Complex analysis of symmetric operators. II: entire operators with deficiency index 1

TL;DR

This paper systematically analyzes entire operators with deficiency index 1 using complex geometry, exploring their characteristic line bundle, curvature, and connections to moment problems and functional models, revealing new insights into their structure and extensions.

Contribution

It introduces a geometric framework for entire operators with deficiency index 1, linking curvature, zero distribution, and functional models, and establishes new results on their extensions and completeness.

Findings

Curvature of the characteristic line bundle relates to zero distribution.

Growth properties of Jacobi operators match Nevanlinna matrix entries.

Mean type determines the obstruction to extension completeness.

Abstract

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.