Jointly separating maps between vector-valued function 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.