## 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 $$\limsup_{n\to\infty} \frac{\log|u_n|} {\lambda_n}< -R$$.

### MSC:

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