Dual truncated Toeplitz operators

Let u be a nonconstant inner function. In this paper, we study the dual truncated Toeplitz operators on the orthogonal complement of the model space Ku2. This is a new class of Toeplitz operator. We show the product of two dual truncated Toeplitz operators DfDg to be zero if and only if either f or g is zero. We give a necessary and sufficient condition for the product of two dual truncated Toeplitz operators to be a finite rank operator. Furthermore, a necessary and sufficient condition is found for the product of two dual truncated Toeplitz operators to be a dual truncated Toeplitz operator. The last two results are different from the classical Toeplitz operator theory.