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 Mφ, when Mφ is defined on the Hardy space H2(D) and φ∈H∞(D) denotes a bounded analytic function in D. Specifically, we show that the commutant of Mφ 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 Mφ is minimal for a large class of symbols φ. Specifically, when φ is an entire function, or more generally, when φ belongs to the Thomson–Cowen class.