Exercise 1, § 2

Let $u$ be an endomorphism of E whose image is closed in E. Let $q$ be an orthoprojector of E and let $v$ be the endomorphism $x\mapsto q(u(x))$ of the image of E. a) The adjoint of $v$ is the endomorphism $x\mapsto q(u^*(x))$ of the image of E.

b) If $u$ is hermitian, then $v$ is hermitian.

c) It is possible that $u$ is normal and that $v$ is not.