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
$\partial {\rm\Omega}$
.