# Differences of Composition Operators on the Space of Bounded Analytic Functions in the Polydisc

Authors: Zhong-Shan Fang; Ze-Hua Zhou
Journal: Abstract and Applied Analysis
 (2008)

---

## Abstract

This paper gives some estimates of the essential norm for the
difference of composition operators induced byφ and
ψ acting on the space, H∞(Dn), of bounded analytic functions
on the unit polydisc Dn, where φ
and ψ are holomorphic self-maps of Dn. As a consequence,
one obtains conditions in terms of the Carathéodory distance
on Dn that characterizes those pairs of holomorphic self-maps
of the polydisc for which the difference of two composition
operators on H∞(Dn) is compact.


---

## Body

## 1. Introduction

Let𝔻n be the unit polydisc of ℂn with boundary ∂𝔻n.
If n=1, we will denote the unit disk 𝔻1 simply by 𝔻. The class of all holomorphic functions on 𝔻n will be denoted by H(𝔻n),
while by H∞(𝔻n),
we denote the space of all bounded analytic functions in the unit polydisc with
the norm ∥f∥∞=supz∈𝔻n|f(z)|.Letφ(z)=(φ1(z),…,φn(z)) and ψ(z)=(ψ1(z),…,ψn(z)) be holomorphic self-maps of 𝔻n.
The composition operator, Cφ,
is defined by(1.1)Cφ(f)(z)=f(φ(z))for any f∈H(𝔻n) and z∈𝔻n.LetX be a Banach space. Recall that the essential
norm of a continuous linear operator T:X→X is the distance from T to the compact operators, that is,(1.2)∥T∥e=inf{∥T−K∥;K:X→Xiscompact}.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.In the past few decades, boundedness, compactness, and
essential norms of composition and closely related operators between various
spaces of holomorphic functions have been studied by many authors (see, e.g.,
the following papers mostly in the settings of the unit ball and the unit
polydisc [1–23] and the references therein). Recently, several papers
focused on studying the mapping properties of the difference of two composition
operators, that is, of an operator of the form(1.3)T=Cφ−Cψ.One of the first results of this
type, in the setting of the Hardy space H2(𝔻),
belongs to Berkson [24]. There, it was shown that if φ is an analytic self-map of the unit disk 𝔻 whose radial limit function φ* satisfies |φ*(ζ)|=1 for ζ∈E⊂∂𝔻, meas(E)>0,
then for any analytic self-map ψ of the disk, ψ≠φ,(1.4)∥Cφ−Cψ∥≥meas(E)2,where meas denotes the normalized Lebesgue measure on ∂𝔻, which means that Cφ is isolated in the operator norm topology.
Some other conditions for isolation in the same setting are obtained in
[15].In [25], MacCluer et al., among other results, characterized
the compactness of the difference of two composition operators on H∞(𝔻) in terms of the Poincaré distance. In
[26], isolated points
and essential components of composition operators on H∞(𝔻) are studied. In [27, 28], the authors have
independently extended the result to H∞(Bn) space, where Bn is the unit ball of ℂn. In [29], Moorhouse showed that if the pseudohyperbolic
distance between the image values φ and ψ converges to zero as z→ζ for every point ζ at which φ and ψ have finite angular derivative, then the
difference Cφ−Cψ yields a compact operator. Differences of
composition operators on the Bloch and the little Bloch space are studied in
[30, 31]. Motivated by these
results, we give some upper and lower estimates of the essential norm for the
difference of composition operators induced by φ and ψ acting on the space H∞(𝔻n),
where φ and ψ are analytic self-maps of 𝔻n.
As a consequence, one obtains conditions in terms of the Carthéodory distance
on 𝔻n that characterize those pairs of holomorphic
self-maps of the polydisc for which the difference of two composition operators
on H∞(𝔻n) is compact.


