BISEPARATING MAPS ON GENERALIZED LIPSCHITZ SPACES

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 intro-duce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uniformly continu-ous functions. Linear biseparating maps between generalized Lipschitz spaces are characterized as weighted composition operators, i.e., of the form Tf(y) = Sy(f(h −1(y)) for a family of vector space isomorphisms Sy: E → F and a homeomorphism h: X → 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