Holomorphic Dirichlet series in the half plane. (English) Zbl 0935.30002

Let \(R\) be a real number, let \(\{\lambda_n\}\) be a strictly increasing sequence of positive real numbers such that \(\lambda_n\to \infty\), and let \[ E_R= \Bigl\{ f(z)= \sum a_ne^{-\lambda_n z}: f\text{ is holomorphic for Re }z> R\Bigr\}. \] Here \(R\) and the sequence \(\{\lambda_n\}\) are considered fixed. The elements of \(E_R\) are determined by the sequence of complex numbers \(\{a_n\}\), and thus \(E_R\) can be considered as a sequence space. For sequence spaces \(A\) and \(B\), define the space of multipliers from \(A\) into \(B\) by \[ (A,B)= \{\{u_n\}: \{u_na_n\}\in B\text{ whenever }\{a_n\}\in A\}. \] In particular, let \(A^\alpha= (A,l^1)\). The author proves that if \(\{u_n\}\) satisfies both conditions \[ \limsup_{n\to\infty} \frac{\log|u_n|}{\lambda_n}< -R \quad\text{ and }\quad \lim_{n\to\infty} \frac{\log n}{\lambda_n}= 0, \] then \(\{u_n\}\in E_R^\alpha\). The condition \(\lim_{n\to\infty} \frac{\log n}{\lambda_n}= 0\) also implies that \(E_R^{\alpha}= E_R\). It is also shown that \(E_R^\alpha= (E_R,l^p)\) and that \((l^p,E_R)= E_R\). This leads to the result that each continuous linear functional in \(E_R^*\) has the form \(F(a)= \sum a_n u_n\), where \(\{u_n\}\) satisfies the condition \(\limsup_{n\to\infty} \frac{\log|u_n|} {\lambda_n}< -R\).


30B50 Dirichlet series, exponential series and other series in one complex variable