关键词: Toeplitz operators, Hardy space, complex symmetry

Abstract: In this paper, we investigate the complex symmetric structure of Toeplitz operators $T_\phi$ on the Hardy space over the bidisk. We first characterize the weighted composition operators, $W_{u,v}$ which are $\mathcal{J}$-symmetric and unitary. As a consequence, we characterize conjugations of the form $A_{u,v}$. In addition, a class of conjugations of the form $C_{\lambda, a}$ is introduced. We show that the class of conjugations $C_{\lambda, a}$ coincides with the class of conjugations $A_{u,v}$; we then characterize the complex symmetry of the Toeplitz operators $T_\phi$ with respect to the conjugation $C_{\lambda, a}$.

Key words: Toeplitz operators, Hardy space, complex symmetry