The compactness of the sum of weighted composition operators on the ball algebra
 CeZhong Tong^{1} and
 ZeHua Zhou^{1}Email author
https://doi.org/10.1186/1029242X201145
© Tong and Zhou; licensee Springer. 2011
Received: 8 April 2011
Accepted: 2 September 2011
Published: 2 September 2011
Abstract
In this paper, we investigate the compactness of the sum of weighted composition operators on the unit ball algebra, and give the characterization of compact differences of two weighted composition operators on the ball algebra. The connectness of the topological space consisting of nonzero weighted composition operators on the unit ball algebra is also studied.
2000 Mathematics Subject Classification. Primary: 47B33; Secondary: 47B38, 46E15, 32A36.
Keywords
1. Introduction
And let H^{∞} = H^{∞} (B_{ N } ) be the set of all bounded holomorphic functions on B_{ N } . We denote by B(A) (B(H^{∞}) resp.) the unit ball of A (H^{∞} (B_{ N } ) resp.).
for z ∈ B_{ N } and f ∈ H(B_{ N } ). If let u ≡ 1, then uC_{ φ } = C_{ φ } ; if let φ = id, then uC_{ φ } = M_{ u } . It is clear that these operators are linear and uC_{ φ } is bounded on A. For some results in this topic see, for example, [1–6], and so on.
Let X be a Banach space of analytic functions, we write C(X), for the space of composition operators on X under the operator norm topology.
Moorhouse [7] considered the compactness of finite sum of composition operators on weighted Bergman spaces in the disk and gave a partial answer to the component structure of $C\left({A}_{\alpha}^{2}\right)$. Kriete and Moorhouse [8] continued to study the finite linear combination of composition operators acting on the Hardy space or weighted Bergman spaces on the disk. Hosokawa and Ohno, [9], and [10], discussed the topological structures of the sets of composition operators and gave a characterization of compact difference on Bloch space in the unit disk. Fang and Zhou [11] then gave a characterization of compact difference between the Bloch space and the set of all bounded analytic functions on the unit polydisk. Fang and Zhou [12] also studied the compact differences of composition operators on the space of bounded analytic functions in polydisk. Hosokawa and coworkers [13] studied the topological components of the topological space of weighted composition operators on the space of bounded analytic functions on the open unit disk in the uniform operator topology. These results were extended to the setting of H^{∞}(B_{ N } ) by Toews [14], and independently by Gorkin and coworkers [15], and the setting of H^{∞}(D^{ N } ) by Fang and Zhou [12], where D^{ N } is the unit polydisk. Bonet and coworkers [16] discussed the same problem for the composition operator on the weighted Banach spaces of holomorphic functions in the unit disk, which was also extended to the unit polydisc by Wolf in [17]. The case of weighted composition operators on the above spaces was treated by Lindström and Wolf [18]. Ohno [19] studied the differences of weighted composition operators on the disk algebra.
Building on these foundations, we study the compactness of the sum of certain class of weighted composition operators on the unit ball algebra, and give the characterization of compact differences of two weighted composition operators on the ball algebra. The connectness of the topological space consisted of nonzero weighted composition operators on the unit ball algebra is also studied.
2. The sum of weighted composition operators on A
In this section, we will give necessary and sufficient conditions for the sum of several weighted composition operators to be compact on A.
Let T be a bounded linear operator on a Banach space. Recall that T is said to be compact if T maps every bounded set into relatively compact one. And if T is a compact operator, T must map every weakly convergent sequence into norm convergent one, and a linear operator with that property is called to be completely continuous. In general, a completely continuous operator may not be always compact.
Toews [14] proved the following lemma:
Lemma 2.1. For any z ∈ B_{ N } , we have
(a): ρ(z; w) = Φ_{ w }(z) for any w ∈ B_{ N }, and
(b): {z : ρ (z, w) < λ} = Φ_{ w }(λB_{ N } ).
Since U Δ(ζ, α) = Δ (Uζ, α) for any unitary transformation U in ${\u2102}^{N}$, we can concentrate on understanding Δ (e_{1}, α), where ${e}_{1}=\left(1,0,\dots ,0\right)\in {\u2102}^{N}$. Next lemma is crucial.
□
For φ_{1}, φ_{2}, ..., φ_{ p } ∈ S(B_{ N } ) ∩ A, we simplify the notations ${\Gamma}_{{\phi}_{j}}$ and ${C}_{{\phi}_{j}}$by Γ _{ j } and C_{ j } respectively. It is obvious that each weighted composition operator is a bounded operator on the ball algebra.
