On an Involution in the Class of Non-negative Hermitian Matrix Measures on a Compact Interval

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two other mathematical objects. The first one is a particular class of holomorphic matrix-valued functions, namely the class R q (C \ [α, β]) (see Notation 6.3). This bijection is expressed via an integral transform (see Theorem 6.4). The other object is of more algebraic nature. More precisely, it is the class F q,∞,α,β of [α, β]-nonnegative definite infinite sequences of complex q × q matrices which is introduced before Theorem 5.3. This bijection is expressed via the sequence of power moments of a measure belonging to M q ([α, β]).
Guided by former investigations [8] in the context of the matricial version of the trigonometric moment problem we are led to a particular involution in the class R q (C\ [α, β]). The main aims of this paper are to translate this involution in the language of M q ([α, β]) on the one side and into the language of F q,∞,α,β on the other side.
In order to realize these goals we start by providing the necessary background material. This is the content of Sects. 2-7. First we formulate the general version of power moment problems on Borel subsets of the real axis in Sect. 2. In the next step we remember in Sect. 3 the concept of Reciprocal Sequence associated with a (finite or infinite) sequence of complex p × q matrices. Taking into account the particular needs of this paper we recall in Sect. 4 (see Theorems 4.4,4.5, and 4.10) some connections between several block Hankel matrices generated by a sequence of complex p × q matrices on the one side and by its associated Reciprocal Sequence on the other side. We also consider connections to associated Schur complements. The content of Sect. 5 lies in the heart of this paper. More precisely, following [13,Proposition 5.5] we characterize in Proposition 5.4 the set of all matricial [α, β]-Hausdorff moment sequences. In the second part of Sect. 5, we summarize some facts on the structure of matricial [α, β]-Hausdorff moment sequences which are taken from [12][13][14][15]. In Sect. 6, we introduce that class of holomorphic q × q matrix-valued functions which plays the key role in our subsequent investigations, namely the class R q (C \ [α, β]). This class turns out to be intimately connected with the set M q ([α, β]) via Stieltjes transform (see Theorem 6.4). In Sect. 7 we reformulate the original matricial moment problem in the language of the class R q (C \ [α, β]).
In Sect. 8, we introduce in Definition 8.1 the first central object of this paper. We associate with each function F ∈ R q (C \ [α, β]) another function F − which turns out to be also a member of R q (C \ [α, β]). It is shown that this construction generates an involution in R q (C \ [α, β]) (see Theorem 8.6). Via [α, β]-Stieltjes transform this involution is reformulated as an involution in M q ([α, β]) (see Theorem 8.7).
The rest of this paper is mainly devoted to express this latter involution in terms of the sequences of power moments. In this regard our considerations are mainly of algebraic nature. In Sect. 9 we associate with a finite or infinite sequence (s j ) κ j=0 of complex p × q matrices a sequence (s − j ) κ j=0 of complex p × q matrices (see Definition 9.12) which turns out to be the second central object of this paper. Our next main goal is to verify that the mapping (s j ) κ j=0 → (s − j ) κ j=0 generates in particular an involution of F q,κ,α,β . For this reason we derive in Sect. 10 several identities between block Hankel matrices generated by sequences from F q,κ,α,β on the one side and their associated [α, β]-reciprocal sequences on the other side. Taking into account the definition of F q,κ,α,β we have to verify four different types of such relations connecting block Hankel matrices. The derivation of these formulas requires a lot of work and skill. It is done in Sect. 10 which provides the basis for all further considerations in this paper. In Sect. 11 we show that the transition from a matricial [α, β]-Hausdorff moment sequence to its [α, β]-reciprocal sequence corresponds to the simple operation of reflecting each of its matricial canonical moments (see Theorem 11.7). Thereby we are able to verify that forming the [α, β]-reciprocal sequence generates an involution in F q,κ,α,β (see Theorem 11.11). In Sect. 12 we show the compatibility of the operation of forming [α, β]-reciprocal functions and [α, β]-reciprocal sequences. The content of the final Sect. 13 can be sketched as follows. In an intermediate step we constructed in [14] a Schur-Nevanlinna (short SN) type algorithm for Non-negative Hermitian q × q measures on [α, β] by translating the original SN type algorithm for matricial Hausdorff moment sequences into the language of measures. In a further step in [15] in turn this algorithm was translated via [α, β]-Stieltjes transform into the class R q (C \ [α, β]). In Sect. 13, we extend these investigations by verifying that the elementary step of the SN type algorithm in the class R q (C \ [α, β]) essentially uses the concept of the [α, β]-reciprocal function (see Theorem 13.6).

