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

Mehmet Çelik; Yunus E. Zeytuncu

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 1, page 207-217
  • ISSN: 0011-4642

Abstract

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On complete pseudoconvex Reinhardt domains in 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 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 z ¯ 1 z ¯ 2 is Hilbert-Schmidt.

How to cite

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Çelik, Mehmet, and Zeytuncu, Yunus E.. "Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains." Czechoslovak Mathematical Journal 67.1 (2017): 207-217. <http://eudml.org/doc/287892>.

@article{Çelik2017,
abstract = {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.},
author = {Çelik, Mehmet, Zeytuncu, Yunus E.},
journal = {Czechoslovak Mathematical Journal},
keywords = {canonical solution operator for $\overline\{\partial \}$-problem; Hankel operator; Hilbert-Schmidt operator},
language = {eng},
number = {1},
pages = {207-217},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains},
url = {http://eudml.org/doc/287892},
volume = {67},
year = {2017},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 207
EP - 217
AB - 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.
LA - eng
KW - canonical solution operator for $\overline{\partial }$-problem; Hankel operator; Hilbert-Schmidt operator
UR - http://eudml.org/doc/287892
ER -

References

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