Multiple summing operators on 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).