Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space H 2

Abstract

We characterize the essentially normal weighted composition operators Cψ,ϕ on the Hardy space H 2 , whenever ϕ is a linear-fractional transformation and ψ ∈ A( ) . Also we investigate the essential normality problem for some other weighted composition operators on H 2 .

Some essentially normal weighted composition operators on the weighted Bergman spaces

“…Then, we characterize the essentially normal weighted composition operators C ψ,ϕ on the weighted Bergman spaces A 2 α , whenever ϕ ∈ LFT(D) is not an automorphism and ψ ∈ H ∞ is continuous at a point ζ ∈ F(ϕ). After that, in Remark 2.7, we state that by a similar proof of Theorem 2.6, we can develop and shorten the second section of [13]. Finally, we characterize the essentially normal weighted composition operators C ψ,ϕ on the weighted Bergman spaces A 2 α , whenever ϕ ∈ Aut(D) and ψ ∈ A(D).…”

“…[11,12]). In [13], Fatehi and Khani Robati showed that for ψ ∈ A(D) and ϕ ∈ LFT(D) − Aut(D) such that ϕ(ζ ) ∈ ∂D, where ζ ∈ ∂D, C ψ,ϕ is essentially normal on H 2 if and only if ψ(ζ ) = 0 or ϕ is parabolic. Furthermore, they obtained a necessary and sufficient condition for a operator C ψ,ϕ on H 2 to be essentially normal, where ϕ ∈ Aut(D) and ψ ∈ A(D) (see also [14]).…”

“…In [13], Fatehi and Khani Robati showed that for ψ ∈ A(D) and ϕ ∈ LFT(D) − Aut(D) such that ϕ(ζ ) ∈ ∂D, where ζ ∈ ∂D, C ψ,ϕ is essentially normal on H 2 if and only if ψ(ζ ) = 0 or ϕ is parabolic. Furthermore, they obtained a necessary and sufficient condition for a operator C ψ,ϕ on H 2 to be essentially normal, where ϕ ∈ Aut(D) and ψ ∈ A(D) (see also [14]). Recently, some authors have investigated that when the commutator [C * ψ , C ϕ ] = C * ψ C ϕ − C ϕ C * ψ is nontrivially compact for linear-fractional self-maps ϕ and ψ of D. Here, non-trivially compact means that [C * ψ , C ϕ ] is compact but non-zero, and C * ψ C ϕ and C ϕ C * ψ are not compact (see [1,15]).…”

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

“…The automorphisms are classified according to the location of their fixed points: elliptic if one fixed point is in D and a second fixed point is in the complement of the closed disk, hyperbolic if both fixed points are in ∂D, and parabolic if there is one fixed point in ∂D of multiplicity two (see [10] and [23]). Let ϕ be an automorphism of D. In [13] and [15], the present authors investigated essentially normal weighted composition operator C ψ,ϕ , when ψ ∈ A(D) and ψ(z) = 0 for each z ∈ D. In this section, we just assume that ψ is analytic on D and we attempt to find all normal weighted composition operators C ψ,ϕ . Also we will show that ψ never vanishes on D.…”

“…Assume that there is ζ ∈ ∂D such that ψ(ζ) = 0. By [13, Theorem 3.2] and [15,Theorem 3.3], ψ is zero on B = {ζ, ϕ(ζ), ϕ 2 (ζ), ...}. It is easy to see that B ⊆ ∂D.…”