Operators in Kerin space and zeros of the Riemann zeta function

Previously the author constructed a specific de Branges space and an operator in it with a spectrum, which is a set of non-trivial zeros of the Riemann zeta function, deployed on a real line. To obtain a self-adjoint operator with such a spectrum (the existence of such an operator in a Hilbert space would prove the Riemann hypothesis), the possibility of constructing an intertwining embedding in weighted $L^2$-spaces is investigated.