Continuity of the norm of a composition operator. (English) Zbl 1053.47019

Let \(ASM({\mathbb{D}})^1\) (respectively, \(ASM({\mathbb{D}})^\infty\)) denote the set of all analytic self-maps of the unit disc, considered as a subset of \(H^1\) (respectively, \(H^\infty \)). Let \(N_i: ASM({\mathbb{D}})^i \to {\mathbb{R}}\) (\(i=1, \infty \)) be given by \(N_i(\varphi )= \| C_\varphi \|\), where \(C_\varphi \) is the composition operator. The authors study the continuity of these maps by showing that \(N_1\) is continuous at any \(\varphi\) which either is an inner map or \(\varphi (0)=0\), and that \(N_\infty \) is continuous at any \(\varphi\) which satisfies \(\varphi ({\mathbb{D}}) \subset r{\mathbb{D}}\) for some \(r<1\). In addition, the authors study similar questions with the operator norm replaced with either the essential norm \(\| \cdot \|_e\) or the Hilbert-Schmidt norm \(\| \cdot \|_{HS}\). They show that if the map \(\varphi \mapsto \| C_\varphi \|_e\) on \(ASM({\mathbb{D}})^\infty\) is continuous at \(\varphi \), then \(C_\varphi\) is compact; and the continuity of \(N_{HS} \) at any \(\varphi\) with \(\| \varphi \|_{\infty } <1\). Finally, the authors characterize when the norm of \(C_\varphi \) is minimal: when \(\varphi (0) \neq 0\), the lower bound \(\sqrt{\frac{1}{1-| \varphi (0)| ^2}}\) for \(\| C_\varphi \|\) is attained only when \(\varphi\) is constant.


47B33 Linear composition operators
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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