Multipliers and essential norm on the Drury-Arveson space

It is well known that for multipliers $f$ of the Drury-Arveson space $H_{n}^{2}$, $\Vert f\Vert _{\infty}$ does not dominate the operator norm of $M_{f}$. We show that in general $\Vert f\Vert _{\infty}$ does not even dominate the essential norm of $M_{f}$. A consequence of this is that there exist multipliers $f$ of $H_{n}^{2}$ for which $M_f$ fails to be essentially hyponormal; i.e., if $K$ is any compact, self-adjoint operator, then the inequality $M_f^\ast M_f - M_fM_f^\ast + K \geq 0$ does not hold.