Compact positive multilinear operators on Banach lattices

Abstract

Let $1<p_1,\dots ,p_n<\infty ,1\le q<\infty$ be such that $\sum _{i=1}^n\frac{1}{p_i}<\frac{1}{q}$ and let $\hat{I}{¼}_1,\dots ,\hat{I}{¼}_n,\hat{I}½$ be arbitrary measures. Generalizing known linear and multilinear results, we prove that all positive $n$-linear operators from ${\ell }_{p_1}\times \cdots \times {\ell }_{p_n}$ to $L_q(\hat{I}½)$ and from $L_{p_1}(\hat{I}{¼}_1)\times \cdots \times L_{p_1}(\hat{I}{¼}_n)$ to ${\ell }_q$ are compact. This result, along with other related ones concerning free Banach lattices, shall emerge as consequences of some facts we prove about $M$-weakly compact multilinear operators on Banach lattices.

19 pages