For any $\zeta \in {\bigcup}_{j=1}^{p}{\Gamma}_{j}$, denote by I(ζ) = {k ∈ {1, 2, ..., p}: ζ ∈ Γ_{ k }}. If k, j ∈ I(ζ), define k ~ j whenever ρ(φ_{ j } (z_{ n } ), φ_{ k } (z_{ n } )) → 0 holds for every sequence {z_{ n } } ⊂ B_{ N } converging to ζ. It is clear that the relation ~ is an equivalent relation on I(ζ). Denote by I(ζ) /~ as the quotient set, and I_{ s } (ζ)'s, which is called index sets of ζ, as the elements of I(ζ)/~. The lower index s ∈ {1, ..., p} is given by the smallest number in Is(ζ).
Next, we are going to study the compactness of ${\sum}_{j=1}^{p}{u}_{j}{C}_{j}$ acting on the ball algebra. In fact, it is difficult to characterize the compactness of the sum of several arbitrary u_{ j }C_{ j } 's. Here, we just consider the sum whose symbols are in a certain class of self mappings of B_{ N } .
Our admissible symbols of self mappings of B_{ N } are constrained by the following condition: Let φ ∈ S(B_{ N } ) ∩ A and Γ _{ φ } as defined above. For every ζ ∈ Γ _{ φ } and any arbitrary small ε > 0, if there exists a Koranyi region Δ(φ (ζ), α) with 1 < α < 2 such that
(*): (N(φ(ζ), ε) ∩ Δ (φ (ζ), α)) ∩ φ (B_{ N } ) ≠ ∅
where N(φ(ζ), ε) is the neighborhood of φ(ζ) with radii ε. We collect all self mappings of B_{ N } satisfied (*) and denote it by A*.
Remark 2.3. If φ ∈ S(B_{ N } ) ∩ A and the closure of φ(B_{ N } ) has no boundary contact with the unit sphere, we say that φ satisfies condition (*) trivially, that is to say φ ∈ A*.
The following theorem is our main theorem.
Theorem 2.4. Let u_{ j } ∈ A and φ_{ j } ∈ A* for j = 1, ..., p. Then${\sum}_{j=1}^{p}{u}_{j}{C}_{j}$is compact on A if and only if and only if${\sum}_{k\in {I}_{s}\left(\zeta \right)}{u}_{k}\left(\zeta \right)=0$for every$\zeta \in {\bigcup}_{j=1}^{p}{\Gamma}_{j}$and every index set I_{ s } (ζ) of ζ.
Proof. To prove the necessity, let ζ be an arbitrary point in ${\bigcup}_{j=1}^{p}{\Gamma}_{j}$, I(ζ) and I_{ s } (ζ)'s are defined as above. We can find out a sequence {w_{ n } } ⊂ B_{ N } satisfying the following properties: as n → ∞,
(1): w_{ n } → ζ;
(2): φ_{ k } (w_{ n } ) → φ_{ k } (ζ) with k ∈ I_{ s } (ζ) and k ≠ s;
(3): φ_{ l } (w_{ n } ) → φ_{ l } (ζ) with l ∉ I_{ s } (ζ);
(4):$\u27e8{\Phi}_{{\phi}_{s}\left({w}_{n}\right)}\left({\phi}_{s}\left(\zeta \right)\right),{\phi}_{s}\left({w}_{n}\right)\u27e9\to \lambda \ne 0$.
The item (4) holds since every φ_{ j } ∈ A* and Lemma 2.2 guarantees the limit is non zero. From the item (3), we can choose the subsequences (also denoted by {w_{ n } }) satisfying
(5):$\u27e8{\Phi}_{{\phi}_{s}\left({w}_{n}\right)}\left({\phi}_{l}\left({w}_{n}\right)\right),{\phi}_{s}\left({w}_{n}\right)\u27e9\to {\sigma}_{ls}\ne 0$.
So f_{ n } converges weakly to 0 in A, thus $\parallel {\sum}_{j=1}^{p}{u}_{j}{C}_{j}{f}_{n}{\parallel}_{\infty}\to 0$ as n → ∞.
Next, we just need to show that G is continuous on $\overline{{B}_{N}}$, and ${\sum}_{j=1}^{p}{u}_{j}{C}_{j}{f}_{{n}_{k}}$ converges uniformly to G on $\overline{{B}_{N}}$. In the following proof, we simplify the subsequence ${f}_{{n}_{k}}$ by f_{ n } .
