Let $\mathcal{H} := \ell^2(\mathbb{Z})$ and denote $e_k := (\delta_{kj})_j \in \mathcal{H}$. Now consider a sequence $(U_n)$ of unitary operators on $\mathcal{H}$ such that for each $j,k\in\mathbb{Z}$ there exists $u_{jk}\in\mathbb{C}$ with $$\left< e_j\mid U_n e_k\right> \longrightarrow u_{jk} \, (n\to \infty).$$ Then does $$\left< e_j\mid U e_k\right> := u_{jk}$$ for $j,k\in\mathbb{Z}$ define a bounded (maybe even unitary) operator $U$? I feel like it comes down to the question whether a sequence of unitary operators can converge weakly to an unbounded operator.
I would appreciate any answer or hint.