On Stability of Square Root Domains for Non-Self-Adjoint Operators Under Additive Perturbations

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, $$dom((T_0+W{)}^{1\mathrm{/}2})=dom(T_0^{1\mathrm{/}2})\mathrm{.}$$ Moreover, assuming in addition that $dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big)$, we prove stability of square root domains in the form $$dom((T_0+W{)}^{1\mathrm{/}2})=dom(T_0^{1\mathrm{/}2})=dom((T_0^{\ast }{)}^{1\mathrm{/}2})=dom(((T_0+W{)}^{\ast }{)}^{1\mathrm{/}2}),$$ which is most suitable for PDE applications. We apply this approach to elliptic second-order partial differential operators of the form $$-div(a\mathrm{\nabla }\cdot )+({\mathbf{B}}_1\cdot \mathrm{\nabla }\cdot )+div({\mathbf{B}}_2\cdot )+V$$ in $L^2(\mathrm{\Omega })$ on certain open sets $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, $n\in \mathbb{N}$, with Dirichlet, Neumann, and mixed boundary conditions on $\mathrm{\partial }\mathrm{\Omega }$, under general hypotheses on the (typically, nonsmooth, unbounded) coefficients and on $\mathrm{\partial }\mathrm{\Omega }$.