Exercise 7, § 2

Let $u$ and $v$ be continuous endomorphisms of E.

a) Prove that Im($u$) $+$ Im($v$) $=$ Im(($u\circ u^*+v\circ v^*$)$^{1/2}$).

b) If $u$ is positive and has closed image, prove that Im($u$) $=$ Im($u^{1/2}$).

c) If $u, v$ are positive with closed image and if $u+v$ also has closed image, one has Im($u+v$) $=$ Im($u$) $+$ Im($v$).