Nonvanishing preservers and compact weighted composition operators between spaces of Lipschitz functions. (English) Zbl 1470.46063

Chen, Dongyang; Li, Lei; Wang, Risheng; Wang, Ya-Shu

Abstr. Appl. Anal. 2013, Article ID 741050, 8 p. (2013).

Summary: We will give the \(\alpha\)-Lipschitz version of the Banach-Stone type theorems for lattice-valued \(\alpha\)-Lipschitz functions on some metric spaces. In particular, when \(X\) and \(Y\) are bounded metric spaces, if \(T : \operatorname{Lip} \left(X\right) \rightarrow \operatorname{Lip} \left(Y\right)\) is a nonvanishing preserver, then \(T\) is a weighted composition operator \(T f = h \cdot f \circ \varphi\), where \(\varphi : Y \rightarrow X\) is a Lipschitz homeomorphism. We also characterize the compact weighted composition operators between spaces of Lipschitz functions.

MSC:

46E40	Spaces of vector- and operator-valued functions
47B33	Linear composition operators

References:

[1]	Garrido, M. I.; Jaramillo, J. A., Variations on the Banach-Stone theorem, Extracta Mathematicae, 17, 3, 351-383 (2002) · Zbl 1027.54021
[2]	Jarosz, K.; Pathak, V. D.; Jarosz, K., Isometries and small bound isomorphisms of function spaces, Function Spaces. Function Spaces, Lecture Notes in Pure and Applied Mathematics, 136, 241-271 (1992), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0804.46030
[3]	Kamowitz, H., Compact weighted endomorphisms of \(C \left(X\right)\), Proceedings of the American Mathematical Society, 83, 3, 517-521 (1981) · Zbl 0509.47026
[4]	Cao, J.; Reilly, I.; Xiong, H., A lattice-valued Banach-Stone theorem, Acta Mathematica Hungarica, 98, 1-2, 103-110 (2003) · Zbl 1027.46025 · doi:10.1023/A:1022861506272
[5]	Chen, J.-X.; Chen, Z. L.; Wong, N., A banach-stone theorem for riesz isomorphisms of banach lattices, Proceedings of the American Mathematical Society, 136, 11, 3869-3874 (2008) · Zbl 1160.46026 · doi:10.1090/S0002-9939-08-09582-8
[6]	Ercan, Z.; Önal, S., Banach-stone theorem for Banach lattice valued continuous functions, Proceedings of the American Mathematical Society, 135, 9, 2827-2829 (2007) · Zbl 1127.46026 · doi:10.1090/S0002-9939-07-08788-6
[7]	Ercan, Z.; Önal, S., The Banach-Stone theorem revisited, Topology and its Applications, 155, 16, 1800-1803 (2008) · Zbl 1166.46309 · doi:10.1016/j.topol.2008.05.018
[8]	Miao, X.; Cao, J.; Xiong, H., Banach-Stone theorems and Riesz algebras, Journal of Mathematical Analysis and Applications, 313, 1, 177-183 (2006) · Zbl 1102.46017 · doi:10.1016/j.jmaa.2005.08.050
[9]	Garrido, M. I.; Jaramillo, J. A., Homomorphisms on function lattices, Monatshefte fur Mathematik, 141, 2, 127-146 (2004) · Zbl 1058.54008 · doi:10.1007/s00605-002-0011-4
[10]	Garrido, M. I.; Jaramillo, J. A., Lipschitz-type functions on metric spaces, Journal of Mathematical Analysis and Applications, 340, 1, 282-290 (2008) · Zbl 1139.46025 · doi:10.1016/j.jmaa.2007.08.028
[11]	Weaver, N., Lattices of Lipschitz functions, Pacific Journal of Mathematics, 164, 179-193 (1994) · Zbl 0797.46007 · doi:10.2140/pjm.1994.164.179
[12]	Jiménez-Vargas, A.; Villegas-Vallecillos, M., Order isomorphisms of little Lipschitz algebras, Houston Journal of Mathematics, 34, 4, 1185-1195 (2008) · Zbl 1169.46010
[13]	Jiménez-Vargas, A.; Campoy, A. M.; Villegas-Vallecillos, M., The uniform separation property and banach-stone theorems for lattice-valued lipschitz functions, Proceedings of the American Mathematical Society, 137, 11, 3769-3777 (2009) · Zbl 1202.46043 · doi:10.1090/S0002-9939-09-09941-9
[14]	Dubarbie, L., Maps preserving common zeros between subspaces of vector-valued continuous functions, Positivity, 14, 4, 695-703 (2010) · Zbl 1220.47042 · doi:10.1007/s11117-010-0046-z
[15]	Araujo, J.; Dubarbie, L., Biseparating maps between Lipschitz function spaces, Journal of Mathematical Analysis and Applications, 357, 1, 191-200 (2009) · Zbl 1169.47024 · doi:10.1016/j.jmaa.2009.03.065
[16]	Li, L.; Leung, D., Order isomorphisms on function space
[17]	Li, L.; Wong, N.-C., Kaplansky theorem of completely regular spaces
[18]	Weaver, N., Lipschitz Algebras (1999), Singapore: World Scientific Pbulishing, Singapore · Zbl 0936.46002
[19]	Kamowitz, H.; Scheinberg, S., Some properties of endomorphisms of Lipschitz algebras, Studia Mathematica, 96, 3, 383-391 (1990)
[20]	Jiménez-Vargas, A.; Villegas-Vallecillos, M., Compact composition operators on noncompact Lipschitz spaces, Journal of Mathematical Analysis and Applications, 398, 1, 221-229 (2013) · Zbl 1279.47042 · doi:10.1016/j.jmaa.2012.08.044