On m-sectorial extensions of sectorial operators

TL;DR

This paper characterizes all m-sectorial extensions of sectorial operators using boundary conditions, providing conditions for such extensions and applying the results to a model with point interactions.

Contribution

It introduces new conditions on boundary pairs that guarantee m-sectorial extensions of sectorial operators and describes all such extensions for a specific model.

Findings

Established conditions for m-sectorial extensions via boundary pairs.

Provided a complete description of m-sectorial extensions in a planar two-point interaction model.

Connected abstract boundary conditions with concrete operator extensions.

Abstract

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.