A Banach function space $X$ for which all operators from $\ell^p$ to $X$ are compact

Abstract

We construct a rearrangement invariant space $X$ on $[0,1]$ with the property that all bounded linear operators from $\ell^p$, $1<p<\infty$, to $X$ are compact, but there exists a non-compact operator from $\ell^\infty$ to $X$. The techniques used allow to give a new proof of the characterization given by Hern\'andez, Raynaud and Semenov of the rearrangement invariant spaces on $[0,1]$ for which the canonical embedding into $L^1([0,1])$ is finitely strictly singular.