Limits in Hilbert space and convergence problem.

Question

Let $\{T_i\}$ be sequence in $B(H)$ with orthonormal basis $\{e_\alpha\}_{\alpha\in I}$ such that for each $x\in H$ $$\lim_i \sum_{\alpha\in I} \left<T_ie_\alpha,x\right> \to 0$$ Can we infer from this that $\lim_i \left<T_ie_\alpha,x\right>=0$ for each $\alpha\in I$ and $x\in H$?

Answer 1

False. Consider $\mathbb R^{2}$ with the standard basis $\{e_1,e_2\}$. Let $T_n=T$ for all $n$ where $T(a,b)=(a-b,0)$. You an check that the hypothesis is satisfied but the conclusion fails when $\alpha =1$ and $x=e_1$.