Compact differences of weighted composition operators on the weighted Bergman spaces

In this paper, we consider the compact differences of weighted composition operators on the standard weighted Bergman spaces. Some necessary and sufficient conditions for the differences of weighted composition operators to be compact are given, which extends Moorhouse’s results in (J. Funct. Anal. 219:70-92, 2005).


Introduction
When u ≡ , uC ϕ is the composition operator C ϕ , in other words, C ϕ (f ) = f • ϕ, f ∈ H(D); when ϕ(z) = z, uC ϕ is the multiplication operator M u , i.e., M u (f ) = u · f , f ∈ H(D). Broadly, one is interested in extracting properties of uC ϕ acting on a given Banach space of holomorphic functions on D (boundedness, compactness, spectral properties, etc.) from function theoretic properties of u and ϕ and vice versa. In the past several decades, weighted composition operators on various spaces of holomorphic functions have been studied extensively, e.g., [-].
As is well known, an early result of Shapiro and Taylor [] in  showed the nonexistence of the angular derivative of the inducing map at any point of the boundary of the unit disk is a necessary condition for the compactness of the composition operator on the Hardy space H  (D). Later, MacCluer and Shapiro [] proved that this condition is a necessary and sufficient condition for the compactness of composition operators on the weighted Bergman spaces A p α (D) (α > -). Using the Nevanlinna counting function, Shapiro [] completely characterized those ϕ which induce compact composition operators on the Hardy space H  (D). With the basic questions such as compactness settled, it is natural to look at the topological structure of composition operators in the operator norm topology and this topic is of continuing interests in the theory of composition operators. Berkson  The standard weighted Bergman space A  α (α > -) is defined as follows: and dA is the area measure on D. As is well known, the Bergman space A  α is a reproducing kernel Hilbert space, the reproducing kernel at z ∈ D is K z (w) =  (-zw) α+ and  K z A  α K z →  weakly as |z| → .
In Section  we recall some related facts and results which are needed in the sequel, and then we prove our main results in Section . Section  deals with the compact perturbations of finite summations of a given weight composition operator.
Constants. Throughout the paper we use the letters C and c to denote various positive constants which may change at each occurrence. Variables indicating the dependency of constants C and c will be often specified in the parentheses. We use the notation X Y or Y X for non-negative quantities X and Y to mean X ≤ CY for some inessential constant C > . Similarly, we use the notation X ≈ Y if both X Y and Y X hold.

Preliminaries
For  < t < ∞ and ξ ∈ T, let t,ξ be a non-tangential approach region at ξ defined by t,ξ := z ∈ D : |zξ | ≤ t  -|z| and t,ξ the boundary curve of t,ξ . Clearly t,ξ has a corner at ξ with angle less than π . A function f is said to have a non-tangential limit at ξ , if lim z→ξ f (z) exists in each nontangential region t,ξ . Let ϕ be a holomorphic self-map of D. We say that ϕ has a finite angular derivative at ξ ∈ T, if there exists a point η ∈ T, such that the non-tangential limit as z → ξ of the difference quotient η-ϕ(z) ξ -z exists as a finite complex value. Write Denote F(ϕ) := {ξ ∈ T : |ϕ (ξ )| < ∞}. For ξ ∈ F(ϕ), by the Julia-Carathéodory theorem in [], we have for any t > . For any z ∈ D, let σ z be the involutive automorphism of D which exchanges  to z, namely, The pseudo-hyperbolic distance on D is defined by Then, for any z, w ∈ D, it is easy to see that Moreover, for any z ∈ D and  < r < , let be the pseudo-hyperbolic disk with 'center' z and 'radius' r. It is well known that, for given  < r < , and where the constants in the estimate above depend only on r and α. In the sequel, we set ρ(z) := ρ(ϕ  (z), ϕ  (z)) for the pseudo-hyperbolic distance of ϕ  (z) and ϕ  (z).
The following lemma is cited from [].
Lemma . For α > -, let ϕ be a holomorphic self-map of D and u a non-negative, bounded, and measurable function on D. Define the measure uλ α by uλ α (E) : For more details as regards Carleson measures, see Section . in [].

