The truncated matrix trigonometric moment problem with an open gap

This paper is a continuation of our previous investigations on the truncated matrix trigonometric moment problem in Ukrainian Math. J., 2011, \textbf{63}, no. 6, 786-797, and Ukrainian Math. J., 2013, \textbf{64}, no. 8, 1199-1214. In this paper we shall study the truncated matrix trigonometric moment problem with an additional constraint posed on the matrix measure $M_{\mathbb{T}}(\delta)$, $\delta\in \mathfrak{B}(\mathbb{T})$, generated by the seeked function $M(x)$: $M_{\mathbb{T}}(\Delta) = 0$, where $\Delta$ is a given open subset of $\mathbb{T}$ (called a gap). We present necessary and sufficient conditions for the solvability of the moment problem with a gap. All solutions of the moment problem with a gap can be constructed by a Nevanlinna-type formula.


Introduction.
This paper is a continuation of our previous investigations on the truncated matrix trigonometric moment problem (briefly TMTMP) by the operator approach in [1], [2]. The truncated matrix trigonometric moment problem consists of finding a non-decreasing C N ×N -valued function M (t) = (m k,l (t)) N −1 k,l=0 , t ∈ [0, 2π], M (0) = 0, which is left-continuous in (0, 2π], and such that 2π 0 e int dM (t) = S n , n = 0, 1, ..., d, where {S n } d n=0 is a prescribed sequence of (N × N ) complex matrices (moments). Here N ∈ N and d ∈ Z + are fixed numbers. Set where S k := S * −k , k = −d, −d + 1, ..., −1, and {S n } d n=0 are from (1). It is well known that the following condition: is necessary and sufficient for the solvability of the moment problem (1) (e.g. [3]). The solvable moment problem (1) is said to be determinate if it has a unique solution and indeterminate in the opposite case. We shall omit here an exposition on the history and recent results for the moment problem (1). All that can be found in [1], [2]. Choose an arbitrary a ∈ N. Denote by S(D; C a×a ) a set of all analytic in D, C a×a -valued functions F ζ , such that F * ζ F ζ ≤ I a , ∀ζ ∈ D. In [2] we obtained a Nevanlinna-type parameterization for all solutions of the moment problem (1): Theorem 1 Let the moment problem (1), with d ∈ N, be given and condition (3), with T d from (2), be satisfied. Suppose that the moment problem is indeterminate. All solutions of the moment problem (1) have the following form: where A ζ , B ζ , C ζ , D ζ , are matrix polynomials defined by the given mo- The scalar polynomial h ζ is also defined by the moments. Here F ζ ∈ S(D; C δ×δ ). Conversely, each function F ζ ∈ S(D; C δ×δ ) generates by relation (4) a solution of the moment problem (1). The correspondence between all functions from S(D; C δ×δ ) and all solutions of the moment problem (1) is one-to-one.
In this paper we shall study the moment problem (1) with an additional constraint posed on the matrix measure M T (δ), δ ∈ B(T), generated by the function M (x) (see the precise definition of M T (δ) and other details below): where ∆ is a given open subset of T (called a gap). Here T is viewed as a metric space with the metric r(z, w) = |z − w|.
We present necessary and sufficient conditions for the solvability of the moment problem (1), (5). All solutions of the moment problem (1),(5) can be constructed by relation (4), where F ζ belongs to a certain subset of S(D; C δ×δ ).
Notations. As usual, we denote by R, C, N, Z, Z + , the sets of real numbers, complex numbers, positive integers, integers and non-negative integers, respectively; D = {z ∈ C : |z| < 1}; D e = {z ∈ C : |z| > 1}; T = {z ∈ C : |z| = 1}; T e = {z ∈ C : |z| = 1}. Let m, n ∈ N. The set of all complex matrices of size (m × n) we denote by C m×n . The set of all complex non-negative Hermitian matrices of size (n × n) we denote by C ≥ n×n . If M ∈ C m×n then M T denotes the transpose of M , and M * denotes the complex conjugate of M . The identity matrix from C n×n we denote by I n . By B(T) we denote a set of all Borel subsets of T.
If H is a Hilbert space then (·, ·) H and · H mean the scalar product and the norm in H, respectively. Indices may be omitted in obvious cases.
By C N we denote the finite-dimensional Hilbert space of complex column vectors of size N with the usual scalar product ( x, y) For a linear operator A in H, we denote by D(A) its domain, by R(A) its range, by Ker A its null subspace (kernel), and A * means the adjoint operator if it exists. If A is invertible then A −1 means its inverse. A means the closure of the operator, if the operator is closable. If A is bounded then A denotes its norm. For a set M ⊆ H we denote by M the closure of M in the norm of H. For an arbitrary set of elements {x n } n∈I in H, we denote by Lin{x n } n∈I the set of all linear combinations of elements x n , and span{x n } n∈I := Lin{x n } n∈I . Here I is an arbitrary set of indices. By E H we denote the identity operator in H, i.e. E H x = x, x ∈ H. In obvious cases we may omit the index H. If H 1 is a subspace of H, then P H 1 = P H For a closed isometric operator V in a Hilbert space H we denote:

