Perturbations of Truncated Toeplitz operators

We investigate the structure of kernels of finite-rank perturbations of truncated Toeplitz operators. By realizing a perturbed truncated Toeplitz operator on a model space as a suitable matrix-valued truncated Toeplitz operator on an enlarged model space, we identify its kernel with a coordinate projection of the kernel of an associated block Toeplitz operator. This realization enables us to transfer the nearly $S^*$-invariant structure of Toeplitz kernels to the perturbed truncated setting. More precisely, for a perturbation of rank at most $n$, we show that the kernel of the perturbed truncated Toeplitz operator is nearly $S^*$-invariant with defect at most $2n+1$. We then extend the method to truncated Toeplitz operators on multiband spaces. In this setting, a perturbation of rank at most $n$ again preserves the finite-defect nearly $S^*$-invariant structure of the kernel, with the explicit upper bound $2n+2$ for the defect.