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

Abstract

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

1. Introduction

Let 𝔻^𝑛 be the unit polydisc of ℂ^𝑛 with boundary 𝜕𝔻^𝑛. If 𝑛=1, we will denote the unit disk 𝔻¹ simply by 𝔻. The class of all holomorphic functions on 𝔻^𝑛 will be denoted by 𝐻(𝔻^𝑛), while by 𝐻^∞(𝔻^𝑛), we denote the space of all bounded analytic functions in the unit polydisc with the norm ‖𝑓‖_∞=sup_{𝑧∈𝔻𝑛}|𝑓(𝑧)|.

Let 𝜑(𝑧)=(𝜑₁(𝑧),…,𝜑_𝑛(𝑧)) and 𝜓(𝑧)=(𝜓₁(𝑧),…,𝜓_𝑛(𝑧)) be holomorphic self-maps of 𝔻^𝑛. The composition operator, 𝐶_𝜑, is defined by𝐶_𝜑(𝑓)(𝑧)=𝑓(𝜑(𝑧)) (1.1)for any 𝑓∈𝐻(𝔻^𝑛) and 𝑧∈𝔻^𝑛.

Let 𝑋 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,‖𝑇‖_𝑒=inf{‖𝑇−𝐾‖;𝐾∶𝑋→𝑋iscompact}. (1.2)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 form𝑇=𝐶_𝜑−𝐶_𝜓. (1.3)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,‖𝐶_𝜑−𝐶_𝜓‖≥meas(𝐸)2, (1.4)where meas denotes the normalized Lebesgue measure on 𝜕𝔻, which means that 𝐶_𝜑 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^∞(𝐵_𝑛) space, where 𝐵_𝑛 is the unit ball of ℂ^𝑛. 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 𝐶_𝜑−𝐶_𝜓 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^∞(𝔻^𝑛), where 𝜑 and 𝜓 are analytic self-maps of 𝔻^𝑛. As a consequence, one obtains conditions in terms of the Carthéodory distance on 𝔻^𝑛 that characterize those pairs of holomorphic self-maps of the polydisc for which the difference of two composition operators on 𝐻^∞(𝔻^𝑛) is compact.

2. Notation and Background

The pseudohyperbolic distance on the unit disk is defined by𝛽(𝑧,𝑤)=|𝑧−𝑤1−𝑧𝑤|,𝑧,𝑤∈𝔻. (2.1)It is easy to see that 0≤𝛽(𝑧,𝑤)≤1.

Definition 2.1. The Poincaré distance 𝜌 on 𝔻 is𝜌(𝑧,𝑤)∶=tanh⁻¹1𝛽(𝑧,𝑤)=2ln1+𝛽(𝑧,𝑤)1−𝛽(𝑧,𝑤) (2.2)for 𝑧,𝑤∈𝔻.

Definition 2.2. The Carathéodory pseudodistance on a domain 𝐺⊂ℂ^𝑛 is given by𝑐_𝐺(𝑧,𝑤)∶=sup{𝜌(𝑓(𝑧),𝑓(𝑤))∶𝑓∈𝐻(𝐺,𝔻)} (2.3)for 𝑧,𝑤∈𝐺, where 𝐻(𝐺,𝔻) denotes the class of holomorphic mappings from 𝐺 to 𝔻.

If we put𝑐^∗_𝐺(𝑧,𝑤)∶=sup{𝛽(𝑓(𝑧),𝑓(𝑤))∶𝑓∈𝐻(𝐺,𝔻)},𝑧,𝑤∈𝐺, (2.4)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 that𝑐_𝐺=tanh⁻¹(𝑐^∗_𝐺)≥𝑐^∗_𝐺. (2.5)Next, we introduce the following pseudodistance on 𝐺:𝑑_𝐺(𝑧,𝑤)∶=sup{|𝑓(𝑧)−𝑓(𝑤)|∶𝑓∈𝐻(𝐺,𝔻)}. (2.6)For the case 𝐺=𝔻, it is known that (see [32])𝑑_𝔻(𝑧,𝑤)=2−21−𝛽(𝑧,𝑤)²𝛽(𝑧,𝑤). (2.7)Hence, the Poincaré metric on 𝔻 is𝜌(𝑧,𝑤)=tanh⁻¹𝛽(𝑧,𝑤)=ln2+𝑑_𝔻(𝑧,𝑤)2−𝑑_𝔻(𝑧,𝑤). (2.8)

