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Complex symmetry of Toeplitz operators on the weighted Bergman spaces
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 855-873 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $U$.
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 $U$.
DOI : 10.21136/CMJ.2022.0210-21
Classification : 32A36, 47B35
Keywords: complex symmetry; Toeplitz operator; weighted Bergman space 
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     title = {Complex symmetry of {Toeplitz} operators on the weighted {Bergman} spaces},
     journal = {Czechoslovak Mathematical Journal},
     pages = {855--873},
     year = {2022},
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Hu, Xiao-He. Complex symmetry of Toeplitz operators on the weighted Bergman spaces. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 855-873. doi: 10.21136/CMJ.2022.0210-21

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