Compact differences of weighted composition operators on the weighted Bergman spaces

Journal of Inequalities and Applications, Jan 2017

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 ). MSC: 47B33, 30D55, 46E15.

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Compact differences of weighted composition operators on the weighted Bergman spaces

Wang et al. Journal of Inequalities and Applications Compact differences of weighted composition operators on the weighted Bergman spaces Maocai Wang 0 Xingxing Yao 1 Fen Chen 1 0 School of Computer, China University of Geosciences , Wuhan, 430074 , China 1 School of Mathematics and Statistics, Wuhan University , Wuhan, 430072 , China 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). weighted Bergman space; weighted composition operator; compact difference - uCϕ(f ) = u · f ◦ ϕ, f ∈ H(D). When u ≡ , uCϕ is the composition operator Cϕ , in other words, Cϕ(f ) = f ◦ ϕ, f ∈ H(D); when ϕ(z) = z, uCϕ is the multiplication operator Mu, i.e., Mu(f ) = u · f , f ∈ H(D). Broadly, one is interested in extracting properties of uCϕ 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 Apα(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, © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 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 [] focused attention on the topological structure with his isolation result on Hp(D) in , which was refined later by Shapiro and Sundberg [], and MacCluer []. In [], Shapiro and Sundberg posed a question: Do the composition operators on H(D) that differ from Cϕ by a compact operator form the component of Cϕ in the operator norm topology? While the same question was answered positively on the weighted Bergman spaces [], this turned out to be not true on the Hardy space []. Some other results on differences of weighted composition operators on spaces of holomorphic functions can be found, for example, in [–]. In relation with the study of the topological structures, the difference or the linear sum of composition operators on various settings has been a very active topic [, , –]. Recently, Moorhouse [] characterized completely the compactness for the difference of two composition operators on the Bergman space over the unit disk, and Al-Rawashdeh and Narayan [] gave a sufficient condition for the same problem on the Hardy space. Here we continue this line to study compact differences of weighted composition operators acting on the standard weighted Bergman spaces. The standard weighted Bergman space Aα (α > –) is defined as follows: Aα := f ∈ H(D) : f Aα = where dλα(z) = π (α + )( – |z|)α dA(z) 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 Kz(w) = (–zw)α+ and Kz Aα Kz →  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. 2 Preliminaries For  < t < ∞ and ξ ∈ T, let t,ξ be a non-tangential approach region at ξ defined by t,ξ := z ∈ D : |z – ξ | ≤ 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 limz→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–ξϕ–(zz) exists as a finite complex value. Write ϕ (ξ ) := ∠ lim η – ϕ(z) z→ξ ξ – z 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 σz be the involutive automorphism of D which exchanges  to z, namely, w ∈ D. The pseudo-hyperbolic distance on D is defined by Then, for any z, w ∈ D, it is easy to see that  – |z|  – ρ(z, w) ≤ | – zw| ≤  + ρ(z, w). Moreover, for