Normal extensions

TL;DR

This paper characterizes all correct extensions of a densely defined minimal operator in a Hilbert space that have equal domains with their adjoints, especially focusing on normal and formally normal operators, exemplified by the Cauchy-Riemann operator.

Contribution

It provides a comprehensive description of all correct extensions with domain-equality properties for minimal operators, extending previous results to normal and formally normal cases.

Findings

All correct extensions with $D(L)=D(L^*)$ are characterized.

All correct normal extensions of a formally normal operator are described.

The Cauchy-Riemann operator serves as an illustrative example.

Abstract

Let $L_0$ be a densely defined minimal linear operator in a Hilbert space $H$. We prove theorem that if there exists at least one correct extension $L_S$ of $L_0$ with the property $D(L_S)=D(L_S^*)$, then we can describe all correct extensions $L$ with the property $D(L)=D(L^*)$. We also prove that if $L_0$ is formally normal and there exists at least one correct normal extension $L_N$, then we can describe all correct normal extensions $L$ of $L_0$. As an example, the Cauchy-Riemann operator is given.