Extrapolation theorems for (p,q)-factorable operators

Abstract

The operator ideal of $(p,q)$-factorable operators can be characterized as the class of operators that factors through the embedding $L^{q\text{'}}\left(\mu \right)\hookrightarrow L^p\left(\mu \right)$ for a finite measure $\mu$, where $p,q\in [1,\infty )$ are such that $1/p+1/q\ge 1$. We prove that this operator ideal is included into a Banach operator ideal characterized by means of factorizations through $r$th and $s$th power factorable operators, for suitable $r,s\in [1,\infty )$. Thus, they also factor through a positive map $L^s(m_1{)}^{*}\to L^r(m_2)$, where $m_1$ and $m_2$ are vector measures. We use the properties of the spaces of $u$-integrable functions with respect to a vector measure and the $u$th power factorable operators to obtain a characterization of $(p,q)$-factorable operators and conditions under which a $(p,q)$-factorable operator is $r$-summing for $r\in [1,p]$.