2009/06/01 by Denny H. Leung, Leung, Denny H.
Mathematics · #47B38 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Rings, Modules, and Algebras #math.FA #msc:47B38
paper · pdf · doi:10.48550/arxiv.0906.0221
arxiv created 2009/06/01 · openalex publication_date 2009/06/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-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.