标题： 二维矩问题在Carleman型条件下的所有解的解析参数化 显示英文标题

标题： An analytical parameterization for all solutions of the two-dimensional moment problem under Carleman-type conditions

摘要： 二维矩问题包括在$\mathbb{R}^2$中找到一个正Borel测度$\mu$，使得$\int_{\mathbb{R}^2} t_1^m t_2^n d\mu = s_{m,n}$,$m,n=0,1,2,...$，其中$s_{m,n}$是给定的实常数（矩）。 我们研究当序列$\{ s_{m,n} \}_{m,n=0}^\infty$是半正定时的情况，并且满足以下Carleman型条件：$$ \sum_{k=1}^\infty \frac{1}{ \sqrt[2k]{ s_{2m,2k} + s_{2m+2,2k} } } = \infty,\quad m=0,1,2,.... $$。在这种情况下，矩问题的所有解由一类解析压缩算子值函数类参数化。 确定矩问题的特殊情况被表征。 我们引入了一对交换对称算子的广义预解的概念。 我们使用此类广义预解的基本性质作为研究上述矩问题的主要工具。 显示英文摘要

摘要： 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: $$ \sum_{k=1}^\infty \frac{1}{ \sqrt[2k]{ s_{2m,2k} + s_{2m+2,2k} } } = \infty,\quad m=0,1,2,.... $$ 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.