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

TL;DR

This paper investigates the relationship between the multiplier algebras of a Hilbert space of analytic functions and its weak product, showing they are the same for certain Besov spaces on the unit ball.

Contribution

It proves that for first order holomorphic Besov Hilbert spaces on the unit ball, the multiplier algebras of the space and its weak product are identical.

Findings

Multiplier algebras of  and  are equal for these spaces.

Weak product space retains the same multipliers as the original space.

Results apply specifically to first order holomorphic Besov Hilbert spaces.

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.