Weak-star generators of zⁿ, n > 1, and transitive operator algebras

A function/ in H∞ is said to be a weak-star generator (w*-gen.) of the function e_n(z) = zⁿ, \z\ < 1, n=, 1 if lim∝ p_∝ ° f = e_n (w*-topology), for some net (p_∝ of complex polynomials. For the case n = 1, f is called a w*- gen. of H^∞ The w*-generators of H^∞ have been defined and characterized by Sarason. It is the purpose of the present paper to give necessary and sufficient conditions for a function to generate e_n. As a result, it follows from our characterization that certain analytic Toeplitz operators have the transitive algebra property.