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

arXiv comment61 pages, to appear in Mathematika