Biseparating maps on generalized Lipschitz spaces

TL;DR

This paper characterizes linear biseparating maps between generalized Lipschitz spaces of vector-valued functions as weighted composition operators, extending understanding of structure and continuity without compactness assumptions.

Contribution

It introduces generalized Lipschitz spaces and characterizes all linear biseparating maps as weighted composition operators, broadening the scope beyond classical Lipschitz spaces.

Findings

Biseparating maps are weighted composition operators.

Characterization holds for unbounded functions and spaces.

Continuity of operators is also analyzed.

Abstract

Let $X, Y$ be complete metric spaces and $E, F$ be Banach spaces. A bijective linear operator from a space of $E$-valued functions on $X$ to a space of $F$-valued functions on $Y$ is said to be biseparating if $f$ and $g$ are disjoint if and only if $Tf$ and $Tg$ are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uniformly continuous functions. Linear biseparating maps between generalized Lipschitz spaces are characterized as weighted composition operators, i.e., of the form $Tf(y) = S_y(f(h^{-1}(y))$ for a family of vector space isomorphisms $S_y: E \to F$ and a homeomorphism $h : X\to Y$. We also investigate the continuity of $T$ and related questions. Here the functions involved (as well as the metric spaces $X$ and $Y$) may be unbounded. Also, the arguments do not require the use of compactification of the spaces $X$ and $Y$.