Under which conditions is $$\langle Ax, x\rangle = \langle x, x\rangle \implies A \text{ is the identity}?$$
I ask because I'd like to show that Isometry on a Hilbert space (Norm preserving) implies Adjoint is a left inverse or in formula $$\|Tx\|= \|x\| \implies \langle T^*Tx, x\rangle = \langle x, x\rangle \implies T^*T = I.$$
Edit: I think I have a solution that is a little bit restrictive:
I need that $T^*T$ to be a compact operator.
$T^*T-I$ is self-adjoint and also compact. Then I can use $||T^*T-I|| = \sup_{||x||=1} \langle T^*T-Ix, x \rangle $ and this supremum is attained since compact. So from this I conclude $||T^*T-I|| = 0 \Rightarrow T^*T-I = 0$ and we are done?