Matricial Moment Problems on Borel Subsets of the Real Axis
In this section, we are going to formulate a certain class of matricial power moment problems. Before doing this, we have to introduce some terminology. We denote by Z the set of all integers. Let N:={n ∈ Z : n ≥ 1}. Furthermore, we write R for the set of all real numbers and C for the set of all complex numbers. In the whole paper, p and q are arbitrarily fixed integers from N. We write C p×q for the set of all complex p × q matrices and C p is short for C p×1 . When using m, n, r , . . . instead of p, q in this context, we always assume that these are integers from N. We write A * for the conjugate transpose of a complex p × q matrix A. Denote by C q×q (resp. C q×q ) the set of all complex Non-negative (resp. positive) Hermitian q × q matrices. Let (X , X) be a measurable space. Each countably additive mapping whose domain is X and whose values belong to C q×q is called a Non-negative Hermitian q × q measure on (X , X). For the integration theory with respect to Non-negative Hermitian measures, we refer to Kats [18] and Rosenberg [20]. Let B R be the σ -algebra of all Borel subsets of R. In the whole paper, stands for a non-empty set belonging to B R . Let B be the σ -algebra of all Borel subsets of and let M q ( ) be the set of all Non-negative Hermitian q × q measures on ( , B ). Observe that M 1 ( ) coincides with the set of ordinary measures on ( , B ) with values in [0, ∞).

Invertibility of Sequences of Complex Matrices and the Concept of the Reciprocal Sequence
In this paper, the Moore-Penrose inverse of a complex matrix plays an important role. For each matrix A ∈ C p×q , there exists a unique matrix X ∈ C q× p , satisfying the four equations Let O p×q be the zero matrix from C p×q and let I q be the identity matrix from C q×q . Sometimes, if the size is clear from the context, we will omit the indices and write O and I , resp. To emphasize that a certain (block) matrix X is built from a sequence (s j ) κ j=0 , we sometimes write X s for X .  Whenever it is clear which sequence is meant, we will simply write S m (resp. S m ) instead of S s m (resp. S s m ).
We recall [17,Definition 4.13]: Definition 3.2 Let (s j ) κ j=0 be a sequence of complex p × q matrices. Then we call the sequence (s j ) κ j=0 defined by s 0 :=s † 0 and, for all j ∈ Z 1,κ , recursively by s j := − s † 0 j−1 =0 s j− s the reciprocal sequence of (s j ) κ j=0 . For each (block) matrix X built from the sequence (s j ) κ j=0 , we denote by X the corresponding matrix built from the Reciprocal Sequence of (s j ) κ j=0 instead of (s j ) κ j=0 . Remark 3.3 (cf. [14,Remark 3.6]). Let (s j ) κ j=0 be a sequence of complex p × q matrices with Reciprocal Sequence (r j ) κ j=0 . Then, for each m ∈ Z 0,κ , the Reciprocal Sequence of (s j ) m j=0 coincides with (r j ) m j=0 . In particular, we have Corollary 3.4 ([14,Cor. 3.19]). If (s j ) κ j=0 is a sequence of complex p × q matrices, then S m S m S m = S m and S m S m S m = S m for all m ∈ Z 0,κ .
In the following, we write R(A):={Ax : x ∈ C q } and N (A):={x ∈ C q : Ax = O p×1 } for the column space and the null space of a complex p × q matrix A, resp.  Given an arbitrary n ∈ N and arbitrary rectangular complex matrices A 1 , A 2 , . . . , A n , we use diag(A j ) n j=1 or diag (A 1 , A 2 , . . . , A n ) to denote the corresponding block diagonal matrix. Furthermore, for arbitrarily given A ∈ C p×q and m ∈ N 0 we write with 2n ≤ κ, by K n :=[s j+k+1 ] n j,k=0 for all n ∈ N 0 with 2n + 1 ≤ κ, and by G n :=[s j+k+2 ] n j,k=0 for all n ∈ N 0 with 2n + 2 ≤ κ, resp. In view of Notation 4. . Let (s j ) κ j=0 be a sequence of complex p × q matrices. Then using Notations 3.1 and 4.3 the identity H n + S n H n S n = y 0,n * p,n,1 + q,n,1 z 0,n holds for all n ∈ N 0 with 2n ≤ κ. Theorem 4.5 ([14,Theorem 4.8]). Let (s j ) κ j=0 be a sequence of complex p × q matrices. Then K n = −S n K n S n for all n ∈ N 0 with 2n + 1 ≤ κ.
In our following consideration, Schur complements in block Hankel matrices play an essential role. For this reason, we recall this construction: is called the Schur complement of A in M. The block Hankel matrix H n admits the following block representations: ([14,Remark 4.9]). If (s j ) κ j=0 is a sequence of complex p × q matrices, then H n = H n−1 y n,2n−1 z n,2n−1 s 2n and H n = s 0 z 1,n y 1,n G n−1 for all n ∈ N with 2n ≤ κ.
j=0 is a sequence of complex p × q matrices, then, by virtue of Remark 4.6, let L 0 :=H 0 and let L n :=H n /s 0 for all n ∈ N with 2n ≤ κ. ([14,Remark 4.11]). If (s j ) κ j=0 is a sequence of complex p × q matrices, then from Remark 4.6 we infer that L n = G n−1 − y 1,n s † 0 z 1,n for all n ∈ N with 2n ≤ κ.
For each n ∈ N 0 , denote by H q,2n the set of all sequences (s j ) 2n j=0 of complex q × q matrices for which the corresponding block Hankel matrix H n = [s j+k ] n j,k=0 is Non-negative Hermitian. Furthermore, denote by H q,∞ the set of all sequences (s j ) ∞ j=0 of complex q × q matrices satisfying (s j ) 2n j=0 ∈ H q,2n for all n ∈ N 0 . The sequences belonging to H q,2n or H q,∞ are said to be R-non-negative definite. Against the background of Remark 4.6, we use in the sequel the following notation: j=0 is a sequence of complex p × q matrices, then let L 0 :=H 0 and let L n :=H n /H n−1 for all n ∈ N with 2n ≤ κ.
A For all ι ∈ N 0 ∪ {∞} and each non-empty set X , let S ι (X ):={(X j ) ι j=0 : X j ∈ X for all j ∈ Z 0,ι }. Obviously, F q,0,α,β coincides with the set of all sequences (s j ) 0 j=0 with s 0 ∈ C q×q . Furthermore, using Notation 5.1, we have for all n ∈ N and for all n ∈ N 0 . j ∈ Z 0,κ and s 2k ∈ C q×q for all k ∈ N 0 with 2k ≤ κ.
Notation 5.14 Let (s j ) κ j=0 be a sequence of complex p × q matrices. Then let the sequence (A j ) κ j=0 be given by A 0 :=s 0 and by A j :=s j − a j−1 . Furthermore, if κ ≥ 1, then let the sequence (B j ) κ j=1 be given by B j :=b j−1 − s j .

