Multipliers and essential norm on the Drury-Arveson space

Fang, Quanlei; Xia, Jingbo

doi:10.1090/S0002-9939-2010-10680-9

Multipliers and essential norm on the Drury-Arveson space
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by Quanlei Fang and Jingbo Xia PDF

Proc. Amer. Math. Soc. 139 (2011), 2497-2504 Request permission

Corrigendum: Proc. Amer. Math. Soc. 141 (2013), 363-368.

Abstract:

It is well known that for multipliers $f$ of the Drury-Arveson space $H_{n}^{2}$, $\|f\|_{\infty }$ does not dominate the operator norm of $M_{f}$. We show that in general $\|f\|_{\infty }$ 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.

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