Complex Symmetry of Toeplitz Composition Operators on the Bergman Space

In this paper, we characterize the conditions for a complex symmetric Toeplitz composition operator \(TψCφ\) on the Bergman space \(A^2(D)\), where the symbol function \(ψ\) is harmonic or \(ψ(z)=Az̄^nz^m+Bz̄^sz^t\) for \(A, B\{0\}\) and \(n-m=t-s\). We also present necessary and sufficient conditions for the commutativity of \(TψCφ\) and \(Cξ,θ\), where \(Cξ,θf(z)=eiξ{f(eiθz̄)}\) for \(ξ, θ ∈ R\).