= Compact factorial triangular operator on l1
{title2=$(T\alpha)_n=\sum_{m>n}(n!/m!)\alpha_m$}
The operator in the title is bounded and compact on the <absolutely summable sequence space>. The <norm> of a <matrix> operator on $\ell^1$ is the supremum of its absolute column sums: the <triangle inequality> gives one bound and coordinate unit vectors give the other. Here the column sum is $c_m=\sum_{n=1}^{m-1}n!/m!\leq2/m$, using $\sum_{n=1}^kn!\leq2k!$. The latter inequality follows by induction. Retaining the first $N$ columns gives a <finite-rank operator> and a remainder of <norm> at most $2/(N+1)$. A <norm> limit of finite-rank operators is a <compact operator>, proving the assertion. This column-decay method can prove compactness even when each row has infinitely many nonzero entries.
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