# Compact Composition Operators between Bloch Type Spaces in the Polydisk

Authors: Zhong-Shan Fang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Wulan et al. (2009), Wulan et al. characterized the compactness of composition operators on the Bloch space in the unit disk by thenth power of the induced analytic function. This paper will generalize the result to the Bloch type space in the polydisk.


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## Body

## 1. Introduction

Let𝔻n be the polydisk of ℂn with boundary ∂𝔻n. The class of all holomorphic functions on the domain 𝔻n will be denoted by H(𝔻n). Let φ(z)=(φ1(z),…,φn(z)) be a holomorphic self-map of 𝔻n. The composition operator is defined as follows:
(1.1)Cφ(f)(z)=f(φ(z));
for any f∈H(𝔻n) and z∈𝔻n.The study of the composition operator dates back to the late 60s. From then on, the boundedness and compactness of composition operators between several spaces of holomorphic functions have been studied extensively. We refer the interested readers to the books in [1–3]. Recently, there has been lot of work for Bloch type spaces. For example, see [4–9], as well as the related references therein. There are still many unsolved problems of interest to numerous mathematicians.Forp>0, the Bloch type space ℬp consists of those f∈H(𝔻n) such that
(1.2)∥f∥p=|f(0)|+supz∈𝔻n∑k=1n‍(1-|zk|2)p|∂f∂zk(z)|<∞;
with this norm, it becomes a Banach space. When p=1, it is the classical Bloch space.In [10], Wulan et al. obtained a new result about the compactness of the composition operators on the Bloch space in the unit disk. We state it as follows.Theorem 1.1.
Letφ be an analytic self-map of the unit disk 𝔻. Then Cφ is compact on the Bloch space if and only if
(1.3)limm→∞∥φm∥1=0,
where φm here means the mth power of φ.Along with the further research, it is natural to consider the higher-dimensional case. The goal of this paper is to extend the above result in the unit disk to the polydisk.Throughout this paper, letℕ be the set of the positive integers, I={i∈ℕ:∥φi∥∞=1} and J={j∈ℕ:∥φj∥∞<1}.


## 2. Some Lemmas

In this section, we present some lemmas which will be used in the proofs of our main results in the next section.The proof of the following lemma can be found in the proof of Theorem  1.1 in [8], see also Lemma  2.2 in [7].Lemma 2.1.
Letp>0, m∈ℕ and 0≤x≤1. And let Hm,p(x)=xm-1(1-x2)p,rm=((m-1)/(m-1+2p))1/2(m≥2). Then for m≥2, Hm,p(x) is decreasing on [rm,rm+1], and
(2.1)limm→∞mpminx∈[rm,rm+1]Hm,p(x)=(2pe)p.Lemma 2.2 (see Corollary 3, [6]).
Letp≥1 and q>0. Suppose that φ=(φ1,…,φn) is a holomorphic self-map of 𝔻n. Then Cφ:ℬp→ℬq is compact if and only if(1)
φl(z)∈ℬq,l∈{1,…,n};(2)
limφ(z)→∂𝔻n∑k,l=1n|(∂φl/∂zk)(z)|(1-|zk|2)q/(1-|φl(z)|2)p=0.Lemma 2.3 (see Corollary 4, [6]).
Let0<p<1 and q>0. Suppose that φ=(φ1,…,φn) is a holomorphic self-map of 𝔻n. Then Cφ:ℬp→ℬq is compact if and only if(3)
Cφ is bounded;(4)
lim|φl(z)|→1∑k,l=1n|(∂φl/∂zk)(z)|(1-|zk|2)q/(1-|φl(z)|2)p=0,l∈{1,…,n}.The following lemma is the crucial criterion for the compactness ofCφ, whose proof is an easy modification of the proof of Proposition  3.11 in [1].Lemma 2.4.
Assume thatφ is a holomorphic self-map of 𝔻n. Then Cφ:ℬp→ℬq is compact if and only if Cφ is bounded and for any bounded sequence {fm}m∈ℕ in ℬp which converges to zero uniformly on compact subsets of 𝔻n, we have
(2.2)∥Cφfm∥q→0,
as m→∞.


