Products of Several Toeplitz Operators

In this paper we give a necessary and sufficient condition for Ta1Ta2…Tan=Ta1a2…an where the Taiʹs are Toeplitz operators on the Hardy space of the unit disk. We then show that Ta1Ta2Ta3Ta4Ta5Ta6=0 if and only if one of ai is identically zero. A criteria for the compactness of Ta1Ta2…Tan−Ta1a2…an is also given.