Definition 5.17
Let (s j ) κ j=0 be a sequence of complex p × q matrices. Let the sequence (f j ) 2κ j=0 be given by f 0 :=A 0 , by f 4k+1 :=A 2k+1 and f 4k+2 :=B 2k+1 for all k ∈ N 0 with 2k + 1 ≤ κ, and by f 4k+3 :=B 2k+2 and f 4k+4 :=A 2k+2 for all k ∈ N 0 with 2k + 2 ≤ κ. Then we call (f j ) 2κ j=0 the F α,β -parameter sequence of (s j ) κ j=0 . In view of (5.5) and (5.3), we have in particular for each j ∈ Z 1,κ the [α, β]-interval parameter sequence of (s j ) κ j=0 . With the Euclidean scalar product ·, · E : C q × C q → C given by x, y E :=y * x, which is C-linear in its first argument, the C-vector space C q becomes a unitary space. Let U be an arbitrary non-empty subset of C q . The orthogonal complement U ⊥ :={v ∈ C q : v, u E = 0 for all u ∈ U} of U is a linear subspace of the unitary space C q . If U is a linear subspace itself, the unitary space C q is the orthogonal sum of U and U ⊥ . In this case, we write P U for the transformation matrix corresponding to the orthogonal projection onto U with respect to the standard basis of C q , i. e., P U is the uniquely determined matrix P ∈ C q×q satisfying the three conditions P 2 = P, P * = P, and R(P) = U.   [12,Definition 10.11]). If (s j ) κ j=0 is a sequence of complex p × q matrices, then we call (m j ) κ j=0 given by m j := 1 2 (a j + b j ) the sequence of [α, β]-interval mid points of (s j ) κ j=0 .

Remark 5.26
Let (s j ) κ j=0 be a sequence of complex p × q matrices with sequence of [α, β]-interval mid points (m j ) κ j=0 . For each k ∈ Z 0,κ , the matrix m k does not depend on matrices s j with j > k. In particular, for each m ∈ Z 0,κ , the sequence of [α, β]-interval mid points of (s j ) m j=0 coincides with (m j ) m j=0 .

