Mathematica Bohemica, online first, 9 pp.

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

Niraj Kumar, Garima Manocha

Received July 16, 2016.   First published May 11, 2017.

Niraj Kumar, Garima Manocha, Department of Mathematics, Netaji Subhas Institute of Technology, Azad Hind Fauz Marg, Sector 3, Dwarka, New Delhi-110078, India, e-mail: nirajkumar2001@hotmail.com, garima89.manocha@gmail.com

Abstract: 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.

Keywords: Dirichlet series; Banach algebra; topological zero divisor; division algebra; continuous linear functional; total set

Classification (MSC 2010): 30B50, 46J15, 17A35

DOI: 10.21136/MB.2017.0066-16

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