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Bounded Toeplitz products on the Bergman space of the polydisk. (English) Zbl 1051.47025

Let \(L_a^2(D^n)\) be the Bergman space on the polydisk \(D^n\). The authors give a necessary and a sufficient condition for the boundedness on \(L_a^2(D^n)\) of the densely defined product of two Toeplitz operators \(T_f T_{\overline{g}}\), with \(f,g \in L_a^2(D^n)\). Both conditions are given in terms of the Berezin transform of the symbols \(f\) and \(g\).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A36 Bergman spaces of functions in several complex variables
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
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References:

[1] F. Nazarov, A counter-example to Sarason’s conjecture, preprint; available at http://www.math.msu.edu/ fedja/prepr.html; F. Nazarov, A counter-example to Sarason’s conjecture, preprint; available at http://www.math.msu.edu/ fedja/prepr.html
[2] Sarason, D., Products of Toeplitz operators, (Khavin, V. P.; Nikol’skiı̆, N. K., Linear and Complex Analysis Problem Book 3, Part I. Linear and Complex Analysis Problem Book 3, Part I, Lecture Notes in Math., 1573 (1994), Springer: Springer Berlin), 318-319
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