What do composition operators know about inner functions? (English) Zbl 0951.47026

Shapiro, Joel H.

Monatsh. Math. 130, No. 1, 57-70 (2000).

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 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 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.

Reviewer: S.Platonov (Petrozavodsk)

MSC:

47B33	Linear composition operators
46J15	Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
30D55	\(H^p\)-classes (MSC2000)

Keywords:

composition operator; inner function; Nevanlinna counting function; Hardy space