Let $A$ and $B$ be two unbounded self-adjoint operators. From this mathoverflow post, for instance, we know that $A + B$ is self-adjoint on $\mathcal{D}(A) \cap \mathcal{D}(B)$ if $A$ and $B$ are commuting and positive operators (Putnam's book is cited as a reference there). My question is: assume $A$ is positive and $A$ and $B$ commute. $B$ is not positive though, but $B$ is a relatively bounded perturbation of $A$. Could we still say $A + B$ is self-adjoint?

Let $A$ and $B$ be two unbounded self-adjoint operators. From this mathoverflow post, for instance, we know that $A + B$ is self-adjoint on $\mathcal{D}(A) \cap \mathcal{D}(B)$ if $A$ and $B$ are commuting and positive operators (Putnam's book is cited as a reference there). My question is: assume $A$ is positive and $A$ and $B$ commute. $B$ is not positive though, but $B$ is a relatively bounded perturbation of $A$. Could we still say $A + B$ is self-adjoint?