Open Access

The compactness of the sum of weighted composition operators on the ball algebra

Journal of Inequalities and Applications20112011:45

https://doi.org/10.1186/1029-242X-2011-45

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 non-zero weighted composition operators on the unit ball algebra is also studied.

2000 Mathematics Subject Classification. Primary: 47B33; Secondary: 47B38, 46E15, 32A36.

Keywords

Weighted composition operatorBall algebraSumDifferenceInequality

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 N-dimensional complex space N . The closure of B N will be written as B N ¯ . Denote by A = A(B N ), the unit ball algebra of all continuous functions on B N ¯ that are holomorphic on B N . Then A is the Banach algebra with the supremum norm
f = sup { | f ( z ) | : z B N } .

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
( C φ f ) ( z ) = f ( φ ( z ) ) ;
the multiplication operator induced by u is defined by
M u f ( z ) = u ( z ) f ( z ) ;
and the weighted composition operator uC φ induced by φ and u is defined by
( u C φ f ) ( z ) = u ( z ) f ( φ ( z ) )

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, [16], 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 ( A α 2 ) . 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 co-workers [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 co-workers [15], and the setting of H(D N ) by Fang and Zhou [12], where D N is the unit polydisk. Bonet and co-workers [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 non-zero 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
Φ w ( z ) = w - P w ( z ) - s w Q w ( z ) 1 - < z , w > ,
where P w ( z ) = < z , w > < w , w > w , Q w ( z ) = z - < z , w > < w , w > w , and s w = 1 - | w | 2 . The induced distance between z and w is defined as
d ( z , w ) = sup f B ( H ) | f ( z ) - f ( w ) | .
Pseudo-hyperbolic metric is defined by
ρ ( z , w ) = sup { | f ( z ) | : f = 1 , f ( w ) = 0 } ,
and a theorem of Bear [20] gives that
d ( z , w ) = 2 - 2 1 - ρ ( z , w ) 2 ρ ( z , w ) .
Since the pseudo-hyperbolic metric ρ(z, w) ≤ 1 we have that
1 - ρ ( z , w ) 2 1 - ρ ( z , w ) 2 1 - 1 - ρ ( z , w ) 2 ρ ( z , w ) 2 d ( z , w ) 2 ρ ( z , w ) .

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
Δ ( ζ , α ) = z B N : | 1 - z , ζ | < α 2 1 - | z | 2 .

Since U Δ(ζ, α) = Δ (, α) for any unitary transformation U in N , we can concentrate on understanding Δ (e1, α), where e 1 = ( 1 , 0 , , 0 ) N . Next lemma is crucial.

Lemma 2.2. For a real number 1 < α < 2 and e1 = (1, 0, ..., 0) inB N , if {w n } is an arbitrary sequence in the Koranyi approach region Δ(e1, α), then
| w n | 2 - e 1 , w n 1 - e 1 , w n > 2 α - 1 > 0 .
On the other hand, for every α > 2, we can find a sequence {w n } Δ(e1, α) and w n e1as n → ∞ satisfying
lim n | w n | 2 - e 1 , w n 1 - e 1 , w n = 0 .
Proof. First, let 1 < α < 2. We have that
| w n | 2 - e 1 , w n 1 - e 1 , w n > 2 α | w n | 2 - e 1 , w n 1 - | w n | 2 2 α 1 - | w n | 2 - 1 - e 1 , w n 1 - | w n | 2 > 2 α 1 - α 2 ( 1 - | w n | 2 ) 1 - | w n | 2 = 2 α - 1 > 0 .
Suppose α > 2. Let w n = ( r n , w n ) , where r n is real with r n → 1- as n → ∞ and | w n | = 1 - r n . It is clear that for sufficient large n
| 1 - e 1 , w n | = 1 - r n < α 2 r n - r n 2 (1) = α 2 1 - r n 2 - 1 + r n = α 2 1 - | w n | 2 .
This means that {w n } Δ(e1, α) for α > 2. Now we compute
lim n | w n | 2 - e 1 , w n 1 - e 1 , w n = lim n r n 2 + 1 - r n - r n 1 - r n = lim n ( 1 - r n ) = 0 .

For φ1, φ2, ..., φ p S(B N ) ∩ A, we simplify the notations Γ φ j and C φ 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 ζ j = 1 p Γ 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 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 j = 1 p u j C j is compact on A if and only if and only if k I s ( ζ ) u k ( ζ ) = 0 for every ζ j = 1 p Γ j and every index set I s (ζ) of ζ.

