On the truncated operator trigonometric moment problem

In this paper we study the truncated operator trigonometric moment problem. All solutions of the moment problem are described by a Nevanlinna-type parameterization. In the case of moments acting in a separable Hilbert space, the matrices of the operator coefficients in the Nevanlinna-type formula are calculated by the prescribed moments. Conditions for the determinacy of the moment problem are given, as well.

The truncated operator trigonometric moment problem consists of finding a non-decreasing [H]-valued function F (t), t ∈ [0, 2π], F (0) = 0, which is strongly left-continuous in (0, 2π] and such that 2π 0 e int dF (t) = S n , n = 0, 1, ..., d, where {S n } d n=0 is a prescribed sequence of bounded operators on H (moments). Here H is a fixed Hilbert space and d ∈ Z + is a fixed number. The operator Stieltjes integrals in (1) are understood as limits of the corresponding integral sums in the strong operator topology. Set S −k = S * k , k = 1, 2, ..., d. The following condition: where {h k } d 0 are arbitrary elements of H, is necessary and sufficient for the solvability of the moment problem (1) (e.g. [1]). The solvable moment problem (1) is said to be determinate if it has a unique solution and indeterminate in the opposite case. The truncated operator trigonometric moment problem was studied in papers [1], [4] (in slightly different statements). The conditions of the solvability were obtained in [1]. In the case of the strict positivity of the corresponding Toeplitz operator, all solutions to the moment problem were described in [4]. For the history of scalar and matrix truncated trigonometric moment problems we refer to [8], [9].
Our aim here is to describe all solutions to the solvable moment problem (1) without any additional conditions. For this purpose we shall develop the operator approach of Szökefalvi-Nagy and Koranyi in [6], [7]. Also our approach is close to the approach of Krein and Krasnoselskii in [5]. All solutions of the moment problem are described by a Nevanlinna-type parameterization. In the case of moments acting in a separable Hilbert space, the matrices of the operator coefficients in the Nevanlinna-type formula are calculated by the prescribed moments. Conditions for the determinacy of the moment problem are given. 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}; T = {z ∈ C : |z| = 1}; T e = {z ∈ C : |z| = 1}. By k ∈ 0, ρ we mean that k ∈ Z + : 0 ≤ k ≤ ρ if ρ ∈ Z + , and k ∈ Z + , if ρ = +∞. In this paper Hilbert spaces are not necessarily separable, operators in them are supposed to be linear. 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. For a linear operator A in H, we denote by D(A) its domain, by R(A) its range, 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.
. By S(D; N, N ′ ) we denote a class of all analytic in a domain D ⊆ C operatorvalued functions F (z), which values are linear non-expanding operators mapping the whole N into N ′ , where N and N ′ are some Hilbert spaces.
2 The solvability and a description of solutions for the moment problem.
Suppose that the moment problem (1) is given and it is solvable. For arbitrary elements {h k } d 0 of H we may write: Here δ is the diameter of a partition of [0, 2π] and {t r } N 0 are points of this partition. Thus, condition (2) is satisfied.