The Class R q (C \ [˛,ˇ])
Notation 6.1 Denote by R q ( + ) the set of all matrix-valued functions F : + → C q×q , which are holomorphic and satisfy Im F(w) ∈ C q×q for all w ∈ + . (I) Im F(z) ∈ C q×q for all z ∈ + .
As already mentioned, the power moments of a measure belonging to M q ([α, β]) exist for each non-negative integer order:   In view of Remark 6.7 we can associate to a given function from the class R q (C \ [α, β]) an auxiliary function, which is intimately connected to the sequence (c j ) κ−2 j=0 introduced in Notation 5.1 (cf. Remark 7.8): Notation 6.13 (cf. [15,Notation 4.19]). Let F ∈ R q (C \ [α, β]) with [α, β]-spectral measureσ F . Then let the function F c : C \ [α, β] → C q×q be defined by

An Equivalent Problem in the Class R q (C \ [˛,ˇ])
According to our interest in the matricial Hausdorff moment problem, we consider the integral transformation (B.1) for the particular case of Non-negative Hermitian measures σ belonging to M q ([α, β]): . The mapping [α,β] : given by σ →S σ , whereS σ is given by (7.1), is well defined and bijective.

The [˛,ˇ]-reciprocal Function
Given F ∈ R q ( + ) it was shown in the paper [8,Theorem 9.4] that the function −F † belongs to R q ( + ), too. The first aim of this section is to find a similar result for the class R q (C \ [α, β]). More precisely, given a function F ∈ R q (C \ [α, β]) we are looking for a function G, which is essentially determined by the Moore-Penrose inverse F † of F, such that G belongs to R q (C \ [α, β]), too. In this regard, it seems natural to expect that the desired construction of G depends not only on F but also on the interval ends α and β. We will see in the sequel that the following notion is the convenient tool we are looking for.
We illustrate our construction by first examples.
We write rank A for the rank of a complex matrix A. The following observation contains essential information on the object introduced in Definition 8.1.
Proof In view of Notation 6.3, the function E : Observe that, by virtue of Definition 8.1 and Remark A.8, the func- Thus, we can apply Proposition C.1 to see that the function G is holomorphic. Let the functions resp. According to Proposition 6.12, then f 1 and f 2 both belong to R q ( + ). Let the functions h 1 , h 2 : resp. From Proposition 6.2 we can conclude that h 1 and h 2 both belong to R q ( + ). Let the functions g 1 , g 2 : + → C q×q be defined by g 1 (w):=(w −α)G(w) and g 2 (w):=(β − w)G(w), resp. In view of Remark A.8, it is readily checked, that g 1 = h 2 and g 2 = h 1 hold true. In particular, g 1 and g 2 both belong to R q ( + ). Since G is holomorphic, we can apply Proposition 6.12 to obtain G ∈ R q (C\ [α, β]). Taking into account F − = MG M and M * = M, then Remark 6.6 yields . Regarding Theorem 6.4, we conclude from Lemma B.1 easily lim y→∞ iy F(iy) = −M and lim y→∞ iyG(iy) = −N .
The combination of Propositions 8.4 and 7.2 leads us to the following notion which translates Definition 8.1 in the language of measures belonging to M q ([α, β]).
We extend now our knowledge on the self-mapping This leads us to the main result of this section.
This shows that the mapping F [α,β] is an involution. Now we translate Theorem 8.6 into the language of M q ([α, β]).
The main aim of our subsequent consideration can be formulated as follows. We are striving for an explicit description of the translation of the involution M [α,β] given in Theorem 8.7 into the language of power moments of measures belonging to M q ([α, β]). For this reason, we recall that Proposition 5.4 tells us that by forming the sequence of power moments a bijective correspondence between M q ([α, β]) and the set F q,∞,α,β of [α, β]-non-negative definite sequences is established. This means, we will work in the sequel mainly in the set F q,∞,α,β and use part of the machinery developed in [12][13][14][15]. We will look for an involution of the set F q,∞,α,β which corresponds to either of the mappings considered in Theorems 8.6 and 8.7.

