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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Bonded projections, duality, and multipliers in spaces of analytic functions


Authors: A. L. Shields and D. L. Williams
Journal: Trans. Amer. Math. Soc. 162 (1971), 287-302
MSC: Primary 46.30; Secondary 30.00
MathSciNet review: 0283559
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Abstract: Let $ \varphi $ and $ \psi $ be positive continuous functions on $ [0,1)$ with $ \varphi (r) \to 0$ as $ r \to 1$ and $ \smallint _0^1\psi (r)\;dr < \infty $. Denote by $ {A_0}(\varphi )$ and $ {A_\infty }(\varphi )$ the Banach spaces of functions f analytic in the open unit disc D with $ \vert f(z)\vert\varphi (\vert z\vert) = o(1)$ and $ \vert f(z)\vert\varphi (\vert z\vert) = O(1),\vert z\vert \to 1$, respectively. In both spaces $ \left\Vert f\right\Vert _\varphi = {\sup _D}\vert f(z)\vert\varphi (\vert z\vert)$. Let $ {A^1}(\psi )$ denote the space of functions analytic in D with $ \left\Vert f\right\Vert _\psi = \smallint {\smallint _D}\vert f(z)\vert\psi (\vert z\vert)\;dx\;dy < \infty $. The spaces $ {A_0}(\varphi ),{A^1}(\psi )$, and $ {A_\infty }(\varphi )$ are identified in the obvious way with closed subspaces of $ {C_0}(D),{L^1}(D)$, and $ {L^\infty }(D)$, respectively. For a large class of weight functions $ \varphi ,\psi $ which go to zero at least as fast as some power of $ (1 - r)$ but no faster than some other power of $ (1 - r)$, we exhibit bounded projections from $ {C_0}(D)$ onto $ {A_0}(\varphi )$, from $ {L^1}(D)$ onto $ {A^1}(\psi )$, and from $ {L^\infty }(D)$ onto $ {A_\infty }(\varphi )$. Using these projections, we show that the dual of $ {A_0}(\varphi )$ is topologically isomorphic to $ {A^1}(\psi )$ for an appropriate, but not unique choice of $ \psi $. In addition, $ {A_\infty }(\varphi )$ is topologically isomorphic to the dual of $ {A^1}(\psi )$. As an application of the above, the coefficient multipliers of $ {A_0}(\varphi ),{A^1}(\psi )$, and $ {A_\infty }(\varphi )$ are characterized. Finally, we give an example of a weight function pair $ \varphi ,\psi $ for which some of the above results fail.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0283559-3
PII: S 0002-9947(1971)0283559-3
Keywords: Banach spaces, analytic functions, projections, topological direct sums, multiplier transforms
Article copyright: © Copyright 1971 American Mathematical Society