Inverses of disjointness preserving operators

Denny H. Leung¹ ; Lei Li² ; Ya-Shu Wang³

¹ Department of Mathematics National University of Singapore Singapore 119076
² School of Mathematical Sciences and LPMC Nankai University Tianjin, 300071, China
³ Department of Applied Mathematics National Chung Hsing University Taichung 402, Taiwan

Studia Mathematica, Tome 234 (2016) no. 3, pp. 217-240

Résumé

A linear operator between (possibly vector-valued) function spaces is disjointness preserving if it maps disjoint functions to disjoint functions. Here, two functions are said to be disjoint if at each point at least one of them vanishes. In this paper, we study linear disjointness preserving operators between various types of function spaces, including spaces of (little) Lipschitz functions, uniformly continuous functions and differentiable functions. It is shown that a disjointness preserving linear isomorphism whose domain is one of these types of spaces (scalar-valued) has a disjointness preserving inverse, subject to some topological conditions on the range space. A representation for a general linear disjointness preserving operator on a space of vector-valued $C^p$ functions is also given.

Détail

DOI : 10.4064/sm8445-5-2016

Keywords: linear operator between possibly vector valued function spaces disjointness preserving maps disjoint functions disjoint functions here functions said disjoint each point least vanishes paper study linear disjointness preserving operators between various types function spaces including spaces little lipschitz functions uniformly continuous functions differentiable functions shown disjointness preserving linear isomorphism whose domain these types spaces scalar valued has disjointness preserving inverse subject topological conditions range space representation general linear disjointness preserving operator space vector valued functions given