# Coefficient multipliers on Banach spaces of analytic functions

### Óscar Blasco

Universidad de Valencia, Spain### Miroslav Pavlović

University of Belgrade, Beograd, Serbia

## Abstract

Motivated by an old paper of Wells [J. London Math. Soc. {\bf 2} (1970), 549-556] we define the space $X⊗Y$, where $X$ and $Y$ are "homogeneous" Banach spaces of analytic functions on the unit disk $D$, by the requirement that $f$ can be represented as $f=∑_{j=0}g_{n}∗h_{n}$, with $g_{n}∈X$, $h_{n}∈Y$ and $∑_{n=1}∥g_{n}∥_{X}∥h_{n}∥_{Y}<∞$. We show that this construction is closely related to coefficient multipliers. For example, we prove the formula $((X⊗Y),Z)=(X,(Y,Z))$, where $(U,V)$ denotes the space of multipliers from $U$ to $V$, and as a special case $(X⊗Y)_{∗}=(X,Y_{∗})$, where $U_{∗}=(U,H_{∞})$. We determine $H_{1}⊗X$ for a class of spaces that contains $H_{p}$ and $ℓ_{p}$ $(1≤p≤2)$, 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.

## Cite this article

Óscar Blasco, Miroslav Pavlović, Coefficient multipliers on Banach spaces of analytic functions. Rev. Mat. Iberoam. 27 (2011), no. 2, pp. 415–447

DOI 10.4171/RMI/642