On rank one singular perturbations of the semigroup of shifts on the semi-axis

In 2013 A.D. Baranov and D.V. Yakubovich constructed a model of singular rank one perturbation of a self-adjoint operator resulting to empty spectrum of the perturbed operator. The peculiarity of this perturbation, marked by the word "singular", is that the domain of the original operator changes. I tried to construct, by analogy, singular rank one perturbations of the semigroup of shifts on the semi-axis, leading to semigroups of contractions.