Equivalences between certain properties of weighted Lipschitz operators

Abstract

We show that for a weighted Lipschitz operator $\omega \hat{f}$, certain linear properties are equivalent. Specifically, we prove that compactness, strict singularity, and strict cosingularity are all equivalent to the property of not fixing any complemented copy of ${\ell }^1$. Then we generalize this result to operators between Lipschitz-free spaces that preserve finitely supported elements, a larger class of operators.

20 pages