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A function      f f       in      H ∞  {H^\infty }       is said to be a weak-star generator (     W ∗  {{\text {W}}^*}      -gen.) of the function      e n ( z ) = z n , | z | &gt; 1 , n ≥  1 {e_n}\left ( z \right ) = {z^n},\,|z| &gt; 1,\,n \geq 1      , if      lim α  p α  ∘  f = e n {\lim _\alpha }{p_\alpha } \circ f = {e_n}       (     W ∗  {{\text {W}}^*}      -topology), for some net (     p α  {p_\alpha }      ) of complex polynomials. For the case      n = 1 n = 1      ,      f f       is called a      W ∗  {{\text {W}}^*}      -gen. of      H ∞  {H^\infty }      . The      W ∗  {{\text {W}}^*}      -generators of      H ∞  {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 {e_n}      . As a result, it follows from our characterization that certain analytic Toeplitz operators have the transitive algebra property.

Weak-star generators of 𝑍ⁿ,𝑛≥1, and transitive operator algebras