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May, 2011 Coefficient multipliers on Banach spaces of analytic functions
Óscar Blasco , Miroslav Pavlović
Rev. Mat. Iberoamericana 27(2): 415-447 (May, 2011).

Abstract

Motivated by an old paper of Wells [J. London Math. Soc. {\bf 2} (1970), 549-556] we define the space $X\otimes Y$, where $X$ and $Y$ are "homogeneous" Banach spaces of analytic functions on the unit disk $\mathbb{D}$, by the requirement that $f$ can be represented as $f=\sum_{j=0}^\infty g_n * h_n$, with $g_n\in X$, $h_n\in Y$ and $\sum_{n=1}^\infty \|g_n\|_X \|h_n\|_Y < \infty$. We show that this construction is closely related to coefficient multipliers. For example, we prove the formula $((X\otimes Y),Z)=(X,(Y,Z))$, where $(U,V)$ denotes the space of multipliers from $U$ to $V$, and as a special case $(X\otimes Y)^*=(X,Y^*)$, where $U^*=(U,H^\infty)$. We determine $H^1\otimes X$ for a class of spaces that contains $H^p$ and $\ell^p$ $(1\le p\le 2)$, and use this together with the above formulas to give quick proofs of some important results on multipliers due to Hardy and Littlewood, Zygmund and Stein, and others.

Citation

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Óscar Blasco . Miroslav Pavlović . "Coefficient multipliers on Banach spaces of analytic functions." Rev. Mat. Iberoamericana 27 (2) 415 - 447, May, 2011.

Information

Published: May, 2011
First available in Project Euclid: 10 June 2011

zbMATH: 1235.42004
MathSciNet: MR2848526

Subjects:
Primary: 30A99 , 42A45
Secondary: 30D55 , 46A45 , ‎46E15

Keywords: analytic functions , ‎Banach spaces , coefficient multipliers , Hardy spaces , Tensor products

Rights: Copyright © 2011 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.27 • No. 2 • May, 2011
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