Spectral functions of products of selfadjoint operators
  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. 
Tomas Ya. Azizov, Mikhail Denisov, Friedrich Philipp
math.SP