May JUN SEP » 3 2016 2017 2018 2 captures 3 Jun 17 - 2 Sep 17

Close Help

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

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

DOI: 10.21136/MB.2017.0066-16

Full text available as PDF.

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.