The TMTMP with an open gap.
Let the moment problem (1), with d ∈ N, be given and condition (3), with T d from (2), be satisfied. Let where γ n,m , S k;s,l ∈ C. Observe that We repeat here some constructions from [1]. Consider a complex linear vector space H, which elements are row vectors u = (u 0 , u 1 , u 2 , ..., u (d+1)N −1 ), with u n ∈ C, 0 ≤ n ≤ (d + 1)N − 1. Addition and multiplication by a scalar are defined for vectors in a usual way. Set ε n = (δ n,0 , δ n,1 , δ n,2 , ..., δ n,(d+1)N −1 ), where δ n,r is Kronecker's delta. In H we define a linear functional B by the following relation: The space H with B form a quasi-Hilbert space ( [4]). By the usual procedure of introducing of the classes of equivalence (see, e.g. [4]), we put two elements u, w from H to the same class of equivalence denoted by Then and span{x n } . Consider the following operator: By [1, Theorem 1] all solutions of the moment problem (1) have the following form where E t is a left-continuous spectral function of the isometric operator A. Conversely, each left-continuous spectral function of A generates by (9) a solution of the moment problem (1). The correspondence between all leftcontinuous spectral functions of A and all solutions of the moment problem (1), established by relation (9), is one-to-one. By [1, Theorem 3] all solutions of the moment problem (1) have the following form where m k,j are obtained from the following relation: generates by relations (10)-(11) a solution of the moment problem (1). The correspondence between all Φ ζ ∈ S(D; N 0 (A), N ∞ (A)) and all solutions of the moment problem (1) is one-to-one.
Observe that the right-hand side of (11) may be written as is a generalized resolvent of the isometric operator A. The correspondence between all generalized resolvents of A and all solutions of the moment problem is one-to-one, as well.
Consider an arbitrary solution M (x) of the moment problem (1). By the construction in [1, pp. 791-793], the corresponding spectral function E t in (9) is generated by the left-continuous orthogonal resolution of unity E t of a unitary operator U 0 in a Hilbert space H 1 ⊇ H. Moreover, the following relation holds: where U is a unitary transformation which maps L 2 (M ) onto H 1 , and U 0 is the operator of multiplication by e it in L 2 (M ).
Denote by E(δ), δ ∈ B(T), the orthogonal spectral measure of the unitary operator U 0 . The spectral measure E(δ) and the resolution of the identity E t are related in the following way: Therefore the spectral function E t and the corresponding spectral measure E(δ), δ ∈ B(T), satisfy the following relation: Relation (9) may be rewritten in the following form: Define the following C ≥ N ×N -valued measure on B(T) (i.e. a C ≥ N ×N -valued function on B(T) which is countably additive): From this definition and relation (13) it follows that Observe that , satisfying the following relation: coincides with the matrix measure M T (δ). In fact, we may consider the following functions: Therefore they coincide on the minimal generated algebra Y , which consists of all finite unions of disjoint sets of the form δ t 2 ,t 1 = {z = e iτ : t 2 ≤ τ < t 1 }, t 1 , t 2 ∈ [0, 2π). Since the Lebesgue continuation is unique, these scalar measures coincide. On the other hand, the entries of M T (δ) and M T (δ) are expressed via f k,j by the polarization formula. Then M T (δ) = M T (δ). During the investigation of the moment problem (1),(5), it is enough to assume that the corresponding moment problem (1) (with the same moments) is indeterminate. In fact, if the corresponding moment problem (1) has no solutions than the moment problem (1), (5) has no solutions, as well. If the corresponding moment problem (1) has a unique solution than this solution can be found explicitly, and then condition (5) may be verified directly.
Choose arbitrary numbers l, m: Let V be a closed isometric operator in a Hilbert space H, and ζ ∈ T. Suppose that ζ −1 is a point of the regular type of V . Consider the following operators (see [5, p. 270]): with the domain D(W ζ ) = P H N 0 N ζ ; Moreover (see [5, p. 271]), since ζ −1 is a point of the regular type of V , S −1 exists and it is defined on the whole N 0 (V ), D(W ζ ) = N 0 (V ), and 2) The continued function F ζ maps isometrically N 0 (V ) on the whole N ∞ (V ), for all ζ ∈ ∆;

