Centralizer of projections

Question

Let $H$ be a Hilbert space and $p, q$ self-adjoint projectors in $B(H)$,
i.e. $$p^2=p=p^* \space \text{ and } \space q^2=q=q^*.$$ Suppose they have the same centralizers $C(p)=C(q)$.
Is it true that $p=\pm q$?
Here $C(x)=\{y\in B(H) : yx=xy\}$.
Maybe I am asking a well known result but I couldn't find anything about this problem in the literature.

Answer 1

Look in Vaughan Jones notes, Exerise 2.1.13. $A$ and $A^\ast$ preserve a subspace $K$, ie $A K \subset K$ and $A^\ast K \subset K$ iff $[A,P_K] = 0$, where $P_K$ is the orthogonal projection onto $K$.