In the early 90's, D. Sarason posed conjectures on the characterization of the boundedness of Toeplitz products on Hardy and Bergman spaces (3). The Hardy space case attracted much attention because of its close relation to the famous two-weight problem for the Hilbert transform in Real Analysis, pointed out by Cruz-Uribe in (1). Unfortunately, the Sarason conjecture for Toeplitz products on Hardy space was shown to be false by F. Nazarov around 2000 [2].
In this talk we will show that Sarason conjecture is also false in the Bergman space. Some aspects of the Bergman space setting are easier, because cancellation plays much less of a role in this setting, unfortunately the opposite happens when we look for a counterexample. We will also provide a characterization of the boundedness of Toeplitz products in the Bergman space in terms of testing conditions. This is a joint work with A. Aleman and S. Pott from Lund University.
1. David Cruz-Uribe, The invertibility of the product of unbounded Toeplitz operators, Int.Eq.Op.Th. 20 n.2 (1994), 231-237. 2. F. Nazarov, A counterexample to Sarason's conjecture, http://www.math.msu.edu/~fedja/prepr.html
3. D. Sarason, Products of Toeplitz Operators, Springer Lecture Notes in Mathematics, vol. 1573 (1994), 318-320
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