Question

Show that an idem potent operator on a Hilbert space H is a

projection on H

iff it is normal.

Answer 1

A linear operator P:H→H is called idempotent if P = P²

Px = x

for orthogonal projection:

$P^2=P,\langle Px,y\rangle =\langle x,Py\rangle$

for  normal operator:

$PP^*=P^*P$

for adjoint:

$\langle Px,y\rangle =\langle x,P^*y\rangle$