Compact differences of composition operators on Bloch and Lipschitz spaces
For \(0<\alpha \leq 1\), let \({\mathcal B}^\alpha\) denote the space of analytic functions on the unit disc \({\mathbb D}\) such that \(\sup_{| z| <1}(1-| z| ^2)^\alpha | f'(z)| <\infty\), that is, the Bloch space for \(\alpha=1\) and the analytic Lipschitz classes for \(0<\alpha<1\). Using the notation \({\mathcal D}^\alpha\phi(z)=(\frac{1-| z| ^2}{1-| \phi(z)| ^2})^\alpha \phi'(z)\), \({\mathcal D}^1= {\mathcal D}\) and \({\mathcal B}^1= {\mathcal B}\), the author shows that, in the case when \(\alpha=1\), the difference of composition operators \(C_\phi -C_\psi\) is (weakly) compact on \(\mathcal B\) if and only if \(\lim_{| \phi(z)| \to 1}{\mathcal D}\phi(z)\rho(\phi(z),\psi(z))=0\) and \(\lim_{| \psi(z)| \to 1}{\mathcal D}\psi(z)\rho(\phi(z),\psi(z))=0\), where \(\rho(w_1,w_2)\) stands for the pseudo-hyperbolic metric, improving the result by \textit{T.\,Hosokawa} and \textit{S.\,Ohno} [J.~Oper.\ Theory 57, No.\,2, 229--242 (2007; Zbl 1174.47019)] where an extra condition was assumed. In the case when \(0<\alpha<1\) and assuming that \(\| \mathcal D\phi\| _\infty<\infty\) and \(\| \mathcal D\psi\| _\infty<\infty\), the (weak) compactness of \(C_\phi- C_\psi\) in \({\mathcal B}^\alpha\) is equivalent to \(\lim_{| \phi(z)| \to 1}{\mathcal D}^\alpha \phi(z)\rho(\phi(z),\psi(z))=0\), \(\lim_{| \psi(z)| \to 1}{\mathcal D}^\alpha \psi(z)\rho(\phi(z),\psi(z))=0\) and \(\lim_{\min\{| \phi(z)| , | \psi(z)| \} \to 1}{\mathcal D}^\alpha \phi(z)-{\mathcal D}^\alpha \psi(z)=0\). It is shown that the extra condition is necessary in the case \(0<\alpha<1\). The author uses a unified approach based on the difference of weighted composition operators acting between two weighted \(H^\infty\)-spaces. Some complementary remarks on the compactness of a single composition operator on Lipschitz spaces are also provided.
scientific article; zbMATH DE number 5281672
scientific article