Czechoslovak Mathematical Journal, Vol. 72, No. 3, pp. 855-873, 2022
Complex symmetry of Toeplitz operators on the weighted Bergman spaces
Xiao-He Hu
Received June 9, 2021. Published online April 26, 2022.
Abstract: We give a concrete description of complex symmetric monomial Toeplitz operators $T_{z^p \bar{z}^q}$ on the weighted Bergman space $A^2(\Omega)$, where $\Omega$ denotes the unit ball or the unit polydisk. We provide a necessary condition for $T_{z^p \bar{z}^q}$ to be complex symmetric. When $p,q \in\mathbb{N}^2$, we prove that $T_{z^p \bar{z}^q}$ is complex symmetric on $A^2(\Omega)$ if and only if $p_1 = q_2$ and $p_2 = q_1$. Moreover, we completely characterize when monomial Toeplitz operators $T_{z^p \bar{z}^q}$ on $A^2(\mathbb{D}_n)$ are $J_U$-symmetric with the $ n \times n$ symmetric unitary matrix $U$.
Keywords: complex symmetry; Toeplitz operator; weighted Bergman space
Affiliations: Xiao-He Hu, College of Mathematics and Information Science, Henan Normal University, 46 East of Construction Road, Xinxiang Henan, 453007, P. R. China, e-mail: huxiaohe94@163.com
