Biseparating Maps on Generalized Lipschitz Spaces
نویسنده
چکیده
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) = Sy(f(h (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 .
منابع مشابه
Automatic Continuity and Weighted Composition Operators between Spaces of Vector-valued Differentiable Functions
Let E and F be Banach spaces. It is proved that if Ω and Ω are open subsets of R and R , respectively, and T is a linear biseparating map between two spaces of differentiable functions A(Ω, E) and A(Ω, F ), then p = q, n = m, and there exist a diffeomorphism h of class C from Ω onto Ω, and a map J : Ω → L(E, F ) of class s−C such that for every y ∈ Ω and every f ∈ A(Ω, E), (Tf)(y) = (Jy)(f(h(y)...
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