Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Various form closures associated with a fixed non-semibounded self-adjoint operator

Andreas Fleige

TL;DR

The paper extends the classical semibounded operator-form correspondence to the non-semibounded setting by employing Kreĭn-space methods. It proves a one-to-one relationship between closed symmetric forms with gap point $0$ and J-non-negative, J-self-adjoint, boundedly invertible Kreĭn-space operators, and analyzes the non-uniqueness of form closures via a detailed triplet construction and regularization concepts. Special attention is given to fixed self-adjoint operators $T$, where the closures of $(T\cdot,\cdot)$ form a rich family corresponding to Kreĭn-space completions, with a unique regular closure when infinity is regular; most other closures are non-regular and admit a regularization that preserves much of the spectral data. The paper then provides explicit model-space examples with a multiplication operator, showing an infinite family of closures parameterized by $\alpha\in[0,2]$, where each closure carries additional information not captured by $T$ alone. Overall, closed symmetric forms in the non-semibounded regime can encode more structural information than the associated self-adjoint operators, with potential implications for spectral theory and physics.

Abstract

If $T$ is a semibounded self-adjoint operator in a Hilbert space $(H, \, (\cdot , \cdot))$ then the closure of the sesquilinear form $(T \cdot , \cdot)$ is a unique Hilbert space completion. In the non-semibounded case a closure is a Kreĭn space completion and generally, it is not unique. Here, all such closures are studied. A one-to-one correspondence between all closed symmetric forms (with ``gap point'' $0$) and all J-non-negative, J-self-adjoint and boundedly invertible Kreĭn space operators is observed. Their eigenspectral functions are investigated, in particular near the critical point infinity. An example for infinitely many closures of a fixed form $(T \cdot , \cdot)$ is discussed in detail using a non-semibounded self-adjoint multiplication operator $T$ in a model Hilbert space. These observations indicate that closed symmetric forms may carry more information than self-adjoint Hilbert space operators.

Various form closures associated with a fixed non-semibounded self-adjoint operator

TL;DR

The paper extends the classical semibounded operator-form correspondence to the non-semibounded setting by employing Kreĭn-space methods. It proves a one-to-one relationship between closed symmetric forms with gap point and J-non-negative, J-self-adjoint, boundedly invertible Kreĭn-space operators, and analyzes the non-uniqueness of form closures via a detailed triplet construction and regularization concepts. Special attention is given to fixed self-adjoint operators , where the closures of form a rich family corresponding to Kreĭn-space completions, with a unique regular closure when infinity is regular; most other closures are non-regular and admit a regularization that preserves much of the spectral data. The paper then provides explicit model-space examples with a multiplication operator, showing an infinite family of closures parameterized by , where each closure carries additional information not captured by alone. Overall, closed symmetric forms in the non-semibounded regime can encode more structural information than the associated self-adjoint operators, with potential implications for spectral theory and physics.

Abstract

If is a semibounded self-adjoint operator in a Hilbert space then the closure of the sesquilinear form is a unique Hilbert space completion. In the non-semibounded case a closure is a Kreĭn space completion and generally, it is not unique. Here, all such closures are studied. A one-to-one correspondence between all closed symmetric forms (with ``gap point'' ) and all J-non-negative, J-self-adjoint and boundedly invertible Kreĭn space operators is observed. Their eigenspectral functions are investigated, in particular near the critical point infinity. An example for infinitely many closures of a fixed form is discussed in detail using a non-semibounded self-adjoint multiplication operator in a model Hilbert space. These observations indicate that closed symmetric forms may carry more information than self-adjoint Hilbert space operators.

Paper Structure

This paper contains 11 sections, 29 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. J-non-negative and boundedly invertible operators in Kreı̆n spaces
  4. Closed symmetric sesquilinear forms
  5. A space triplet associated with a J-non-negative, J-self-adjoint and boundedly invertible Kreı̆n space operator
  6. Relations between closed symmetric sesquilinear forms and J-self-adjoint, J-non-negative operators in Kreı̆n spaces
  7. Families of form closures and of Kreı̆n space completions associated with a fixed form $(T\cdot,\cdot)$
  8. Examples with multiplication operators in "model spaces"
  9. A regular closed form
  10. Non-regular closed forms
  11. Conclusions for forms associated with the multiplication operator

Key Result

Theorem 2.1

Let $A$ be a J-self-adjoint, J-non-negative and boundedly invertible operator in the Kreı̆n space $(K, \, [\cdot , \cdot])$ and let $E(\Delta)$ be given as constructed above for $\Delta \in \Sigma$. Then, $E$ defines a mapping from $\Sigma$ to $L(K)$ with the following properties ($\Delta, \Delta' \ where $\sigma$ denotes the spectrum.

Theorems & Definitions (53)

  • Theorem 2.1: Langer
  • Theorem 2.2: First Representation Theorem
  • Theorem 2.3: Second Representation Theorem
  • Theorem 2.4
  • Proposition 2.5
  • Lemma 2.6
  • proof
  • Lemma 3.1
  • proof
  • Example 1
  • ...and 43 more