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Approximation in operator algebras on bounded analytic functions


Author: M. W. Bartelt
Journal: Trans. Amer. Math. Soc. 170 (1972), 71-83
MSC: Primary 46J10
DOI: https://doi.org/10.1090/S0002-9947-1972-0361791-9
MathSciNet review: 0361791
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Abstract: Let $ B$ denote the algebra of bounded analytic functions on the open unit disc in the complex plane. Let $ (B,\beta )$ denote $ B$ endowed with the strict topology $ \beta $. In 1956, R. C. Buck introduced $ [\beta :\beta ]$, the algebra of all continuous linear operators from $ (B,\beta )$ into $ (B,\beta )$. This paper studies the algebra $ [\beta :\beta ]$ and some of its subalgebras, in the norm topology and in the topology of uniform convergence on bounded subsets. We also study a special class of operators, the translation operators. For $ \phi $ an analytic map of the open unit disc into itself, the translation operator $ {U_\phi }$ is defined on $ B$ by $ {U_\phi }f(x) = f(\phi x)$. In particular we obtain an expression for the norm of the difference of two translation operators.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0361791-9
Keywords: Bounded analytic functions, operator algebras, strict topology, translation operators
Article copyright: © Copyright 1972 American Mathematical Society

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