Proof. To prove the necessity, let ζ be an arbitrary point in j = 1 p Γ 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 ks;

(3): φ l (w n ) → φ l (ζ) with l I s (ζ);

(4): Φ φ s ( w n ) ( φ s ( ζ ) ) , φ s ( w n ) λ 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): Φ φ s ( w n ) ( φ l ( w n ) ) , φ s ( w n ) σ l s 0 .

For such a sequence {w n }, define the functions
f n ( z ) = ( Φ φ s ( w n ) ( z ) , φ s ( w n ) - 1 ) ( Φ φ s ( w n ) ( z ) , φ s ( w n ) - λ ) (1) l I s ( ζ ) ( Φ φ s ( w n ) ( z ) , φ s ( w n ) - σ l s ) .
Then f n A and ||f n || ≤ 2p+1. We claim that f n (z) → 0 for all z B N ¯ as n → ∞. Indeed, if z = φ s (ζ), the item (4) guarantees f n (φ s (ζ)) → 0; and if zφ s (ζ), we have
lim n Φ φ s ( w n ) ( z ) , φ s ( w n ) = lim n φ s ( w n ) , φ s ( w n ) z , φ s ( w n ) φ s ( w n ) , φ s ( w n ) φ s ( w n ) , φ s ( w n ) 1 z , φ s ( w n ) = lim n | φ s ( w n ) | 2 z , φ s ( w n ) 1 z , φ s ( w n ) = 1.

So f n converges weakly to 0 in A, thus j = 1 p u j C j f n 0 as n → ∞.

On the other hand,
j = 1 p u j C j f n j = 1 p u j ( w n ) f n ( φ j ( w n ) ) (1) k I s ( ζ ) u k ( w n ) | λ | l I s ( ζ ) σ l s - l I s ( ζ ) | u l ( w n ) | | f n ( φ l ( w n ) ) | - k I s ( ζ ) \ { s } | u k ( w n ) | | f n ( φ s ( w n ) ) - f n ( φ k ( w n ) ) | k I s ( ζ ) u k ( w n ) | λ | l I s ( ζ ) σ l s - l I s ( ζ ) | u l ( w n ) | | f n ( φ l ( w n ) ) | - 2 k I s ( ζ ) \ { s } u k f n ρ ( φ s ( w n ) , φ k ( w n ) ) .
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
k I s ( ζ ) u k ( ζ ) = 0 .
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 { f n k } 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
s u p z B N | g ( z ) | 1 .
Now define a function G on B N ¯ by setting
G ( z ) = l I ( z ) u l C l g ( z ) z j = 1 p Γ j ; j = 1 p u j C j g ( z ) otherwise .

Next, we just need to show that G is continuous on B N ¯ , and j = 1 p u j C j f n k converges uniformly to G on B N ¯ . In the following proof, we simplify the subsequence f n k by f n .

We first prove that G is continuous on B N ¯ . Indeed, it is obvious that G is continuous on B N ¯ \ j = 1 p Γ j .

For ζ j = 1 p Γ j , let I s (ζ)'s be its index sets and I ( ζ ) = s I s ( ζ ) . 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
j = 1 p u j C j g ( z n ) - l I ( ζ ) u l C l g ( ζ ) (1)  s g k I s ( ζ ) u k ( z n ) + 2 k I s ( ζ ) \ { s } u k g ρ ( φ s ( z n ) , φ k ( z n ) ) + l I ( ζ ) ( u l C l g ( z n ) - u l C l g ( ζ ) ) .

So, we have lim z n ζ j = 1 p u j C j g ( z n ) = G ( ζ ) , and this holds for every ζ j = 1 p Γ j . Thus G is continuous on B N ¯ .

Next, we shall show that j = 1 p u j C j f n converges uniformly to G on B N ¯ . Suppose not, and we may assume that, for ε > 0, j = 1 p u j C j f n - G > ε > 0 . Then there exists a sequence {z n } B N such that
| j = 1 p u j C j f n ( z n ) - G ( z n ) | > ε for every n .
(2.1)

This implies that max j { 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
| j = 1 p ( u j C j ) f n ( z n ) - G ( z n ) | (1) s f n k I s ( ζ ) u k ( z n ) + 2 k I s ( ζ ) \ { s } u k f n ρ ( φ s ( z n ) , φ k ( z n ) ) + l I ( ζ ) u l ( z n ) [ f n ( φ l ( z n ) ) - g ( φ l ( z n ) ) ] .

