Hilbert–Pólya Operators in Krein Spaces

We construct some class of selfadjoint operators in the Krein spaces consisting of functions on the straight line \( \{\operatorname{Re}s=\frac{1}{2}\} \). Each of these operators is a rank-one perturbation of a selfadjoint operator in the corresponding Hilbert space and has eigenvalues complex numbers of the form \( \frac{1}{s(1-s)} \), where \( s \) ranges over the set of nontrivial zeros of the Riemann zeta-function.