A perturbation problem for the shift semigroup

Motivated by B. Tsirelson's construction of $E_0$-semigroups of type III, we investigate a $C_0$-semigroup acting on the space of square integrable functions of the half line whose difference from the shift semigroup is a Hilbert-Schmidt operator. We give a description of such semigroups in terms of analytic functions on the right-half plane and construct several examples.