I'd like to know whether a certain unbounded operator on a Hilbert space is the generator of a strongly continuous contraction semigroup. (Such operators are known as maximally dissipative operators.)

• Let $A$ be an unbounded **skew-adjoint** operator on $H$.

• Let $P$ be an unbounded **positive self-adjoint** operator on $H$.

Suppose that $\mathcal D:=\mathcal D(A)\cap \mathcal D(P)$ is a common dense core for $A$ and $P$, and let $A-P$ be the strong sum of $A$ and $-P$ (the closure of $(A-P)|_{\mathcal D}$).

Is $A-P$ the generator of a strongly continuous contraction semigroup on $H$ (i.e., is $A-P$ a maximally dissipative operator)? Are there reasonable extra assumptions that I could add which would imply that $A-P$ is the generator of a strongly continuous contraction semigroup?