Some essentially normal weighted composition operators on the weighted Bergman spaces

Abstract

First of all, we obtain a necessary and sufficient condition for a certain operator T w C ϕ to be compact on A 2 α . Next, we give a short proof for Proposition 2.5 which was proved by MacCluer, Narayan and Weir. Then, we characterize the essentially normal weighted composition operators C ψ,ϕ on the weighted Bergman spaces A 2 α , when ϕ ∈ LFT(D) is not an automorphism and ψ ∈ H ∞ is continuous at a point ζ which ϕ has a finite angular derivative. After that we find some nontrivially essentially normal weighted composition operators, when ϕ ∈ LFT(D) is not an automorphism. In the last section, for ϕ ∈ Aut(D) and ψ ∈ A(D), we characterize the essentially normal weighted composition operators C ψ,ϕ on A 2 α and investigate some essentially normal weighted composition operators C ψ,ϕ on H 2 and A 2 α . Finally, we find some non-trivially essentially normal weighted composition operators C ψ,ϕ on H 2 and A 2 α , when ϕ ∈ Aut(D) and ψ ∈ A(D).

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

“…Then C ψ,ϕ is essentially normal. Since ψ never vanishes on ∂D, we conclude from [13,Theorem 2.6] and [13,Remark 2.7] that ϕ is a parabolic non-automorphism and the result follows from Proposition 3.4 and Equation (2).…”

Essentially hyponormal weighted composition operators on the Hardy and weighted Bergman spaces

“…From now on, unless otherwise stated, we assume that σ, h and g are given as above. In [13] and [16], the adjoint of C ϕ , modulo the ideal of compact operators on H 2 and A 2 α , was obtained, when ϕ ∞ = 1 but ϕ is not an automorphism; then in [11] the present authors gave another proof for the form of this adjoint.…”

“…Since σ e,γ (C ϕn ) is a compact set, by Lemma 3.1, there is λ ∈ σ e,γ (C ϕn ) such that |λ| = r e,γ (C ϕn ) = ϕ ′ (ζ) −nγ/2 . Since ϕ(ζ) = ζ, [13, Corollary 2.2] and [11,Proposition 2.3] imply that σ e,γ (C n ψ,ϕ ) = σ e,γ (T ψ·ψ•ϕ...ψ•ϕ n−1 C ϕn ) = ψ(ζ) n σ e,γ (C ϕn ). We may now apply Lemma 2.1 and [1, Lemma 5.1] to observe that |µ| ≤ |ψ(ζ)| n r γ (C ϕn ) for each µ in σ p,γ (C n ψ,ϕ ∂σ γ (C n ψ,ϕ ) ⊆ σ ap,γ (C n ψ,ϕ ) ⊆ σ p,γ (C n ψ,ϕ ) ∪ σ e,γ (C n ψ,ϕ ).…”

“…This may be seen as follows. By[11, Proposition 2.3] and [13, Corollary 2.2], σ e,γ (C ψ,ϕ C |ϕ ′ (ζ)| −γ σ e,γ (C σ•ϕ ). By the fact which was stated before Proposition 2.8, we can see that r e,γ (C ψ,ϕ C * ψ,ϕ ) = |ψ(ζ)| 2 |ϕ ′ (ζ)| −γ r e,γ (C σ•ϕ ).…”

Fredholmness of multiplication of a weighted composition operator with its adjoint on H 2 $H^{2}$ and A α 2 $A_{\alpha}^{2}$

“…[ 2 ] and [ 16 ]), . By [ 23 , Corollary 2.2] and [ 4 , Proposition 2.3] there is a compact operator K such that Also, [ 23 , Theorem 3.1], [ 23 , Proposition 3.6], and [ 24 , Theorem 3.2] imply that there is a compact operator such that From the fact that and equation ( 2 ) we can infer that . Then is not Fredholm.…”