Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains

Czechoslovak Mathematical Journal

Mehmet Çelik, Yunus E. Zeytuncu¹ 

关键词: canonical solution operator for $\overline{\partial}$-problem;    Hankel operator;    Hilbert-Schmidt operator;   

DOI  :  10.21136/CMJ.2017.0471-15

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

  On complete pseudoconvex Reinhardt domains in $\mathbb{C}^2$, we show that there is no nonzero Hankel operator with anti-holomorphic symbol that is Hilbert-Schmidt. In the proof, we explicitly use the pseudoconvexity property of the domain. We also present two examples of unbounded non-pseudoconvex domains in $\mathbb{C}^2$ that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols. In the first example the Bergman space is finite dimensional. However, in the second example the Bergman space is infinite dimensional and the Hankel operator $H_{\bar{z}_1 \bar{z}_2}$ is Hilbert-Schmidt.

【 授权许可】

Unknown