Complex symmetric weighted composition operators on Hγ(D)

We study the complex symmetric structure of weighted composition operators of the form Wψ,φ on the Hilbert space Hγ(D) of holomorphic functions over the open unit disk D with reproducing kernels Kw(γ)=(1−w‾z)−γ, where γ∈N. First, we consider conjugations on Hγ(D) of the form Au,vf=u⋅f∘v‾‾ (such conjugations are also known as weighted composition conjugations) and characterize them into two classes, denoted by C1 and C2. Then, we obtain explicit conditions for Wψ,φ when it is C1-symmetric and C2-symmetric respectively.