Berezin transform of products of Toeplitz operators on the Hardy space

Xia, Jingbo

doi:10.1090/proc/16268

Berezin transform of products of Toeplitz operators on the Hardy space
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by Jingbo Xia;

Proc. Amer. Math. Soc. 152 (2024), 4269-4276

DOI: https://doi.org/10.1090/proc/16268

Published electronically: August 27, 2024

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Abstract:

Let $H^2(S)$ be the Hardy space on the unit sphere in $\mathbf {C}^n$. We show that there are Toeplitz operators $T_f$ and $T_g$ on $H^2(S)$ such that the product $T_fT_g$ is not compact and yet $\|T_fT_gk_z\|$ tends to $0$ as $|z| \rightarrow 1$. Consequently, the Berezin transform $\langle T_fT_gk_z,k_z\rangle$ tends to $0$ as $|z| \rightarrow 1$.

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