Biseparating maps on generalized Lipschitz spaces

Denny H. Leung. "Biseparating maps on generalized Lipschitz spaces." Studia Mathematica 196.1 (2010): 23-40. <http://eudml.org/doc/286158>.

@article{DennyH2010,
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 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 $Tf(y) = S_\{y\}(f(h^\{-1\}(y)))$ for a family of vector space isomorphisms $S_\{y\}: E → F$ 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.},
author = {Denny H. Leung},
journal = {Studia Mathematica},
keywords = {Banach space; generalized modulus of continuity; generalized Lipschitz space; biseparating operator; Lipschitz normal space},
language = {eng},
number = {1},
pages = {23-40},
title = {Biseparating maps on generalized Lipschitz spaces},
url = {http://eudml.org/doc/286158},
volume = {196},
year = {2010},
}

TY - JOUR
AU - Denny H. Leung
TI - Biseparating maps on generalized Lipschitz spaces
JO - Studia Mathematica
PY - 2010
VL - 196
IS - 1
SP - 23
EP - 40
AB - 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 $Tf(y) = S_{y}(f(h^{-1}(y)))$ for a family of vector space isomorphisms $S_{y}: E → F$ 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.
LA - eng
KW - Banach space; generalized modulus of continuity; generalized Lipschitz space; biseparating operator; Lipschitz normal space
UR - http://eudml.org/doc/286158
ER -