We first prove that G is continuous on $\overline{{B}_{N}}$. Indeed, it is obvious that G is continuous on $\overline{{B}_{N}}\backslash {\bigcup}_{j=1}^{p}{\Gamma}_{j}$.
For $\zeta \in {\bigcup}_{j=1}^{p}{\Gamma}_{j}$, let I_{ s } (ζ)'s be its index sets and $I\left(\zeta \right)={\bigcup}_{s}{I}_{s}\left(\zeta \right)$. Suppose that {z_{ n } } be a sequence in B_{ N } converging to ζ such that φ_{ j } (z_{ n } ) → φ_{ j } (ζ) j = 1, ..., p. So for each I_{ s } (ζ).

ρ(φ_{ k }(z_{ n }), φ_{ s }(z_{ n })) → 0 where k ∈ I_{ s }(ζ), and

ρ (φ_{ l }(z_{ n }), φ_{ s }(z_{ n })) ↛ 0 where l ∉ I(ζ).
So, we have $\underset{{z}_{n}\to \zeta}{lim}{\sum}_{j=1}^{p}{u}_{j}{C}_{j}g\left({z}_{n}\right)=G\left(\zeta \right)$, and this holds for every $\zeta \in {\bigcup}_{j=1}^{p}{\Gamma}_{j}$. Thus G is continuous on $\overline{{B}_{N}}$.
This implies that max_{ j }{φ_{ j } (z_{ n } )} → 1 as n → ∞. Here we may assume that z_{ n } → ζ ∈ ∂B_{ N } , and define I(ζ), I_{ s } (ζ) as before. Similarly as the analysis above, we have that

ρ(φ_{ k }(z_{ n }), φ_{ s }(z_{ n })) → 0 where k ∈ I_{ s }(ζ),

ρ (φ_{ l }(z_{ n }); φ_{ s }(z_{ n })) ↛ 0 where l ∉ I_{ s }(ζ).
When l ∉ I(ζ), note that the limit of φ_{ l } (z_{ n } ) is strictly less than 1, and f_{ n } converges to g on compact subsets of B_{ N }, thus f_{ n } (φ_{ l } (z_{ n } ))  g(φ_{ l } (z_{ n } )) → 0. By the preliminary conditions we know that $\left{\sum}_{k\in {I}_{s}\left(\zeta \right)}{u}_{k}\left(\zeta \right)\right=0$ for each index sets I_{ s } (ζ). Together with the above two items, we have ${\sum}_{j=1}^{p}{u}_{j}{C}_{j}{f}_{n}\left({z}_{n}\right)G\left({z}_{n}\right)\to 0$. This fact contradicts (2.1). Thus, we have proved the sufficient condition. □
As a corollary of the sum theorem, we state the characterization of the compact differences of two weighted composition operators on A.
Theorem 2.5 Let u, v ∈ A and φ, ψ ∈ A*. Then uC_{ φ }  vC_{ ψ } is compact on A if and only if the following three conditions hold:
(a): If ζ ∈ Γ _{ φ }and lim_{z→ζ}ρ(φ(z), ψ(z)) ≠ 0, then u(ζ) = 0;
(b): If ζ ∈ Γ _{ ψ }and lim_{z→ζ}ρ(φ(z), ψ(z)) ≠ 0, then v(ζ) = 0;
(c): If ζ ∈ Γ _{ φ } ∩ Γ_{ ψ }, then u(ζ) = v(ζ).
Corollary 2.6 Let u ∈ A and φ ∈ A*, the associated weighted composition operator uC_{ φ } act compactly on A if and only if u(ζ) = 0 for every ζ ∈ Γ_{ φ }.
Proof. Just set v(z) = u(z) and ψ(z) = φ(z) in the proof of Theorem 2.4. □
Corollary 2.7 Let φ, ψ ∈ A*. Then the following conditions are equivalent:
(i): C_{ φ }  C_{ ψ } is compact on A;
(ii): C_{ φ }  C_{ ψ } is completely continuous on A
(iii): Γ _{ φ } = Γ_{ ψ }.
Now, we illustrate Theorem 2.4 with the following example.
Then, ${\sum}_{j=1}^{5}{u}_{j}{C}_{j}$ is a compact operator on A.
