The two-dimensional moment problem consists of finding a positive Borel measure \(\mu \) in \(\mathbb {R}^2\) such that \(\int _{\mathbb {R}^2} t_1^m t_2^n d\mu = s_{m,n}\) , \(m,n=0,1,2,...\) , where \(s_{m,n}\) are prescribed real constants (moments). We study this moment problem in the case when the sequence \(\{ s_{m,n} \}_{m,n=0}^\infty \) is positive semi-definite, and the following Carleman-type conditions hold: \(\begin{aligned} \sum _{k=1}^\infty \frac{1}{ \root 2k \of { s_{2m,2k} + s_{2m+2,2k} } } = \infty ,\quad m=0,1,2,.... \end{aligned}\) In this case all solutions of the moment problem are parameterized by a class of analytic contractive operator-valued functions. The special case of the determinate moment problem is characterized. We introduce a notion of a generalized resolvent for a pair of commuting symmetric operators. We use basic properties of such generalized resolvents as a main tool in studying the above moment problem.
