Complex symmetry of Toeplitz operators on the weighted Bergman spaces

Hu, Xiao-He

Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 855-873

Résumé

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 $U$.

Détail

DOI : 10.21136/CMJ.2022.0210-21

Classification : 32A36, 47B35
Keywords: complex symmetry; Toeplitz operator; weighted Bergman space