Let ${\cal H}$ be a separable Hilbert space over $\mathbb{C}$, with inner product denoted $\langle\cdot,\cdot\rangle$. Suppose that $A$ and $B$ are densely defined but not-necessarily-bounded operators on ${\cal H}$, with these properties:
They both have $0$ as an eigenvalue, and all other elements of their spectra are real numbers $\geq 1$. The purpose of the condition $\geq 1$ on the rest of the spectrum is to make the following approximations easier to express.
They almost commute with each other, in the sense that the norm of the vector $[A,B]\psi$ is much less than the norm of the original vector $\psi\in {\cal H}$ whenever $AB$ and $BA$ are both defined on $\psi$.
Let $P_A$ and $P_B$ be the projectors onto the zero-eigenvalue eigenspaces of $A$ and $B$, respectively. How can we prove that these projection operators almost commute with each other? More specifically, can we prove that $$ \big\|[P_A,P_B]\big\|^2\leq \kappa \frac{\big\langle [A,B]\psi,\,[A,B]\psi\big\rangle}{\langle\psi,\,\psi\rangle} $$ for all $\psi\in{\cal H}$ on which $AB$ and $BA$ are defined, for some constant $\kappa$ that is independent of $A$ and $B$?