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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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