Taiwanese Journal of Mathematics

Norm-attaining Composition Operators on Lipschitz Spaces

Antonio Jiménez-Vargas

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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 $0 \lt \alpha \lt 1$.

Article information

Taiwanese J. Math., Volume 23, Number 1 (2019), 129-144.

Received: 23 January 2018
Accepted: 28 May 2018
First available in Project Euclid: 9 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B33: Composition operators 47B38: Operators on function spaces (general)

composition operator compact operator norm-attaining operator Lipschitz function little Lipschitz function


Jiménez-Vargas, Antonio. Norm-attaining Composition Operators on Lipschitz Spaces. Taiwanese J. Math. 23 (2019), no. 1, 129--144. doi:10.11650/tjm/180508. https://projecteuclid.org/euclid.twjm/1528509850

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