| Letφbe a non-trivial analytic self-map of the unit ball BN,ψis a analytic function defined on BN,the weighted composition operator with symbolsφandψis defined by Cφ,ψf=ψf(?)φ.This paper presents a general formula for the adjoint of a weighted composition operator available for all admissibleφandψin any Hilbert spaces of analytic functions with reproducing kernels. That is,supposeφis an holomorphic self-map of BN and kw is an reproducing kernel of Hilbert space H of analytic functions in the unit ball. Then, if Cφ,ψ is bounded on H,we haveAt the base of this, some improved explicit expressions for the adjoint of Cφ,ψ induced by specificφandψare obtained, and the characterization for Cφ,ψ to be self-adjoint on the weighted Bergman space is also given in this paper.
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