The Definition of [˛,ˇ]-reciprocal Sequence
In the rest of this paper we are going to construct a particular self-mapping of the set of finite or infinite complex p × q matrices which will turn out in the case p = q to generate that involution of F q,∞,α,β which is compatible with either of the involutions considered in Theorems 8.6 and 8.7. This section contains the first step of this plan namely, we introduce the object which will turn out to play a key role in our subsequent considerations. As the set F q,∞,α,β stands in the background of what we are doing our concrete way reflects some experiences from [12][13][14][15]. First we introduce some notation. Notation 9.1 For each n ∈ N 0 , let T q,n :=[δ j,k+1 I q ] n j,k=0 , where δ jk is the Kronecker delta.
In particular, we have T q,0 = O q×q and T q,n = O q×nq O q×q I nq O nq×q for all n ∈ N. For each n ∈ N 0 , the block matrix T q,n has the shape of the first of the two matrices in Notation 3.1 with the matrix O q×q in the q × q block main diagonal. Thus, det (I (n+1)q − zT q,n ) = 1 holds true for all z ∈ C. Hence, the following matrix-valued function is well defined: [14,Remark 8.4]). Let n ∈ N 0 and let z ∈ C. In view of Notation A.14, then R q,n (z) is a block Toeplitz matrix belonging to L p,n and [R q,n (z)] * is a block Toeplitz matrix belonging to U q,n . In particular, det R q,n (z) = 1.
Given a sequence (s j ) κ j=0 of complex p × q matrices and a bounded interval [α, β] of R we have introduced in Notation 5.1 three particular sequences associated with (s j ) κ j=0 . These three sequences correspond to the intervals [α, ∞), (−∞, β], and [α, β], respectively. Now we slightly modify each of these sequences. We start with the case of the interval [α, ∞).

Some Identities for Block Hankel Matrices Associated with the [˛,ˇ]-reciprocal Sequence of [˛,ˇ]-non-negative Definite Sequences
Roughly speaking, a central step of our plan is to verify that the [α, β]-reciprocal sequence (s − j ) ∞ j=0 corresponding to a sequence (s j ) ∞ j=0 ∈ F q,∞,α,β belongs to F q,∞,α,β , too. In order to realize this we will need several identities which express block Hankel matrices built from the sequence (s − j ) ∞ j=0 with block Hankel matrices built from (s j ) ∞ j=0 . Reflecting the definition of an [α, β]-non-negative definite sequence we have to take into account four different forms of block Hankel matrices. We start with some auxiliary results taken mainly from [14].
. Let (s j ) κ j=0 be a sequence of complex p × q matrices. For all n ∈ Z 0,κ and z ∈ C then R p,n (z)S n = S n R q,n (z) and Remark 10.7 (cf. [14,Remark 5.19]). Let (s j ) κ j=0 be a sequence of complex p × q matrices and let m ∈ Z 0,κ . Then D m is a block Toeplitz matrix belonging to L p,m and D m is a block Toeplitz matrix belonging to U q,m . In particular, det D m = 1 and det D m = 1.
The next result is the first central observation in this section. In the following, we use the equivalence relation "∼" introduced in Notation A.17:  ⊕ H α,n−1,β ).

24)
Proof Denote by (r j ) 2n+1 j=0 the Reciprocal Sequence of (c j ) 2n+1 j=0 . According to Definition 9.12, we have d j = −s 0 r j s 0 for all j ∈ Z 0,2n+1 . Regarding Proof Let m ∈ N 0 and n ∈ N with m + 2n ≤ κ. Consequently, we obtain Regarding Notation 9.2, the proof is complete.

Remark 10.13
Let (s j ) κ j=0 be a sequence of complex p × q matrices and let ζ, ω ∈ C. According to Lemma 10.12 and Notation 4.1, for all n ∈ N with 2n ≤ κ, then We start now from a sequence (s j ) 2n+1 j=0 . We form the sequences (u j ) 2n j=0 given by u j := − αd j + d j+1 where the 1st equality is due to Notation 4.1, the 3rd equality is due to (9.1), the 4th equality can be seen using (3.1), the 5th equality is due to Notation 5.1, and the last equality is due to (3.3) and Notations 9.
follows. Regarding Notation 5.1, the combination of (10.31) and (10.32) yields Applying Remark 10.13 to the sequence (b j ) 2n j=0 , we can conclude then j=0 . Let the sequence (v j ) 2n j=0 be given by v j :=βd j − d j+1 . Then where the 1st equality is due to Notation 4.1, the 3rd equality is due to (9.1), the 4th equality can be seen using (3.1), the 5th equality is due to Notation 5.1, and the last equality is due to (3.3) and Notations 9.2, 9.1, 3.1 and 4.1. Now suppose n ≥ 1. By virtue of Notation 4.1, we see  In the final part of this section we start from a sequence (s j ) 2n+2 j=0 . We form the sequence (w j ) 2n j=0 given by Proof Denote by (r j ) 2n+2 j=0 the Reciprocal Sequence of (c j ) 2n+2 j=0 . According to Definition 9.12, we have d j = −s 0 r j s 0 for all j ∈ Z 0,2n+2 . Regarding In view of Notation 9.2 furthermore  Taking additionally into account Notation 9.2, (10.60), and (10.57), hence and analogously follow. From (10.48) and (10.60) we get      and We first prove (10.97) in the case n = 0. We have where the 1st equality is due to Notation 4.1, the 3rd equality is due to (9.1) and (9.2), the 4th equality can be seen using (3.1), the 5th equality is due to Notation 4.1, and the last equality is due to ( as well as In view of (10.108) and (