Conversely, suppose that the moment problem (1) with d ∈ N is given and condition (2) is satisfied. Like it was done in [7] we consider abstract symbols ε j , j = 0, 1, ..., d, and form a formal sum h: where h j ∈ H. If α ∈ C, then we set αh = d j=0 (αh j )ε j . If where g j ∈ H, then we set h + g = d j=0 (h j + g j )ε j . A set of all formal sums of type (3) becomes a complex linear vector space B. Let h, g ∈ B have the form as in (3), (4). Let The functional Φ is sesquilinear and it has the properties Φ(h, g) = Φ(g, h), Φ(h, h) ≥ 0. If Φ(h − g, h − g) = 0, we put elements h and g to the same equivalence class denoted by [h] or [g]. A set of all equivalent classes we denote by L. By the completion of L we obtain a Hilbert space H. Set Observe that Set Consider a linear operator A 0 with D(A 0 ) = D 0 : Let us check that A 0 is well-defined. Suppose that an element h ∈ D 0 has two representations: Therefore A 0 is isometric. Set A = A 0 . By the induction argument it may be checked that be its strongly left-continuous orthogonal resolution of the identity. We may write Consider the following operator I: H → H: It is readily checked that I is linear. Moreover, since then I is bounded. By (9) we may write where E t is a strongly left-continuous spectral function of A (corresponding to A). Thus, is a solution of the moment problem (1). We conclude that each strongly left-continuous spectral function of A generates a solution of the moment problem (1) by relation (12). Let F (t) be an arbitrary solution of the moment problem (1). We shall check that F (t) can be constructed by relation (12). By C 00 (H; [0, 2π]) we denote a set of all strongly continuous H-valued functions f (t), t ∈ [0, 2π], which take their values in finite-dimentional subspaces of H (depending on f ). For arbitrary f, g ∈ C 00 (H; [0, 2π]) we set (see [2]) Here, as usual, the limit does not depend on the choice of partitions and points t k . It is easy to see that in the case of f, g ∈ C 00 (H; [0, 2π]) the limit in (13) exists and reduces to a finite sum of scalar Stieltjes-type integrals.
Introducing classes of the equivalence with respect to Ψ and by the completion we obtain a Hilbert space L 2 = L 2 (H; [0, 2π]; d F (t)). Consider two operator polynomials of the following form: Since p, q ∈ C 00 (H; [0, 2π]) then the corresponding classes [p], [q] belong to L 2 (H; [0, 2π]; d F (t)). As usual in such situations, we shall say that p, q belong to L 2 (H; [0, 2π]; d F (t)). Then x h j ,j , h j ∈ H, which maps P (H; [0, 2π]; d F (t)) on the whole L(⊆ H). Let us check that W 0 is well-defined. In fact, suppose that p, q from (14) belong to the same class of the equivalence. Then Therefore W 0 is well-defined. Moreover, W 0 is linear and relation (15) By the well-known inversion formula we conclude that F (t) = I * E t I.
Theorem 1 Let the truncated operator trigonometric moment problem (1) with d ∈ N be given and condition (2) hold. Let an operator A 0 in a Hilbert space H be constructed as in (8), A = A 0 . All solutions of the moment problem have the following form:

where I is defined by (10) and E t is a strongly left-continuous spectral function of A. On the other hand, each strongly left-continuous spectral function of A generates by (17) a solution of the moment problem. Moreover, the correspondence between all strongly left-continuous spectral functions of A and all solutions of the moment problem (1) is one-to-one.
Proof. It remains to check that different left-continuous spectral functions of A generate different solutions of the moment problem (1). Set L 0 := {x h,0 } h∈H . Choose an arbitrary element x ∈ L, x = d j=0 x h j ,j , h j ∈ H. For arbitrary ζ ∈ T e \{0} there exists the following representation: Here v and y may depend on the choice of ζ. In fact, for an arbitrary element u ∈ D(A 0 ), u = d−1 j=0 x g j ,j , g j ∈ H, we may write Consider the following system of equations: We can find g d−1 , then g d−2 , ..., g 0 . Consider u with this choice of g j and set v : By (19),(20) we see that Then relation (18) holds. Suppose to the contrary that two different strongly left-continuous spectral functions E j,t , j = 1, 2, generate the same solution of the moment problem: I * E 1,t I = I * E 2,t I. For arbitrary f, g ∈ H we have: Multiplying by 1 1−ζe it and integrating we get where R j,ζ is a generalized resolvent corresponding to E j,t , j = 1, 2. Let R j,ζ is generated by a unitary extension A j of A in a Hilbert space H j ⊇ H, j = 1, 2. Since for arbitrary f ∈ D(A), ζ ∈ T e and j = 1, 2 we have Choose an arbitrary ζ ∈ T e \{0}. We may write: Choose an arbitrary element w ∈ L and ζ ∈ T e \{0}. By (18) we may write: For an arbitrary w ∈ L using (18) we may write: (22),(24) we get R 1,ζ w = R 2,ζ w, ζ ∈ T e \{0}. Therefore E 1,t = E 2,t . This contradiction completes the proof. ✷ Notice that relation (17) is equivalent to the following relation: From the latter relation it follows that (26) By virtue of Chumakin's formula for the generalized resolvents of an isometric operator (see [3]) we conclude that the following formula: establishes a one-to-one correspondence between all functions Φ ∈ S(D; H ⊖ D(A), H ⊖ R(A)) and all solutions of the moment problem (1). We shall need the following proposition which is close to Frobenius's inversion formula for matrices.
