Multiple summing operators on $l_{p} $ spaces

Dumitru Popa¹

¹ Department of Mathematics Ovidius University of Constanţa Bd. Mamaia 124 900527 Constanţa, Romania

Studia Mathematica, Tome 225 (2014) no. 1, pp. 9-28

Résumé

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 ($1 \leq s \leq 2$).

Détail

DOI : 10.4064/sm225-1-2

Keywords: maurey rosenthal factorization theorem obtain characterization multiple summing operators product spaces characterization multiple s summing operators product spaces values hilbert space characterized boundedness natural multilinear functional leq leq these results there exist many natural multiple s summing operators times rightarrow none associated linear operators s summing leq leq further geq there exist natural bounded multilinear operators times cdots times rightarrow which none associated multilinear operators multiple s summing leq leq