The composition operators on the space of Dirichlet series with square summable coefficients
Let \({\mathcal H}\) be the space of Dirichlet series \(\sum^\infty_{n=1} a_nn^{-s}\) with square summable coefficients. Each function in \({\mathcal H}\) is holomorphic on the half-plane \(\mathbb{C}_{1/2}= \{s\in\mathbb{C}: \text{Re }s> {1\over 2}\}\). \({\mathcal H}\) is a Hilbert space under the norm \(\|f\|^2= \sum^\infty_{n=1}|a_n|^2\). The authors completely solve the difficult problem for which analytic mappings \(\Phi: \mathbb{C}_{1/2}\to \mathbb{C}_{1/2}\) the composition operator \({\mathcal C}_\Phi(f)= f\circ\Phi\) is a bounded linear operator on \({\mathcal H}\). Their study involves the space \({\mathcal D}\) of analytic functions on some half-plane \(\mathbb{C}_\theta\) which admit representation by a convergent Dirichlet series. Their main result is the following: An analytic function \(\Phi: \mathbb{C}_{1/2}\to \mathbb{C}_{1/2}\) defines a bounded composition operator \({\mathcal C}_\Phi:{\mathcal H}\to{\mathcal H}\) if and only if: (a) it is of the form \(\Phi(s)= c_0s+ \varphi(s)\), where \(c_0\in \{0,1,2,\dots\}\) and \(\varphi\in {\mathcal D}\) for some \(\theta\in \mathbb{R}\); and (b) \(\Phi\) has an analytic extension to \(\mathbb{C}_0\), also denoted by \(\Phi\), such that (i) \(\Phi(\mathbb{C}_0)\subseteq \mathbb{C}_0\) if \(c_0> 0\) and (ii) \(\Phi(\mathbb{C}_0)\subseteq \mathbb{C}_{1/2}\) if \(c_0= 0\). The paper is a fine piece of mathematical work.
scientific article; zbMATH DE number 1548320