A lifting theorem for operators between Lipschitz spaces

2026/05/04 by Leandro Candido
Mathematics · #math.FA

paper · pdf

Abstract

We prove that every bounded linear operator between Lipschitz spaces admits a lifting along the de Leeuw embedding. More precisely, given pointed metric spaces M and N and ε>0, every bounded linear operator S:Lip₀(M)→ Lip₀(N) admits a lifting \mathfrakS:C(β\widetildeM)→ C(β\widetildeN) such that ‖\mathfrakS‖≤ ‖S‖+ε and \mathfrakS(\varPhi_M(f))=\varPhi_N(S(f)) for every f∈ Lip₀(M). Moreover, compact operators admit compact liftings.

Related

• Compact and weakly compact Lipschitz operators
• Lipschitz geometry of operator spaces and Lipschitz-free operator spaces
• Equivariant liftings in Lipschitz-free spaces
• Closed linear spaces consisting of strongly norm attaining Lipschitz mappings
• Lipschitz Embeddings of Metric Spaces into c₀