Exercise 8, § 2

Let $F$ be a complex Hilbert space and let $u: E\rightarrow F$ be a continuous linear mapping with closed image.

a) Prove that there exists a unique continuous linear mapping $v\in$ $\mathscr{L}(F; E)$ satisfying the following conditions:

(i) $u\circ v\circ u=u$ and $v\circ u\circ v=v$;

(ii) $u\circ v$ and $v\circ u$ are self-adjoint.

This mapping is denoted by $u^{\dagger}$ (Moore-Penrose pseudo-inverse).

b) Prove that Im($u$) $=$ Im($u\circ u^{\dagger}$). Prove that for $x\in E$ and $y\in$ Im($u$), the conditions $u(x) =y$ and $x-u^{\dagger}(y)\in$ Im(1$_E-u^{\dagger}\circ u$) are equivalent.