Hilbert–Pólya operators in Krein spaces

We construct some class of selfadjoint operators in the Krein spaces consisting of functions on the straight line $\{\operatorname{Re}s=\frac12\}$. 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 $\frac1{s(1-s)}$, where $s$ ranges over the set of nontrivial zeros of the Riemann zeta-function.