Complex symmetry of weighted composition operators on the space \(\mathcal{H}_{\alpha, \beta}^2(\mathbb{D})\) (Q1999146)

The authors investigate the complex structure of the weighted composition operator \(W_{\phi,\varphi}\) on the space \(\mathcal{H}^2_{\alpha,\beta}(\mathbb{D})\). Here, \(\mathcal{H}^2_{\alpha,\beta}(\mathbb{D})\) is a subspace of codimension one of the Hardy space \(\mathcal{H}^2(\mathbb{D})\) for which \(\lbrace \alpha+\beta z, z^2, z^3, \dots \rbrace \) form an orthonormal basis where \(\alpha,\beta \in \mathbb{C}\) with \(|\alpha|^2+|\beta|^2=1\) and \(\alpha \neq 0\). The complex symmetric structure of \(W_{\phi,\varphi}\) is studied for different symbols \(\phi\) and \(\varphi\). Isometric properties of the complex symmetric operator \(W_{\phi,\varphi}\) on \(\mathcal{H}^2_{\alpha,\beta}(\mathbb{D})\) are discussed in the last chapter.