Jointly separating maps between vector-valued function spaces

TL;DR

This paper characterizes jointly separating linear operators between vector-valued function spaces, revealing their structure and how they induce homeomorphisms between underlying spaces.

Contribution

It provides a comprehensive description of jointly separating maps for various classes of vector-valued function spaces, including Lipschitz, absolutely continuous, and differentiable functions.

Findings

Characterization of the form of jointly separating operators.

Application to operators on Banach function algebras.

Induction of homeomorphisms between underlying spaces.

Abstract

Let $X$ and $Y$ be compact Hausdorff spaces, $E$ and $F$ be real or complex Banach spaces, and $A(X,E)$ be a subspace of $C(X,E)$. In this paper we study linear operators $S,T: A(X,E) \lo C(Y,F)$ which are jointly separating, in the sense that $\coz(f) \cap \coz(g) = \emptyset$ implies that $\coz(Tf) \cap \coz(Sg)=\emptyset$. Here $\coz(\cdot)$ denotes the cozero set of a function. We characterize the general form of such maps between certain class of vector-valued (as well as scalar-valued) spaces of continuous functions including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions. The results can be applied for a pair $T:A(X) \lo A(X)$ and $S:A(X,E) \lo A(X,E)$ of linear operators, where $A(X)$ is a regular Banach function algebra on $X$, such that $f\cdot g=0$ implies $Tf \cdot Sg=0$, for $f\in A(X)$ and $g\in A(X,E)$. If $T$ and $S$ are jointly separating bijections between Banach algebras of scalar-valued functions of this class, then they induce a homeomorphism between $X$ and $Y$ and, furthermore, $T^{-1}$ and $S^{-1}$ are also jointly separating maps.