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February, 2019 Norm-attaining Composition Operators on Lipschitz Spaces
Antonio Jiménez-Vargas
Taiwanese J. Math. 23(1): 129-144 (February, 2019). DOI: 10.11650/tjm/180508


Every composition operator $C_{\varphi}$ on the Lipschitz space $\operatorname{Lip}_0(X)$ attains its norm. This fact is essentially known and we give in this paper a sequential characterization of the extremal functions for the norm of $C_{\varphi}$ on $\operatorname{Lip}_0(X)$. We also characterize the norm-attaining composition operators $C_{\varphi}$ on the little Lipschitz space $\operatorname{lip}_0(X)$ which separates points uniformly and identify the extremal functions for the norm of $C_{\varphi}$ on $\operatorname{lip}_0(X)$. We deduce that compact composition operators on $\operatorname{lip}_0(X)$ are norm-attaining whenever the sphere unit of $\operatorname{lip}_0(X)$ separates points uniformly. In particular, this condition is satisfied by spaces of little Lipschitz functions on Hölder compact metric spaces $(X,d^{\alpha})$ with $0 \lt \alpha \lt 1$.


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Antonio Jiménez-Vargas. "Norm-attaining Composition Operators on Lipschitz Spaces." Taiwanese J. Math. 23 (1) 129 - 144, February, 2019.


Received: 23 January 2018; Accepted: 28 May 2018; Published: February, 2019
First available in Project Euclid: 9 June 2018

zbMATH: 07021721
MathSciNet: MR3909993
Digital Object Identifier: 10.11650/tjm/180508

Primary: 47B33, 47B38

Rights: Copyright © 2019 The Mathematical Society of the Republic of China


Vol.23 • No. 1 • February, 2019
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