Weak convergence of Hilbert Schmidt operators

Question

Let $\{T_i\}$ be a sequence of positive operators in $B_2(H)$ converging to $T$ with respect to the weak topology on $B_2(H)$ (space of Hilbert Schmidt operators on $H$). Does $\{T_ix\}$ converges to $Tx$ in norm (or weakly) for each $x\in H$?

I know that $T$ must be positive since positive operators form a convex set and convex weakly closed sets are norm closed.

Answer 1

Fix $x,y\in H$. Then the functional $T\mapsto \langle Tx,y\rangle$ is continuous. Then the weak convergence $T_i\rightharpoonup T$ implies $$ \langle T_i x,y\rangle \to \langle Tx,y\rangle \quad \forall x,y\in H, $$ which is equivalent to $$ T_ix \rightharpoonup Tx \quad \forall x\in H. $$

Answer 2

The injection $B_2(H) \rightarrow B(H)$ is linear continuous between Hilbert spaces, so it maps weakly convergent sequences to weakly convergent sequences. In other words, $T_i$ converges weakly.