Hamburger and Stieltjes Moment Problems for Operators

Solutions to some operator-valued, unidimensional, Hamburger and Stieltjes moment problems in this paper are given. Necessary and sufficient conditions on some sequences of bounded operators being Hamburger, respectively, Stieltjes operator-valued moment sequences are obtained. The determinateness of the operator-valued Hamburger and Stieltjes moment sequence is studied.

The spectral function () is called an orthogonal spectral function if every () is an orthogonal projection [1, page 322].
In both cases (1) and (2), the operator-valued measures () or () are called the representing measures for the sequence {  } +∞ =0 .Necessary and sufficient conditions for representing scalar sequences or operator-valued sequences, in one or several variables, as Hamburger or Stieltjes moment sequences with respect to scalar, respectively, operatorvalued, positive measures, represent the subject of many outstanding papers such as [1][2][3][4]. .., to quote only few of them.
In the present paper, in Section 3, we give a necessary and sufficient condition on a sequence of bounded, selfadjoint operators to be a Hamburger operator-valued, unidimensional moment sequence.In Section 4, we discuss the uniqueness of the representing measures of the operatorvalued Hamburger moment sequence both in (1) and (2) forms.In Section 5, we give some necessary and sufficient conditions on a sequence of positive operators to be a Stieltjes operator-valued, unidimensional moment sequence with respect to a positive, operator-valued measure.The positive representing measures in Sections 3 and 5 are obtained by applying Kolmogorov's theorem of decomposition of the positive definite kernels.

Preliminaries
Let  ∈ R denote the real variable in the real Euclidean space; for H an arbitrary complex Hilbert space, (H) represents the algebra of bounded operators an H; we denote with  ⋅ : N → {0, 1}, the function for  a Hilbert space, (H, ) represents the set of bounded operators from H in .We consider the C-vector space of vectorial functions:  = { : {0, 1, . . ., , . . ., } → H, (⋅) = ∑ ∈N  ⋅ (),  with finite support, () ∈ H}.We define also the convolution  *  1⋅ ∈  as and make the convention: *  1⋅ (0) = 0 H .We have  *  1⋅ = ∑ ∈N  (+1)⋅ (),  with finite support.In Section 3, a necessary and sufficient condition on a sequence of self-adjoint operators to be a Hamburger operator-valued moment sequence is given.In Section 5, we give necessary and sufficient conditions on a sequence of positive operators to be a Stieltjes operator-valued moment sequence.In Section 4, the problem of the uniqueness of the represented measures in Sections 3 and 5 is studied.The representing measures in Sections 3 and 5 are obtained by applying Kolmogorov's theorem on decomposition of the positive kernels.Classical Kolmogorov's theorem for the decomposition of positive kernels is as follows: "Let Γ :  ×  → () be a nonnegativedefinite function where  is an arbitrary set and  a Hilbert space, namely, ∑  ,=1 ⟨Γ(  ,   )  ,   ⟩  ≥ 0, for any finite number of points  1 , . . .,   ∈  and any vectors  1 , . . .,   ∈ .In this case, there exists a Hilbert space  (essentially unique) and a function ℎ :  → (, ) such that Γ(, ) = ℎ() * ℎ() for any ,  ∈ ." We apply this theorem for a particular set  and a particular positive-definite operator-valued function to give an integral representation as Hamburger operator-valued moment sequence and Stieltjes operator-valued moment sequence, respectively, to some sequences of self-adjoint and positive operators, respectively.

An Operator-Valued Hamburger Moment Sequence Main Result
Let Γ = {Γ  } ∈N be a sequence of bounded self-adjoint operators, acting on an arbitrary complex, separable Hilbert space; that is, Γ  ∈ (H), Γ  = Γ *  , for all  ∈ N, Γ 0 = Id H , subject on the following conditions: for any finite vectors' sequence {  } ∈⊂N ⊂ H, there exists another vector sequence {   } ∈⊂N ⊂ H such that the following two equations are satisfied; (A) and for any finite vectors' sequence {  } ∈⊂N ⊂ H, there exists another vectors' sequence Proposition 1.Let Γ = {Γ  } ∈N be a sequence of bounded, selfadjoint operators, acting on an arbitrary complex, separable Hilbert space H, subject on the conditions: Γ 0 = Id H , (A) and (B) satisfied.The following statements are equivalent.
(i) We have: for all sequences {  }  ∈ H with finite support.
Moreover, if the representing measure is that associated with a self-adjoint extension of a symmetric operator with deficiency indices (0,0), the self-adjoint extension is the canonical closure of the given operator and is defined on the whole space.Indeed, if  : () → H is symmetric with ( ± i) = H and  ⊃ , the canonical closure of , it follows that H ⊇ ( ± i) ⊃ ( ± i) are closed subspaces in H; that is ( ± i) = H.In this case the canonical closure of  is the smallest self-adjoint extension of  and is defined on the whole space H (as in Section 3 of this paper, Proposition 1).The same arguments are in [4,