Products of Toeplitz operators on the Bergman space
Résumé
When is the product of two Toeplitz operators again a Toeplitz operator in the context of the Bergman space in the unit disk? This problem was satisfactorily solved by A. Brown and P. R. Halmos [J. Reine Angew. Math. 213 (1963/1964), 89–102 in 1963 for the Hardy space H2. P. R. Ahern and Zˇ . Cˇ ucˇkovic´ obtained a corresponding result for the Bergman space with the additional assumption that the symbols of the operators are bounded and harmonic. In this paper the authors deal with some special types of operators with symbols eip'(r) where p is an integer, (r, ) are polar coordinates in the complex plane and ' belongs to L1(D), D being the unit disk. These symbols give rise to Toeplitz operators that are only densely defined, possibly unbounded. So any such function is called a T-function if the associated Toeplitz operator is bounded. Theorem 6.1 of the authors gives a necessary and sufficient condition for a product of two Toeplitz operators given by two T-functions eip'1(r) and e−is'2(r) to be again a Toeplitz operator assuming p s > 0 are integers. They also provide examples to show that even in the class of such special operators, general products need not be Toeplitz without additional hypotheses.