It is easy to see that for 𝑧,𝑤∈𝐺,𝑑_𝐺(𝑧,𝑤)=sup{|𝑔(𝑓(𝑧))−𝑔(𝑓(𝑤))|∶𝑔∈𝐻(𝔻,𝔻),𝑓∈𝐻(𝐺,𝔻)}=sup_{𝑓∈𝐻(𝐺,𝔻)}𝑑_𝔻(𝑓(𝑧),𝑓(𝑤)). (2.9)

Since the map 𝑡→ln((2+𝑡)/(2−𝑡)) is strictly increasing on [0,2), it follows thatln2+𝑑_𝐺2−𝑑_𝐺=sup_{𝑓∈𝐻(𝐺,𝔻)}ln2+𝑑_𝔻(𝑓(𝑧),𝑓(𝑤))2−𝑑_𝔻(𝑓(𝑧),𝑓(𝑤))=sup_{𝑓∈𝐻(𝐺,𝔻)}𝜌(𝑓(𝑧),𝑓(𝑤))=𝑐_𝐺(𝑧,𝑤), (2.10)or equivalently for any domain 𝐺 and any 𝑧,𝑤∈𝐺,𝑑_𝐺(𝑧,𝑤)=2−21−(tanh𝑐_𝐺(𝑧,𝑤))²tanh𝑐_𝐺=(𝑧,𝑤)2−21−(𝑐^∗_𝐺(𝑧,𝑤))²𝑐^∗_𝐺.(𝑧,𝑤) (2.11)

It is well known that 𝑐^∗_𝔻^𝑛(𝑧,𝑤)=max_{1≤𝑗≤𝑛}𝛽(𝑧_𝑗,𝑤_𝑗) (see [33, Corollary 2.2.4]). So we have𝑑_𝔻^𝑛(𝑧,𝑤)=2−21−(max_{1≤𝑗≤𝑛}𝛽(𝑧_𝑗,𝑤_𝑗))²max_{1≤𝑗≤𝑛}𝛽(𝑧_𝑗,𝑤_𝑗). (2.12)

