Compact composition operators on certain analytic Lipschitz spaces (Q1954007)

In this paper, the authors study necessary conditions for a composition operator on certain Lipschitz spaces of analytic functions on the closed unit disc to be compact. For \(\phi\) a selfmap of \(\overline{\mathbb D}\), it was proved in [\textit{F. Behrouzi} and the second author, Bull. Iran. Math. Soc. 30, No. 1, 1--11 (2004; Zbl 1070.46036)] that \(\phi(\overline{\mathbb D}) \subseteq \mathbb D\) is a sufficient condition for the compactness of the composition operator \(C_{\phi}\) on \(\mathrm{Lip}_{A}(\overline{\mathbb D}, \alpha)\) and on \(\mathrm{Lip}^n(\overline{\mathbb D}, \alpha)\) when \(0< \alpha \leq 1\), where \(\mathrm{Lip}_A(\overline{\mathbb D}, \alpha)\) is given by \(\mathrm{Lip}(\overline{\mathbb D}, \alpha)\) intersecting with a Banach function algebra \(A\) of continuous complex-valued functions. In the same paper, the condition was shown to be necessary when \(\alpha= 1\). In the paper under review, the authors modify the problem to study the compactness of the operator \(C_{\varphi}\) on a Bloch type space for some \(\varphi: \mathbb D \to \mathbb D\), and provide a different approach to show that \(\phi(\overline{\mathbb D})\subseteq \overline{\mathbb D}\) is necessary for the compactness of \(C_{\phi}\) on \(\mathrm{Lip}_A(\overline{\mathbb D}, \alpha)\) in the case \(0< \alpha< 1\). By looking at an equivalent problem in the compactness of a weighted composition operator \(\varphi' C_{\varphi}\) on a Bloch type space, the authors show that \(\phi(\overline{\mathbb D})\subseteq \overline{\mathbb D}\) is also necessary for the compactness of \(C_{\phi}\) on \(\mathrm{Lip}^n(\overline{\mathbb D}, \alpha)\) in the case \(0< \alpha< 1\). Their approach yields a characterisation of the compactness of \(C_{\varphi}\) on Zygmund type spaces as well. Knowing a function which is continuous on \(\overline{\mathbb D}\) and analytic on \(\mathbb D\) is in \(\mathrm{Lip}(\partial\mathbb D, \alpha)\) if and only if it is in \(\mathrm{Lip}(\overline{\mathbb D}, \alpha)\), the same results with different approaches can be found in [\textit{B. R. Choe} et al., Integral Equations Oper. Theory 56, No. 3, 357--380 (2006; Zbl 1114.47028)].