Complex symmetry of Toeplitz operators on the weighted Bergman spaces

Xiao-He Hu

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 3, page 855-873
  • ISSN: 0011-4642

Abstract

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We give a concrete description of complex symmetric monomial Toeplitz operators T z p z ¯ q on the weighted Bergman space A 2 ( Ω ) , where Ω denotes the unit ball or the unit polydisk. We provide a necessary condition for T z p z ¯ q to be complex symmetric. When p , q ∈ ℕ 2 , we prove that T z p z ¯ q is complex symmetric on A 2 ( Ω ) if and only if p 1 = q 2 and p 2 = q 1 . Moreover, we completely characterize when monomial Toeplitz operators T z p z ¯ q on A 2 ( 𝔻 n ) are J U -symmetric with the n × n symmetric unitary matrix U .

How to cite

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Hu, Xiao-He. "Complex symmetry of Toeplitz operators on the weighted Bergman spaces." Czechoslovak Mathematical Journal 72.3 (2022): 855-873. <http://eudml.org/doc/298381>.

@article{Hu2022,
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 $U$.},
author = {Hu, Xiao-He},
journal = {Czechoslovak Mathematical Journal},
keywords = {complex symmetry; Toeplitz operator; weighted Bergman space},
language = {eng},
number = {3},
pages = {855-873},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Complex symmetry of Toeplitz operators on the weighted Bergman spaces},
url = {http://eudml.org/doc/298381},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Hu, Xiao-He
TI - Complex symmetry of Toeplitz operators on the weighted Bergman spaces
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 3
SP - 855
EP - 873
AB - 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 $U$.
LA - eng
KW - complex symmetry; Toeplitz operator; weighted Bergman space
UR - http://eudml.org/doc/298381
ER -

References

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