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


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