Mathematica Bohemica, Vol. 143, No. 1, pp. 1-9, 2018
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.
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
References: [1] L. H. Khoi: Coefficient multipliers for some classes of Dirichlet series in several complex variables. Acta Math. Vietnam. 24 (1999), 169-182. MR 1710776 | Zbl 0942.32001 [2] N. Kumar, G. Manocha: A class of entire Dirichlet series as an FK-space and a Fréchet space. Acta Math. Sci., Ser. B, Engl. Ed. 33 (2013), 1571-1578. DOI 10.1016/S0252-9602(13)60105-8 | MR 3116603 | Zbl 1313.30007 [3] N. Kumar, G. Manocha: On a class of entire functions represented by Dirichlet series. J. Egypt. Math. Soc. 21 (2013), 21-24. DOI 10.1016/j.joems.2012.10.008 | MR 3040754 | Zbl 1277.30004 [4] N. Kumar, G. Manocha: Certain results on a class of entire functions represented by Dirichlet series having complex frequencies. Acta Univ. M. Belii, Ser. Math. 23 (2015), 95-100. MR 3373834 | Zbl 1336.30004 [5] R. Larsen: Banach Algebras - An Introduction. Pure and Applied Mathematics 24. Marcel Dekker, New York (1973). MR 0487369 | Zbl 0264.46042 [6] R. Larsen,: Functional analysis - An Introduction. Pure and Applied Mathematics 15. Marcel Dekker, New York (1973). MR 0461069 | Zbl 0261.46001 [7] R. K. Srivastava: Some growth properties of a class of entire Dirichlet series. Proc. Natl. Acad. Sci. India, Sect. A 61 (1991), 507-517. MR 1169262 | Zbl 0885.30004 [8] R. K. Srivastava: On a paper of Bhattacharya and Manna. Internal Report (1993), IC/93/417.
Affiliations: 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
