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.



Spectrum of the operator PAP, with A self-adjoint and P strictly positive
The only reasonable condition I can think of is that $P = g(A)$ where $g \ge 1$ is a bounded continuous function on $\mathbb R$.


It is instructive to consider the case where $A$ is the multiplication operator $(A f)(x) = x f(x)$ on $L^2({\mathbb R})$ and $P = g(A)$ for some bounded measurable function $g \ge 1$. Then $\sigma(PAP)$ is the essential range of $\{ x g(x)^2: x \in {\mathbb R}\}$. For suitable discontinuous $g$, this could be any closed subset $S$ of the plane that intersects $(-\infty,N)$, $(-1/N,0)$, $(0,1/N)$ and $(N,\infty)$ for all $N$. 


