Let $H^2$ denote the standard Hardy space on the unit disk $\mathbb D$ and let $\mathbb T=\partial \mathbb D$. For $\varphi\in L^{\infty}(\mathbb T)$ the Toeplitz operator $T_{\varphi}$ on $H^2$ is given by $T_{\varphi}f=P_{+}(\varphi f)$, where $P_{+}$ is the orthogonal projection of $L^2(\mathbb T)$ onto $H^2$. It is a consequence of Hitt's Theorem that $\ker T_{\varphi}= fK_I$, where $K_I= H^2\ominus IH^2$ is the model space corresponding to the inner function $I$ such that $I(0)=0$ and $f$ is an outer function of unit $H^2$ norm that acts as an isometric multiplier from $K_I$ onto $f K_{I}$. The sufficient and necessary condition for the space $fK_I$ to be the kernel of a Toeplitz operator was given by E. Hayashi (1990). In 1994 D. Sarason gave another proof of this condition based on de Branges-Rovnyak spaces theory. If $M= fK_I$ is a kernel of a Toeplitz operator, then also we have $M=\ker T_{\frac{\overline{If}}{f}}$. In the talk we consider the case when $fK_I\varsubsetneq \ker T_{\frac{\overline{If}}{f}}$ and describe the space $\ker T_{\frac{\bar I\bar f}{f}}\ominus fK_I$ in the case when this space is finite dimensional. We use Sarason's approach and the structure of de Branges-Rovnyak spaces generated by nonextreme functions.

The talk is mainly based on the papers:

[1] Nowak, M. T., Sobolewski, P., So³tysiak, A., Wo³oszkiewicz-Cyll, M., On kernels of Toeplitz operators. Anal. Math. Phys. 10 (2020), no. 4

[2] Nowak, M. T., Sobolewski, P., So³tysiak, A. The orthogonal complement of M(a) in H(b). Studia Math. 260 (2021), no. 3