Revista Matemática Iberoamericana

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Volume 27, Issue 2, 2011, pp. 415–447
DOI: 10.4171/RMI/642

Published online: 2011-08-31

Coefficient multipliers on Banach spaces of analytic functions

Óscar Blasco[1] and Miroslav Pavlović[2]

(1) Universidad de Valencia, Spain
(2) University of Belgrade, Beograd, Serbia

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.

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

Blasco Óscar, Pavlović Miroslav: Coefficient multipliers on Banach spaces of analytic functions. Rev. Mat. Iberoam. 27 (2011), 415-447. doi: 10.4171/RMI/642