Equivalences between certain properties of weighted Lipschitz operators

We show that for a weighted Lipschitz operator ωf, 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 ¹. Then we generalize this result to operators between Lipschitz-free spaces that preserve finitely supported elements, a larger class of operators.