An Involution in the Set F q,∞,˛,Ť
he central aim of this section is to verify that the operation of forming the [α, β]reciprocal sequence in F q,∞,α,β is an involution. In the first step we show that this operation is a self-mapping of F q,∞,α,β .
In view of Remark A.18, consequently the assertions follow.
We come now to the first main result of this section.
Proof From Theorem 11.7 we can conclude that (s − j ) κ j=0 belongs to F q,κ,α,β and that q 0 = e 0 and q j = P R(d j−1 ) − e j for all j ∈ Z 1,κ hold true. According to Proposition 5.16 we have d j ∈ C q×q for all j ∈ Z 0,κ . In particular, P R(d j−1 ) = I q for all j ∈ Z 1,κ follows. Consequently, we obtain q j = I q − e j for all j ∈ Z 1,κ . The application of Proposition 5.24 completes the proof.
Corollary 11.9 Assume κ ≥ 1, let k ∈ Z 1,κ , and let (s j ) κ Proof In view of Theorem 11.7 the sequence (s − j ) κ j=0 belongs to F q,κ,α,β and the [α, β]-interval parameter sequence (q j ) κ j=0 of (s − j ) κ j=0 fulfills q j = P R(d j−1 ) − e j for all j ∈ Z 1,κ . According to Proposition 5.28 we have e j = 1 2 P R(d j−1 ) for all j ∈ Z k,κ . Denote by (l j ) κ j=0 the sequence of [α, β]-interval lengths of (s − j ) κ j=0 . Corollary 11.5 yields then l j = d j for all j ∈ Z 0,κ . Consequently, we obtain for all j ∈ Z k,κ . The application of Proposition 5.28 completes the proof. j=0 . According to Definition 5.29, we have t j = s − j ands j = s j for all j ∈ Z 0,m . Regarding Remark 9.13, the last identity implies r j = s − j for all j ∈ Z 0,m . Consequently, t j = r j holds true for all j ∈ Z 0,m . Proposition 5.30 yields that (s j ) ∞ j=0 belongs to F q,∞,α,β and is [α, β]-central of order m + 1. Thus, we can infer from Corollary 11.9 that (r j ) ∞ j=0 is [α, β]-central of order m + 1. Since, again by Proposition 5.30, the sequence (t j ) ∞ j=0 is [α, β]-central of order m + 1 as well, and the initial elements of the sequences (t j ) ∞ j=0 and (r j ) ∞ j=0 coincide up to j = m, the proof now can be easily completed by mathematical induction, taking into account Definition 5.27 and Remark 5.26.
Finally, we turn our attention to the second main result of this section.

β is well defined and an involution.
Proof This can be easily seen using Corollary 11.8 and Theorem 11.11.