## 2. Notation and Background

The pseudohyperbolic distance on the unit disk is
defined by(2.1)β(z,w)=|z−w1−z¯w|,z,w∈𝔻.It is easy to see that 0≤β(z,w)≤1.Definition 2.1.
The Poincaré distanceρ on 𝔻 is(2.2)ρ(z,w):=tanh−1β(z,w)=12ln1+β(z,w)1−β(z,w)for z,w∈𝔻.Definition 2.2.
The  Carathéodory pseudodistance on a domainG⊂ℂn is given by(2.3)cG(z,w):=sup{ρ(f(z),f(w)):f∈H(G,𝔻)}for z,w∈G, where H(G,𝔻) denotes the class of holomorphic mappings from G to 𝔻.If we put(2.4)cG∗(z,w):=sup{β(f(z),f(w)):f∈H(G,𝔻)},z,w∈G,then by the monotonicity of the
function h(x)=ln((1+x)/(1−x)) on [0,1) and the inequality h(x)≥2x, x∈[0,1),
we have that(2.5)cG=tanh−1(cG∗)≥cG∗.Next, we introduce the following
pseudodistance on G:(2.6)dG(z,w):=sup{|f(z)−f(w)|:f∈H(G,𝔻)}.For the case G=𝔻,
it is known that (see [32])(2.7)d𝔻(z,w)=2−21−β(z,w)2β(z,w).Hence, the Poincaré metric on 𝔻 is(2.8)ρ(z,w)=tanh−1β(z,w)=ln2+d𝔻(z,w)2−d𝔻(z,w).It is easy to see that forz,w∈G,(2.9)dG(z,w)=sup{|g(f(z))−g(f(w))|:g∈H(𝔻,𝔻),f∈H(G,𝔻)}=supf∈H(G,𝔻)d𝔻(f(z),f(w)).Since the mapt→ln((2+t)/(2−t)) is strictly increasing on [0,2), it follows that(2.10)ln2+dG2−dG=supf∈H(G,𝔻)ln2+d𝔻(f(z),f(w))2−d𝔻(f(z),f(w))=supf∈H(G,𝔻)ρ(f(z),f(w))=cG(z,w),or equivalently for any domain G and any z,w∈G,(2.11)dG(z,w)=2−21−(tanhcG(z,w))2tanhcG(z,w)=2−21−(cG∗(z,w))2cG∗(z,w).It is well known thatc𝔻n∗(z,w)=max1≤j≤nβ(zj,wj) (see [33, Corollary 2.2.4]). So we have(2.12)d𝔻n(z,w)=2−21−(max1≤j≤nβ(zj,wj))2max1≤j≤nβ(zj,wj).Before formulating and proving the main theorem, we
give some notations. For anyδ∈(0,1),
define(2.13)Eδj:={z∈𝔻n:|φj(z)|∨|ψj(z)|>1−δ},and we put Eδ=∪j=1nEδj, where ∨ means the maximum of two real
numbers.Lemma 2.3  (see [34]).
Let{zn} be a sequence in 𝔻 with |zn|→1 as n→∞.
Then, there is a subsequence {znj} of {zn},
a positive number M,
and a sequence of functions fm∈H∞(𝔻) such that(i)
fm(znj)=δmj,(ii)
∑m|fm(z)|≤M<∞foranyz∈𝔻,(the symbol δmj is equal to 1 if m=j and 0, otherwise.)Lemma 2.4.
LetΩ be a domain in ℂn, f∈H(Ω).
If a compact set K and a neighborhood G of K satisfy K⊂G⊂⊂Ω (that is, G is relative compact in Ω) and η=dist(K,∂G)>0,
then(2.14)supz∈K|∂f∂zj(z)|≤nηsupz∈G|f(z)|,for each j∈{1,…,n}.Proof.
Sinceη=dist(K,∂G)>0 for any a∈K,
the polydisc(2.15)Pa={(z1,…,zn)∈ℂn:|zj−aj|<ηn,j=1,…,n}is contained in G.
Using Cauchy's inequality, we have(2.16)|∂f∂zj(a)|≤nηsupz∈∂0Pa|f(z)|≤nηsupz∈G|f(z)|,as desired (where ∂0Pa is the distinguished boundary of Pa).Lemma 2.5.
For fixed0<δ<1,
let Fδ={z∈𝔻n:max1≤j≤n|zj|>1−δ}. Then,(2.17)limr→1sup∥f∥∞=1supz∈Fδc|f(z)−f(rz)|=0for any f in the unit ball of H∞(𝔻n) (where Fδc denotes the complement of Fδ relative to 𝔻n).Proof.
We have(2.18)supz∈Fδc|f(z)−f(rz)|=supz∈Fδc|∑j=1n(f(rz1,rz2,…,rzj−1,zj,…,zn)−f(z1,rz2,…,rzj,zj+1,…,zn))|≤supz∈Fδc∑j=1n∫r1|zj||∂f∂zj(rz1,…,rzj−1,tzj,zj+1,…,zn)|dt≤(1−r)supz∈Fδc∑j=1n|∂f∂zj(z)|.Consider Fδ/2c,
then Fδc⊂Fδ/2c and dist(Fδ/2c,∂𝔻n)=δ/2.
From Lemma2.4, we have that for each j∈{1,…,n}(2.19)supz∈Fδc|∂f∂zj(z)|≤2nδsupz∈Fδ/2c|f(z)|.From this and (2.18), it follows
that(2.20)supz∈Fδc|f(z)−f(rz)|≤2(1−r)nnδ∥f∥∞.
Taking the supremum in (2.20) over the unit ball in H∞(𝔻n), then letting r→1 in (2.20), the lemma follows.


