 Absolutely p-summing operator

For $1\le p<\infty$, a bounded linear operator $T:E\to F$ between Banach spaces is absolutely p-summing if $({\sum }_j\Vert T{x}_j{\Vert }^p{)}^{1/p}\le C{\sup }_{\varphi \in B_{E^{\ast }}}({\sum }_j|\varphi (x_j){|}^p{)}^{1/p}$ for every finite family. The least $C$ is its p-summing norm ${\pi }_p(T)$. Direct substitution gives the ideal inequality ${\pi }_p(BTA)\le \Vert B\Vert {\pi }_p(T)\Vert A\Vert$ for compatible bounded linear operators.