Products of asymmetric truncated Toeplitz operators

This paper is concerned with asymmetric truncated Toeplitz operators, which are compressions of multiplication operators acting between two model spaces. We give necessary and sufficient conditions so that the product of two asymmetric truncated Toeplitz operators is also an asymmetric truncated Toeplitz operator. Then we define analogue classes to the Sedlock classes, which consist on the algebras of truncated Toeplitz operators, and deal with the case of two inner functions such that one divides the other. Some examples of the product of rank-one asymmetric truncated Toeplitz operators involving inner functions are also given.