On bounded complex Jacobi matrices and related moment problems in the complex plane

In this paper we consider the following moment problem: find a positive Borel measure μ on ℂ subject to conditions ∫ zn dμ = sn, n∈ℤ+, where sn are prescribed complex numbers (moments). This moment problem may be viewed (informally) as an extension of the Stieltjes and Hamburger moment problems to the complex plane. A criterion for the moment problem for the existence of a compactly supported solution is given. In particular, such moment problems appear naturally in the domain of complex Jacobi matrices. For every bounded complex Jacobi matrix its associated functional S has the following integral representation: S(p) = ∫ℂ p(z) dμ, with a positive Borel measure μ in the complex plane. An interrelation of the associated to the complex Jacobi matrix operator A0, acting in l2 on finitely supported vectors, and the multiplication by z operator in L2μ is discussed.