Suppose that $K$ is a Hilbert space and $A$, $B$ are two unitary operators on $K$ such that
$$P_HAy=-P_HBy, y\in K,$$
for some subspace $H$ of $K$. Consider the following subspaces of $K$:
$$H'=\{P_{H^\perp}Ax:~x\in H\},~H''=\{P_{H^\perp}Bx:~x\in H\}.$$
Suppose that $\psi:H'\to H''$, defined by
$$\psi(P_{H^\perp}Ax)=P_{H^\perp}Bx$$
is unitary.
Let $M$ be any positive operator on $H$. It is also given that
$$\|P_{H^\perp}Ax\|^2=\|P_{H^\perp}Bx\|^2=\|Mx\|^2, x\in H.$$
From this can we conclude that $\langle P_{H^\perp}Ax, P_{H^\perp}Bx\rangle = \|Mx\|^2$?