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Register a new userOn m-sectorial extensions of sectorial operators
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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 $langle mathbf{Z},Xrangle$, where $mathbf{Z}$ is a $m$-accretive linear relation in $mathcal{H}$ and $X:mathrm{dom}(mathbf{Z})tooverline{mathrm{ran}(S_{F})}$ is a linear operator such that: [ |Xe|^2leqslantmathrm{Re}(mathbf{Z}(e),e)_{mathcal{H}}quadforall einmathrm{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 $langle mathbf{Z},Xrangle$ 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.
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