where H 1 , H 2 are subspaces of H, and the operator M has the following block representation: Suppose that A has a bounded inverse which is defined on the whole H 1 . Then the following assertions hold.
(i) If the operator H = D−CA −1 B has a bounded inverse which is defined on the whole H 2 , then M has a bounded inverse which is defined on the whole H, and M −1 has the following block representation with respect to decomposition (28): Notice that Therefore R(H) = H 2 . Since an operator H −1 is closed and defined on the whole H 2 , then H −1 is bounded. Applying assertion (i) we conclude that M −1 has a block representation (30) with respect to decomposition (28). ✷ Return to our constructions for the solvable moment problem (1) By (27) we conclude that the following relation establishes a one-to-one correspondence between all functions Φ ∈ S(D; H ⊖ D(A), H ⊖ R(A)) and all solutions of the moment problem (1). By the substitution of expressions from (32) we obtain that relation (33) takes the following form: Finally, we obtain that the following relation Theorem 2 Let the truncated operator trigonometric moment problem (1) with d ∈ N be given and condition (2) hold. Let an operator A 0 in a Hilbert space H be constructed as in (8) Proof. The proof follows from the preceding considerations. ✷ Corollary 1 Let the truncated operator trigonometric moment problem (1) with d ∈ N be given and condition (2) hold. Let an operator A 0 in a Hilbert space H be constructed as in (8) On the other hand, if both defect numbers of A are non-zero than we may choose unit nonzero vectors h ∈ H ⊖ D(A), g ∈ H ⊖ R(A). Set Φ 1 (ζ) = 0, Φ 2 (ζ) = (·, h)g, ζ ∈ D. Functions Φ 1 , Φ 2 generate different solutions of the moment problem. ✷ Let the truncated operator trigonometric moment problem (1) with d ∈ N be given. Suppose that condition (2) holds and the Hilbert space H is separable, H = {0}. Let C = {f k } ω−1 k=0 , 1 ≤ ω ≤ +∞, is an orthonormal basis in H. Let us calculate the matrices of operators A, B, C, D in (37) with respect to some proper bases. As a consequence, we can obtain the matrix of the operator appearing on the right of (36) with respect to C using the prescribed moments.
Observe that H is separable. In fact, an arbitrary element x of L has the following form: x = d j=0 x h j ,j , h j ∈ H. Let W be a dense subset of H (which is not supposed to be countable). Choose an arbitrary ε > 0. There exist elements y j ∈ W, 0 ≤ j ≤ d, such that The above orthonormal bases can be used to construct the matrices of operators on the right in (36). Here relation (7) will be used intensively.
In the case of the determinate moment problem the right-hand side of (36) is equal to A(ζ). Thus, in this case we can use C and an orthonormal basis A, constructed by the orthogonalization of {x f k ,j } k∈0,ω−1, 0≤j≤d .
On the truncated operator trigonometric moment problem.

S.M. Zagorodnyuk
In this paper we study the truncated operator trigonometric moment problem. All solutions of the moment problem are described by a Nevanlinnatype parameterization. In the case of moments acting in a separable Hilbert space, the matrices of the operator coefficients in the Nevanlinna-type formula are calculated by the prescribed moments. Conditions for the determinacy of the moment problem are given, as well.