 

Chaman Kumar Sahu
Multipliers of the Hilbert spaces of Dirichlet series
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Published: 
March 5, 2023. 
Keywords: 
Dirichlet series, reproducing kernel Hilbert space, multiplier, Mobius function, additive function. 
Subject [2020]: 
Primary 30B50, 46E22; Secondary 11Z05. 


Abstract
In this paper we describe the multiplier algebra of the Hilbert space of Dirichlet series associated with a diagonal Dirichlet series kernel. In particular, we obtain a sufficient condition on "weights" (i.e., the coefficients of the diagonal kernel), which shows that the multiplier algebra is isometrically isomorphic to the space of all bounded holomorphic functions on a domain beyond the common domain of functions in the Hilbert space, that are representable by a convergent Dirichlet series on some right halfplane. As an application, we describe the multiplier algebra when the weight is an additive function satisfying certain inequalities. Moreover, we recover a result of Stetler that describes the multipliers when the weight is multiplicative. The proof of the main result is a refinement of the techniques of Stetler.


Acknowledgements
I am grateful to Sameer Chavan for suggesting this topic. Needless to say, this paper would not have seen the present form without his continuous support. I am also thankful to the anonymous referee for several valuable inputs improving the presentation of the paper.


Author information
Chaman Kumar Sahu:
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur, India
chamanks@iitk.ac.in

