Inner functions in the hyperbolic little Bloch class
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 \textit{K. Madigan} and \textit{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.
scientific article; zbMATH DE number 1548257
scientific article