Lorentz sequence spaces

It is shown that the condition
$$ \sup\limits_n\Bigl\{n^{1/2}\Bigl(\sum_{j\le n}c_j^2\Bigr)^{1/2}\Bigr/\sum_{j\le n}c_j\Bigr\}<\infty $$
on the normalizing sequence $\{c_j\}_{j<\infty}$ of the Lorentz sequence space $\Lambda(c)$ is a necessary and sufficient condition for having each bounded linear operator acting from an arbitrary $\mathscr L_\infty$-space into $\Lambda(c)$ be 2-absolutely summing.