Toeplitz Operators on Dirichlet-Type Space of Unit Ball

Abstract

We construct a function $u$ in $L^2\left({\mathbb{B}}_n, dV\right)$ which is unbounded on any neighborhood of each boundary point of ${\mathbb{B}}_n$ such that Toeplitz operator $T_u$ is a Schatten $p$-class operator on Dirichlet-type space $\mathcal{D}\left({\mathbb{B}}_n, dV\right)$. Then, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Dirichlet-type space $\mathcal{D}\left({\mathbb{B}}_n, dV\right)$. We determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. We investigate the zero-product problem for several Toeplitz operators with radial symbols. Furthermore, the corresponding commuting problem of Toeplitz operators whose symbols are of the form ${\xi }^{k}u$ is studied, where $k\hspace{0.17em}\in \hspace{0.17em}{\mathbb{Z}}^n$, $\xi \hspace{0.17em}\in \hspace{0.17em}\partial {\mathbb{B}}_n$, and $u$ is a radial function.