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Introducing the Polylogarithmic Hardy Space

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Abstract

In this paper we investigate the reproducing kernel Hilbert space where the polylogarithm appears as the kernel function. This investigation begins with the properties of functions in this space, and here a connection to the classical Hardy space is shown through the Bose–Einstein integral equation. Next we consider function theoretic operators over the polylogarithmic Hardy space, such as multiplication and Toeplitz operators. It is determined that there are only trivial densely defined multiplication operators (and therefore only trivial bounded multipliers) over this space, which makes this space the first for which this has been found to be true. In the case of Toeplitz operators, a connection between a certain subset of these operators and the number theoretic divisor function is established. Finally, the paper concludes with an operator theoretic proof of the divisibility of the divisor function.

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  1. University of Florida, Gainesville, FL, USA

    Joel A. Rosenfeld

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  1. Joel A. Rosenfeld
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Correspondence to Joel A. Rosenfeld.

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The author would like to express his appreciation for the patience and guidance of his mentor, Dr. Michael T. Jury, throughout this project.

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Rosenfeld, J.A. Introducing the Polylogarithmic Hardy Space. Integr. Equ. Oper. Theory 83, 589–600 (2015). https://doi.org/10.1007/s00020-015-2256-z

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  • DOI: https://doi.org/10.1007/s00020-015-2256-z

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