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 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.

**[1]**Peter L. Duren,*Theory of 𝐻^{𝑝} spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655****[2]**P. L. Duren, B. W. Romberg, and A. L. Shields,*Linear functionals on 𝐻^{𝑝} spaces with 0<𝑝<1*, J. Reine Angew. Math.**238**(1969), 32–60. MR**0259579****[3]**P. L. Duren and A. L. Shields,*Coefficient multipliers of 𝐻^{𝑝} and 𝐵^{𝑝} spaces*, Pacific J. Math.**32**(1970), 69–78. MR**0255825****[4]**G. H. Hardy and J. E. Littlewood,*Some properties of fractional integrals. II*, Math. Z.**34**(1932), no. 1, 403–439. MR**1545260**, 10.1007/BF01180596**[5]**E. Landau,*Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie*, Springer-Verlag, Berlin, 1929.**[6]**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), 276–280. MR**0276744****[8]**J. H. Shapiro, A. L. Shields and G. D. Taylor,*The second duals of some function spaces*(preprint).**[9]**A. Zygmund,*On the preservation of classes of functions*, J. Math. Mech. 8 (1959), 889-895; erratum**9**(1959), 663. MR**0117498****[10]**-,*Trigonometric series*, Vols. 1, 2, 2nd ed., Cambridge Univ. Press, London, 1968. MR**38**#4882.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0283559-3

Keywords:
Banach spaces,
analytic functions,
projections,
topological direct sums,
multiplier transforms

Article copyright:
© Copyright 1971
American Mathematical Society