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Let Bn be the open unit ball in C n . We characterize the spectra of pointwise multipliers Mu acting on Banach spaces of analytic functions on Bn satisfying some general conditions. These spaces include Bergman-Sobolev spaces A p α,β , Bloch-type spaces Bα, weighted Hardy spaces H p w with Muckenhoupt weights and Hardy-Sobolev Hilbert spaces H 2 β . Moreover, we describe the essential spectra of multipliers in most of the aforementioned spaces, in particular, in those spaces for which the set of multipliers is a subset of the ball algebra.2010 Mathematics Subject Classification. Primary 47B35; 47B38.

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We show that a weighted composition operator W ψ , φ {W_{\psi ,\varphi }} is compact on BMOA precisely when it is unconditionally converging. As a direct consequence to this result we deduce that all weakly compact or completely continuous weighted composition operators on BMOA are compact. The proof of this result is based on a new simplified function-theoretic characterization of the compactness of weighted composition operators on BMOA.