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Inner functions in the hyperbolic little Bloch class. (English) Zbl 0976.30018
The hyperbolic derivative of an analytic self-map $$\varphi\:{\mathcal D}\to{\mathcal D}$$ of the unit disk is given by $$|\varphi'|/(1-|\varphi|^2)$$. An analytic self-map $$\varphi$$ of $${\mathcal D}$$ induces a composition operator $$C_\varphi\:H({\mathcal D})\to H({\mathcal D})$$ defined by $$C_\varphi f=f\circ \varphi$$. Here $$H({\mathcal D})$$ is the space of analytic functions on $${\mathcal D}$$. An analytic function $$f$$ on $${\mathcal D}$$ is said to belong to the Bloch space $${\mathcal B}$$ provided that $$\sup_{z\in{\mathcal D}}(1-|z|^2)|f'(z)|<\infty$$. Similarly, $$f\in{\mathcal B}_0$$, the little Bloch space, if $$\lim_{|z|\to 1}(1-|z|^2)|f'(z)|=0$$. The hyperbolic Bloch class $${\mathcal B}^h$$ is defined in the following way: $$\varphi\in{\mathcal B}^h$$ if $$\varphi\:{\mathcal D}\to{\mathcal D}$$ is analytic and $\sup_{z\in{\mathcal D}}\tau_{\varphi}(z)= \sup_{z\in{\mathcal D}}\frac{(1-|z|^2)|\varphi'(z)|}{1-|\varphi(z)|^2}<\infty.$ Similarly, $$\varphi\in{\mathcal B}_0^h$$, the hyperbolic little Bloch class, if $$\varphi\in{\mathcal B}^h$$ and $\lim_{|z|\to 1}\tau_{\varphi}(z)=\lim_{|z|\to 1}\frac{(1-|z|^2)|\varphi'(z)|}{1-|\varphi(z)|^2}=0.$ It is well known that $$C_\varphi\:{\mathcal B}\to{\mathcal B}$$ is bounded for every analytic self-map of $${\mathcal D}$$, while $$C_\varphi\:{\mathcal B}_0\to{\mathcal B}_0$$ is bounded if and only if $$\varphi\in{\mathcal B}_0$$. Recently K. Madigan and A. Matheson [Trans. Am. Math. Soc. 347, No. 7, 2679-2687 (1995; Zbl 0826.47023)] proved that $$C_\varphi\:{\mathcal B}_0\to{\mathcal B}_0$$ is compact if and only if $$\varphi\in{\mathcal B}_0^h$$. Earlier in [Trans. Am. Math. Soc. 348, No. 6, 2331-2348 (1996; Zbl 0857.47020)] the author showed that if $$\varphi$$ is univalent and in $${\mathcal B}_0^h$$, then $$C_\varphi\:L_a^p\to H^q$$ is compact for all $$0<p<q<\infty$$. Here $$L_a^p$$ and $$H^q$$ are the classical Bergman and Hardy spaces. It is known that $${\mathcal B}_0^h$$ contains no finite Blaschke products which are special cases of inner functions. First in this paper the author shows that for an inner function $$\varphi$$ the expression $$\tau_{\varphi}(z)$$ satisfies $\int_{\mathcal D}\frac{\tau_{\varphi}(z)^2}{1-|z|^2} dA(z)=\infty.$ Then, in the main theorem of this paper, he shows that $${\mathcal B}_0^h$$ contains inner functions. This implies that $$\varphi\in{\mathcal B}_0^h$$ does not even imply that $$C_\varphi$$ is compact on $$H^2$$, since an inner function cannot induce such an operator. Further, by the above mentioned Madigan and Matheson result it is obtained that there exists an inner function $$\varphi$$ such that $$C_\varphi\:\mathcal B_0\to \mathcal B_0$$ is compact.

##### MSC:
 30D45 Normal functions of one complex variable, normal families 47B33 Linear composition operators
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