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), 287302
MSC:
Primary 46.30; Secondary 30.00
MathSciNet review:
0283559
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Abstract: Let and be positive continuous functions on with as and . Denote by and the Banach spaces of functions f analytic in the open unit disc D with and , respectively. In both spaces . Let denote the space of functions analytic in D with . The spaces , and are identified in the obvious way with closed subspaces of , and , respectively. For a large class of weight functions which go to zero at least as fast as some power of but no faster than some other power of , we exhibit bounded projections from onto , from onto , and from onto . Using these projections, we show that the dual of is topologically isomorphic to for an appropriate, but not unique choice of . In addition, is topologically isomorphic to the dual of . As an application of the above, the coefficient multipliers of , and are characterized. Finally, we give an example of a weight function pair for which some of the above results fail.
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 P. L. Duren, Theory of spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
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 P. L. Duren, B. W. Romberg and A. L. Shields, Linear functionals on spaces with , J. Reine Angew. Math. 238 (1969), 3260. MR 0259579 (41:4217)
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 P. L. Duren and A. L. Shields, Coefficient multipliers of and spaces, Pacific J. Math. 32 (1970), 6978. MR 41 #485. MR 0255825 (41:485)
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 G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z. 34 (1932), 403439. MR 1545260
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 E. Landau, Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie, SpringerVerlag, Berlin, 1929.
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 J. Lindenstrauss and A. Pełczyúski, Contributions to the theory of the classical Banach spaces (preprint).
 [7]
 L. A. Rubel and A. L. Shields, The second duals of certain spaces of analytic functions, J. Austral. Math. Soc. 11 (1970), 276280. MR 0276744 (43:2484)
 [8]
 J. H. Shapiro, A. L. Shields and G. D. Taylor, The second duals of some function spaces (preprint).
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 A. Zygmund, On the preservation of classes of functions, J. Math. Mech. 8 (1959), 889895; erratum, ibid. 9 (1960), 663. MR 22 #8277. MR 0117498 (22:8277)
 [10]
 , Trigonometric series, Vols. 1, 2, 2nd ed., Cambridge Univ. Press, London, 1968. MR 38 #4882.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197102835593
PII:
S 00029947(1971)02835593
Keywords:
Banach spaces,
analytic functions,
projections,
topological direct sums,
multiplier transforms
Article copyright:
© Copyright 1971
American Mathematical Society