Compact difference
Let ϕ ∈ S(D) and u ∈ H(D). If the weighted composition operator uC ϕ is bounded on For ϕ ∈ S(D), by the Schwarz-Pick theorem in [], for any z ∈ D.
The following lemma can be obtained by modifying Lemma . in [] (e.g., at the third line on p. in []). See also Proposition . in [] in a different form for the unit ball case. Here, we give a more elementary proof for convenience.
Lemma . Let ϕ  and ϕ  be holomorphic self-maps of D. Then, for any ξ ∈ F(ϕ  ), the following holds: Proof First we notice that If ϕ  has no finite angular derivative at ξ , namely, then, for any t > , by (.), we have If ϕ  has finite angular derivative at ξ and ϕ  (ξ ) = ϕ  (ξ ), then it follows clearly that If ϕ  has finite angular derivative at ξ and ϕ  (ξ ) = ϕ  (ξ ), then Consequently, we get the desired result.
To further study compact differences of weighted composition operators on A  α , we define F u (ϕ) as It is easy to check that F u (ϕ) ⊆ F(ϕ) if u is bounded. To avoid the trivial case, in the sequel we assume In the following we take the test functions First note that {g w } is bounded in A  α . Indeed, note that Again it is well known that g w (z) →  uniformly on any compact subset of D, and hence that g w →  weakly as |w| → . We now give some necessary conditions for the difference of weighted composition operators to be compact.
Theorem . Let ϕ  , ϕ  be holomorphic self-maps of D and let u  , u  be bounded holomorphic functions on D such that neither α , then the following statements are true: for any t > . Using the submean value type inequality in [] and equation (.), then, for a given  < r < , for all t > . Since u  , u  are bounded holomorphic functions on D and ξ ∈ F(ϕ  ), then it follows from our assumption ξ ∈ F u  (ϕ  ) that So () is obtained, and thus F u  (ϕ  ) ⊆ F u  (ϕ  ) by Lemma .. Similarly, we have F u  (ϕ  ) ⊆ F u  (ϕ  ). Thus the proof for () is complete.
To prove (), we assume that there exists a sequence {z n } with |z n | →  such that Without loss of generality, we may further assume that Due to (.) and the boundedness of u  on D, by passing to a subsequence if necessary, we can suppose that for some constants a  ∈ (, ], a  > , a  > . Then lim n→∞ |u  (z n )|  = a  by the obtained facts () and (). Actually, we may further assume that for some a  > . We put f n := K z n / K z n A  α , where K z n is the reproducing kernel function at z n ∈ D in A  α for each n ≥ . So f n →  weakly as n → ∞. We will arrive at a contradiction to the compactness of The contradiction implies (), which completes the proof.
To give a sufficient condition for the compact difference of weighted composition operators, we need the following fact from [], pp.-: for any ε >  small enough and To simplify our sufficient condition, we use the following simple lemma.

Lemma . Let ϕ  , ϕ  be holomorphic self-maps of D and let u  , u  be bounded holomorphic functions on D.
If and ξ / ∈ F(ϕ  ) by the assumption lim z→ξ |u  (z)u  (z)| =  and ξ ∈ F u  (ϕ  ). Hence by Note that and (.) implies This leads to a contradiction to the assumption. Thus F u  (ϕ  ) = F u  (ϕ  ).
We are now ready to give our sufficiency theorem.
Theorem . Let ϕ  , ϕ  be holomorphic self-maps of D and let u  , u  be bounded holomorphic functions on D. If the following hold: Proof Assume that {f n } is any bounded sequence in A  α such that f n →  (n → ∞) uniformly on each compact subsets of D. Given ε > , we put Now we can write for each n.
Let χ Q be the characteristic function of Q , then by the assumption (), So by Lemma ., for the second term of the right-hand side of (.), and H  := Q\H  . Also, for the first term of the right-hand side of (.), we have for all n and We now claim that Indeed, if either (.) or (.) fails, then we will arrive at a contradiction to (.), and thus the desired is obtained. To this end, we assume that there exist some η ∈ T and a sequence z n ∈ H  satisfying z n → η such that If (.) holds, then η ∈ F u  (ϕ  ). Thus η ∈ F u  (ϕ  ) due to the fact that F u  (ϕ  ) = F u  (ϕ  ) by Lemma .. If (.) holds, then by z n ∈ H  and (.). Thus we also have η ∈ F u  (ϕ  ). This leads to a contradiction to (.). So our claim holds. Thus by Lemma ., we have as n → ∞. Therefore the proof is complete.

Compact perturbation
In the final section, we consider the compact perturbation of finite summations of a given weighted composition operator.
To end the proof, we assume that {f n } is any bounded sequence in A  α such that f n →  (n → ∞) uniformly on each compact subset of D. For  ≤ i ≤ N , Let χ D i and χ E i be the characteristic functions of D i and E i , respectively, then it is obvious from the assumption () that for fixed i and each k = i. Indeed, if (.) fails for some k = i, then there exist ξ ∈ T and z n ∈ D i satisfying z n → ξ such that lim n→∞ u k (z n )   -|z n |   -|ϕ k (z n )|  > .
then ζ ∈ F u (ϕ) = N k= F u k (ϕ k ). Since this ζ / ∈ F u k (ϕ k ) when k = i by (.), then ζ ∈ F u i (ϕ i ), which contradicts the definition of H i . So (.) holds. We now claim that Indeed, the argument for (.) is similar to (.) and we omit it. To prove that (.) holds, we assume that there exist some η ∈ T and a sequence {z n } ⊆ H i such that z n → η and lim n→∞ u(z n )   -|z n |   -|ϕ i (z n )|  > .
Note that because of {z n } ⊆ H i and (.). Then η ∈ F u (ϕ), which contradicts (.). Thus again by Lemma ., we have as n → ∞. So