## 3. Main Theorems

In [6, 8], the authors characterized the boundedness and compactness of composition operators between different Bloch type spaces in the polydisk. In this section, we will give some new results about the old problems.Theorem 3.1.
Letp≥1 and q>0. Suppose that φ=(φ1,…,φn) is a holomorphic self-map of 𝔻n. Then Cφ:ℬp→ℬq is compact if and only if(5)
φl(z)∈ℬq,l∈{1,…,n};(6)
limm→∞mp-1∥φim∥q=0,i∈I;(7)
limφ(z)→∂𝔻n∑k=1n|(∂φj/∂zk)(z)|(1-|zk|2)q=0,j∈J.Proof.
For anyl∈{1,…,n}, set
(3.1)al=limφ(z)→∂𝔻n∑k=1n‍|∂φl∂zk(z)|(1-|zk|2)q(1-|φl(z)|2)p.
For the proof of the sufficiency, by Lemma 2.2, we only have to show that al=0 for every l∈{1,2,…,n}. There are two cases to consider.
Case
1. If j∈J with ∥φj∥∞<1, we have
(3.2)limφ(z)→∂𝔻n∑k=1n‍|∂φj∂zk(z)|(1-|zk|2)q≤aj≤limφ(z)→∂𝔻n∑k=1n‍|∂φj∂zk(z)|(1-|zk|2)q(1-∥φj∥∞2)p.
Therefore, for any j∈J, limφ(z)→∂𝔻n∑k=1n|(∂φj/∂zk)(z)|(1-|zk|2)q=0 is equivalent to aj=0.
Case
2. If i∈I with ∥φi∥∞=1, then for each m≥2, let
(3.3)Am,i={z=(z1,…,zn)∈𝔻n:rm≤|zi|≤rm+1},
where rm=((m-1)/(m-1+2α))1/2.
For each fixedi∈I and every ε>0, there exists a δ0 with 0<δ0<1 such that
(3.4)∑k=1n‍|∂φi∂zk(z)|(1-|zk|2)q(1-|φi(z)|2)p>ai-ε,
whenever dist(φ(z),∂𝔻n)<δ0.
Sincerm→1 as m→∞, we may choose sufficiently large m such that rm>1-δ0. If φ(z)∈Am,i, then
(3.5)rm≤|φi(z)|≤rm+1,1-rm+1<1-|φi(z)|<1-rm<δ0,
thus
(3.6)dist(φi(z),∂𝔻)<δ0.
There exists wi with |wi|=1 such that
(3.7)dist(φi(z),wi)=dist(φi(z),∂𝔻)<δ0.
Let w=(φ1(z),…,φi-1(z),wi,φi+1(z),…,φn(z)). Then
(3.8)dist(φ(z),∂𝔻n)≤dist(φ(z),w)≤dist(φi(z),wi)<δ0.
So we have
(3.9)supφ(z)∈Am,i∑k=1n‍|∂φi∂zk(z)|(1-|zk|2)q(1-|φi(z)|2)p>ai-ε.
Letting ε→0 and by Lemma 2.1, we have
(3.10)ai≤limm→∞supφ(z)∈Am,i∑k=1n‍|∂φi∂zk(z)|(1-|zk|2)q(1-|φi(z)|2)p≤limm→∞supφ(z)∈Am,i∑k=1n‍|∂φi∂zk(z)|mp|φi|m-1(1-|zk|2)qmp|φi|m-1(1-|φi(z)|2)p≤limm→∞supφ(z)∈Am,i∑k=1n|(∂φi/∂zk)(z)|mp|φi|m-1(1-|zk|2)qlimm→∞infφ(z)∈Am,imp|φi|m-1(1-|φi(z)|2)p≤(e2p)plimm→∞mp-1∥φim∥q.
From which and (6), we know ai=0. Combining the two cases, thus (2) holds. Note that conditions (5) and (1) are the same, it follows from Lemma 2.2 that Cφ:ℬp→ℬq is compact.
Now we turn to prove the necessity. The result (5) follows by Lemma2.2. Using Lemma 2.1, we see that
(3.11)limm→∞mp-1∥zlm∥p=(2pe)p,
for any l∈{1,…,n}.
