A Birman–Kreĭn–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 $V\ge \varepsilon >0$ are assumed to be symmetric. We develop an analog to the Birmin–Kreĭn–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 $\psi \mapsto {\text{ Re }}\langle \psi ,A_D\psi \rangle$ 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. All of our results will be presented in a way that emphasizes their connection to the classical BKVG-theory and that shows how the BKVG formulas generalize when considering sectorial operators $A_\pm =\pm iS+V$ instead of just a strictly positive symmetric operator V. Applications to second order ordinary differential operators are discussed.