A Birman-Krein-Vishik-Grubb theory for sectorial 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 $V\ge \epsilon >0$ 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 $\overline{A_{+}}\subset A_D\subset A_{-}^{\ast }$. Here, $D$ is an auxiliary operator from $\ker (A_{-}^{\ast })$ to $\ker (A_{+}^{\ast })$ that parametrizes the different extensions $A_D$. After this, we will give a criterion for when the quadratic form $ψ\mapsto\mbox{Re}\langleψ,A_Dψ\rangle$ is closable and show that the selfadjoint operator $\hat{V}$ that corresponds to the closure is an extension of $V$. We will show how $\hat{V}$ 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.

arXiv comment22 pages, a few slight modifications and some more references added