Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Friedrichs and Kreĭn type extensions in terms of representing maps

Seppo Hassi, Henk de Snoo

Abstract

A semibounded operator or relation $S$ in a Hilbert space with lower bound $m \in {\mathbb R}$ has a symmetric extension $S_{\rm f}=S {\, \widehat + \,} (\{0\} \times {\rm mul\,} S^*)$, the weak Friedrichs extension of $S$, and a selfadjoint extension $S_{\rm F}$, the Friedrichs extension of $S$, that satisfy $S \subset S_{\rm f} \subset S_{\rm F}$. The Friedrichs extension $S_{\rm F}$ has lower bound $γ$ and it is the largest semibounded selfadjoint extension of $S$. Likewise, for each $c \leq γ$, the relation $S$ has a weak Kreĭn type extension $S_{{\rm k},c}=S {\, \widehat + \,} (\ker (S^*-c) \times \{0\})$ and Kreĭn type extension $S_{{\rm K},c}$ of $S$, that satisfy $S \subset S_{{\rm k},c} \subset S_{{\rm K},c}$. The Kreĭn type extension $S_{{\rm K},c}$ has lower bound $c$ and it is the smallest semibounded selfadjoint extension of $S$ which is bounded below by $c$. In this paper these special extensions and, more generally, all extremal extensions of $S$ are constructed in terms of a representing map for ${\mathfrak t}(S)-c$ and their properties are being considered.

Friedrichs and Kreĭn type extensions in terms of representing maps

Abstract

A semibounded operator or relation in a Hilbert space with lower bound has a symmetric extension , the weak Friedrichs extension of , and a selfadjoint extension , the Friedrichs extension of , that satisfy . The Friedrichs extension has lower bound and it is the largest semibounded selfadjoint extension of . Likewise, for each , the relation has a weak Kreĭn type extension and Kreĭn type extension of , that satisfy . The Kreĭn type extension has lower bound and it is the smallest semibounded selfadjoint extension of which is bounded below by . In this paper these special extensions and, more generally, all extremal extensions of are constructed in terms of a representing map for and their properties are being considered.
Paper Structure (9 sections, 22 theorems, 191 equations)

This paper contains 9 sections, 22 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Semibounded relations and representing maps
  3. Semibounded forms induced by semibounded relations
  4. A characterization of the Friedrichs extensions
  5. Forms and representing maps induced by inverses of semibounded relations
  6. A characterization of the Kreı̆n type extensions
  7. Extremal semibounded extensions
  8. Symmetric relations whose domain and range are orthogonal
  9. Symmetric relations defined by semibounded quadratic forms

Key Result

Proposition 2.3

The closed form ${\mathfrak s}(S) \in \mathbf{F}({\mathfrak H})$ defined in ss extends the form ${\mathfrak t}(S) \in \mathbf{F}({\mathfrak H})$ in s-ip and its lower bound is $c$. Consequently, the form ${\mathfrak t}(S)$ is closable and Furthermore, the closure $\bar{{\mathfrak t}}(s) \in \mathbf{F}({\mathfrak H})$ is given by

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • ...and 35 more