A study of various results for a class of entire Dirichlet series with complex frequencies

• Main Authors: Niraj Kumar, Garima Manocha
• Format: Article
• Language: English
• Published: Institute of Mathematics of the Czech Academy of Science 2018-04-01
• Series: Mathematica Bohemica
• Subjects: Dirichlet series; Banach algebra; topological zero divisor; division algebra; continuous linear functional; total set
• Online Access: http://mb.math.cas.cz/full/143/1/mb143_1_1.pdf

Let $F$ be a class of entire functions represented by Dirichlet series with complex frequencies $\sum a_k {\rm e}^{\langle\lambda^k, z\rangle}$ for which $(|\lambda^k|/{\rm e})^{|\lambda^k|} k!|a_k|$ is bounded. Then $F$ is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. $F$ is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to $F$ have also been established.