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 $M_{\phi }$, when $M_{\phi }$ is defined on the Hardy space $H^2(\mathbb{D})$ and $\phi \in H^{\infty }(\mathbb{D})$, denotes a bounded analytic function on $\mathbb{D}$. Specifically we show that the commutant of $M_{\phi }$ is minimal if and only if the polynomials on $\phi$ are weak-star dense in $H^{\infty }(\mathbb{D})$, that is, $\phi$ 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 $M_{\phi }$ is minimal, for a large class of symbols $\phi$. Namelly, when $\phi$ is an entire function, or more generally when $\phi$ belongs to the Thomson-Cowen's class.