## 3. Main Theorem

In this section, we will state our main result and
give its proof.Theorem 3.1.
Letφ,ψ:𝔻n→𝔻n and Cφ−Cψ:H∞(𝔻n)→H∞(𝔻n).
Then, (3.1)1MΨ≤∥Cφ−Cψ∥e≤4−41−Ψ2Ψ,where Ψ:=max1≤k≤nlimδ→0supz∈Eδkmax1≤j≤nβ(φj(z),ψj(z)) and M is a positive constant.Proof.
First,
we consider the upper estimate. For fixedr∈(0,1),
it is easy to check that both Crφ and Crψ are compact operators. Therefore,(3.2)∥Cφ−Cψ∥e≤∥Cφ−Cψ−Crφ+Crψ∥.Now, for any 0<δ<1,(3.3)∥Cφ−Cψ−Crφ+Crψ∥=sup∥f∥∞=1∥(Cφ−Cψ−Crφ+Crψ)f∥∞=sup∥f∥∞=1supz∈𝔻n|f(φ(z))−f(ψ(z))−f(rφ(z))+f(rψ(z))|≤sup∥f∥∞=1supz∈Eδ|f(φ(z))−f(ψ(z))−f(rφ(z))+f(rψ(z))|+sup∥f∥∞=1supz∈Eδc|f(φ(z))−f(ψ(z))−f(rφ(z))+f(rψ(z))|=I1+I2.From Lemma 2.5, we can choose r sufficiently close to 1 such that I2 is sufficiently small.
Applying the Schwarz-Pick lemma on the functionϕ(z)=rz,  r∈(0,1),
and by the monotony of the function f(x)=(2−21−x2)/x,
we obtain(3.4)I1≤sup∥f∥∞=1supz∈Eδ(|f(φ(z))−f(ψ(z))|+|−f(rφ(z))+f(rψ(z))|)=supz∈Eδsup∥f∥∞=1(|f(φ(z))−f(ψ(z))|+|−f(rφ(z))+f(rψ(z))|)≤supz∈Eδ(d𝔻n(φ(z),ψ(z))+d𝔻n(rφ(z),rψ(z)))≤2supz∈Eδ2−2(1−max1≤j≤n(β(φj(z),ψj(z)))2)1/2max1≤j≤nβ(φj(z),ψj(z))=4−4(1−supz∈Eδmax1≤j≤n(β(φj(z),ψj(z)))2)1/2supz∈Eδmax1≤j≤nβ(φj(z),ψj(z))(3.5)≤4−4(1−max1≤k≤nsupz∈Eδkmax1≤j≤n(β(φj(z),ψj(z)))2)1/2max1≤k≤nsupz∈Eδkmax1≤j≤nβ(φj(z),ψj(z)).By direct calculation, it is
easy to check that(3.6)limδ→0max1≤k≤nsupz∈Eδkmax1≤j≤nβ(φj(z),ψj(z))=max1≤k≤nlimδ→0supz∈Eδkmax1≤j≤nβ(φj(z),ψj(z)).From which, and letting δ→0 in (3.5), the upper estimate in (3.1) follows.
Now, we turn to the lower estimate.
Let(3.7)ak=limδ→0supz∈Eδkmax1≤j≤nβ(φj(z),ψj(z)),k=1,…,n.If we set δm=1/m,
then δm→0 as m→∞,
and there exists zm∈Eδmj and some j such that(3.8)limm→∞β(φj(zm),ψj(zm))=ak.
Sincezm∈Eδmj and δm→0,
we have |φj(zm)|→1 or |ϕj(zm)|→1.
Without loss of generality, we can assume |φj(zm)|→1.
Let wm=φj(zm),
by Lemma  2.3, we have that there is a subsequence of wm (we may denote it again by wm), a positive number M,
and a sequence of functions fm∈H∞(𝔻) such that(i)
