## 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.

### MSC:

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

### Keywords:

composition operators; compactness; Bloch space

Zbl 1174.47019
Full Text:

### References:

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