Compact differences of composition operators on Bloch and Lipschitz spaces. (English) Zbl 1146.47016

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 T.Hosokawa and 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.


47B33 Linear composition operators
30D45 Normal functions of one complex variable, normal families
47B07 Linear operators defined by compactness properties


Zbl 1174.47019
Full Text: DOI


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