[Banach] Abstract of a paper by Denny H. Leung

This is an announcement for the paper "Biseparating maps on generalized Lipschitz spaces" by Denny H. Leung.

Abstract: Let $X, Y$ be complete metric spaces and $E, F$ be Banach spaces. If $A(X,E)$ and $A(Y,F)$ stand for certain spaces of functions from $X$ to $E$ and from $Y$ to $F$ respectively, a bijective linear operator $T: A(X,E) \to A(Y,F)$ is said to be biseparating if $f$ and $g \in A(X,E)$ are disjoint if and only if $Tf$ and $Tg$ are disjoint. When $A(X,E)$ and $A(Y,F)$ are either the space of Lipschitz functions of order $\alpha$, the space of little Lipschitz functions of order $\alpha$, or the space of uniformly continuous functions, every linear biseparating map between them is characterized as a weighted composition operator, 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 the possibility of having biseparating maps between different classes of spaces. 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 $Y$.

Archive classification: math.FA

Mathematics Subject Classification: 47B38

The source file(s), Lipschitz3.tex: 62347 bytes, is(are) stored in gzipped form as 0906.0221.gz with size 18kb. The corresponding postcript file has gzipped size 118kb.

Submitted from: matlhh@nus.edu.sg

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