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

Yicao Wang

TL;DR

This work develops a unified geometric framework for entire symmetric operators with deficiency index one, centering on the characteristic line bundle $F$ and its curvature $\omega(\lambda)$. By constructing a canonical functional model and linking Weyl data to reproducing-kernel spaces (including de Branges and de Branges-Rovnyak realizations), the paper connects spectral properties to geometric invariants, and demonstrates how growth, completeness, and moment-problem phenomena can be read off from $F$ and its height function $h_T(r)$. A key result is that the curvature $\omega(u)$ on the real axis is a complete unitary invariant and that the mean type of non-self-adjoint extensions is the sole obstruction to completeness, with almost all extensions being complete. This framework unifies several existing models and provides a canonical means to measure growth, via $T_F(r)$, which aligns with Nevanlinna characteristics of the Hamburger moment problem. The treatment yields new insights into indeterminate Hamburger moment problems, including a direct identification of growth data with the Nevanlinna matrix entries and a Weyl-class determination of the Jacobi operator within its unitary equivalence class.

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.

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

TL;DR

This work develops a unified geometric framework for entire symmetric operators with deficiency index one, centering on the characteristic line bundle and its curvature . By constructing a canonical functional model and linking Weyl data to reproducing-kernel spaces (including de Branges and de Branges-Rovnyak realizations), the paper connects spectral properties to geometric invariants, and demonstrates how growth, completeness, and moment-problem phenomena can be read off from and its height function . A key result is that the curvature on the real axis is a complete unitary invariant and that the mean type of non-self-adjoint extensions is the sole obstruction to completeness, with almost all extensions being complete. This framework unifies several existing models and provides a canonical means to measure growth, via , which aligns with Nevanlinna characteristics of the Hamburger moment problem. The treatment yields new insights into indeterminate Hamburger moment problems, including a direct identification of growth data with the Nevanlinna matrix entries and a Weyl-class determination of the Jacobi operator within its unitary equivalence class.

Abstract

This paper is a continuation of our previous work \cite{wang2024complex}. It mainly deals with entire operators with deficiency index 1 \emph{systematically} from the complex-geometric viewpoint proposed in \cite{wang2024complex}. We pay special attention to the characteristic line bundle of . We investigate its curvature in detail and demonstrate how it is connected to the height function of and to the distribution of zeros of elements in the canonical model Hilbert space which consists of certain holomorphic sections of . 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 can be derived from our canonical model by restricting to certain subsets of 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.
Paper Structure (22 sections, 61 theorems, 272 equations)

This paper contains 22 sections, 61 theorems, 272 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric operator with deficiency indices (1,1) and its Weyl curve
  4. A canonical functional model
  5. Boundary value problems and value distribution theory
  6. Geometry of the characteristic line bundle
  7. The structure of characteristic functions
  8. Contractive Weyl function
  9. Weyl function
  10. de Branges function
  11. Some other existing models compared with the canonical model
  12. de Branges model
  13. de Branges-Rovnyak model
  14. Herglotz model
  15. Kempf-Martin model
  16. ...and 7 more sections

Key Result

Theorem 2.2

If $T$ is a simple symmetric operator in the Hilbert space $H$, then $T$ is densely defined if and only if $\varpi(w)$ doesn't have an angular tangent line at $w=1$ in the sense of Caratheodory.

Theorems & Definitions (138)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 128 more