Considering two bounded linear operators $S,T$ in $\mathcal{B}(X)$, where $X$ is a complex Hilbert space.
If $\def\norm#1#2{\langle {#1},{#2}\rangle} \norm{Sx}{x} = \norm{Tx}{x}$ for all $x\in X$, do we have $S=T$?
Furthermore, for another bounded linear operator $A$, if $\norm{Ax_n}{x_n}$ converges to $\norm{A x}{x}$ for all $x_n\to x$, does the convergence has a name?
Let $X=\mathbf{R}^2$, $S=0$, $T=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$. In the complex case the polarization identity expresses $(Sx,y)$ via $(S(x\pm y),x\pm y)$ and $(S(x\pm iy),x\pm iy)$, so the criterion is true.
As for the second question, from the preceding discussion we see that this is just weak convergence.