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 T f and T g 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 T f (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 .2000 Mathematics Subject Classification. 46E15; 47B38; 47B33.

Compact composition operators on noncompact Lipschitz spaces

“…and Weaver [3] in their studies of the representation of surjective linear isometries of Lip(X , d) by means of weighted composition operators, and those of Araujo and Dubarbie [4] and Leung [5] on biseparating maps of Lipschitz spaces. Our approach lies in tackling the problem for pointed Lipschitz spaces Lip 0 that generalize the Lipschitz spaces Lip.…”

How Lipschitz Functions Characterize the Underlying Metric Spaces

“…In [15], Leung defined a new class of spaces, the generalized Lipschitz function space Lip Σ (X, E). We say that σ : [0, +∞) → [0, +∞] is a modulus function if σ is nondecreasing, σ(0) = 0, and σ is continuous at 0.…”

“…In the sequel, all generalized Lipschitz function spaces Lip Σ (X, E) are assumed to be Lipschitz normal. In particular, Z(Lip Σ (X)) = Z(X) ( [15,Lemma 3]).…”

“…In 2009, Araujo and Dubarbie [7] showed that if there is a linear biseparating map between spaces of vector-valued bounded Lipschitz functions on complete metric spaces X and Y , then X and Y are Lipschitz homeomorphic. In 2010, Leung [15] extended this to generalized Lipschitz function spaces, and provided a (topological) homeomorphism between X and Y . On the other hand, Jiménez-Vargas, Villegas-Vallecillos, and Wang [13,14] worked on the same problem for spaces of vector-valued little Lipschitz functions defined on locally compact metric spaces X and Y and showed that X and Y are locally Lipschitz homeomorphic if the biseparating map is continuous.…”

Nonlinear biseparating maps

“…Moreover, generalization to disjointness preserving or biseparating maps allows for extension to function spaces that are neither algebras nor lattices, and even to vector-valued functions. Copius research has been devoted to the study of disjointness preserving and biseparating maps on various function spaces; see, e.g., [1]- [8], [13], [16], [20]- [26], [29]. As far as the authors are aware of, the study of biseparating maps thus far has been confined to linear or at least additive maps.…”