Discontinuous linear operator on $\ell^{2}$

Question

Let $e_{n} = (0, 0, \ldots, 0, 1, 0, \ldots)$ where $1$ is in the $n$th position. Then $\{e_{n}\}$ is an orthonormal basis for the Hilbert space $\ell^{2}(\mathbb{N})$. Does there exists a linear operator $T: \ell^{2} \rightarrow \ell^{2}$ which is not continuous but also satisfies $\sum_{n, m = 1}^{\infty}|\langle Te_{n}, e_{m} \rangle|^{2} < \infty$?