对称算子的复分析. II: 亏指数为1的整算子
Complex analysis of symmetric operators. II: entire operators with deficiency index 1
AI总结：
本文从复几何观点系统研究亏指数为1的整算子，通过特征线丛曲率与高度函数及零点分布的联系，应用于Hamburger矩问题，并证明平均类型是非自伴扩张完备性的唯一障碍。
摘要：
本文是我们先前工作\cite{wang2024complex}的延续。它主要从\cite{wang2024complex}中提出的复几何观点出发，系统地处理亏指数为1的整算子$T$。我们特别关注$T$的特征线丛$F$。我们详细研究其曲率，并展示它如何与$T$的高度函数以及典范模型希尔伯特空间中元素零点的分布相联系，该空间由$F$的某些全纯截面组成。此研究应用于一个不确定的Hamburger矩问题，以证明相应的Jacobi算子的增长性质与用Nevanlinna矩阵元素定义的增长性质一致。我们还展示了如何通过将$F$限制到$\mathbb{C}$的某些子集并选择合适的平凡化，从我们的典范模型推导出$T$的各种函数模型。这使得这些模型之间的相互关系更加透明。通过引入一般非自伴扩张的平均类型并使用de Branges-Rovnyak模型，我们证明了平均类型是此类扩张完备性的唯一障碍。我们还证明了不完备扩张的测度为零。文中还包含一些其他新结果和旧结果的新证明。
英文摘要：
This paper is a continuation of our previous work \cite{wang2024complex}. It mainly deals with entire operators $T$ with deficiency index 1 \emph{systematically} from the complex-geometric viewpoint proposed in \cite{wang2024complex}. We pay special attention to the characteristic line bundle $F$ of $T$. We investigate its curvature in detail and demonstrate how it is connected to the height function of $T$ and to the distribution of zeros of elements in the canonical model Hilbert space which consists of certain holomorphic sections of $F$. This study is applied to an indeterminate Hamburger moment problem to show the growth property of the associated Jacobi operator coincides with that defined in terms of entries of the Nevanlinna matrix. We also show how various functional models for $T$ can be derived from our canonical model by restricting $F$ to certain subsets of $\mathbb{C}$ and choosing suitable trivializations. This makes the interrelationships among these models much more transparent. By introducing the mean type of a generic non-self-adjoint extension and using the de Branges-Rovnyak model, we show the mean type is the only obstruction to completeness of such an extension. We also prove that the measure of incomplete extensions is zero. Some other new results and new proofs of old results are also included.
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