Coefficient multipliers on Banach spaces of analytic functions

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 $f={\sum }_{j=0}^{\infty }{g}_n\ast h_n$, with $g_n\in X$, $h_n\in Y$ and ${\sum }_{n=1}^{\infty }\Vert g_n{\Vert }_X\Vert h_n{\Vert }_Y<\infty$. 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 $(X\otimes Y{)}^{\ast }=(X,Y^{\ast })$, where $U^{\ast }=(U,H^{\infty })$. 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.