Normal, Cohyponormal and Normaloid Weighted Composition Operators on the Hardy and Weighted Bergman Spaces

Abstract

Abstract. If ψ is analytic on the open unit disk D and ϕ is an analytic self-map of D, the weighted composition operator C ψ,ϕ is defined by C ψ,ϕ f (z) = ψ(z)f (ϕ(z)), when f is analytic on D. In this paper, we study normal, cohyponormal, hyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces. First, for some weighted Hardy spaces H 2 (β), we prove that if C ψ,ϕ is cohyponormal on H 2 (β), then ψ never vanishes on D and ϕ is univalent, when ψ ≡ 0 and ϕ is not a constant function. Moreover, for ψ = Ka, where |a| < 1, we investigate normal, cohyponormal and hyponormal weighted composition operators C ψ,ϕ . After that, for ϕ which is a hyperbolic or parabolic automorphism, we characterize all normal weighted composition operators C ψ,ϕ , when ψ ≡ 0 and ψ is analytic on D. Finally, we find all normal weighted composition operators which are bounded below.

Quasinormal and hyponormal weighted composition operators on $H^2$ and $A^2_α$ with linear fractional compositional symbol

“…In [19], the first two authors found all normal weighted composition operators W ψ,ϕ when ϕ ∈ Aut(D) and ψ is analytic on D. In the following proposition, we add the assumption that ψ is bounded away from zero on D, and by the similar idea which was used in [19], we characterize all invertible quasinormal weighted composition operators. Proposition 2.3.…”