Quasicompact and Riesz composition operators on Banach spaces of Lipschitz functions on pointed metric spaces (Q2193484)

Let \((X,d)\) be a metric space, \(x_0\in X\) a given point and \(\alpha\in \, ]0,1]\). Consider \(B\), the Banach algebra of Lipschitz bounded complex functions \(f\) on \((X,d^\alpha)\) endowed with the usual Lipschitz norm, that is, \[\|f\|_{L}=\sup\{|f(x)|:x\in X\} + \sup \left\{\frac{|f(x)-f(y)|}{d^\alpha(x,y)}:x,y\in X,\; x\neq y \right\},\] and \(A:=\operatorname{ker}(\delta_{x_0})\) the ideal in \(B\) of the functions vanishing at \(x_0\). The subject of the paper is the study of composition operators \(C_\varphi\) acting on \(A\) that arise from Lipschitz mappings \(\varphi:(X,d) \to (X,d)\) that fix \(x_0\). Let \(L(\varphi)=\sup \left\{\frac{d(\varphi(x)),\varphi((y))}{d(x,y)}:x,y\in X, \;x\neq y \right\} \) denote the Lipschitz constant of \(\varphi\). If \((X,d)\) is compact, \(\alpha\in \,]0,1[,\) and \(L(\varphi_k)<1\) for some \(k\)-times iteration of \(\varphi,\; \varphi_k,\) then the authors show that the essential spectral radius of \(C_\varphi\) coincides with \(\lim_n L(\varphi_n)^{\alpha/n}.\) If the existence of iterations of \(\varphi\) with Lipschitz constants less than \(1,\) can be assured for a suitable finite decomposition of \(X\) by open sets, a formula is proved for \(r_e(C_\varphi)\). Such formula becomes an estimate if \(\alpha=1\). Next, the authors compare properties of \(C_\varphi\) and its natural extension \(\tilde{C}_\varphi\) acting on \(B\). For instance, under the assumptions quoted above, \(r_e(\tilde{C}_\varphi)= r_e(C_\varphi)\). If, additionally, \((X,d)\) is connected, \(\tilde{C}_\varphi\) is quasicompact (resp. Riesz) if and only if \(C_\varphi\) is quasicompact (resp. Riesz). The final section is devoted to spectral properties of \(C_\varphi\). Among the results: if \((X,d)\) is compact, \(\alpha\in\, ]0,1[,\) and \(C_\varphi\) is quasicompact, then its spectrum lies inside \(\{\lambda: |\lambda|\leq r_e(C_\varphi)\}\cup \bigcup_{n\in L}\{\lambda: \lambda^n =1\}\) where \(L\) is the (so proved) finite set of \(n\in \mathbb{N}\) such that \(\varphi\) has a fixed point of order \(n,\) that is, a fixed point for \(\varphi_n\) that is not fixed for any \(\varphi_m\) if \(m<n\).