Compatibility of Forming [˛,ˇ]-reciprocal Functions with Forming [˛,ˇ]-reciprocal Sequences
In this section we finalize our preceding considerations. More precisely, we show that if F ∈ R q (C\ [α, β]) has an [α, β]-spectral measure with sequence of power moments (s j ) ∞ j=0 then the sequence of power moments of the [α, β]-spectral measure of F − is exactly (s − j ) ∞ j=0 . This is a consequence of the following result.
β]-modification of (s j ) ∞ j=0 introduced in Definition 9.8. Then c 0 = −s 0 , c 1 = (α +β)s 0 −s 1 , and c j :=c j−2 for all j ∈ Z 2,∞ . According to Notation 6.13, consequently Therefore, the matrix-valued function X : j=0 the Reciprocal Sequence of (c j ) ∞ j=0 . Using Proposition 6.8, for all z ∈ C ρ we obtain R(X (z)) = R(E(z)) = R(F(z)) = R(σ F ([α, β])) = R(s 0 ) = R(c 0 ) and analogously N (X (z)) = N (c 0 ). Hence, we can conclude from Lemma C.3 that the matrix-valued function Y is holomorphic in C ρ with for all z ∈ C ρ . Regarding Definition 8.1 and Remark A.8, we have for all z ∈ C ρ . In view of Definition 9.12 consequently for all z ∈ C ρ follows. Due to Proposition 11.1 the sequence (s − j ) ∞ j=0 belongs to F q,∞,α,β . Proposition 8.4 yields F − ∈ R q (C \ [α, β]). In particular, F is holomorphic in C \ [α, β]. Therefore, we can apply Proposition 7.7 to obtain , completing the proof in the case κ = ∞. Now we consider the case κ < ∞. Then m:=κ belongs to N 0 . Regarding Remark 6.7, denote byŝ j := [α,β] x jσ F (dx) for all j ∈ N 0 the power moments of Since the assertion is already proved for κ = ∞, we see that and thus s j =ŝ j for all j ∈ Z 0,m . From Remark 9.13 we can conclude thenŝ  S(z, q) is the Stieltjes transform of the measure with canonical moments q = (q 1 , q 2 , . . . ) given by q n :=1 − p n . We conclude now some consequences of Theorem 12.1.
j=0 ] is also well defined. Furthermore, using Theorem 8.6, it is readily checked that these mappings are mutual inverses.
is well defined and bijective.
follows. Thus the mapping under consideration is well defined. Taking into account Theorem 11.11, the same reasoning shows that the map- is also well defined. Furthermore, using Theorem 8.7, it is readily checked that these mappings are mutual inverses.
j=0 is the sequence of power moments of ν [α,β] given in Remark 12.10.
Proof First observe that in view of δ = 0 the left identity in (5.4) implies P R(d 0 ) = P R(s 0 ) . According to Theorem 11.7, the sequence (s − j ) κ j=0 belongs to F q,κ,α,β and the [α, β]-interval parameter sequence (q j ) κ j=0 of (s − j ) κ j=0 fulfills q 0 = e 0 and q j = P R(d j−1 ) − e j for all j ∈ Z 1,κ . By virtue of Theorem 5.22 statement (i) is then equivalent to the coincidence of the sequences (q j ) κ j=0 and (e j ) κ j=0 which in turn is equivalent to e j = 1 2 P R(d j−1 ) being fulfilled for all j ∈ Z 1,κ . Because of Proposition 5.28 and P R(d 0 ) = P R(s 0 ) the latter is equivalent to statement (ii) as well as to statement (iii).
Remark 5.8 yields s 0 ∈ C q×q . In view of Proposition 5.4 and Definition 5.32 we can conclude from Remark 12.10 that the (scalar) sequence (γ j ) ∞ j=0 belongs to F 1,∞,α,β and that the [α, β]-interval parameter sequence ( j ) ∞ j=0 of (γ j ) ∞ j=0 fulfills 0 = 1 and j = 1/2 for all j ∈ N. Thus we can apply Lemma 5.23 to see that the (matricial) sequence (γ j s 0 ) ∞ j=0 belongs to F q,∞,α,β and that the [α, β]-interval parameter sequence (p j ) ∞ j=0 of (γ j s 0 ) ∞ j=0 fulfills p 0 = 0 s 0 = s 0 and p j = j P R(s 0 ) =  and (s j ) ∞ j=0 coincide. Due to Proposition 12.11 and M = s 0 the latter is equivalent to s j = γ j M being fulfilled for all j ∈ N 0 , where (γ j ) ∞ j=0 is the sequence of power moments of ν [α,β] . In view of M ∈ C q×q , we can conclude from Lemma

