On Toeplitz operators on Bergman spaces of the unit polydisk

Le, Trieu

doi:10.1090/S0002-9939-09-10060-6

On Toeplitz operators on Bergman spaces of the unit polydisk
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Proc. Amer. Math. Soc. 138 (2010), 275-285 Request permission

Abstract:

We study Toeplitz operators on the Bergman space $A^2_{\vartheta }$ of the unit polydisk $\mathbb {D}^n$, where $\vartheta$ is a product of $n$ rotation-invariant regular Borel probability measures. We show that if $f$ is a bounded Borel function on $\mathbb {D}^n$ such that $F(w)=\lim \limits _{\substack {z\rightarrow w\\z\in \mathbb {D}^n}}f(z)$ exists for all $w\in \partial \mathbb {D}^n$, then $T_f$ is compact if and only if $F=0$ a.e. with respect to a measure $\gamma$ associated with $\vartheta$ on the boundary $\partial \mathbb {D}^n$ . We also discuss the commuting problem: if $g$ is a non-constant bounded holomorphic function on $\mathbb {D}^n$, then what conditions does a bounded function $f$ need to satisfy so that $T_f$ commutes with $T_g$?

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