 Solution

Compact. Its column indexed by $m$ has entries $n!/m!$ for $1\le n<m$, and zero otherwise. The sum of that column is $c_m=(1!+\cdots +(m-1)!)/m!\le 2/m$ for $m\ge 2$, by the elementary induction $1!+\cdots +k!\le 2k!$. For operators from ${\ell }^1$ the supremum of the absolute column sums is the operator norm: one inequality is the triangle inequality, and coordinate unit vectors give the reverse. Keeping just the first $N$ input columns is finite rank and its remainder norm is at most ${\sup }_{m>N}{c}_m\le 2/(N+1)$. Hence it converges to $T$ in norm, proving compactness.