any z ∈ D and  < r < , let  – ρ(z, w)  – |z|  + ρ(z, w)  + ρ(z, w) ≤  – |w| ≤  – ρ(z, w) , w ∈ Er(z), λα Er(z) ≈  – |z| +α, z ∈ D, be the pseudo-hyperbolic disk with ‘center’ z and ‘radius’ r. It is well known that, for given  < r < , 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 uλα by uλα(E) := E u(z) dλα(z) on all Borel subsets E ⊆ D. If |zli|→m u(z)  ––|ϕ|(zz|)| = , then uλα ◦ ϕ– is a compact α-Carleson measure and the inclusion map Iα : Aα → L(uλα ◦ ϕ–) is compact. For more details as regards Carleson measures, see Section . in []. 3 Compact difference Let ϕ ∈ S(D) and u ∈ H(D). If the weighted composition operator uCϕ is bounded on Aα (α > –), then the adjoint (uCϕ)∗ of uCϕ satisfies (uCϕ)∗Kz(w) = u(z)Kϕ(z)(w), z, w ∈ D.  ––|ϕ|(zz|)| <  –+ ||ϕϕ(())|| < ∞ 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  – |ϕ(z)| =  – |ϕ(z)|  – ϕ(z)ϕ(z)  – |z| where (z) = ϕ(z)(ϕ(z)–ϕ(z)) , and –|z| limz→iξnf (z) ≥ limz→iξnf   – |–ϕ|z(|z)| . then, for any t > , by (.), we have (z) = ∞. (z) = ∞. (z) = . (z) = ∞. Consequently, we get the desired result. To further study compact differences of weighted composition operators on Aα , we define Fu(ϕ) as It is easy to check that Fu(ϕ) ⊆ F(ϕ) if u is bounded. To avoid the trivial case, in the sequel we assume Fui (ϕi) = ∅, i = , , i.e., neither uCϕ nor uCϕ is compact on Aα . In the following we take the test functions gw(z) := w, z ∈ D. Ic(w) = ( – |z|)α  D | – wz|α++c dA(z) ≈ ( – |w|)c , |w| →  for c >  by Lemma .. in [], and then dA(z) < ∞. for any t > . Using the submean value type inequality in [] and equation (.), then, for a given  < r < , So by (.) u(w) – u(w) =  for all t > . Since u, u are bounded holomorphic functions on D and ξ ∈ F(ϕ), then it follows from our assumption ξ ∈ Fu (ϕ) that = , So () is obtained, and thus Fu (ϕ) ⊆ Fu (ϕ) by Lemma .. Similarly, we have Fu (ϕ) ⊆ Fu (ϕ). Thus the proof for () is complete. To prove (), we assume that there exists a sequence {zn} with |zn| →  such that nl→im∞ ρ(zn) u(zn)   – –|ϕ|z(nz|n)| + u(zn)   – |–ϕ|z(nz|n)| Without loss of generality, we may further assume that for all n. Then linm→s∞up ρ(zn) u(zn)   – –|ϕ|z(nz|n)| > . 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 limn→∞ |u(zn)| = a by the obtained facts () and (). Actually, we may further assume that for some a > . We put fn := Kzn / Kzn Aα , where Kzn is the reproducing kernel function at zn ∈ D in Aα for each n ≥ . So fn →  weakly as n → ∞. We will arrive at a contradiction to the compactness of uCϕ – uCϕ by showing (uCϕ – uCϕ )∗fn  (n → ∞) in Aα . In fact, notice that u(zn)u(zn) for all n. Then 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  f ∈ Aα , 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 then Fu (ϕ) = Fu (ϕ). zl→imξ u(z) – u(z) =  for any ξ ∈ Fu (ϕ) ∪ Fu (ϕ) |z|→  – |ϕ(z)| + u(z)   – |–ϕ|z(|z)| lim ρ(z) u(z)   – |z| = , Proof If Fu (ϕ) = Fu (ϕ), we may assume that ξ ∈ Fu (ϕ) but ξ ∈/ Fu (ϕ). Then ξ ∈ F(ϕ), and ξ ∈/ F(ϕ) by the assumption limz→ξ |u(z) – u(z)| =  and ξ ∈ Fu (ϕ). Hence by Lemma ., = .  – ρ(z) = ( – |ϕ(z)|)( – |ϕ(z)|) , | – ϕ(z)ϕ(z)| and (.) implies ≤ . t→∞ zz∈→tξ,ξ  – |ϕ(z)| + u(z)   – |–ϕ|z(|z)| lim lim ρ(z) u(z)   – |z| = . We are now ready to give our sufficiency theorem. Proof Assume that {fn} is any bounded sequence in Aα such that fn →  (n → ∞) uniformly on each compact subsets of D. Given ε > , we put Q := D\Q. Now we can write for each n. Let χQ be the characteristic function of Q , then by the assumption (), = . So by Lemma ., for the second term of the right-hand side of (.), as n → ∞. For any ξ ∈ Fu (ϕ), by the assumption (), there exists δ(ξ ) >  such that |u(z) – u(z)| < ε whenever |z – ξ | < δ(ξ ). We decompose Q into two parts, Q := H + H, where H := Q ∩ ( ξ∈Fu (ϕ){z ∈ D : |z – ξ | < δ(ξ )}) and H := Q\H. Also, for the first term of the right-hand side of (.), we have i= Hi i= H Note that by (.) for all n. Also, by the definition of H, we can easily get |u – u| fn(ϕ)  dλα ≤ ε Cϕ fn Aα for all n and We now claim that nl→im∞ u(zn)   – |–ϕ|z(nz|n)| > , nl→im∞ u(zn)   – |–ϕ|z(nz|n)| > . nl→im∞ u(zn)   – –|ϕ|z(nz|n)| > ,  – |zn|  – ε  – |zn|  – |ϕ(zn)| ≥  + ε  – |ϕ(zn)| 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 zn ∈ H satisfying zn → η such that If (.) holds, then η ∈ Fu (ϕ). Thus η ∈ Fu (ϕ) due to the fact that Fu (ϕ) = Fu (ϕ) by Lemma .. If (.) holds, then by zn ∈ H and (.). Thus we also have η ∈ Fu (ϕ). This leads to a contradiction to (.). So our claim holds. Thus by Lemma ., we have i= H i= D as n → ∞. Therefore the proof is complete. The following, given in [] and [], respectively, are immediate consequences of Theorems . and .. Corollary . Let ϕ, ϕ be holomorphic self-maps of D, then Cϕ – Cϕ is compact on Aα –|z| –|z| if and only if lim|z|→ ρ(z)( –|ϕ(z)| + –|ϕ(z)| ) = . Corollary . Let u, u be bounded holomorphic functions on D, then Mu – Mu is compact on Aα if and only if u = u. 4 Compact perturbation In the final section, we consider the compact perturbation of finite summations of a given weighted composition operator. () limz→ξ ρi(z)(|u(z)| –|–ϕ|(zz|)| + |ui(z)| –|–ϕ|iz(|z)| ) = , and () limz→ξ |u(z) – ui(z)| =  for any ξ ∈ Fui (ϕi) for every i = , , . . . , N , then uCϕ – iN= uiCϕi is compact on Aα . Proof Define Di := {z ∈ D : |ui(z)| –|–ϕ|iz(|z)| ≥ |uj(z)| –|–ϕ|jz(z|)| , for all j = i} for each i = , , . . . , N . Fix ε >  and denote Ei := {z ∈ Di, ρi(z) ≤ ε} and Ei := Di\Ei. To end the proof, we assume that {fn} is any bounded sequence in Aα such that fn →  (n → ∞) uniformly on each compact subset of D. For  ≤ i ≤ N , k=i Di k=i Di = . Let χDi and χEi be the characteristic functions of Di and Ei, respectively, then it is obvious from the assumption () that nl→im∞ ui(zn)   – –|ϕ|iz(znn|)| >  by the definition of Di, which implies ξ ∈ Fui (ϕi). So Fui (ϕi) ∩ Fuk (ϕk) = ∅ when i = k, which contradicts our assumption. Then the last three terms of (.) tend to  as n → ∞ by Lemma .. In the following, we consider the first term of (.) by a similar argument to the proof of Theorem .. For any ξ ∈ Fui (ϕi), there exists δ(ξ ) >  such that Hi := Ei ∩ and Hi := Ei\Hi. Note that j= Hij Clearly, by (.) for all n. Also, by the definition of Hi, we can easily get |u – ui| fn(ϕi)  dλα ≤ ε Cϕi fn Aα for all n. Moreover, Indeed, if (.) fails, there exist some ζ ∈ T and a sequence {zn} ⊆ Hi satisfying zn → ζ such that N k= Fuk (ϕk ). Since this ζ ∈/ Fuk (ϕk ) when k = i by (.), then ζ ∈ Fui (ϕi), which contradicts the definition of Hi. So (.) holds. 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Maocai Wang, Xingxing Yao, Fen Chen. Compact differences of weighted composition operators on the weighted Bergman spaces, Journal of Inequalities and Applications, 2017, 2, DOI: 10.1186/s13660-016-1277-8