Orthogonal projectors on non-orthogonal subspaces 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!)
 A: Such a formula is available, and it simplifies to $\,P_{\,V+W} =  P_{\,V} + P_{\,W}\,$ if $\,V\perp W$.
In the general case where $V$ and $W$ are oblique, the two summands
$P_{\,V}$ and $P_{\,W}$ require additional "pre- and post-processing" factors.
It is assumed throughout that $V\oplus W$ is a closed subspace of $H$ (this is necessary if $H$ is infinite-dimensional). Only then one has
$$H=V\oplus W \oplus (V\oplus W)^\perp\,.$$
If $\,Z:=(V\oplus W)^\perp=\{0\}$, then $\,P_{\,V+W} =\mathbb{1}_H$.
Now assume $Z\neq\{0\}$. It is exploited that
the (oblique) projection $\,P_{\,V,\,W\oplus Z}\,$ onto $V$ along $\,W\oplus Z\,$, which is determined by $\,\operatorname{im}(P_{\,V,\,W\oplus Z})=V\,$
and $\,\ker (P_{\,V,\,W\oplus Z})=W\oplus Z$, can be expressed in terms of the orthogonal projectors $P_{\,V}, P_{\,W}$ as
$$P_{\,V,\,W\oplus Z}\:=\:\big(\mathbb 1-P_{\,V}P_{\,W}\big)^{-1}\,P_{\,V}\,
\big(\mathbb 1-P_{\,V}P_{\,W}\big)$$
see for instance this math.SE post .
After swapping $V$ and $W$ we analogously have
$\,P_{\,W,\,V\oplus Z}=(\mathbb 1-P_{\,W}P_{\,V})^{-1}P_{\,W}
(\mathbb 1-P_{\,W}P_{\,V})\,$.
Then let
$$\,P_{\,V+W}\::=\: P_{\,V,W\oplus Z} + P_{\,W,V\oplus Z}$$
and observe that $\,P_{\,V+W}$


*

*is idempotent as 
$\,P_{\,V,W\oplus Z}\,P_{\,W,V\oplus Z} =0= P_{\,W,V\oplus Z}\,P_{\,V,W\oplus Z}$,

*is self-adjoint because equal to the identity on $V+W$, and equal to zero on $Z$,

*has $\,\operatorname{im}(P_{\,V+W})=V+W$.


Hence $P_{\,V+W}$ is the (one and only) orthogonal projector onto
$\,V+W=V\oplus W$.
