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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Approximation in operator algebras on bounded analytic functions

Author: M. W. Bartelt
Journal: Trans. Amer. Math. Soc. 170 (1972), 71-83
MSC: Primary 46J10
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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Keywords: Bounded analytic functions, operator algebras, strict topology, translation operators
Article copyright: © Copyright 1972 American Mathematical Society