On kernels of Toeplitz operators

Abstract: Let $H^2$ denote the standard Hardy space on the unit disk $\mathbb D$ and let $\mathbb T=\partial \mathbb D$. Every $f(z)=\sum_{n=0}^{\infty}a_nz^n\in H^2$ has a nontangential limit $f(e^{i\theta})$ a.e. on $\mathbb {T}=\partial\mathbb {D}$ and this boundary function $f(e^{i\theta})$ is in $L^2(\mathbb {T})$. Furthermore, if $\{c_n\}$ are Fourier coefficients of $f(e^{i\theta})$ then $c_n=a_n$ for $n\geq 0$ and $c_n=0$ for $n<0$. Actually, the space $H^2$ can be identified with a closed subspace of $L^2(\mathbb {T})$ whose Fourier coefficients with negative indices vanish.

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 $f K_{I}$. However, not all spaces $fK_{I}$, where $f$ and $K_I$ are as above, can be kernels of Toeplitz operators. 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 try to describe the space $\ker T_{\frac{\bar I\bar f}{f}}\ominus fK_I$. We use Sarason's approach and the structure of de Branges-Rovnyak spaces generated by nonextreme functions.

The talk is based on joint work with P. Sobolewski, A. So{\l}tysiak and M. Wo{\l}oszkiewicz-Cyll.

complex variablesdynamical systems

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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