Minimal commutant and double commutant property for analytic Toeplitz operators

Abstract

In this paper we study the minimality of the commutant of an analytic Toeplitz operator $M_φ$, when $M_φ$ is defined on the Hardy space $H^2(\mathbb{D})$ and $φ\in H^\infty(\mathbb{D})$, denotes a bounded analytic function on $\mathbb{D}$. Specifically we show that the commutant of $M_φ$ is minimal if and only if the polynomials on $φ$ are weak-star dense in $H^\infty(\mathbb{D})$, that is, $φ$ is a weak-star generator of $H^\infty(\mathbb{D})$. We use our result to characterize when the double commutant of an analytic Toeplitz operator $M_φ$ is minimal, for a large class of symbols $φ$. Namelly, when $φ$ is an entire function, or more generally when $φ$ belongs to the Thomson-Cowen's class.