As we know, in the category of Banach space, we have that $\ell^1=c_0^*$. Indeed, for any $(x_n)\in \ell^1$
$$\|(x_n)\|_{\ell^1}=\sum |x_n|=\sup \{|\sum_1^{\infty } t_nx_n| : (t_n)\in B\}$$ where $B$ is the unit ball of $c_0$. It means
$$\|(x_n)\|_{\ell^1}<\infty ~\Longleftrightarrow ~\sum |x_n|<\infty$$
It is supposed to extend this fact for operators:
Let $H$ be a Hilbert space and $\{x_n\}$ be a sequence of operators in $B(H)$. Let us put
$$\|(x_n)\|^{op}_{\ell}:=\sup \{\|\sum_1^{\infty } t_nx_n\| : (t_n)\in B\}$$
Q. What is yor suggestion to complete the following: $$\|(x_n)\|^{op}_{\ell}<\infty \Longleftrightarrow ~~?$$