Some properties of composition operators on entire Dirichlet series with real frequencies (Q765743)

Let \(\sum^\infty_{n=1} a_n e^{-\lambda_nz}\) (\(a_n, z\in\mathbb{C}\), \(\lambda_n\in\mathbb{R}\)) be a Dirichlet series with \(\limsup_{n\to+\infty}\log|a_n|/\lambda_n= -\infty\). This series defines an entire function \(f(z)\). The set \(H\) of these functions with the norm defined by the inner product \((f,g)=\sum^\infty_{n=1} a_n\overline b_n\) for \(f,g\in H\), \(f(z)= \sum^\infty_{n=1} a_n e^{-\lambda_nz}\), \(g(z)= \sum^\infty_{n=1} b_n e^{-\lambda_nz}\), is a normed non-complete space \(H^2\). The subspace \(\mathcal{H}^2\) of the functions which have Ritt order zero and logarithmic orders finite is dense in \(H^2\). Every series in \(\mathcal{H}^2\) has finite ordinary order and, if \(\varphi\) is an entire function, the composition operator \(C_\varphi(f)= \varphi\) of maps \(\mathcal{H}^2\) into itself if and only if \(\varphi= az+ b\), \(b\in\mathbb{C}\) and \(a\geq 1\) satisfies the following condition: for every \(k\in\mathbb{N}\), there exists \(N(k)\) such that \(a=\lambda_{N(k)}/\lambda_k\). Furthermore, \(C_\varphi\) is bounded on \(H^2\) if and only if also the condition \(\text{Re\,}b\geq 0\) is satisfied.