Spectral functions of products of selfadjoint operators

Abstract

AbstractGiven 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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb R$\end{document} with singularities if there exists a polynomial p ≠ 0 such that the symmetric operator Gp(AG) is non‐negative. This result generalizes a well‐known theorem for definitizable operators in Krein spaces.