Stroethoff, Karel; Zheng, Dechao Bounded Toeplitz products on the Bergman space of the polydisk. (English) Zbl 1051.47025 J. Math. Anal. Appl. 278, No. 1, 125-135 (2003). 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 \(g\). Reviewer: Nikolaj L. Vasilevskij (México, D.F.) Cited in 2 ReviewsCited in 19 Documents 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) Keywords:Toeplitz operator; Bergman space; polydisk PDFBibTeX XMLCite \textit{K. Stroethoff} and \textit{D. Zheng}, J. Math. Anal. Appl. 278, No. 1, 125--135 (2003; Zbl 1051.47025) Full Text: DOI 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 [3] Stroethoff, K., The Berezin transform and operators on spaces of analytic functions, (Zemánek, J., Linear Operators. Linear Operators, Banach Center Publications, 38 (1997), Polish Academy of Sciences: Polish Academy of Sciences Warsaw), 361-380 · Zbl 0890.47014 [4] Stroethoff, K.; Zheng, D., Products of Hankel and Toeplitz operators on the Bergman space, J. Funct. Anal., 169, 289-313 (1999) · Zbl 0945.47019 [5] Zheng, D., The distribution function inequality and products of Toeplitz operators and Hankel operators, J. Funct. Anal., 138, 477-501 (1996) · Zbl 0865.47019 [6] Zhu, K., Operator Theory in Function Spaces (1990), Marcel Dekker: Marcel Dekker New York This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.