A Hilbert-Schmidt integral operator and the Weil distribution

TL;DR

This paper introduces a positive operator related to the Weil distribution, demonstrating that its product with a convolution operator forms a trace class Hilbert-Schmidt operator with nonnegative eigenvalues, and provides a trace formula connecting to number theory.

Contribution

It presents a novel positive operator and establishes its connection to the Weil distribution through trace class Hilbert-Schmidt operators, offering new insights into their spectral properties.

Findings

The product operator is a trace class Hilbert-Schmidt operator.

The operator has nonnegative eigenvalues.

A trace formula relates the operator to the Weil distribution.

Abstract

In this paper, a positive operator is given. It is shown that the product of this positive operator and the convolution operator is a trace class Hilbert-Schmidt integral operator and has nonnegative eigenvalues. A formula is given for the trace of this product operator. It seems that this product operator is the closest trace class integral operator which has nonnegative eigenvalues and is related to the Weil distribution. A relation is given between the trace of the product operator and the Weil distribution.