Biseparating maps on generalized Lipschitz spaces

Denny H. Leung ¹

¹ Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543

Studia Mathematica, Tome 196 (2010) no. 1, pp. 23-40

Résumé

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 $Y$.

Détail

DOI : 10.4064/sm196-1-3

Keywords: complete metric spaces banach spaces bijective linear operator space e valued functions space f valued functions nbsp said biseparating disjoint only disjoint introduce class generalized lipschitz spaces which includes special cases classes lipschitz little lipschitz uniformly continuous functions linear biseparating maps between generalized lipschitz spaces characterized weighted composition operators form h family vector space isomorphisms homeomorphism investigate continuity related questions here functions involved metric spaces may unbounded arguments require compactification spaces nbsp