Surjective Toeplitz operators.
Résumé
The authors establish a criterion for the surjectivity of noninjective bounded Toeplitz operators on $H^2\coloneq H^2(T)$ and show that the canonical right inverse of a surjective Toeplitz operator is the product of three Toeplitz operators. The details are as follows. Assume $\varphi$ is unimodular and the Toeplitz operator $T_\varphi$ is not injective. Then ${\rm Ker}\,T_\varphi = H^2 \ominus IH^2$ with some inner function $I$ and there is a unique function $g\in{\rm Ker}\,T_\varphi$ of unit norm for which ${\rm Re}\,g(0)$ is maximal. One has $g=a/(1-b)$ with certain $a$ and $b$ in the unit ball of $H^\infty$, and $b=Ib_0$. Put $g_0=a/(1-b_0)$. The main result of the paper says that $T_\varphi$ is surjective if and only if $|g_0|^2$ is a Helson-Szegö weight and that in this case the right inverse of $T_\varphi$ whose range is orthogonal to ${\rm Ker}\,T_\varphi$ equals $T_gT_{I-\overline{b_0}}\,T_{1/\overline{a}}$. The proofs involve de Branges-Rovnyak spaces and results by E. Hayashi, D. Hitt, and T. Nakazi. The authors' characterization of surjectivity is of great depth but, in the authors' words, "it is unclear how useful it might be in analyzing specific Toeplitz operators or classes of Toeplitz operators. This matter awaits further investigation."