Before formulating and proving the main theorem, we give some notations. For any 𝛿∈(0,1), define𝐸^𝑗_𝛿∶={𝑧∈𝔻^𝑛∶|𝜑_𝑗(𝑧)|∨|𝜓_𝑗(𝑧)|>1−𝛿}, (2.13)and we put 𝐸_𝛿=∪^𝑛_𝑗=1𝐸^𝑗_𝛿, where ∨ means the maximum of two real numbers.Lemma 2.3 (see [34]). Let {𝑧_𝑛} be a sequence in 𝔻 with |𝑧_𝑛|→1 as 𝑛→∞. Then, there is a subsequence {𝑧_𝑛𝑗} of {𝑧_𝑛}, a positive number 𝑀, and a sequence of functions 𝑓_𝑚∈𝐻^∞(𝔻) such that(i)𝑓_𝑚(𝑧_𝑛𝑗)=𝛿^𝑗_𝑚,(ii)∑_𝑚|𝑓_𝑚(𝑧)|≤𝑀<∞forany𝑧∈𝔻,(the symbol 𝛿^𝑗_𝑚 is equal to 1 if 𝑚=𝑗 and 0, otherwise.)Lemma 2.4. Let Ω be a domain in ℂ^𝑛, 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, thensup_𝑧∈𝐾|𝜕𝑓𝜕𝑧_𝑗√(𝑧)|≤𝑛𝜂sup_𝑧∈𝐺|𝑓(𝑧)|, (2.14)for each 𝑗∈{1,…,𝑛}.Proof. Since 𝜂=dist(𝐾,𝜕𝐺)>0 for any 𝑎∈𝐾, the polydisc𝑃_𝑎={(𝑧₁,…,𝑧_𝑛)∈ℂ^𝑛∶|𝑧_𝑗−𝑎_𝑗𝜂|<√𝑛,𝑗=1,…,𝑛} (2.15)is contained in 𝐺. Using Cauchy's inequality, we have|𝜕𝑓𝜕𝑧_𝑗√(𝑎)|≤𝑛𝜂sup_{𝑧∈𝜕0𝑃𝑎}√|𝑓(𝑧)|≤𝑛𝜂sup_𝑧∈𝐺|𝑓(𝑧)|, (2.16)as desired (where 𝜕₀𝑃_𝑎 is the distinguished boundary of 𝑃_𝑎).Lemma 2.5. For fixed 0<𝛿<1, let 𝐹_𝛿={𝑧∈𝔻^𝑛∶max_{1≤𝑗≤𝑛}|𝑧_𝑗|>1−𝛿}. Then,lim_𝑟→1sup_{‖𝑓‖∞=1}sup_{𝑧∈𝐹𝑐𝛿}|𝑓(𝑧)−𝑓(𝑟𝑧)|=0 (2.17)for any 𝑓 in the unit ball of 𝐻^∞(𝔻^𝑛) (where 𝐹^𝑐_𝛿 denotes the complement of 𝐹_𝛿 relative to 𝔻^𝑛).Proof. We havesup_{𝑧∈𝐹𝑐𝛿}|𝑓(𝑧)−𝑓(𝑟𝑧)|=sup_{𝑧∈𝐹𝑐𝛿}|^𝑛_𝑗=1(𝑓(𝑟𝑧₁,𝑟𝑧₂,…,𝑟𝑧_𝑗−1,𝑧_𝑗,…,𝑧_𝑛)−𝑓(𝑧₁,𝑟𝑧₂,…,𝑟𝑧_𝑗,𝑧_𝑗+1,…,𝑧_𝑛))|≤sup_{𝑧∈𝐹𝑐𝛿}^𝑛_𝑗=1¹_𝑟|𝑧_𝑗||𝜕𝑓𝜕𝑧_𝑗(𝑟𝑧₁,…,𝑟𝑧_𝑗−1,𝑡𝑧_𝑗,𝑧_𝑗+1,…,𝑧_𝑛)|𝑑𝑡≤(1−𝑟)sup_{𝑧∈𝐹𝑐𝛿}^𝑛_𝑗=1|𝜕𝑓𝜕𝑧_𝑗(𝑧)|. (2.18)Consider 𝐹^𝑐_𝛿/2, then 𝐹^𝑐_𝛿⊂𝐹^𝑐_𝛿/2 and dist(𝐹^𝑐_𝛿/2,𝜕𝔻^𝑛)=𝛿/2.
From Lemma 2.4, we have that for each 𝑗∈{1,…,𝑛}sup_{𝑧∈𝐹𝑐𝛿}|𝜕𝑓𝜕𝑧_𝑗2√(𝑧)|≤𝑛𝛿sup_{𝑧∈𝐹𝑐𝛿/2}|𝑓(𝑧)|. (2.19)From this and (2.18), it follows thatsup_{𝑧∈𝐹𝑐𝛿}√|𝑓(𝑧)−𝑓(𝑟𝑧)|≤2(1−𝑟)𝑛𝑛𝛿‖𝑓‖_∞. (2.20)
Taking the supremum in (2.20) over the unit ball in 𝐻^∞(𝔻^𝑛), 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. Let 𝜑,𝜓∶𝔻^𝑛→𝔻^𝑛 and 𝐶_𝜑−𝐶_𝜓∶𝐻^∞(𝔻^𝑛)→𝐻^∞(𝔻^𝑛). Then, 1𝑀Ψ≤‖𝐶_𝜑−𝐶_𝜓‖_𝑒≤4−41−Ψ²Ψ, (3.1)where Ψ∶=max_{1≤𝑘≤𝑛}lim_𝛿→0sup_{𝑧∈𝐸𝑘𝛿}max_{1≤𝑗≤𝑛}𝛽(𝜑_𝑗(𝑧),𝜓_𝑗(𝑧)) and 𝑀 is a positive constant.

