Weakly compact weighted composition operators on pointed Lipschitz spaces

‎Let (X,d) be a pointed compact metric space with the base point x_{۰} and let ‎‎\Lip((X,d),x_{۰})‎ ‎‎‎(\lip((X,d),x_{۰}))‎ denote the pointed (little) Lipschitz space on ‎‎(X,d)‎‎. ‎In ‎this ‎paper,‎ ‎we prove that every weakly compact composition operator u C_{\varphi} on \Lip((X,d)‎, ‎x_{۰}) is compact provided that \lip((X,d),x_{۰}) has the uniform separation property‎, ‎{\varphi} is a base point preserving Lipschitz self-map of X and u \in \Lip(X,d) with u(x) \neq۰ for all x \in X \backslash \{x_{۰}\}.‎‎‎‎

Keywords:

Compact operator‎ , ‎Lipschitz space‎ , ‎weakly compact operator‎ , ‎Weighted composition operator