Kernels of perturbed Hankel operators

In the classical Hardy space $H^2\left(D\right)$, it is well-known that the kernel of the Hankel operator is invariant under the action of shift operator S and sometimes nearly invariant under the action of backward shift operator $S^{⁎}$. It appears in this paper that kernels of finite rank perturbations of Hankel operators are almost shift invariant as well as nearly $S^{⁎}$-invariant with finite defect. This allows us to obtain a structure of the kernel in several important cases by applying a recent theorem due to Chalendar, Gallardo, and Partington.