For anyi∈I, we consider the test functions fm(z1,z2,…,zn)=zim/∥zim∥p. It is clear that ∥fm∥p=1 and fm→0 uniformly on compact subsets of 𝔻n as m→∞. If Cφ is compact, then
(3.12)0=limm→∞∥Cφfm∥q=limm→∞∥φim∥q∥zim∥p=limm→∞mp-1∥φim∥qmp-1∥zim∥p=(e2p)plimm→∞mp-1∥φim∥q.
This shows that (6) holds.
For anyj∈J, from the discussion of Case 1 in the proof of the sufficiency, it follows that aj=0. And this fact implies that condition (7) holds. Now the proof of the theorem is completed.Remark 3.2.
Forn=1, if p=q=1, it is well known that Cφ is always bounded on the Bloch space, which implies that (5) is true. Note that in this case, conditions (6) and (7) are the same. Thus, we immediately obtain the result on the Bloch space in [10] by Theorem 3.1.Theorem 3.3.
Let0<p<1 and q>0. Suppose that φ=(φ1,…,φn) is a holomorphic self-map of 𝔻n. Then Cφ:ℬp→ℬq is compact if and only if(8)
Cφ is bounded;(9)
limm→∞mp-1∥φim∥q=0,i∈I.Proof.
Suppose first thatCφ:ℬp→ℬq is compact. It is clear that Cφ:ℬp→ℬq is bounded. Taking the test functions fm(z)=zim/∥zim∥p, and using the same arguments as in the proofs of Theorem 3.1, we obtain
(3.13)limm→∞mp-1∥φim∥q=0,
for any i∈I.
This proves the necessity.
Conversely, by Lemma2.4, it suffices to show that (4) holds. In fact, for any j∈J with ∥φj∥∞<1, there exists some positive number δ close enough to 1, such that the set Ej={z∈𝔻n:|φj(z)|>δ} is empty. Without loss of generality, we may assume that
(3.14)lim|φj(z)|→1∑k=1n‍|∂φj∂zk(z)|(1-|zk|2)q(1-|φj(z)|2)p=0.
For any i∈I, and m≥2, let
(3.15)Am,i={z=(z1,…,zn)∈𝔻n:rm≤|zi|≤rm+1},
where rm=((m-1)/(m-1+2α))1/2.
Therefore, we have(3.16)lim|φi(z)|→1∑k=1n‍|∂φi∂zk(z)|(1-|zk|2)q(1-|φi(z)|2)p≤limm→∞supφ(z)∈Am,i∑k=1n‍|∂φi∂zk(z)|(1-|zk|2)q(1-|φi(z)|2)p≤limm→∞supφ(z)∈Am,i∑k=1n‍|∂φi∂zk(z)|mp|φi|m-1(1-|zk|2)qmp|φi|m-1(1-|φi(z)|2)p≤limm→∞supφ(z)∈Am,i∑k=1n|(∂φi/∂zk)(z)|mp|φi|m-1(1-|zk|2)qlimm→∞infφ(z)∈Am,imp|φi|m-1(1-|φi(z)|2)p≤(e2p)plimm→∞mp-1∥φim∥q.
This along with condition (9) yields that
(3.17)lim|φi(z)|→1∑k=1n‍|∂φi∂zk(z)|(1-|zk|2)q(1-|φi(z)|2)p=0,
for any i∈I.
Combining the results of the two cases forj∈J and i∈I, we get (4). By Lemma 2.3, we know that Cφ:ℬp→ℬq is compact. This completes the proof of the theorem.



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*Source: 732709-2012-10-24.xml*