## 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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