Positive $p$-summing operators on $L_ p$-spaces

Abstract:

It is shown that for any Banach space $B$ every positive $p$-summing operator from ${L^{p’}}(\mu )$ in $B$, $1/p + 1/p’ = 1$, is also cone absolutely summing. We also prove here that a necessary and sufficient condition that $B$ has the Radon-NikodṔm property is that every positive $p$-summing operator $T:{L^{p’}}(\mu ) \to B$ is representable by a function $f$ in ${L^p}(\mu ,B)$.