Closed-range composition operators on $\mathbb{A}^{2}$

Closed-Range Composition Operators on Weighted Bergman Spaces

“…Let {I ν } ν be an enumeration of the components of T\σ B and, for each ν, let ω ν denote the number of radians through which B wraps I ν (which might be infinite). If there exists ν 0 such that B(I ν0 ) = T, then, by the proof of [1], Lemma 3.1, is an open subarc of T for all ν. Therefore, if ∪ ν B(I ν ) = T, then, by the compactness of T, there is a positive integer N such that ∪ N ν=1 B(I ν ) = T. But then we could find a compact subset K of ∪ N ν=1 I ν such that B(K) = T. Now if ε > 0 is sufficiently small, then this compact set K is contained in the closure of Ω ε := {z ∈ D : 1−|z| 2 1−|B(z)| 2 > ε} and hence, by Theorem 2.3 and the proof of Lemma 2.1, we would find that C B is closed-range on every A p α -space.…”

“…Now, by Lemma 2.1, we need only show that (3) implies (1) and that (1) implies (2). As in the proof of [1], Theorem 2.4, we can use versions of Lemmas 2.1, 2.2 and 2.3 in [1] to reduce to the case that ϕ is nonconstant and fixes zero. And we restrict our attention to C ϕ on A p α,0 := {f ∈ A p α : f (0) = 0}.…”

“…We proceed as in the proof of [1], Theorem 2.4 and decompose D\{z ∈ D : ϕ (z) = 0} into at most countably many semi-closed polar rectangles R n , over each of which ϕ is univalent. Letting S n = ϕ(U ∩ R n ), we find that…”

“…Other implications of Theorem 2.3 arise from the fact that the result is independent of p and α; we mention two here. All of the examples constructed in [1] and [2] for the classical Bergman space also apply to any of our A p α -spaces. Secondly, in [10], Zorboska has characterized the closed-range composition operators on A 2 α in terms of…”

“…In a recent paper (cf., [1]), the authors give another, quite tractable condition that is necessary and sufficient for C ϕ to be closed-range on A 2 . It involves the set G ε := ϕ(Ω ε ), where Ω ε := {z ∈ D : 1−|z| 2 1−|ϕ(z)| 2 > ε} and ε is some positive constant.…”