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 $$\mathcal{H}\odot\mathcal{H}=\left{h=\sum_{n=1}^\infty f_n g_n : \sum_{n=1}^\infty |f_n|{\mathcal{H}}|g_n|{\mathcal{H}} <\infty\right}.$$ 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.