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A different proof for the non-existence of Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on the Bergman space. (English) Zbl 1220.47040
Summary: We show that there are no (nontrivial) Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on the Bergman space of the unit ball \(B^{2}(B^{l})\) in \(\mathbb{C}^l\), \(l\geq 2\). The result dates back to [K. Zhu, Proc. Am. Math. Soc. 109, No. 3, 721–730 (1990; Zbl 0731.47028)]. However, we give a different proof. The methodology can be easily applied to other, more general settings. Especially, as indicated in the section containing generalizations, the new methodology allows to prove some robustness results for existing ones.

MSC:
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A15 Entire functions of several complex variables
32A36 Bergman spaces of functions in several complex variables
46E15 Banach spaces of continuous, differentiable or analytic functions
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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