Suppose that I have two self-adjoint operators $A$ and $B$ such that $\mathcal{D}(A)\cap\mathcal{D}(B)$ is dense and $B$ positive. Then $A\pm iB$ (with domains $\mathcal{D}(A)\cap\mathcal{D}(B)$) are closable. What are generic conditions so that $(A+iB)^\*$ is the closure of $(A-iB)|\_{\mathcal{D}(A)\cap\mathcal{D}(B)}$? $A$ or $B$ bounded suffices, but does this hold in general?


NB: What I'm really interested in is whether $0$ is in the resolvent set of the closure of $A+iB$, which follows easily from the above condition.



For self-adjoint $A$ and $B$, when is $(A+iB)^*$ the closure of $A-iB$?
In general this is not true. Let $\Omega$ be a smooth bounded domain, let $A$ be $-\Delta$ with Neumann conditions and let $B$ be $-\Delta$ with Dirichlet conditions. The $(A+iB)$ is $(-1-i)\Delta$ with domain $H^2\_0(\Omega)$. Its adjoint is $(-1+i)\Delta$ with domain $H^2(\Omega)$, which is not the closure of $A-iB$.


