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A theorem of Nehari type on weighted Bergman spaces of the unit ball. (English) Zbl 1162.32003

The following result is proved:
Let \(S\) be a bounded linear operator acting on the weighted Bergman space \(A_{\alpha}^{2}(B_n) := L^2(B_n, c_{\alpha} (1 - |z|)^{\alpha} dV)\cap H(B_n) = L^2(B_n, dV_{\alpha} )\cap H(B_n)\), \(\alpha > -1\), and \(T_{\psi} f := P(\psi f)\) be a Toeplitz operator on \(A_{\alpha}^{2}(B_n)\) with symbol \(\psi\in L^{\infty}(B_n)\).
If the relations \(S T_{z_i} = T_{\overline{z}_i} S, i = 1, \ldots, n\) hold, then there exists \(\phi\in L^{\infty}(B_n)\) such that \(S\) coincides with the Hankel operator \(H_{\phi}\) on \(A_{\alpha}^{2}(B_n)\) (\(H_{\phi}(f) := P(J(\phi f))\)).
Here \(P\) is the weighted Bergman projection from \(L^2(B_n, dV_{\alpha})\) on \(A_{\alpha}^{2}(B_n)\), and \(J(f(z)) :=f(\overline{z})\) is the unitary operator on \(L^2(B_n, dV_{\alpha})\).

MSC:

32A36 Bergman spaces of functions in several complex variables
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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References:

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