Proof. First, note that Γ_{1} = {(1, 0, ..., 0)} = {e_{1}}, Γ_{2} = {e_{1}}, Γ_{3} = {e_{1}}, Γ_{4} = {±e_{1}}, Γ_{2} = ∅. And φ_{1}(e_{1}) = φ_{3}(e_{1}) = e_{1}, φ_{2}(e_{1}) = (0, 1, 0, ..., 0) = (e_{2}), φ_{4}(±e_{1}) = e_{1}. Then, φ_{ j } ∈ A* for each j = 1, ..., 5. Take φ_{1} for example, the sequence {(1  1/n, 0, ..., 0)} converges to φ(e_{1}) radially, and they have preimages {(1  2/n, 0, ..., 0)} of φ_{1} in B_{ N } . Hence, φ_{1} satisfies (*). Then, we have I_{1}(e_{1}) = {1, 3, 4}, I_{2}(e_{1}) = {2} and I(e_{1}) = I_{4}(e_{1}) = {4}. Compute that
I_{1}(e_{1}):u_{1}(e_{1}) + u_{3}(e_{1}) + u_{4}(e_{1}) = 2  3 + 1 = 0,
I_{2}(e_{1}):u_{2}(e_{1}) = 0,
I(e_{1}):u_{4}(e_{1}) = 0.
By Theorem 2.4, ${\sum}_{j=1}^{5}{u}_{j}{C}_{j}$ is compact on A. □
3. Connectness of weighted composition operators on A
Shapiro and Sundberg [22] first discovered the relationship between compact differences and topological structure of the collection of composition operators on Hardy space. In this section, we will investigate the connectness of the topological space consisted of all weighted composition operators act on the unit ball algebra, and we use the notion $\mathcal{C}\left(A\right)$ to represent that topological space. It is trivial that
Because we can connect any uC_{ φ } and 0 (the weighted composition operator uC_{ φ } when u ≡ 0) by the continuous path ${T}_{t}:\left[0,1\right]\to \mathcal{C}\left(A\right)$ defined by t ↦ tuC_{ φ } . Indeed, for many function spaces, we denote by S generally, the weighted composition operators topological space $\mathcal{C}\left(S\right)$ is connected by the same path constructed above. So, we are interested in the connectness of the nonzero weighted composition operators topological space, and here we denote by ${\mathcal{C}}_{w}\left(S\right)$. In this setting, the connectness of ${\mathcal{C}}_{w}\left(S\right)$ depends on the function space S.
Theorem 3.1${\mathcal{C}}_{w}\left(A\right)$is also a connected topological space.
where u_{ t } (z) = tu(z) + (1  t)v(z) and φ_{ t } (z) = tφ(z) + (1  t)ψ(z). To avoid u_{ t } = 0 and φ_{ t } = 0 for some t ∈ [0, 1], we first assume that
(1): v ≠ θu for every θ < 0;
(2): ψ ≠ κφ for every κ < 0 with κφ ∈ S(B_{ N } ), where ψ = (ψ_{1}, ..., ψ_{ N } ) and φ = (φ_{1}, ..., φ_{ N } ).
So, it converges to 0 as s → t.
So, it converges to 0 as s → t, since φ_{ t } , φ_{ s } ∈ A (See Lemma 6 in [14]). Here, we obtain that {T(t)} is a continuous path which connects any different uC_{ φ } and vC_{ ψ } in ${\mathcal{C}}_{w}\left(A\right)$.
Next, for any given $u{C}_{\phi}\in {\mathcal{C}}_{w}\left(A\right)$, we need to find out continuous pathes from uC_{ φ } to either θuC_{ φ } or uC_{ κφ } where θ < 0 and κ < 0 with κφ ∈ S(B_{ N } ) ∩ A.
as t  s → 0. Thus, θuC_{ φ } , with any θ < 0, can be connected continuously with uC_{ φ } .
By the same method of computation as in (3.1), Y_{1}(t) and Y_{2}(t) can be shown to be two continuous pathes in ${\mathcal{C}}_{w}\left(A\right)$ from uC_{ φ } and uC_{φ}to uC_{ μ } .
Thus, the space ${\mathcal{C}}_{w}\left(A\right)$ is a connected topological space. □
Corollary 3.2 The topological space consisting of composition operators acting on the unit ball algebra is connected.
From the last result, we know that two weighted composition operators can be in the same component in ${\mathcal{C}}_{w}\left(A\right)$ without compact differences.
Declarations
Acknowledgements
The authors would like to thank the referees for the useful comments and suggestion which improved the presentation of this paper.
Authors’ Affiliations
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