Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 207-217
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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.
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.
DOI :
10.21136/CMJ.2017.0471-15
Classification :
32A36, 47B10, 47B35
Keywords: canonical solution operator for $\overline {\partial }$-problem; Hankel operator; Hilbert-Schmidt operator
Keywords: canonical solution operator for $\overline {\partial }$-problem; Hankel operator; Hilbert-Schmidt operator
@article{10_21136_CMJ_2017_0471_15,
author = {\c{C}elik, Mehmet and Zeytuncu, Yunus E.},
title = {Hilbert-Schmidt {Hankel} operators with anti-holomorphic symbols on complete pseudoconvex {Reinhardt} domains},
journal = {Czechoslovak Mathematical Journal},
pages = {207--217},
year = {2017},
volume = {67},
number = {1},
doi = {10.21136/CMJ.2017.0471-15},
mrnumber = {3633007},
zbl = {06738513},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0471-15/}
}
TY - JOUR AU - Çelik, Mehmet AU - Zeytuncu, Yunus E. TI - Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains JO - Czechoslovak Mathematical Journal PY - 2017 SP - 207 EP - 217 VL - 67 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0471-15/ DO - 10.21136/CMJ.2017.0471-15 LA - en ID - 10_21136_CMJ_2017_0471_15 ER -
%0 Journal Article %A Çelik, Mehmet %A Zeytuncu, Yunus E. %T Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains %J Czechoslovak Mathematical Journal %D 2017 %P 207-217 %V 67 %N 1 %U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0471-15/ %R 10.21136/CMJ.2017.0471-15 %G en %F 10_21136_CMJ_2017_0471_15
Çelik, Mehmet; Zeytuncu, Yunus E. Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 207-217. doi: 10.21136/CMJ.2017.0471-15
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