Complex symmetry of weighted composition operators on the space $\mathcal{H}_{\alpha, \beta}^2(\mathbb{D})$

Abstract

We investigate the complex symmetric structure of the weighted composition operator $W_{\psi,\phi}$ on the subspace $\mathcal{H}_{\alpha, \beta}^2(\mathbb{D})$ of the Hardy space $\mathcal{H}^2(\mathbb{D})$ of codimension one. We provide characterizations of symbols $\phi$ and $\psi$ such that $W_{\psi,\phi}$ is complex symmetric with respect to a special conjugation. In addition, we discuss isometric properties of the complex symmetric operator $W_{\psi,\phi}$ on $\mathcal{H}_{\alpha, \beta}^2(\mathbb{D})$.