Composition operators on Hardy spaces of a half-plane (Q2893266)

The authors prove that a composition operator is bounded on the Hardy space \(H^2\) of the half-plane \(\{\Re z>0\}\) if and only if the inducing map fixes the point at infinity non-tangentially and has a finite angular derivative \(\lambda>0\). This was first shown by \textit{V. Matache} [``Composition operators on Hardy spaces of a half-plane'', Proc. Am. Math. Soc. 127, No. 5, 1483--1491 (1999; Zbl 0916.47022)]. In such a case the norm, the essential norm and the spectral radius of \(C_\phi\) coincide with \(\sqrt \lambda\). It is also shown that \(\|C_\phi\|=\sup\{ \|C^*_\phi k_z\|/\|k_z\|\}\) where \(k_z(w)=\frac{1}{w+\bar z}\). The key observation used in the proof is that the existence of the angular derivative at infinity can be described using the positivity of a certain kernel.