One property of the unitary operator is that it preserves the norm of the state-vectors: $$ \langle \Psi | U^\dagger U | \Psi \rangle = \langle \Psi | \Psi \rangle $$ If $U$ is unitary.
Is the inverse statement also true? For an operator that will satisfy above equation for all $|\Psi \rangle$, will that operator be unitary?