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 $\{\mathcal{H},\Gamma\}$ is a boundary pair of $S$, then there is a bijective correspondence between all $m$-accretive extensions $\tilde{S}$ of $S$ and all pairs $\langle \mathbf{Z},X\rangle$, where $\mathbf{Z}$ is a $m$-accretive linear relation in $\mathcal{H}$ and $X:\mathrm{dom}(\mathbf{Z})\to\overline{\mathrm{ran}(S_{F})}$ is a linear operator such that: $$\|Xe\|^2\leqslant\mathrm{Re}(\mathbf{Z}(e),e)_{\mathcal{H}}\quad\forall e\in\mathrm{dom}(\mathbf{Z}).$$ 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 $\langle \mathbf{Z},X\rangle$ 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.