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Norms and essential norms of composition operators from H∞to general weighted Bloch spaces in the polydisk
Journal of Inequalities and Applications volume 2011, Article number: 46 (2011)
Abstract
Let Un be the unit polydisk of and φ a holomorphic self-map of Un. H∞(Un) and denote the space of bounded holomorphic functions and the space of general weighted Bloch functions defined on Un, respectively, where α > 0. This paper gives some estimates of the norm and essential norm of the composition operator C φ induced by φ from H∞(Un) to . As applications, some characterizations of the boundedness and compactness of C φ from H∞(Un ) to are obtained. Moreover, we also characterize the weak compactness of the composition operator C φ .
MSC(2000):
Primary 47B33; Secondary 32A37
1 Introduction
Let D be a bounded homogeneous domain in , H (D) the class of all holomorphic functions on D. For φ, a holomorphic self-map of D, the linear operator defined by
is called the composition operator with symbol φ. The study of composition operators is fundamental in the study of Banach and Hilbert spaces of holomorphic functions. We refer to the books [1] and [2] for an overview of some classical results on the theory of composition operators.
Let K (z, z) be the Bergman kernel function of D, and the Bergman metric H z (u, u) in D is defined by
where z = (z1, ..., z n ) ∈ D and . A function f ∈ H (D) is said to be a Bloch function if is finite, where
. By fixing a base point z0 ∈ D, the Bloch space of all Bloch functions on D is a Banach space under the norm [3]. For convenience, we assume the bounded homogeneous domain D to contain the origin and take z0 = 0. In [3], Timoney proved that the space H∞(D) of bounded holomorphic functions on a bounded homogeneous domain D is a subspace of and for each , where C D is a constant depending only on the domain D and .
Let Un be the unit polydisk of . Timony [3] showed that if and only if
and is equivalent to the Bloch norm . This characterization was the starting point for introducing α-Bloch spaces. For α > 0, the α -Bloch space is defined as follows.
Recently, Li and Liu [4] introduced the notation of general weighted Bloch spaces (Stević called these the logarithmic Bloch-type spaces in [5]) in polydisk. For α > 0, a function f ∈ H (Un) is said to belong to the general weighted Bloch space if
It is easy to show that is a Banach space with the norm
Composition operators on various Bloch-type spaces have been studied extensively by many authors. For the unit disk , Madigan and Matheson [6] proved that C φ is always bounded on . They also gave some sufficient and necessary conditions that C φ is compact on . Since then, there were many authors generalizing the results in [6] to the unit ball, polydisk and other classical symmetric domains, see, for example, [7–17]. At the same time, there were also many papers dealing with the composition operators between Bloch-type spaces and bounded holomorphic function spaces, refer to [18, 19] and the references therein for the details. Specially, Li and Liu [4] stated and proved the corresponding boundedness and compactness characterizations for C φ from H∞(Un) to . But there is a little gap in the proof [[4], line 17, p. 1637]. In this paper, we apply methods developed by Montes-Rodriguez [9] to give some estimates of the norm and essential norm of C φ from H∞(Un) to . Recall that the essential norm ||T|| e of a bounded operator T between Banach spaces X and Y is defined as the distance from T to the space of compact operators from X to Y. Notice that ||T|| e = 0 if and only if T is compact, so that estimates on || T || e lead to conditions for T to be compact. For convenience, we define || T|| e = || T|| = ∞ for any unbounded linear operator T. As an application of our estimates, we obtain the main results in [4] with new proofs. In addition, we also show the equivalence of the compactness and weak compactness of .
Throughout the remainder of this paper, C will denote a positive constant, the exact value of which will vary from one occurrence to the others.
2 The norm of C φ
In this section, we give the following estimate of the norm of .
Theorem 1. Let α > 0 and φ = (φ1, ..., φ n ) be a holomorphic self-map of the unit polydisk Un, then
Here and in the sequel, the symbol A ≲ B(or B ≳ A) means that A ≤ CB for some positive constant C independent of A and B. A ~ B means that A ≲ B and B ≲ A.
Proof. For the lower estimate: If , then the result is trivially true. Now suppose is bounded. For any fixed w ∈ Un and k ∈ {1, ..., n}, take the following test function
Then, f ∈ H∞ (Un) and || f ||∞ ≤ 2α. Fix any δ ∈ (0, 1). If | φ k ( w )| ≥ δ, then
If | φ k (w)| < δ, then
That is, for any w ∈ Un,
Since k ∈ {1, ..., n} is arbitrary, so
For the upper estimate: . We also assume , since for the other case nothing needs to be proven. For any f ∈ H∞(Un),
where is used in the last line above.
Since
Taking the supremum over all f ∈ H∞(Un) with || f ||∞ ≤ 1, we have
which completes the proof.
The following corollary is obtained immediately from Theorem 1.
