A weighted composition operator C ψ,ϕ takes an analytic map f on the open unit disc of the complex plane to the analytic map ψ · f • ϕ where ϕ is an analytic map of the open unit disc into itself and ψ is an analytic map on the open unit disc. This paper studies the invertibility of such operators. The two maps ψ and ϕ are characterized when C ψ,ϕ acts on the Hardy-Hilbert space of the unit disc H 2 (D). Depending upon the nature of the fixed points of ϕ spectra are then investigated.

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Abstract. A weighted composition operator C ψ,ϕ takes an analytic map f on the open unit disk of the complex plane to the analytic map ψf • ϕ, where ϕ is an analytic map of the open unit disk into itself and ψ is an analytic map on the open unit disk. This paper studies how the compactness of C ψ,ϕ depends on the interaction between the two maps ψ and ϕ.

Abstract. For ψ analytic in the open unit disk and ϕ an analytic map from the unit disk into itself, the weighted composition operator C ψ,ϕ is the operator on the weighted Hardy space H 2 (β) given by (C ψ,ϕ f )(z) = ψ(z)f (ϕ(z)). This paper discusses the spectrum of C ψ,ϕ when it is compact on a certain class of weighted Hardy spaces and when the composition map ϕ has a fixed point inside the open unit disk.

Abstract:Let be an open simply connected proper subset of the complex plane and an analytic self map of . If f is in the Hardy-Smirnov space defined on , then the operator that takes f to f ı is a composition operator. We show that for any , analytic self maps that induce bounded Hermitian composition operators are of the form .w/ D aw C b where a is a real number. For ceratin , we completely describe values of a and b that induce bounded Hermitian composition operators.