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Hilbert-Schmidt Hankel operators on the Bergman space. (English) Zbl 0731.47028
Let $$B_ n\subset {\mathbb{C}}^ n$$ be the open unit ball with (normalized) Lebesgue measure $$\nu$$. $$L^ 2_ a(B_ n)$$ denotes the Bergman space. P is the orthogonal projection from $$L^ 2(B_ n,\nu)$$ onto $$L^ 2_ a(B_ n)$$. The Hankel operator with symbol $$f\in L^ 2(B_ n,\nu)$$ is the densely defined operator $$H_ f: L^ 2_ a(B_ n)\to L^ 2_ a(B_ n)^{\perp}$$ by the equality $H_ fg=(I-P)(fg),\quad g\in L^ 2_ a(B_ n).$ It is proved that if $$n\geq 2$$, then there are no nonzero Hilbert-Schmidt Hankel operators on $$L^ 2_ a(B_ n)$$ with antiholomorphic symbols in $$B_ n$$.

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 47B38 Linear operators on function spaces (general) 30H05 Spaces of bounded analytic functions of one complex variable
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