Georgia Institute of Technology College of Sciences

Following Kato, we define the sum, $H=H_0+V$, of two linear operators, $H_0$ and $V$, in a fixed Hilbert space in terms of its resolvent. In an abstract theorem, we present conditions on $V$ that guarantee $\text{dom}(H_0^{1/2})=\text{dom}(H^{1/2})$ (under certain sectorality assumptions on $H_0$ and $H$). Concrete applications to non-self-adjoint Schr\"{o}dinger-type operators--including additive perturbations of uniformly elliptic divergence form partial differential operators by singular complex potentials on domains--where application of the abstract theorem yields $\text{dom}(H^{1/2})=\text{dom}((H^{\ast})^{1/2})$, will be presented. This is based on joint work with Fritz Gesztesy and Steve Hofmann.