If $U(t)$ and $U^0(t)$ are unitary operators, $|\psi⟩$ and $|\phi⟩$ two vectors and we have the limit between these vectors below as follows
$\lim_{t \to -\infty} ∥U(t) |\psi⟩ - U^0(t)|\phi⟩∥=0$.
How do I go from the above equation to the following equation
$\lim_{t \to -\infty} ∥ |\psi⟩ - U^{\dagger}(t)U^0(t)|\phi⟩∥=0$.
I think I just take the square root of $U^{\dagger}(t)$ and multiply the first expression by that. Is my reasoning right?
Unitary implies $U^{\dagger}(t) U(t) = I$, where $I$ is the identity operator. Hence, just multiply both sides by $U^{\dagger}$ on the left and distribute by linearity. You are allowed to absorb $U^{\dagger}$ into the norm since unitary operations preserve norms. I.e. $$ \|U(t) \psi - U^0 (t) \phi \| = x, \\ \|\psi - U^{\dagger}(t)U^0 (t)\phi\| = \|U^{\dagger}(t)\| x. $$ Note that $x \to 0$ as $t \to -\infty$, and since $U^{\dagger}(t)$ is bounded (Unitary operators are bounded) , we get $\|U^{\dagger}(t)\| x \to 0$ as well.
Also note that one can prove the reverse direction by the same argument.