Let T be a (unbounded) selfadjoint operator in $B(H)$, the bounded operator acting on Hilbert space $H$.

Def: A compression of T is an operator $pTp$, where $p$ is a projection in $B(H)$.

I am thinking the relation between the resolution of identity (spectrum decomposition) of $pTp$ and that of $T$. I guess there should be something like $e_{pTp}(-\infty, t)$ $\le $ $e_T(-\infty, t)$ $\le$ $e_{pTp}(-\infty, t) + (I-p)$.. But I do not know how to prove it. I think someone here might help me out.