We characterize all weighted composition operators v C ψ vC that commute with a J μ J -symmetric weighted composition operator u C φ uC on the reproducing kernel Hilbert space H γ H of analytic functions on the unit disk D D. It turns out these commuting operators v C ψ vC are necessarily J μ J -symmetric. Furthermore, we obtain characterization (s) for the commuting operator v C ψ vC to be self-adjoint, normal or unitary.
Ching-on Lo (Thu,) studied this question.