Exercise 9, § 2

Let $u$ and $v$ be endomorphisms of $E$ with closed image. If $u+v$ also has closed image, define $u:v=u\circ (u+v)^{\dagger}\circ v$, where $(u+v)^{\dagger}$ is the Moore-Penrose pseudo-inverse of $u+v$.

a) Prove that $u:v$ is self-adjoint.

b) We have $u:v=v:u$ and Im($u:v$) $=$ Im($u$)$\cap$ Im($v$).

c) Assume that $u$ and $v$ are orthogonal projectors. Prove that $2u:v$ is the orthogonal projector whose image is Im($u$)$\cap$ Im($v$).