Operators commuting with complex symmetric weighted composition operators on $H^2$

Abstract: In this paper, we initially study when an anti-linear Toeplitz operator is in the commutant of a composition operator. Primarily, we investigate weighted composition operators $W_{g,\psi}$ commuting with complex symmetric weighted composition operators $W_{f,\varphi}$ on the Hardy space $H^2(\mathbb{D})$. In particular, we give the descriptions of the symbols $g$ and $\psi$ such that the inducing weighted composition operator $W_{g,\psi}$ commutes with the complex symmetric weighted composition operator $W_{f,\varphi}$ with the conjugation $\mathcal{J}$. Furthermore, we subsequently demonstrate that these weighted composition operators are normal and complex symmetric in accordance with the properties of the fixed point of the associated symbol $\varphi$.