Theorem 2 Let V be a closed isometric operator in a Hilbert space H, and ∆ be an open subset of T, ∆ = T, such that
3) The operator F ζ − W ζ is invertible for all ζ ∈ ∆, and where W ζ is from (21).
As it follows from Remark 4.14 in [5, p. 274], conditions (22),(23) are necessary for the existence of at least one generalized resolvent of V , which admits an analytic continuation on T e ∪ ∆. Proof. By Corollary 4.7 in [5, p. 268] we may write: Applying P H M∞(V ) to the both sides of the latter equality we obtain relation (23). ✷ By Proposition 3 and Theorem 2 we get the following result. 2) The continued function F ζ maps isometrically N 0 (V ) into N ∞ (V ), for all ζ ∈ ∆; 3) The operator F ζ −W ζ is invertible for all ζ ∈ ∆, where W ζ is from (21).
Let us return to the investigation of the moment problem (1). At first, we shall obtain some necessary conditions for the solvability of the moment problem (1), (5). We shall use the orthonormal bases constructed in [2].

Proposition 4 Let the indeterminate moment problem (1) with d ∈ N be given and the operator A in a Hilbert space H be constructed as in (8). Let ∆ be an open subset of
be an orthonormal basis in M ζ (A), obtained by the Gram-Schmidt orthogonalization procedure from the following sequence: Here ζ ∈ ∆ = {z ∈ C : z ∈ ∆}. The case τ = 0 means that A ζ = ∅, and Suppose that there exists a solution M (x), x ∈ [0, 2π], of the moment problem (1), such that M T (∆) = 0, where M T (δ), δ ∈ B(T), is the corresponding matrix measure. By Proposition 1 we get E(∆) = 0, where E(δ), δ ∈ B(T), is the corresponding spectral measure. By Proposition 2 this means that the corresponding generalized resolvent R z (A) admits an analytic continuation on a set T e ∪ ∆. In this case, as it was noticed after Theorem 2 relation (22) holds for ζ ∈ ∆. ✷ Consider the indeterminate moment problem (1), such as in Proposition 4, and suppose that condition (b) of Proposition 4 is satisfied. Set L := Lin{g 0 (ζ), g 1 (ζ), ..., g τ −1 (ζ), x 0 , x 1 , ..., x N −1 }, ζ ∈ ∆. Notice that L = H. In fact, this follows from the following inclusion, which may be checked by the induction argument: {x n } kN +N −1 n=0 ⊆ L, k = 0, 1, ..., d.
Observe that the first τ elements are already orthonormal. During the orthogonalization of the rest N elements we shall obtain an orthonormal set For the operator A in the Hilbert space H and an arbitrary ζ ∈ ∆, we may construct the operators S ζ , Q ζ from (20) with V = A. Let M S ζ (M Q ζ ) be the matrix of the operator S ζ (Q ζ ) with respect to the bases A ζ + , A 3 (respectively, to the bases A ζ + , A ′ 3 ): Denote by W ζ the matrix of the operator W ζ from (21) with respect to the bases A 3 , A ′ 3 . Then Definition 1 Choose an arbitrary a ∈ N, ∆ ⊆ T, and let Y ζ be an arbitrary C a×a -valued function, ζ ∈ ∆. By S(D; C a×a ; ∆; Y ) we denote a set of all functions G ζ from S(D; C a×a ) which satisfy the following conditions: A) G ζ admits a continuation on D ∪ ∆, and the continued function G ζ is continuous (i.e. each entry of G ζ is continuous); We denote by S(D; N 0 (A), N ∞ (A); ∆; W ) a set of all functions Φ ζ from S(D; N 0 (A), N ∞ (A)), which satisfy conditions 1)-3) of Theorem 3 with V = A and ∆ instead of ∆.
Consider a transformation T which for an arbitrary function Φ ζ ∈ S(D; N 0 (A), N ∞ (A)) put into correspondence the following C δ×δ -valued function F ζ : Proof. By Proposition 4 we obtain that condition (22) holds for V = A and with ∆ instead of ∆.
Suppose that the moment problem (1) has a solution M (x), x ∈ [0, 2π], such that M T (∆) = 0. As in the proof of Proposition 4 we conclude that the corresponding generalized resolvent R z (A) admits an analytic continuation on T e ∪ ∆. Let Φ ζ be the function from S(D; N 0 (A), N ∞ (A)) which corresponds to R z (A) by Chumakin's formula. By Theorem 3 we obtain that Φ ζ belongs to S(D; N 0 (A), N ∞ (A); ∆; W ). By (29) we get S(D; C δ×δ ; ∆; W ) = ∅.