An Application of Kadets–Pełczyński Sets to Narrow Operators

A known analogue of the Pitt compactness theorem for function spaces asserts that if $1 \leq p < 2$ and $p < r < \infty$, then every operator $T:L_p \to L_r$ is narrow. Using a technique developed by M. I. Kadets and A. Pełczyński, we prove a similar result. More precisely, if $1 \leq p \leq 2$ and $F$ is a Köthe–Banach space on $[0,1]$ with an absolutely continuous norm containing no isomorph of $L_p$ such that $F \subset L_p$, then every regular operator $T: L_p \to F$ is narrow.