Revista Matemática Iberoamericana

Coefficient multipliers on Banach spaces of analytic functions

Óscar Blasco and Miroslav Pavlović

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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.

Article information

Source
Rev. Mat. Iberoamericana, Volume 27, Number 2 (2011), 415-447.

Dates
First available in Project Euclid: 10 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1307713033

Mathematical Reviews number (MathSciNet)
MR2848526

Zentralblatt MATH identifier
1235.42004

Subjects
Primary: 42A45: Multipliers 30A99: None of the above, but in this section
Secondary: 30D55 46E15: Banach spaces of continuous, differentiable or analytic functions 46A45: Sequence spaces (including Köthe sequence spaces) [See also 46B45]

Keywords
Banach spaces analytic functions coefficient multipliers tensor products Hardy spaces

Citation

Blasco, Óscar; Pavlović, Miroslav. Coefficient multipliers on Banach spaces of analytic functions. Rev. Mat. Iberoamericana 27 (2011), no. 2, 415--447. https://projecteuclid.org/euclid.rmi/1307713033


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