Joint Compact-Friendly Multiplication Operators

Let $X$ be a Banach function space and $M=(M_{h_1},\dots,M_{h_N})$ an $N-$tuple of positive multiplication operators. In this article, we characterize joint compact-friendly operators $M=(M_{h_1},\dots,M_{h_N})$ acting on $X$. Concretely, we will see that $M=(M_{h_1},\dots,M_{h_N})$ is joint compact-friendly if and only if $h_1,\dots,h_N$ have a common flat.