It is a well known fact that if(f) $V,W$ are orthogonal subspaces of a Hilbert space $H$, then their orthogonal projectors satisfy $$P_{\,V+W} = P_V + P_W,$$
where $P_{\,V+W}$ is the projector on $V+W$.
What happens if $V,W$ are not orthogonal, but we still take orthogonal projectors, and still $V\cap W= \{0\}$?
I am looking for a formula of the type $$P_{V+W} = P_V + P_W + A(V,W)$$
where $A$ is some operator, depending for example on the angle between the subspaces.
Is there such a formula?
(Feel free to modify tags appropriately!)