Minimal commutant and double commutant property for analytic Toeplitz operators

We study when the commutant $\{M_\varphi\}'$ and the double commutant $\{M_\varphi\}''$ are minimal where $M_\varphi$ is an analytic Toeplitz operator induced in the Hardy space $H^2(\mathbb{D})$ by an analytic function $\varphi$ bounded on the unit disk $\mathbb{D}$. Our work centers on the existing connection between the minimality of $\{M_\varphi\}'$ and the density on $H^2(\mathbb{D})$ of the polynomials on $\varphi$. This connection continues being true for the minimality of the double commutant $\{M_\varphi\}''$ when $\varphi$ is in the Thomson-Cowen's class, but the density now is given in term on some subspaces of $H^2(\mathbb{D})$. If we denote $\gamma(t)$ denotes the unit circle and $\varphi\in H^\infty(\overline{\mathbb{D}})$, along the way, some geometric conditions are discovered in terms of the winding number $n(\varphi(\gamma),a)$ that don't guarantee the minimality of the double commutant of $M_\varphi$ (joint work with Marí­a José González).