Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on a class of unbounded complete Reinhardt domains

Le He; Yanyan Tang

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 4, page 1097-1112
  • ISSN: 0011-4642

Abstract

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We consider a class of unbounded nonhyperbolic complete Reinhardt domains D n , m , k μ , p , s : = ( z , w 1 , ⋯ , w m ) ∈ ℂ n × ℂ k 1 × ⋯ × ℂ k m : ∥ w 1 ∥ 2 p 1 e - μ 1 ∥ z ∥ s + ⋯ + ∥ w m ∥ 2 p m e - μ m ∥ z ∥ s < 1 , where s , p 1 , ⋯ , p m , μ 1 , ⋯ , μ m are positive real numbers and n , k 1 , ⋯ , 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 μ , 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.

How to cite

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He, Le, and Tang, Yanyan. "Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on a class of unbounded complete Reinhardt domains." Czechoslovak Mathematical Journal 74.4 (2024): 1097-1112. <http://eudml.org/doc/299639>.

@article{He2024,
abstract = {We consider a class of unbounded nonhyperbolic complete Reinhardt domains \[ D\_\{n,m,k\}^\{\mu ,p,s\}:=\Big \lbrace (z,w\_1,\cdots ,w\_m)\in \mathbb \{C\}^\{n\}\times \mathbb \{C\}^\{k\_1\}\times \cdots \times \mathbb \{C\}^\{k\_m\}\colon \frac\{\Vert w\_1\Vert ^\{2p\_1\}\}\{\{\rm e\}^\{-\mu \_1\Vert z\Vert ^\{s\}\}\}+\cdots +\frac\{\Vert w\_m\Vert ^\{2p\_m\}\}\{\{\rm e\}^\{-\mu \_m\Vert z\Vert ^\{s\}\}\}<1\Big \rbrace , \] 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.},
author = {He, Le, Tang, Yanyan},
journal = {Czechoslovak Mathematical Journal},
keywords = {unbounded complete Reinhardt domain; Hankel operator; Hilbert-Schmidt operator},
language = {eng},
number = {4},
pages = {1097-1112},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on a class of unbounded complete Reinhardt domains},
url = {http://eudml.org/doc/299639},
volume = {74},
year = {2024},
}

TY - JOUR
AU - He, Le
AU - Tang, Yanyan
TI - Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on a class of unbounded complete Reinhardt domains
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 4
SP - 1097
EP - 1112
AB - We consider a class of unbounded nonhyperbolic complete Reinhardt domains \[ D_{n,m,k}^{\mu ,p,s}:=\Big \lbrace (z,w_1,\cdots ,w_m)\in \mathbb {C}^{n}\times \mathbb {C}^{k_1}\times \cdots \times \mathbb {C}^{k_m}\colon \frac{\Vert w_1\Vert ^{2p_1}}{{\rm e}^{-\mu _1\Vert z\Vert ^{s}}}+\cdots +\frac{\Vert w_m\Vert ^{2p_m}}{{\rm e}^{-\mu _m\Vert z\Vert ^{s}}}<1\Big \rbrace , \] 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.
LA - eng
KW - unbounded complete Reinhardt domain; Hankel operator; Hilbert-Schmidt operator
UR - http://eudml.org/doc/299639
ER -

References

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