A short proof of the boundedness of the composition operators on the Hardy space H²

Abstract

Following an idea of J. Shapiro, we give a simple proof of the fact that an element of the Gordon Hedenmalm class \(\Phi\) such that \(\Phi(\infty)=\infty\) defines a contractive composition operator \(C_\Phi\) on the space \(\mathcal{H}^2\) of Dirichlet series.