What do composition operators know about inner functions?
Let \(H^2\) be the Hardy space of the disc, i.e. the space of all functions holomorphic in the open unit disc \(\mathbb{U}\) whose Taylor expansions about the origin have square summable coefficient sequences. \(H^2\) is, in the obvious norm, a Hilbert space. Given a holomorphic function \(\varphi:\mathbb{U}\to\mathbb{U}\) we define the \textit{composition operator} \(C_\varphi\) on the space of all functions holomorphic on \(\mathbb{U}\) by \(C_\varphi f=f\circ\varphi\). It is obvious that \(C_\varphi\) is linear, and it is known that \(C_\varphi\) is a bounded operator on \(H^2\). The paper under review studies properties of composition operators \(C_\varphi\) that characterize those self-maps \(\varphi\) of \(\mathbb{U}\) that are \textit{inner}, i.e. have radial limits of modulus one at almost every point of \(\partial\mathbb{U}\). The author obtains several interesting results in this directions. For example: if \(\varphi(0)\neq 0\), then \(\varphi\) is inner if and only if \(\|C_\varphi\|= \sqrt{\frac{1+|\varphi(0)|}{1-|\varphi(0)|}}\). The case \(\varphi(0)=0\) requires the special characterization. The other result is a characterization of inner functions in terms of the asymptotic behavior of the Nevanlinna counting function.
scientific article; zbMATH DE number 1466227