Hilbert–Pólya Operators in Krein Spaces

We construct some class of selfadjoint operators in the Krein spaces consisting of functions onthe straight line $ \{\operatorname{Re}s=\frac{1}{2}\} $ .Each of these operators is a rank-one perturbation of a selfadjoint operatorin the corresponding Hilbert spaceand 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.