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Weak-Star Generators of $Z^n, n \geq 1$, and Transitive Operator Algebras

Mohamad A. Ansari
Proceedings of the American Mathematical Society
Vol. 103, No. 1 (May, 1988), pp. 131-136
DOI: 10.2307/2047539
Stable URL: http://www.jstor.org/stable/2047539
Page Count: 6
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Weak-Star Generators of $Z^n, n \geq 1$, and Transitive Operator Algebras
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Abstract

A function $f$ in $H^\infty$ is said to be a weak-star generator (w$^\ast$-gen.) of the function $e_n(z) = z^n, |z| < 1, n \geq 1$, if $\lim_\alpha p_\alpha \circ f = e_n$ (w$^\ast$-topology), for some net $(p_\alpha)$ of complex polynomials. For the case $n = 1, f$ is called a w$^\ast$-gen. of $H^\infty$. The w$^\ast$-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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