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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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


Author: Mohamad A. Ansari
Journal: Proc. Amer. Math. Soc. 103 (1988), 131-136
MSC: Primary 46J15; Secondary 30H05, 47B35
DOI: https://doi.org/10.1090/S0002-9939-1988-0938656-1
MathSciNet review: 938656
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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},\,\vert z\vert < 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.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0938656-1
Article copyright: © Copyright 1988 American Mathematical Society