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The compactness of the sum of weighted composition operators on the ball algebra
Journal of Inequalities and Applications volume 2011, Article number: 45 (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.
1. Introduction
Let H(B_{ N } ) be the class of all holomorphic functions on B_{ N } and S(B_{ N } ), the collection of all holomorphic self mappings of B_{ N } , where B_{ N } is the unit ball in the Ndimensional complex space {\u2102}^{N}. The closure of B_{ N } will be written as \overline{{B}_{N}}. Denote by A = A(B_{ N } ), the unit ball algebra of all continuous functions on \overline{{B}_{N}} that are holomorphic on B_{ N } . Then A is the Banach algebra with the supremum norm
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 u, φ ∈ A with φ ∈ S(B_{ N } ), recall that the composition operator C_{ φ } induced by φ is defined by
the multiplication operator induced by u is defined by
and the weighted composition operator uC_{ φ } induced by φ and u is defined by
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.
If φ ∈ S(B_{ N } )∩A, denote by Γ _{ φ } = {ζ ∈ ∂B_{ N } : φ (ζ) = 1}. For z, w ∈ B_{ N } , the involution automorphism between z and w is given by
where {P}_{w}\left(z\right)=\frac{<z,w>}{<w,w>}w, {Q}_{w}\left(z\right)=z\frac{<z,w>}{<w,w>}w, and {s}_{w}=\sqrt{1w{}^{2}}. The induced distance between z and w is defined as
Pseudohyperbolic metric is defined by
and a theorem of Bear [20] gives that
Since the pseudohyperbolic metric ρ(z, w) ≤ 1 we have that
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 } ).
For ζ ∈ ∂B_{ N } and α > 1, we define a Koranyi approach region by
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.
Lemma 2.2. For a real number 1 < α < 2 and e_{1} = (1, 0, ..., 0) in ∂B_{ N }, if {w_{ n } } is an arbitrary sequence in the Koranyi approach region Δ(e_{1}, α), then
On the other hand, for every α > 2, we can find a sequence {w_{ n } } ⊂ Δ(e_{1}, α) and w_{ n } → e_{1}as n → ∞ satisfying
Proof. First, let 1 < α < 2. We have that
Suppose α > 2. Let {w}_{n}=\left({r}_{n},{w}_{n}^{\prime}\right), where r_{ n } is real with r_{ n } → 1^{} as n → ∞ and \left{w}_{n}^{\prime}\right=\sqrt{1{r}_{n}}. It is clear that for sufficient large n
This means that {w_{ n } } ⊂ Δ(e_{1}, α) for α > 2. Now we compute
□
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)=0for 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.
For such a sequence {w_{ n } }, define the functions
Then f_{ n } ∈ A and f_{ n } _{∞} ≤ 2^{p+1}. We claim that f_{ n } (z) → 0 for all z\in \overline{{B}_{N}} as n → ∞. Indeed, if z = φ_{ s } (ζ), the item (4) guarantees f_{ n } (φ_{ s } (ζ)) → 0; and if z ≠ φ_{ s } (ζ), we have
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 → ∞.
On the other hand,
We have that ρ(φ_{ s } (w_{ n } ), φ_{ k } (w_{ n } )) → 0 for every k ∈ I_{ s } (ζ) \ {s}. And from the item (5), we know that f_{ n } (φ_{ l } (w_{ n } )) → 0 for every l ∉ I_{ s } (ζ). Hence we have
To verify the sufficiency, we cannot use the "weak convergence theorem" (Proposition 3.11 of [21]), because the unit ball algebra is not closed in the compact open topology. Here, we use the definition of compact operator to illustrate that the sum of weighted composition operators is compact. Let f_{ n } ∈ A and f_{ n } _{∞} = 1. By the normal family argument, there exists a subsequence \left\{{f}_{{n}_{k}}\right\}of {f_{ n } } and a function g analytic on B_{ N } such that {f}_{{n}_{k}} converges to g uniformly on compact subsets of B_{ N } . Here, we have
Now define a function G on \overline{{B}_{N}} by setting
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).
Then, we compute
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}}.
Next, we shall show that {\sum}_{j=1}^{p}{u}_{j}{C}_{j}{f}_{n} converges uniformly to G on \overline{{B}_{N}}. Suppose not, and we may assume that, for ε > 0, \parallel {\sum}_{j=1}^{p}{u}_{j}{C}_{j}{f}_{n}G{\parallel}_{\infty}>\epsilon >0. Then there exists a sequence {z_{ n } } ⊂ B_{ N } such that
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).
Hence
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.
Example. If φ_{ j } (j = 1, ..., 5) are analytic self maps of B_{ N }with following formulas:
and
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
is a connected topological space.
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.
Proof. Let distinct uC_{ φ } , v{C}_{\psi}\in {\mathcal{C}}_{w}\left(A\right). We construct a path
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 } ).
We are going to prove that T(t) is a continuous path which connects uC_{ φ } and vC_{ ψ } , and here we simply denote by {C}_{t}={C}_{{\phi}_{t}}. For any s, t ∈ [0, 1],
The second term:
So, it converges to 0 as s → t.
The first term:
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.
It is obvious that uC_{ φ } and θuC_{ φ } can be connected. Then, we construct
to connect uC_{ φ } and uC_{ φ } continuously: for s, t ∈ [0, 1],
as t  s → 0. Thus, θuC_{ φ } , with any θ < 0, can be connected continuously with uC_{ φ } .
Similarly as the analysis above, we note that φ cannot be equal to κφ multiplied by any negative real numbers. Thus, uC_{φ}and uC_{ κφ } can be connected. To connect uC_{φ}and uC_{ φ } , we put μ = iφ ∈ S(B_{ N } ), and
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.
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Tong, CZ., Zhou, ZH. The compactness of the sum of weighted composition operators on the ball algebra. J Inequal Appl 2011, 45 (2011). https://doi.org/10.1186/1029242X201145
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DOI: https://doi.org/10.1186/1029242X201145
Keywords
 Weighted composition operator
 Ball algebra
 Sum
 Difference
 Inequality