Corollary 2. Let φ = (φ 1 , ..., φ n ) be a holomorphic self-map of Un and α > 0. Then, is bounded if and only if
3 The essential norm of C φ
This section mainly gives the following estimate of the essential norm of C φ from H∞(Un) to .
Theorem 3. Let φ = (φ1, ..., φ n ) be a holomorphic self-map of Un and α > 0, then
Proof. For the lower estimate:
It is trivial when C φ is unbounded. So we assume that C φ is bounded. By Corollary 2,
Take , then ||f m ||∞ = 1 and f m (z) converge to zero uniformly on any compact subset of Un. So for any compact operator . Then
where A m = {z ∈ Un : r m ≤ |φ1(z)| ≤ rm + 1}, . Since y = mxm-1(1 - x2), x ∈ [0, 1), is increasing on [0, r m ] and decreasing on [r m , 1), . From (2), we have
It is from (1) that
Then, for any ε > 0, there is δ0 ∈ (0, 1) such that
whenever dist (φ(z), ∂Un) < δ0. Again r m ↑ 1, so for m large enough,
So ||C φ − K|| ≳ a1 − ε. Since K is arbitrary,
Similarly, considering the functions , l = 2, ..., n, we also have
Thus,
Again ε is arbitrary, and we obtain the desired lower estimate.
For the upper estimate:
If , then the estimate is trivial too. Now we suppose , then is bounded by Corollary 2. Define the operators K m (m ≥ 2) as follows
It is easy to see that K m : H∞(Un) → H∞(Un) is compact since K m maps every bounded sequence in H∞( Un) converging to zero on compact subsets of Un to the sequence converging to zero in norm of H∞(Un). In addition, ||I − K m : H∞(Un) → H∞(Un)|| ≤ 2. Therefore, is compact. Then
Where and .
Fix δ ∈ (0, 1) and let G1 = {z ∈ Un : dist(φ(z), ∂Un) < δ}, G2 = Un\G1 = {z ∈ Un : dist(φ(z), ∂Un) ≥ δ}, which is a compact subset of Un.
Where
And
It is clear that the sequence of operators {I − K m } m satisfies for each f ∈ H (Un), and the space H(Un) endowed with the compact open topology τ is a Fréchet space. Further, D j : (H (Un ), τ ) → (H (Un), τ ) defined by is a contionuous linear operator. Therefore, by the Banach-Steinhaus theorem, the sequence {D j ° (I − K m )} m converges to zero uniformly on compact subsets of (H (Un), τ ). Since, by Montel's normal theorem, the closed unit ball of H∞(Un) is a compact subset of (H (Un), τ ), we conclude that
Thence, J2 → 0 (as m → ∞).
Similarly, we know that
Consequently,
Thus
The proof is complete.
As an application, we have the following corollary.
Corollary 4. Let φ = (φ1, ..., φ n ) be a holomorphic self-map of Un and α > 0. Then the following are equivalent.
-
(1)
is compact.
-
(2)
is weakly compact.
-
(3)
.
Proof. (1)⇒ (2) is obvious, and (3)⇒ (1) follows immediately from Theorem 3. So it suffices to prove (2)⇒ (3). Now assume that is weakly compact. If (3) is not true, then there is a sequence {zj} ⊂Un and ε0 > 0 such that wj = φ(zj) → ∂Un (as j → ∞) together with
for each j ≥ 1. Since C φ is weakly compact, C φ is bounded. Then, by Corollary 2,
Extracting a subsequence of {zj}, if needed, we may assume that exists for every l and
From (3), there are k0 and l0 such that , i.e.,
If , define . Then, the sequence {f j } j ⊂ H∞(Un) is bounded and converges to zero uniformly on any compact subset of Un. That is, {f j } weakly converges to zero in H∞( Un). Because H∞(Un) has Dunford-Pettis property (See Theorem 5.3 in [20] for H∞(U), and note the proof there works also for H∞(Un)), the weak compactness of implies that . But this is impossible since using (4) we may estimate that for each j ≥ 1,
If . Since wj → ∂Un, there is l1 ∈ {1, ..., n}\{l0} such that . If there exists k1 such that
then as in the last paragraph above we obtain the desired contradiction using the following test functions:
Thus, we may assume that
for each k. We now define the test functions h j as follows
Then, ||h j ||∞ ≲ 1 and h j converge to zero uniformly on any compact subset of Un. But for any j large enough
the inequalities in the third and fourth lines above follow from (5), and the last line is due to. This contradicts again , which completes the proof.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (No.10901158,11071190).
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Wang, M., Wu, H. Norms and essential norms of composition operators from H∞to general weighted Bloch spaces in the polydisk. J Inequal Appl 2011, 46 (2011). https://doi.org/10.1186/1029-242X-2011-46
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DOI: https://doi.org/10.1186/1029-242X-2011-46