Spectral functions of products of selfadjoint operators

TL;DR

This paper extends the theory of spectral functions for products of selfadjoint operators, showing conditions under which such operators have spectral functions with singularities, generalizing known results in Krein spaces.

Contribution

It introduces a new criterion involving polynomials that ensures the existence of spectral functions for operator products, broadening the scope of spectral theory.

Findings

Operators AG have spectral functions with singularities under certain polynomial conditions.

Generalizes the definitizable operators theorem to broader classes of selfadjoint operator products.

Provides conditions for the non-negativity of symmetric operators derived from Gp(AG).

Abstract

Given two possibly unbounded selfadjoint operators A and G such that the resolvent sets of AG and GA are non-empty, it is shown that the operator AG has a spectral function on IR with singularities if there exists a non-zero polynomial p such that the symmetric operator Gp(AG) is non-negative. This result generalizes a well-known theorem for definitizable operators in Krein spaces.