In this paper, we study the minimality of the commutant of an analytic Toeplitz operator \(M_\varphi \) , when \(M_\varphi \) is defined on the Hardy space \(H^2(\mathbb {D})\) and \(\varphi \in H^\infty (\mathbb {D})\) denotes a bounded analytic function in \(\mathbb {D}\) . Specifically, we show that the commutant of \(M_\varphi \) is minimal if and only if the polynomials on \(\varphi \) are weak star dense in \(H^\infty (\mathbb {D})\) , that is, \(\varphi \) 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 \(M_\varphi \) is minimal for a large class of symbols \(\varphi \) . Specifically, when \(\varphi \) is an entire function, or more generally, when \(\varphi \) belongs to the Thomson–Cowen class.
