A closer look at a Poisson-like condition on the Drury-Arveson space

Abstract:

Let $\mathcal {M}$ be the collection of the multipliers of the Drury-Arveson space $H^2_n$, $n \geq 2$. In a recent paper [Adv. Math. 335 (2018), pp. 372–404], Aleman et al. showed that for $f \in H^2_n$, the condition $\sup _{|z|<1}\text {Re}\langle f,K_zf\rangle < \infty$ is sufficient for the membership $f \in \mathcal {M}$. We show that this condition is not necessary for $f \in \mathcal {M}$. Moreover, we show that the condition $\sup _{|z|<1}\text {Re}\langle f,K_zf\rangle$ $<$ $\infty$ only captures a nowhere dense subset of $\mathcal {M}$.