Complex Symmetry of Toeplitz Operators over the Bidisk

In this paper, we investigate the complex symmetric structure of Toeplitz operators T ϕ on the Hardy space over the bidisk. We first characterize the weighted composition operators, W u, v which are J -symmetric and unitary. As a consequence, we characterize conjugations of the form A u, v . In addition, a class of conjugations of the form C λ, a is introduced. We show that the class of conjugations C λ, a coincides with the class of conjugations A u, v ; we then characterize the complex symmetry of the Toeplitz operators T ϕ with respect to the conjugation C λ, a .