Subnormal Toeplitz operators and the kernels of their self-commutators (Q1036179)

Let \({\mathbb T}\) be the unit circle. A function \(\varphi\in L^\infty({\mathbb T})\) is said to be of bounded type if there are bounded analytic functions functions \(\psi_1,\psi_2\) in the open unit disk such that \(\varphi(t)=\psi_1(t)/\psi_2(t)\) for almost all \(t\in{\mathbb T}\). Let \(T_\varphi\) be a Toeplitz operator acting on the Hardy space \(H^2({\mathbb T})\). The main result of the present paper says that if \(\varphi\in L^\infty({\mathbb T})\), \(\varphi\) and \(\overline{\varphi}\) are of bounded type and the kernel of \(T_\varphi^* T_\varphi-T_\varphi T_\varphi^*\) is invariant for \(T_\varphi\), then the operator \(T_\varphi\) is normal or analytic.