fm(wk)=δmk,(ii)
∑m|fm(z)|≤M<∞foranyz∈𝔻.
Now, for anyz∈𝔻n,
we define f˜m(z):=fm(zj),
where zj is the jth component of z,
then ∑m|f˜m(z)|≤M<∞.
Next we claim thatf˜m converge weakly to 0.
Let λ∈H∞(𝔻n)∗.
For any natural N,
there exist some unimodular sequences αm such that(3.9)∑m=0N|λ(f˜m)|=∑m=0Nαmλ(f˜m)=λ(∑m=0Nαmf˜m)≤∥λ∥∥∑m=0Nαmf˜m∥∞≤∥λ∥M.Thus, λ(f˜m)→0 as m→∞,
that is, f˜m converge weakly to 0.
Set(3.10)gm(z)=f˜m(z)Mzj−ψj(zm)1−ψj(zm)¯zj,m∈ℕ.Then, ∥gm∥∞≤1 and similarly to f˜m, it is easy to see
that
gm converge weakly to 0.
Thus, for any compact operator K,
we have ∥Kgm∥∞→0 as m→∞.
Now, we have(3.11)J=∥Cφ−Cψ−K∥≥limsupm→∞∥(Cφ−Cψ−K)gm∥∞≥limsupm→∞(∥(Cφ−Cψ)gm∥∞−∥Kgm∥∞)=limsupm→∞supz∈𝔻n|gm(φ(z))−gm(ψ(z))|≥1Mlimsupm→∞supz∈𝔻n|φj(z)−ψj(zm)1−ψj(zm)¯φj(z)f˜m(φ(z))−ψj(z)−ψj(zm)1−ψj(zm)¯ψj(z)f˜m(ψ(z))|≥1Mlimsupm→∞|φj(zm)−ψj(zm)||1−ψj(zm)¯φj(zm)|=1Mlimm→∞β(φj(zm),ψj(zm))=1Mak.
Then, we have(3.12)∥Cφ−Cψ∥e≥1Mmax1≤k≤nlimδ→0supz∈Eδkmax1≤j≤nβ(φj(z),ψj(z)),finishing the proof of the
theorem.Corollary 3.2.
The
operatorCφ−Cψ is compact if and only if (3.13)max1≤k≤nlimδ→0supz∈Eδkmax1≤j≤nβ(φj(z),ψj(z))=0.Proof.
By using
the inequality(1−1−x2)/x≤x (0<x≤1) and
the fact that T is compact if and only if ∥T∥e=0,
the corollary follows by Theorem 3.1.Example 3.3.
Letn=2, φ(z)=(z1,(1/2)z2), and ψ(z)=(z1,(1/3)z2).
Then, β(φ1(z),ψ1(z))=0 and β(φ2(z),ψ2(z))=(1/6)(|z2|/(1−(1/6)|z2|2)).
A direct calculation shows that(3.14)max1≤k≤2limδ→0supz∈Eδkmax1≤j≤2β(φj(z),ψj(z))=15>0;so by Corollary 3.2, Cφ−Cψ is not compact.Example 3.4.
Letn=2, p>1, 0<c≤1, φ(z)=((z1+1)/2,(1/2)z2), and(3.15)ψ(z)=(z1+12+c(1−z12)p,12z2),where we choose the usual branch
of the logarithm of w1, Rew1>0,
in order to define ((1−z1)/2)p.
By  [35], ψ is a self-map of 𝔻2,
whenever c is small. Moreover, max1≤j≤2β(φj(z),ψj(z))=β(φ1(z),ψ1(z)).
By Corollary 3.2 and the proof of Example 1 of [25], we have, for these c,(1)
if1<p≤2,
then Cφ−Cψ is noncompact;(2)
if2<p<∞,
then Cφ−Cψ is compact.



---

*Source: 983132-2009-01-08.xml*