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Continuous Operators from Spaces of Lipschitz Functions.

Christian Bargetz1, Jerzy Kąkol2,3, Damian Sobota4

  • 1Department of Mathematics, Universität Innsbruck, Innsbruck, Austria.

Results in Mathematics
|December 5, 2024
PubMed
Summary

This study investigates continuous operators between Lipschitz function spaces and their preduals, showing surjections are rare under weaker topologies. It also characterizes the Schur property for Lipschitz-free spaces.

Keywords:
Grothendieck spacesLipschitz-free spacesSpaces of Lipschitz functionscontinuous operatorscontinuous surjectionsdensityweak topologies

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Area of Science:

  • Functional Analysis
  • Metric Space Theory
  • Operator Theory

Background:

  • Lipschitz functions on metric spaces form Banach spaces with rich structures.
  • Understanding operator properties between function spaces is crucial in analysis.
  • Predual spaces offer alternative perspectives on Banach space properties.

Purpose of the Study:

  • To investigate the existence of continuous linear operators between Lipschitz spaces and related Banach spaces.
  • To explore conditions under which surjective operators do not exist.
  • To characterize the Schur property of Lipschitz-free spaces.

Main Methods:

  • Analysis of continuous (linear and nonlinear) operators.
  • Topological considerations on Banach spaces (norm vs. weaker topologies).
  • Investigation of metric space embeddings and their implications for operator theory.

Main Results:

  • Proved that continuous surjections between Lipschitz spaces and C(K)-spaces are generally non-existent when weaker topologies are used.
  • Established conditions for the existence of continuous operators onto specific Banach spaces.
  • Provided criteria for a metric space M implying that its Lipschitz space is not a Grothendieck space.
  • Derived a new characterization of the Schur property for Lipschitz-free spaces.

Conclusions:

  • The existence of continuous operators is highly dependent on the chosen topology.
  • Metric space properties significantly influence the operator theoretic characteristics of associated function spaces.
  • The Schur property of Lipschitz-free spaces is linked to weak sequential homeomorphisms with specific discrete spaces.