A Birman-Krein-Vishik-Grubb theory for sectorial operators

TL;DR

This paper extends the classical Birman-Krein-Vishik-Grubb theory to sectorial operators, constructing and classifying their maximally accretive extensions and analyzing their quadratic forms and real parts.

Contribution

It develops a novel framework for sectorial operators analogous to BKVG theory, including criteria for quadratic form closability and extension dependence on auxiliary operators.

Findings

Constructed all maximally accretive extensions of sectorial operators

Provided criteria for quadratic form closability and associated selfadjoint extensions

Applied the theory to second order differential operators

Abstract

We consider densely defined sectorial operators $A_\pm$ that can be written in the form $A_\pm=\pm iS+V$ with $\mathcal{D}(A_\pm)=\mathcal{D}(S)=\mathcal{D}(V)$, where both $S$ and $V\geq \varepsilon>0$ are assumed to be symmetric. We develop an analog to the Birmin-Krein-Vishik-Grubb (BKVG) theory of selfadjoint extensions of a given strictly positive symmetric operator, where we will construct all maximally accretive extensions $A_D$ of $A_+$ with the property that $\overline{A_+}\subset A_D\subset A_-^*$. Here, $D$ is an auxiliary operator from $\ker(A_-^*)$ to $\ker(A_+^*)$ that parametrizes the different extensions $A_D$. After this, we will give a criterion for when the quadratic form $\psi\mapsto\mbox{Re}\langle\psi,A_D\psi\rangle$ is closable and show that the selfadjoint operator $\widehat{V}$ that corresponds to the closure is an extension of $V$. We will show how $\widehat{V}$ depends on $D$, which --- using the classical BKVG-theory of selfadjoint extensions --- will allow us to define a partial order on the real parts of $A_D$ depending on $D$. Applications to second order ordinary differential operators are discussed.