Equivalences between certain properties of weighted Lipschitz operators

TL;DR

The paper proves that for weighted Lipschitz operators between Lipschitz-free spaces, a suite of norm and structural properties—compactness, weak compactness, strict (co)singularity, and the absence of complemented copies of $\ell^1$—are all equivalent. The core result leverages Pełczyński’s framework and the Lipschitz-free/Lipshitz-operator duality by showing $\omega\widehat{f}$ is compact if and only if it is not fixing any complemented copy of $\ell^1$, with corresponding adjoint statements for $\omega C_f$. A key contribution is extending these equivalences to a broader class of operators that preserve finitely supported elements, thereby unifying several notions of compactness and operator regularity within the Lipschitz-free setting. The findings illuminate how metric and combinatorial properties of the underlying spaces translate into operator-theoretic characteristics, with potential implications for the analysis of $L^1$-type structures and transport-like operators in Lipschitz-free spaces.

Abstract

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

Figures (1)

• Figure 1: Diagram of implications

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: McShane-Whitney's extension theorem
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Proposition 2.5
• Definition 2.6
• Definition 2.7
• Remark 2.8