Sum of Toeplitz products on the Hardy space over the polydisk (Q1746529)

Let $P$ be the orthogonal projection of the space $L^2(\mathbb{T}^N)$ onto the Hardy space $H^2(\mathbb{D}^N)$ over the polydisk $\mathbb{D}^N$, and let $T_u$ be the Toeplitz operator with symbol $u\in L^\infty(\mathbb{T}^N)$ given by $T_u f=P(uf)$ for all $f\in H^2(\mathbb{D}^N)$. The paper deals with criteria for a finite sum $\sum_{m=1}^M T_{f_m}T_{g_m}$ of products of Toeplitz operators on $H^2(\mathbb{D}^N)$ to be zero. \par For each $i\in\hat{N}:=\{1,2,\dots,N\}$, let $\partial_i=\partial_{z_i}$ and $\overline{\partial}_i=\partial_{\overline{z}_i}$. A~twice differentiable function $f$ defined on an open set in $\mathbb{C}^N$ is called $N$-harmonic if $\partial_i\overline{\partial}_if=0$ for each $i\in\hat{N}$. Given $f_m,g_m\in L^\infty(\mathbb{T}^N)$ for $m=1,2,\dots,M$, it is proved that $\sum_{m=1}^M T_{f_m}T_{g_m}=0$ on $H^2(\mathbb{D}^N)$ if and only if the Berezin transform of $\sum_{m=1}^M T_{f_m}T_{g_m}$ is $N$-harmonic on $\mathbb{D}^N$ and $\sum_{m=1}^M f_m(\xi)g_m(\xi)=0$ for almost all $\xi\in\mathbb{T}^N$. \par Let $\partial_I=\partial_{i_1}\partial_{i_2}\cdots\partial_{i_k}$ for $I=\{i_1,i_2,\dots,i_k\}\subset\hat{N}$, $I^c=\hat{N}\setminus I$, $z_I=(z_{i_1},z_{i_2},\dots,z_{i_k})$, $(z_I,z_{I^c})=(z_1,z_2,\dots,z_N)$, and let $\partial_\emptyset f=f$ for $I=\emptyset$. Given $f\in L^1(\mathbb{T}^N)$, the partial harmonic extension of $f$ with respect to a set $I\subset\hat{N}$ is defined by \[ \widetilde{f}_I(z_I,\xi_{I^c})=\int_{\mathbb{T}^{|I|}}f(\xi_I,\xi_{I^c})\prod_{i\in I} \frac{1-|z_i|^2}{|1-\overline{z}_i\xi_i|^2}\:d\sigma(\xi_I), \] where $d\sigma(\xi_I)$ is the Haar measure on $\mathbb{T}^{|I|}$ and $|I|$ is the cardinal number of $I$. In particular, the following criterion is obtained: if $f_m,g_m\in L^\infty(\mathbb{T}^N)$ for all $m=1,2,\dots,M$, then $\sum_{m=1}^M T_{f_m}T_{g_m}=0$ on $H^2(\mathbb{D}^N)$ if and only if \[ \sum_{m=1}^M \partial_I\widetilde{f}_m(z_I,\xi_{I^c}) \overline{\partial}_I \widetilde{g}_m(z_I,\xi_{I^c})=0 \] for all $I\subset\hat{N}$, all $z_I\in\mathbb{D}^{|I|}$ and almost all $\xi_{I^c}\in\mathbb{T}^{N-|I|}$.