Let $A:D(A) \subseteq H \to H$ and $B:D(B) \subseteq H \to H$ be linear operators on a Hilbert space $H$ such that $A$ is a closed densely defined operator and $B$ is relatively bounded with respect to $A$ with relative bound $0$. We have to show that $D((A+B)^*)= D(A^*)$.
Since $B$ is $A$-bounded with $A$-bound $0$ and $A$ is closed, we have that $A+B$ is a densely defined closed operator with $D(A+B)=D(A)$. We know that $D(A^*) \subseteq D((A+B)^*)$ because $A^*+B^* \subseteq (A+B)^*$, but how can we show that $D((A+B)^*) \subseteq D(A^*)$.
Can you give me any hint or a reference for the adjoint of a relatively bounded perturbation, please?