I am looking for citations in a book or article for the following fact:
Let $T_n$ be a sequence of operators in Hilbert space and $T$ also be an operator in Hilbert space. If $T_n$ is convergent in a weak operator topology to $T$ and $T_n^*T_n$ is also convergent in a weak operator topology to $T^*T$, then $T_n$ is convergent in a strong operator topology to $T$.
$\def\Re{\operatorname{Re}}$ It's a rather straightforward computation: \begin{align} \|T_nx-Tx\|^2 &=\|T_nx\|^2+\|Tx\|^2-2\Re\langle T_nx,Tx\rangle\\[0.2cm] &=\langle T_n^*T_nx,x\rangle+\|Tx\|^2-2\Re\langle T_nx,Tx\rangle\\[0.2cm] &\xrightarrow[n\to\infty]{}\langle T^*Tx,x\rangle+\|Tx\|^2-2\Re\langle Tx,Tx\rangle\\[0.2cm] &=\|Tx\|^2+\|Tx\|^2-2\|Tx\|^2=0. \end{align}
