Standard models of abstract intersection theory for operators in Hilbert space

TL;DR

This paper introduces a standard model of abstract intersection theory for certain Hilbert space operators, establishing a connection between the existence of such models and the Riemann hypothesis for associated L-functions.

Contribution

It presents a new standard model of abstract intersection theory and proves its equivalence to the Riemann hypothesis and semi-simplicity for operators, under weaker conditions than previous work.

Findings

Existence of a standard model implies the Riemann hypothesis for the operator.

The Riemann hypothesis and semi-simplicity are equivalent to the existence of a standard model.

Results apply to Dirichlet L-functions, including the Riemann zeta-function.

Abstract

For an operator in a possibly infinite-dimensional Hilbert space of a certain class, we set down axioms of an abstract intersection theory, from which the Riemann hypothesis regarding the spectrum of that operator follows. In our previous paper [BU] we constructed a GNS (Gelfand-Naimark-Segal) model of abstract intersection theory. In this paper we propose another model, which we call a standard model of abstract intersection theory. We show that there is a standard model of abstract intersection theory for a given operator if and only if the Riemann hypothesis and semi-simplicity hold for that operator. (For the definition of semi-simplicity of an operator in Hilbert space, see the definition in Introduction.) We show this result under a condition for a given operator which is much weaker than the condition in the previous paper. The operator satisfying this condition can be constructed by the method of automorphic scattering in [U]. Combining this with a result from [U], we can show that an Dirichlet $L$-function, including the Riemann zeta-function, satisfies the Riemann hypothesis and its all nontrivial zeros are simple if and only if there is a corresponding standard model of abstract intersection theory. Similar results can be proven for GNS models since the same technique of proof for standard models can be applied.