Multiple summing operators on $l_{p} $ spaces
Volume 225 / 2014
Abstract
We use the Maurey–Rosenthal factorization theorem to obtain a new characterization of multiple $2$-summing operators on a product of $l_{p} $ spaces. This characterization is used to show that multiple $s$-summing operators on a product of $l_{p} $ spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional ($1 \leq s \leq 2$). We use these results to show that there exist many natural multiple $s$-summing operators $T:l_{4/3}\times l_{{4}/{3}}\rightarrow l_{2} $ such that none of the associated linear operators is $s$-summing ($1 \leq s \leq 2$). Further we show that if $n\geq 2$, there exist natural bounded multilinear operators $T:l_{{2n}/{(n+1)}}\times \cdots \times l_{{2n}/{(n+1)}}\rightarrow l_{2} $ for which none of the associated multilinear operators is multiple $s$-summing ($1 \leq s \leq 2$).
