2015/09/29 by Fatehi, Mahsa, Shaabani, Mahmood Haji
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1509.08632
If ψ is analytic on the open unit disk \mathbbD and φ is an analytic self-map of \mathbbD, the weighted composition operator Cψ,φ is defined by Cψ,φf(z)=ψ(z)f (φ(z)), when f is analytic on \mathbbD. 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 H2(β), we prove that if Cψ,φ is cohyponormal on H2(β), then ψ never vanishes on \mathbbD and φ is univalent, when ψ\not ≡ 0 and φ is not a constant function. Moreover, for ψ=Ka, where |a| < 1, we investigate normal, cohyponormal and hyponormal weighted composition operators Cψ,φ. After that, for φ which is a hyperbolic or parabolic automorphism, we characterize all normal weighted composition operators Cψ,φ, when ψ\not ≡ 0 and ψ is analytic on \mathbbD. Finally, we find all normal weighted composition operators which are bounded below.