An operator A is said to be positive if $\displaystyle {<\psi|A|\psi>}$ $\displaystyle \ge$ 0 for all states |$\displaystyle \psi$> in the space.

PROVE: If C is an arbitrary operator the products $\displaystyle A=C^\dagger C$ and $\displaystyle B=CC^\dagger$ are Hermitian and positive.

Since C is an arbitrary operator then $\displaystyle {C|\psi>} = {|\phi>}$ for some $\displaystyle |\phi>$ in the space.

So $\displaystyle <\psi|C^\dagger C|\psi> = <\phi|\phi>$

Then $\displaystyle A^\dagger = (C^\dagger C)^\dagger$

Which gives $\displaystyle <\psi|(C^\dagger C)^\dagger|\psi> = <\psi|C C^\dagger|\psi> = (|\phi> |(\psi^*) >< (\psi) | <\phi|)^* = <\phi|\phi>$

Can someone verify if I got this correct so far or if I totally botched this up?

Thanks