Some Complex Symmetric Operators on Weighted Hardy Spaces

In this paper, we study the complex symmetry of weighted composite operators and Toeplitz operators on the weighted Hardy space H²(β) by constructing conjugate-linear involutions. When the weight coefficients are β_n=bⁿ, b≥1, we combine the operator J: $\mathit{J}f(z)=\overline{f(\overline{z})}$ with the bounded weighted composite operator W_{u, v} to give W_{u, v}J, and characterize when it is a conjugate-linear isometric involution. Furthermore, we obtain a necessary and sufficient condition for when the bounded weighted composite operator W_{ψ, φ}, with respect to W_{u, v}J, is complex symmetric. On the classical Hardy space, i.e. β_n≡1, we use the idea of permutation method to construct an involutional linear isometric operator C_σ, and characterize the complex symmetry of Toeplitz operators and weighted composite operators with respect to C_σ.