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Keywords:
unbounded complete Reinhardt domain; Hankel operator; Hilbert-Schmidt operator
unbounded complete Reinhardt domain; Hankel operator; Hilbert-Schmidt operator
Summary:
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.
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