma 5.2.Let be a positive integer and let The following statements are equivalent.
(i) The first (resp., third) identity of Lemma 4.1 is satisfied.
Proof.If we fix v ∈ C \ P α, then, in view of (2.1) and forming the adjoint with respect to the + 2 points α 0 ,α 1 ,...,α ,α −1 , the first identity of Lemma 4.1 is equal to ( Since, by fixing now the point u ∈ C \ P α, and adjoining, this relation is equal to we obtain the equivalence of the first and the second identity of Lemma 4.1.Similarly, one can conclude that the third and the fourth identity of Lemma 4.1 are equivalent. Lemma 5.3.Let X ,Y ,X # ,Y # ∈ q×q α, for each integer ∈ {1, 2,...,τ} and let X 0 ,Y 0 ,X # 0 ,Y # 0 be the rational matrix-valued functions defined as in (2.4) and (2.6) for some nonsingular complex q × q matrices X 0 , Y 0 satisfying the condition X * 0 X 0 = Y 0 Y * 0 .The following Andreas Lasarow 21 statements are equivalent.
Proof.Using the same arguments as in the proof of Theorem 4.2, one can inductively show that (i) implies (ii).It remains to verify that (ii) implicates also (i).Note that the choice of X 0 , Y 0 supplies immediately that the identities of Theorem 4.2 are also satisfied for j = 0 (cf. the proof of Theorem 4.2).Thus, for each integer ∈ {1, 2,...,τ} and points u,v ∈ C \ P α, , from (4.1), (ii), Remark 5.1, the fourth identity of Corollary 4.3, and the fourth identity of Theorem 4.2, it follows that (5.3) Consequently, with respect to the fourth kind of identities it is shown that (ii) yields (i).
Similarly by a straightforward calculation, one can prove this implication referring to the first, second, and third kind of identities, respectively.

On particular measures corresponding to the dual pairs
We study now an inverse question to Theorem 2.2.By using similar arguments as in [26, Section 3], it is not hard to accept that if [(X j ) τ j=0 ,(Y j ) τ j=0 ] is some Szegö pair of rational matrix-valued functions, then there exists a nonnegative Hermitian q × q Borel measure F on T so that [(X j ) τ j=0 ,(Y j ) τ j=0 ] is exactly a Szegö pair of orthonormal systems corresponding to (α j ) τ j=0 and F. The following considerations are to explain that the construction of such a measure occurring already in [26,Section 3] includes actually a simultaneous answer referring to the dual pair [(X # j ) τ j=0 ,(Y # j ) τ j=0 ] of [(X j ) τ j=0 ,(Y j ) τ j=0 ].For technical reasons we prove before two useful results.Here and in the sequel, if A is a complex q × q matrix, then e A stands for the real part of A, that is, e A := (1/2)(A + A * ).Lemma 6.1.For each integer j ∈ {0, 1,...,τ} and point z ∈ T, if [(X j ) τ j=0 ,(Y j ) τ j=0 ] is a Szegö pair of rational matrix-valued functions generated by some [(α j ) τ j=0 ;(E ) τ =1 ;X 0 ,Y 0 ], where (E ) τ =1 is a sequence of strictly contractive q × q matrices and X 0 , Y 0 are nonsingular q × q matrices fulfilling X * 0 X 0 = Y 0 Y * 0 .Proof.Let j ∈ {0, 1,...,τ} and z ∈ T. In view of Corollary 3.2 and (2.1) it follows that the matrices X By using some well-known results on weak convergence of nonnegative Hermitian q × q Borel measures (see, e.g., [20]) we study now a little the case of Szegö pairs of infinite sequences of rational matrix-valued functions.Note that one says that a sequence (F n ) ∞ n=0 of nonnegative Hermitian q × q Borel measures on T converges weakly to a nonnegative Hermitian q × q Borel measure F on T if lim n→∞ T hF n (dz) = T hF(dz) (6.26) for each bounded, continuous, and real-valued function h on T.