A mapping $T$ on a Hilbert space $H$ is said to be a linear operator if $T(\alpha x + y) = \alpha T(x) + T(y)$ for any element $x , y \in H$ and every complex number $\alpha$. We define the operator norm as follows:
$\|T\| = \inf \{ c>0 : \| T(x) \| \ge c \|x\| \}$,
also this norm is defined with
$\| T\| = \sup \{ \| T(x) \| : \|x\| = 1 \} = \sup \{ \| T(x) \| : \|x\| \le 1 \} = \sup \{ | <T(x) ,y> | : \|x\| = \|y\| = 1 \} $.
The operator $T$ is called projection if $T$ is idempotent and hermitian, i.e. $T^2 = T$ and $T^* = T$ respectively. Now I want to show that :
If $T$ is idempotent and $\|T\| \le 1$ then $T$ is a projection.
In fact I have to show that $T=T^*$ but I can't prove it. Please say me how can I say $T=T^*$ ?