In this paper, we find complex symmetric composition operators on the classical Hardy space H 2 whose symbols are linear-fractional but not automorphic. In doing so, we answer a recent question of Noor, and partially answer the original problem posed by Garcia and Hammond.

Abstract. We completely characterize the spectrum of a weighted composition operator W ψ,ϕ on H 2 (D) when ϕ has Denjoy-Wolff point a with 0 < |ϕ ′ (a)| < 1, the iterates, ϕn, converge uniformly to a, and ψ is in H ∞ and continuous at a. We also give bounds and some computations when |a| = 1 and ϕ ′ (a) = 1 and, in addition, show that these symbols include all linear fractional ϕ that are hyperbolic and parabolic non-automorphisms. Finally, we use these results to eliminate possible weights ψ so that W ψ,ϕ is seminormal.

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When ϕ is an analytic self-map of the unit disk with Denjoy-Wolff point a ∈ D, and ρ(W ψ,ϕ ) = ψ(a), we give an exact characterization for when W ψ,ϕ is normaloid. We also determine the spectral radius, essential spectral radius, and essential norm for a class of non-powercompact composition operators whose symbols have Denjoy-Wolff point in D. When the Denjoy-Wolff point is on ∂D, we give sufficient conditions for several new classes of normaloid weighted composition operators.

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