# On bounded complex Jacobi matrices and related moment problems

## TL;DR

This paper explores the extension of classical moment problems to the complex plane through bounded complex Jacobi matrices, providing solvability conditions and linking operator theory with measure representation.

## Contribution

It introduces conditions for solving complex moment problems associated with bounded Jacobi matrices and connects operator theory with measure representations in the complex plane.

## Key findings

- Provided sufficient conditions for the solvability of complex moment problems.
- Established criteria for the existence of integral representations with positive measures.
- Discussed the relationship between operators associated with Jacobi matrices and multiplication operators in $L^2$ spaces.

## Abstract

In this paper we study the linear functional $S$ on complex polynomials which is associated to a bounded complex Jacobi matrix $J$. The associated moment problem is considered: find a positive Borel measure $\mu$ on $\mathbb{C}$ subject to conditions $\int z^n d\mu = s_n$, where $s_n$ are prescribed complex numbers (moments). This moment problem may be viewed as an extension of the Stieltjes and Hamburger moment problems to the complex plane. Sufficient conditions for the solvability of the moment problem are provided. As a corollary, we obtain conditions for the existence of an integral representation $S(p) = \int_\mathbb{C} p(z) d\mu$, with a positive Borel measure $\mu$. An interrelation of the associated to the complex Jacobi matrix operator $A_0$, acting in $l^2$ on finite vectors, and the multiplication by z operator in $L^2_\mu$ is discussed as well.