Equivalences between certain properties of weighted Lipschitz operators

TL;DR

This paper establishes equivalences between key properties of weighted Lipschitz operators, linking compactness, strict singularity, and cosingularity to their action on complemented copies of ll^1, and extends these results to a broader class of operators.

Contribution

It proves the equivalence of several important properties for weighted Lipschitz operators and generalizes the results to operators preserving finitely supported elements.

Findings

Compactness, strict singularity, and strict cosingularity are equivalent for weighted Lipschitz operators.

These properties are characterized by the operators not fixing any complemented ll^1.

The results are extended to operators between Lipschitz-free spaces that preserve finitely supported elements.

Abstract

We show that for a weighted Lipschitz operator $\omega\widehat{f}$, 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.