# 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!)

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 $$\,V+W=V\oplus W$$.