Abstract
Given pointed metric spaces X and Y, we characterize the basepoint-preserving Lipschitz maps \(\phi \) from Y to X inducing an isometric composition operator \(C_\phi \) between the Lipschitz spaces \(\mathrm {Lip}_0(X)\) and \(\mathrm {Lip}_0(Y)\), whenever X enjoys the peak property. This gives an answer to a question posed by Weaver in his book [Lipschitz algebras. Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018].
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References
Aliaga, R.J., Guirao, A.J.: On the preserved extremal structure of Lipschitz-free spaces. Studia Math. 245(1), 1–14 (2019)
Colonna, F.: Characterisation of the isometric composition operators on the Bloch space. Bull. Aust. Math. Soc. 72, 283–290 (2005)
García-Lirola, L., Procházka, A., Rueda Zoca, A.: A characterisation of the Daugavet property in spaces of Lipschitz functions. J. Math. Anal. Appl. 464(1), 473–492 (2018)
A. Jiménez-Vargas, MR3792558 (this is the review of [8])
Martín, M., Vukotić, D.: Isometries of the Bloch space among the composition operators. Bull. Lond. Math. Soc. 39, 151–155 (2007)
Mayer-Wolf, E.: Isometries between Banach spaces of Lipschitz functions. Isr. J. Math. 38, 58–74 (1981)
Weaver, N.: Lipschitz Algebras. World Scientific Publishing Co., River Edge (1999)
Weaver, N.: Lipschitz algebras, Second edn. World Scientific Publishing Co., Hackensack (2018)
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Jiménez-Vargas, A. Isometric Composition Operators on Lipschitz Spaces. Mediterr. J. Math. 17, 52 (2020). https://doi.org/10.1007/s00009-020-1488-6
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DOI: https://doi.org/10.1007/s00009-020-1488-6