Minimal commutant and double commutant property for analytic Toeplitz operators

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 H²(D) and φ∈ H^∞(D), denotes a bounded analytic function on 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^∞(D), that is, φ is a weak-star generator of H^∞(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.