Let $A$ be an unbounded self-adjoint operator with spectrum $\sigma(A)=\mathbb R$ in a Hilbert space $\mathcal H$. Let $P$ be a bounded operator in $\mathcal H$ satisfying $P\ge1$ and $$ {\rm Domain}(AP) \equiv\big[\varphi\in\mathcal H:P\varphi\in{\rm Domain}(A)\big] ={\rm Domain}(A). $$ Finally, suppose that the operator $$ H:=PAP,\quad{\rm Domain}(H):={\rm Domain}(A), $$ is self-adjoint in $\mathcal H$.

Question: Can we find conditions on $A$ and $P$ guaranteeing that the spectrum of $H$ is the same of that of $A$; that is, guaranteeing that $\sigma(H)=\sigma(A)=\mathbb R$ ?

Even though operators like $H$ appear in Spectral Theory, I haven't been able to find much information on their spectrum in the literature. I am only aware of the paper:

Hladnik, Milan, Omladic, Matjaz, Spectrum of the product of operators. Proc. Amer. Math. Soc. 102 (1988), no. 2, 300–302,

whose results are too general to answer the present question.