The kernel of a Toeplitz operator is nearly \(S^{*}\) -invariant where \(S^{*}\) is the backward shift on the Hardy space of the unit disk. Recently, the question when the kernel of a finite rank perturbation of a Toeplitz operator is nearly \(S^{*}\) -invariant with finite defect is studied in Liang and Partington (Integral Equ Oper Theory 92, Paper No. 35, 2020) where the multiplicity of S is one and in Chattopadhyay et al. (Adv. Oper. Theory 6, Paper No. 49, 2021) where the multiplicity of S is finite. This question is answered affirmatively for several important classes of Toeplitz operators in Liang and Partington (2020), Chattopadhyay et al. (2021). In this paper we give a complete answer to this question even when S is of infinite multiplicity. Furthermore, our approach is general enough to cover related questions on almost \(S^{*}\) -invariant and almost S-invariant kernels and to include related operators such as Hankel operators and product of Toeplitz and Hankel operators.
