Weak-star generators of $Z^ n,\;n\geq 1,$ and transitive operator algebras

Abstract:

A function $f$ in ${H^\infty }$ is said to be a weak-star generator (${{\text {W}}^*}$-gen.) of the function ${e_n}\left ( z \right ) = {z^n}, |z| < 1, n \geq 1$, if ${\lim _\alpha }{p_\alpha } \circ f = {e_n}$ (${{\text {W}}^*}$-topology), for some net (${p_\alpha }$) of complex polynomials. For the case $n = 1$, $f$ is called a ${{\text {W}}^*}$-gen. of ${H^\infty }$. The ${{\text {W}}^*}$-generators of ${H^\infty }$ 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.