Multipliers for Dirichlet series in the complex plane (Q1300184)

Let \(0<\lambda_n \nearrow+ \infty\) be a given sequence with \(\limsup_{n\to \infty}\log n/ \lambda_n< +\infty\). A Dirichlet series \(\sum a_ne^{-\lambda_nz}\), \(a_n\in\mathbb{C}\), converges absolutely in \(\mathbb{C}\) iff \[ \limsup_{n\to\infty} {\log|a_n|\over\lambda_n} =-\infty,\tag{1} \] and it converges absolutely in \(\{z\in\mathbb{C}: \text{Re} z>R\}\) iff \[ \limsup_{n \to \infty} {\log|a_n|\over \lambda_n}\leq R.\tag{2} \] Denote by \(E_{-\infty} \) and \(E_R\) the spaces of sequences \(\{a_n\}\) satisfying (1) and (2), respectively. For two sequence spaces \(A\) and \(B\), \((A,B)\) denotes the space of multipliers from \(A\) to \(B\), that is, of sequences \(\{u_n\}\) with \(\{u_n a_n\}\in B\) whenever \(\{a_n\} \in\mathbb{A}\). In the paper, a complete description is obtained for the multiplier spaces \((E_{-\infty},\ell^p)\), \((\ell^p,E^{-\infty})\), and \((E_{R_1},E_{R_2})\), \(R_1,R_2\geq-\infty\).