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

If $\mathcal{H}$ denotes a Hilbert space of analytic functions on a region $\mathrm{\Omega }\subseteq {\mathbb{C}}^d$, then the weak product is defined by $$\mathcal{H}\odot \mathcal{H}=\{h=\sum _{n=1}^{\mathrm{\infty }}{f}_n{g}_n:\sum _{n=1}^{\mathrm{\infty }}\mathrm{\parallel }{f}_n{\mathrm{\parallel }}_{\mathcal{H}}\mathrm{\parallel }{g}_n{\mathrm{\parallel }}_{\mathcal{H}}<\mathrm{\infty }\}\mathrm{.}$$ 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.