Introducing the Polylogarithmic Hardy Space

TL;DR

This paper introduces a new polylogarithmic Hardy space, explores its properties, and reveals unique operator behaviors, including trivial multiplication operators and a link between Toeplitz operators and the divisor function.

Contribution

It is the first to identify that only trivial multiplication operators exist in this space and connects Toeplitz operators to number theory, expanding the understanding of Hardy spaces.

Findings

Only trivial densely defined multiplication operators exist in this space.

A connection between certain Toeplitz operators and the divisor function is established.

An operator theoretic proof of the divisor function's divisibility is provided.

Abstract

In this paper we investigate the reproducing kernel Hilbert space where the polylogarithm appears as kernel functions. 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 found. Finally, the paper concludes with an operator theoretic proof of the divisibility of the divisor function.