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ON STABILITY OF SQUARE ROOT DOMAINS FOR NON-SELF-ADJOINT OPERATORS UNDER ADDITIVE PERTURBATIONS

Abstract

Assuming $T_{0}$ to be an m-accretive operator in the complex Hilbert space ${\mathcal{H}}$ , we use a resolvent method due to Kato to appropriately define the additive perturbation $T=T_{0}+W$ and prove stability of square root domains, that is,

$$\begin{eqnarray}\text{dom}((T_{0}+W)^{1/2})=\text{dom}(T_{0}^{1/2}).\end{eqnarray}$$
Moreover, assuming in addition that $\text{dom}(T_{0}^{1/2})=\text{dom}((T_{0}^{\ast })^{1/2})$ , we prove stability of square root domains in the form
$$\begin{eqnarray}\text{dom}((T_{0}+W)^{1/2})=\text{dom}(T_{0}^{1/2})=\text{dom}((T_{0}^{\ast })^{1/2})=\text{dom}(((T_{0}+W)^{\ast })^{1/2}),\end{eqnarray}$$
which is most suitable for partial differential equation applications. We apply this approach to elliptic second-order partial differential operators of the form
$$\begin{eqnarray}-\text{div}(a{\rm\nabla}\,\cdot )+(\vec{B}_{1}\cdot {\rm\nabla}\,\cdot )+\text{div}(\vec{B}_{2}\,\cdot )+V\end{eqnarray}$$
in $L^{2}({\rm\Omega})$ on certain open sets ${\rm\Omega}\subseteq \mathbb{R}^{n}$ , $n\in \mathbb{N}$ , with Dirichlet, Neumann, and mixed boundary conditions on $\partial {\rm\Omega}$ , under general hypotheses on the (typically, non-smooth, unbounded) coefficients and on $\partial {\rm\Omega}$ .

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