Connection to Schur-Nevanlinna Type Algorithm
In this section we will show that the elementary step of the SN type algorithm for the class R q (C \[α, β]) developed in [15] can be written in a simple way using the [α, β]reciprocal function. We now present the corresponding detailed considerations. Let (s j ) κ j=0 and (t j ) κ j=0 be sequences of complex p × q and q × r matrices, resp. As usual, the Cauchy product (x j ) κ j=0 of (s j ) κ j=0 and (t j ) κ j=0 is given by x j := j =0 s t j− . Definition 13.1 ([14, Definition 8.14]). Suppose κ ≥ 1. Let (s j ) κ j=0 be a sequence of complex p × q matrices. Denote by (g j ) κ−1 j=0 the (−∞, β]-modification of (a j ) κ−1 j=0 and by (x j ) κ−1 j=0 the Cauchy product of (b j ) κ−1 j=0 and (g j ) κ−1 j=0 . Then we call the sequence (t j ) κ−1 j=0 given by t j := − a 0 s † 0 x j a 0 the F α,β -transform of (s j ) κ j=0 .  Consequently, we obtain Funding Open Access funding enabled and organized by Projekt DEAL.
Data availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Notation A.14 For each n ∈ N 0 denote by L p,n (resp., U p,n ) the set of all lower (resp., upper) p × p block triangular matrices belonging to C (n+1) p×(n+1) p with matrices I p on its block main diagonal.
Remark A.15 ([14,Remark A.20]). For each n ∈ N 0 the sets L p,n and U p,n are both subgroups of the general linear group of invertible complex (n + 1) p × (n + 1) p matrices. Using the classes L p,n and U q,n , we can introduce an equivalence relation for complex (n + 1) p × (n + 1)q matrices: Notation A. 17 If n ∈ N 0 and A, B are two complex (n + 1) p × (n + 1)q matrices, then we write A ∼ n, p×q B if there exist matrices L ∈ L p,n and U ∈ U q,n such that B = LAU. If the corresponding (block) sizes are clear from the context, we will omit the indices and write A ∼ B.
Lemma A. 19 Let , m ∈ N 0 , let A 1 , B 1 be two complex ( + 1) p × ( + 1)q matrices, and let A 2 , B 2 be two complex (m + 1) p × (m + 1)q matrices. Then the following statements are equivalent: Proof (i)⇒(ii): Because of (i) and Notation A.17 there exist matrices L ∈ L p,n and U ∈ U q,n such that B 1 ⊕ B 2 = L(A 1 ⊕ A 2 )U. According to Remark A.16 then L = L 1 O X L 2 and U = U 1 Y O U 2 with matrices L 1 ∈ L p, , X ∈ C (m+1) p×( +1) p , L 2 ∈ L p,m , and U 1 ∈ U q, , Y ∈ C ( +1)q×(m+1)q , U 2 ∈ U q,m . A straightforward calculation yields Consequently, B 1 = L 1 A 1 U 1 . Since Remark A.15 shows that the matrix L 1 is invertible, we can furthermore conclude A 1 Y = O. Hence, B 2 = L 2 A 2 U 2 follows. In view of Notation A.17, thus (ii) holds true.
(ii)⇒(i): Because of (ii) and Notation A.17 there exist matrices L 1 ∈ L p, and U 1 ∈ U q, with B 1 = L 1 A 1 U 1 and L 2 ∈ L p,m and U 2 ∈ U q,m with B 2 = L 2 A 2 U 2 . According to Remark A.16 then L:=L 1 ⊕ L 2 belongs to L p,n and U:=U 1 ⊕U 2 belongs to U q,n . A straightforward calculation yields B 1 ⊕ B 2 = L(A 1 ⊕ A 2 )U. In view of Notation A.17, hence (i) holds true.

G:
=C\ is an open subset of C. Observe that, for each t ∈ , the function h t : G → C defined by h t (z):=1/(t − z) is holomorphic. Consider an arbitrary z ∈ G and let d z := inf x∈ |x − z|. Then d z > 0 and 1/|t − z| ≤ 1/d z for all t ∈ . Consequently, the function g z : → C defined by g z (t):=1/(t − z) belongs to L(σ ). For each closed disk K ⊆ G, we have with d K := inf (x,w)∈ ×K |x − w|, furthermore d K > 0 and 1/|t − z| ≤ 1/d K . By means of that, one can check that the matrix-valued function S σ : G → C q×q defined byŜ (ii) R(F(z)) = R(F(w)) and N (F(z)) = N (F(w)) for all z, w ∈ G.
(iii) For all z ∈ G the sequence ( 1 n! F (n) (z)) ∞ n=0 belongs to the class D p×q,∞ introduced in Notation 3.5.
If (i) is fulfilled and z 0 ∈ G, then ( 1 n! G (n) (z 0 )) ∞ n=0 is exactly the Reciprocal Sequence of ( 1 n! F (n) (z 0 )) ∞ n=0 introduced in Definition 3.2. Next, we give analogous results for power series expansions at z 0 = ∞. To that end, let ρ ∈ (0, ∞) and suppose that the improper open annulus C ρ :={z ∈ C : |z| > ρ} is entirely contained in G. Furthermore, let a holomorphic matrix-valued function F : G → C p×q be given, admitting the series representation F(z) = ∞ n=0 z −n C n (C.1) for all z ∈ C ρ with certain complex p × q matrices C 0 , C 1 , C 2 , . . . ([15,Remark F.5]). Let F : G → C p×q be holomorphic, admitting the series representation (C.1) for all z ∈ C ρ with certain complex p × q matrices C 0 , C 1 , C 2 , . . . Let U ∈ C r × p and let V ∈ C q×s . Then H :=U F V is holomorphic and H (z) = ∞ n=0 z −n (UC n V ) for all z ∈ C ρ . Lemma C.3 ([15, Lemma F.7]). Let F : G → C p×q be holomorphic and let (C n ) ∞ n=0 be a sequence of complex p × q matrices such that (C.1) and furthermore R(F(z)) = R(C 0 ) and N (F(z)) = N (C 0 ) hold true for all z ∈ C ρ . Let G : C ρ → C q× p be defined by G(z):=[F(z)] † and denote by (D n ) ∞ n=0 the Reciprocal Sequence of (C n ) ∞ n=0 . Then G is holomorphic and G(z) = ∞ n=0 z −n D n for all z ∈ C ρ .