Czechoslovak Mathematical Journal, Vol. 74, No. 4, pp. 1097-1112, 2024
Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on a class
of unbounded complete Reinhardt domains
Le He, Yanyan Tang
Received February 16, 2024. Published online October 17, 2024.
Abstract: We consider a class of unbounded nonhyperbolic complete Reinhardt domains $D_{n,m,k}^{\mu,p,s}:=\Big\{(z,w_1,\cdots,w_m)
\in\mathbb{C}^n \times \mathbb{C}^{k_1} \times \cdots \times\mathbb{C}^{k_m}\colon\frac{\| w_1\|^{2p_1}}{{\rm e}^{-\mu_1\| z\|^s}}+\cdots+\frac{\| w_m\|^{2p_m}}{{\rm e}^{-\mu_m\| z\|^s}}<1\Big\}$, where $s$, $p_1,\cdots,p_m$, $\mu_1,\cdots, \mu_m$ are positive real numbers and $n$, $k_1,\cdots,k_m$ are positive integers. We show that if a Hankel operator with anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space $A^2(D_{n,m,k}^{\mu,p,s})$, then it must be zero. This gives an example of high dimensional unbounded complete Reinhardt domain that does not admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols.
Affiliations: Le He, School of Mathematics and Physics, Wuhan Institute of Technology, 206 Guanggu 1st Road, Wuhan 430070, Hubei, P. R. China, e-mail: hele2014@whu.edu.cn; Yanyan Tang (corresponding author), School of Mathematics and Statistics, Henan University, 85 Minglun Street, Kaifeng 475001, Henan, P. R. China, e-mail: yanyantang@whu.edu.cn
