A remark on the multipliers on spaces of weak products of functions

Abstract

If $\mathcal{H}$ denotes a Hilbert space of analytic functions on a region $\Omega \subseteq {\mathbb{C}}^d$, then the weak product is defined by $$\mathcal{H}\odot \mathcal{H}=\left\{h=\sum _{n=1}^{\infty }{f}_n{g}_n:\sum _{n=1}^{\infty }\Vert f_n{\Vert }_{\mathcal{H}}\Vert g_n{\Vert }_{\mathcal{H}}<\infty \right\}.$$ We prove that if $\mathcal{H}$ is a first order holomorphic Besov Hilbert space on the unit ball of ${\mathbb{C}}^d$, then the multiplier algebras of $\mathcal{H}$ and of $\mathcal{H}\odot \mathcal{H}$ coincide.

arXiv commentv1: 6 pages. To appear Concr. Oper