Let $\ell^2$, $\ell^\infty$ denote the usual sequence spaces and let $L(\ell^2,\ell^2)$ the Banach space of bounded linear operators from $\ell^2$ to $\ell^2$ as well as $L(\ell^2,\ell^\infty)$ the Banach space of bounded linear operators from $\ell^2$ to $\ell^\infty$, equipped with the operator norm. Clearly, there holds

$$ L(\ell^2,\ell^2)\subseteq L(\ell^2,\ell^\infty). $$

My question: How can I characterize the closure $\overline{L(\ell^2,\ell^2)}$ with respect to $L(\ell^2,\ell^\infty)$.