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A new functional model for contractions

Wang Yicao

TL;DR

This paper introduces an intrinsic, geometry-based framework for functional models of completely non-unitary contractions by aligning contraction theory with the Weyl-curve formalism from symmetric-operator extension theory. It constructs a canonical model space as a reproducing-kernel Hilbert space of holomorphic sections over a holomorphic vector bundle, avoiding the traditional dilation to a larger Hilbert space and the explicit use of the Sz.-Nagy–Foias characteristic function. Central contributions include the identification of Weyl-curve data and contractive Weyl functions as the governing boundary information, a canonical boundary quadruple that yields explicit model operators, and a constructive link between marked Nevanlinna discs and contractions. The work provides a unifying perspective that recasts contractions in terms of strong symplectic geometry and Nevanlinna data, with the Weyl curve playing the geometric role traditionally played by $\Theta_T(\lambda)$, and offers new avenues for analyzing unitary parts and invariant subspaces through geometric data.

Abstract

The paper presents a new functional model for completely non-unitary contractions on a Hilbert space. This model is based on the observation that the theory of contractions shares a common geometric basis with the extension theory of symmetric operators recently developed by the author in \cite{wang2024complex}. Compared with the now classical Sz.-Nagy-Foias model and the de Branges-Rovnyak model, ours is intrinsic in the sense that we need not construct a bigger space $H$ including the model space $\mathfrak{H}$ and realize the model operator on $\mathfrak{H}$ as the compression of a minimal unitary dilation on $H$. Our model space $\mathfrak{H}$ is constructed in a canonical and conceptually more direct manner and doesn't depend on the Sz. Nagy-Foias characteristic function explicitly. We also show how a contraction can be constructed from a marked Nevanlinna disc, which is the geometric analogue of the characteristic function.

A new functional model for contractions

TL;DR

This paper introduces an intrinsic, geometry-based framework for functional models of completely non-unitary contractions by aligning contraction theory with the Weyl-curve formalism from symmetric-operator extension theory. It constructs a canonical model space as a reproducing-kernel Hilbert space of holomorphic sections over a holomorphic vector bundle, avoiding the traditional dilation to a larger Hilbert space and the explicit use of the Sz.-Nagy–Foias characteristic function. Central contributions include the identification of Weyl-curve data and contractive Weyl functions as the governing boundary information, a canonical boundary quadruple that yields explicit model operators, and a constructive link between marked Nevanlinna discs and contractions. The work provides a unifying perspective that recasts contractions in terms of strong symplectic geometry and Nevanlinna data, with the Weyl curve playing the geometric role traditionally played by , and offers new avenues for analyzing unitary parts and invariant subspaces through geometric data.

Abstract

The paper presents a new functional model for completely non-unitary contractions on a Hilbert space. This model is based on the observation that the theory of contractions shares a common geometric basis with the extension theory of symmetric operators recently developed by the author in \cite{wang2024complex}. Compared with the now classical Sz.-Nagy-Foias model and the de Branges-Rovnyak model, ours is intrinsic in the sense that we need not construct a bigger space including the model space and realize the model operator on as the compression of a minimal unitary dilation on . Our model space is constructed in a canonical and conceptually more direct manner and doesn't depend on the Sz. Nagy-Foias characteristic function explicitly. We also show how a contraction can be constructed from a marked Nevanlinna disc, which is the geometric analogue of the characteristic function.

Paper Structure

This paper contains 9 sections, 21 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Strong symplectic Hilbert spaces and related Hermitian symmetric spaces
  3. Symmetric operators revisited
  4. Symmetric operator and its Weyl curve
  5. A canonical functional model for symmetric operators
  6. A functional model for contractions
  7. Some preliminaries
  8. Characteristic vector bundles and the functional model
  9. From marked Nevanlinna discs to contractions

Key Result

Proposition 2.7

For an isotropic subspace $L$ of $H$, $L^{\bot_s}/L$ has a canonical structure of strong symplectic Hilbert space.

Theorems & Definitions (57)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • Example 2.9
  • ...and 47 more