JournalsrmiVol. 27, No. 2pp. 415–447

Coefficient multipliers on Banach spaces of analytic functions

  • Óscar Blasco

    Universidad de Valencia, Spain
  • Miroslav Pavlović

    University of Belgrade, Beograd, Serbia
Coefficient multipliers on Banach spaces of analytic functions cover
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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 XYX\otimes Y, where XX and YY are "homogeneous" Banach spaces of analytic functions on the unit disk D\mathbb{D}, by the requirement that ff can be represented as f=j=0gnhnf=\sum_{j=0}^\infty g_n * h_n, with gnXg_n\in X, hnYh_n\in Y and n=1gnXhnY<\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 ((XY),Z)=(X,(Y,Z))((X\otimes Y),Z)=(X,(Y,Z)), where (U,V)(U,V) denotes the space of multipliers from UU to VV, and as a special case (XY)=(X,Y)(X\otimes Y)^*=(X,Y^*), where U=(U,H)U^*=(U,H^\infty). We determine H1XH^1\otimes X for a class of spaces that contains HpH^p and p\ell^p (1p2)(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.

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