Range preserving maps between the spaces of continuous functions with values in a locally convex space

TL;DR

This paper characterizes maps between spaces of continuous functions into a locally convex space that preserve the range of differences, extending understanding of structure-preserving transformations in functional analysis.

Contribution

It provides a complete characterization of range-preserving maps between spaces of continuous functions with values in a locally convex space.

Findings

Identifies conditions under which such maps preserve the range of differences.

Establishes a structural description of these maps in terms of composition and linear operators.

Extends previous results from scalar to locally convex space-valued functions.

Abstract

Let $C(X,E)$ be the linear space of all continuous functions on a compact Hausdorff space $X$ with values in a locally convex space $E$. We characterize maps $T:C(X,E)\to C(Y,E)$ which satisfy $\mathrm{Ran}(TF-TG)\subset\mathrm{Ran}(F-G)$ for all $F,G\in C(X,E)$.