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 | k I s ( ζ ) u k ( ζ ) | = 0 for each index sets I s (ζ). Together with the above two items, we have j = 1 p u j C j f n ( z n ) - G ( z n ) 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 limzζρ(φ(z), ψ(z)) ≠ 0, then u(ζ) = 0;

(b): If ζ Γ ψ and limzζρ(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:
φ 1 ( z 1 , , z N ) = z 1 + 1 2 , z 2 2 , , z N 2 , φ 2 ( z 1 , , z N ) = z 2 2 , z 1 + 1 2 , , z N 2 , φ 3 ( z 1 , , z N ) = z 1 + 3 4 , z 2 4 , , z N 4 , φ 4 ( z 1 , , z N ) = z 1 2 + 1 2 , 0 , , 0 , φ 5 ( z 1 , , z N ) = z 1 2 , z 2 2 , , z N 2 ,
and
u 1 ( z 1 , , z N ) = 2 z 1 , u 2 ( z 1 , , z 2 ) = z 2 + + z N , u 3 ( z 1 , , z N ) = - 3 z 1 , u 4 ( z 1 , , z N ) = ( z 1 + 1 ) 2 , u 5 1 .

Then, j = 1 5 u j C j is a compact operator on A.

Proof. First, note that Γ1 = {(1, 0, ..., 0)} = {e1}, Γ2 = {e1}, Γ3 = {e1}, Γ4 = {±e1}, Γ2 = . And φ1(e1) = φ3(e1) = e1, φ2(e1) = (0, 1, 0, ..., 0) = (e2), φ4e1) = e1. Then, φ j A* for each j = 1, ..., 5. Take φ1 for example, the sequence {(1 - 1/n, 0, ..., 0)} converges to φ(e1) radially, and they have pre-images {(1 - 2/n, 0, ..., 0)} of φ1 in B N . Hence, φ1 satisfies (*). Then, we have I1(e1) = {1, 3, 4}, I2(e1) = {2} and I(-e1) = I4(-e1) = {4}. Compute that

I1(e1):u1(e1) + u3(e1) + u4(e1) = 2 - 3 + 1 = 0,

I2(e1):u2(e1) = 0,

I(-e1):u4(-e1) = 0.

By Theorem 2.4, 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 C ( A ) to represent that topological space. It is trivial that

C ( A )
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 : [ 0 , 1 ] C ( A ) defined by t tuC φ . Indeed, for many function spaces, we denote by S generally, the weighted composition operators topological space C ( S ) is connected by the same path constructed above. So, we are interested in the connectness of the non-zero weighted composition operators topological space, and here we denote by C w ( S ) . In this setting, the connectness of C w ( S ) depends on the function space S.

Theorem 3.1 C w ( A ) is also a connected topological space.

Proof. Let distinct uC φ , v C ψ C w ( A ) . We construct a path
T ( t ) : [ 0 , 1 ] C w ( A ) by t u t C φ t

where u t (z) = tu(z) + (1 - t)v(z) and φ t (z) = (z) + (1 - t)ψ(z). To avoid u t = 0 and φ t = 0 for some t [0, 1], we first assume that

(1): vθu for every θ < 0;

(2): ψκφ for every κ < 0 with κφ S(B N ), where ψ = (ψ1, ..., ψ N ) and φ = (φ1, ..., 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 φ t . For any s, t [0, 1],
u t C t - v t C s u t ( C t - C s ) + ( u t - u s ) C s .
The second term:
( u t - u s ) C s | t - s | u + v C s | t - s | u + v .

So, it converges to 0 as st.

The first term:
u t ( C t - C s ) u t C t - C s = u t sup f B ( A ) sup z B N | ( C t - C s ) f ( z ) | u t sup f B ( H ) sup z B N | f ( φ t ( z ) ) - f ( φ s ( z ) ) | = u t sup z B N sup f B ( H ) | f ( φ t ( z ) ) - f ( φ s ( z ) ) | 2 u t sup z B N ρ ( φ t ( z ) , φ s ( z ) ) .
(3.1)

So, it converges to 0 as st, 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 C w ( A ) .

Next, for any given u C φ C w ( A ) , 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
R ( t ) = e i π t u C φ : [ 0 , 1 ] C w ( A )
to connect uC φ and -uC φ continuously: for s, t [0, 1],
e i π t u C φ - e i π s u C φ e i π t - e i π s u | | C φ | | π | t - s | u C φ 0

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-φand uC κφ can be connected. To connect uC-φand uC φ , we put μ = S(B N ), and
Y 1 ( t ) = u C t φ + ( 1 - t ) μ : [ 0 , 1 ] C w ( A ) ; Y 2 ( t ) = u C t ( - φ ) + ( 1 - t ) μ : [ 0 , 1 ] C w ( A )

By the same method of computation as in (3.1), Y1(t) and Y2(t) can be shown to be two continuous pathes in C w ( A ) from uC φ and uC-φto uC μ .

Thus, the space C w ( A ) 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 C w ( A ) 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

(1)
Department of Mathematics, Tianjin University

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© Tong and Zhou; licensee Springer. 2011

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