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

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 $Vgeq varepsilon>0$ 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 $overline{A_+}subset A_Dsubset A_-^*$. 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 $psimapstombox{Re}langlepsi,A_Dpsirangle$ 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.