On m-sectorial extensions of sectorial operators

In our article [15] description in terms of abstract boundary conditions of all $m$-accretive extensions and their resolvents of a closed densely defined sectorial operator $S$ have been obtained. In particular, if $\{\mathcal{H},\mathrm{\Gamma }\}$ is a boundary pair of $S$, then there is a bijective correspondence between all $m$-accretive extensions $\tilde{S}$ of $S$ and all pairs $⟨\mathbf{Z},X⟩$, where $\mathbf{Z}$ is a $m$-accretive linear relation in $\mathcal{H}$ and $X:\mathrm{d}\mathrm{o}\mathrm{m}(\mathbf{Z})\to \overline{\mathrm{r}\mathrm{a}\mathrm{n}(S_F)}$ is a linear operator such that: $$\mathrm{\parallel }Xe{\mathrm{\parallel }}^2⩽\mathrm{R}\mathrm{e}(\mathbf{Z}(e),e{)}_{\mathcal{H}}\mathrm{\forall }e\in \mathrm{d}\mathrm{o}\mathrm{m}(\mathbf{Z})\mathrm{.}$$ As is well known the operator $S$ admits at least one $m$-sectorial extension, the Friedrichs extension. In this paper, assuming that $S$ has non-unique $m$-sectorial extension, we established additional conditions on a pair $⟨\mathbf{Z},X⟩$ guaranteeing that corresponding $\tilde{S}$ is $m$-sectorial extension of $S$. As an application, all $m$-sectorial extensions of a nonnegative symmetric operator in a planar model of two point interactions are described.