Proof. First, we consider the upper estimate. For fixed 𝑟∈(0,1), it is easy to check that both 𝐶_𝑟𝜑 and 𝐶_𝑟𝜓 are compact operators. Therefore,‖𝐶_𝜑−𝐶_𝜓‖_𝑒≤‖𝐶_𝜑−𝐶_𝜓−𝐶_𝑟𝜑+𝐶_𝑟𝜓‖. (3.2)Now, for any 0<𝛿<1,‖𝐶_𝜑−𝐶_𝜓−𝐶_𝑟𝜑+𝐶_𝑟𝜓‖=sup_{‖𝑓‖∞=1}‖(𝐶_𝜑−𝐶_𝜓−𝐶_𝑟𝜑+𝐶_𝑟𝜓)𝑓‖_∞=sup_{‖𝑓‖∞=1}sup_{𝑧∈𝔻𝑛}|𝑓(𝜑(𝑧))−𝑓(𝜓(𝑧))−𝑓(𝑟𝜑(𝑧))+𝑓(𝑟𝜓(𝑧))|≤sup_{‖𝑓‖∞=1}sup_{𝑧∈𝐸𝛿}|𝑓(𝜑(𝑧))−𝑓(𝜓(𝑧))−𝑓(𝑟𝜑(𝑧))+𝑓(𝑟𝜓(𝑧))|+sup_{‖𝑓‖∞=1}sup_{𝑧∈𝐸𝑐𝛿}|𝑓(𝜑(𝑧))−𝑓(𝜓(𝑧))−𝑓(𝑟𝜑(𝑧))+𝑓(𝑟𝜓(𝑧))|=𝐼₁+𝐼₂. (3.3)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 function 𝜙(𝑧)=𝑟𝑧, 𝑟∈(0,1), and by the monotony of the function 𝑓(𝑥)=(2−2√1−𝑥²)/𝑥, we obtain𝐼₁≤sup_{‖𝑓‖∞=1}sup_{𝑧∈𝐸𝛿}(|𝑓(𝜑(𝑧))−𝑓(𝜓(𝑧))|+|−𝑓(𝑟𝜑(𝑧))+𝑓(𝑟𝜓(𝑧))|)=sup_{𝑧∈𝐸𝛿}sup_{‖𝑓‖∞=1}(|𝑓(𝜑(𝑧))−𝑓(𝜓(𝑧))|+|−𝑓(𝑟𝜑(𝑧))+𝑓(𝑟𝜓(𝑧))|)≤sup_{𝑧∈𝐸𝛿}(𝑑_𝔻^𝑛(𝜑(𝑧),𝜓(𝑧))+𝑑_𝔻^𝑛(𝑟𝜑(𝑧),𝑟𝜓(𝑧)))≤2sup_{𝑧∈𝐸𝛿}2−2(1−max_{1≤𝑗≤𝑛}(𝛽(𝜑_𝑗(𝑧),𝜓_𝑗(𝑧)))²)^1/2max_{1≤𝑗≤𝑛}𝛽(𝜑_𝑗(𝑧),𝜓_𝑗=(𝑧))4−4(1−sup_{𝑧∈𝐸𝛿}max_{1≤𝑗≤𝑛}(𝛽(𝜑_𝑗(𝑧),𝜓_𝑗(𝑧)))²)^1/2sup_{𝑧∈𝐸𝛿}max_{1≤𝑗≤𝑛}𝛽(𝜑_𝑗(𝑧),𝜓_𝑗≤(𝑧))4−4(1−max_{1≤𝑘≤𝑛}sup_{𝑧∈𝐸𝑘𝛿}max_{1≤𝑗≤𝑛}(𝛽(𝜑_𝑗(𝑧),𝜓_𝑗(𝑧)))²)^1/2max_{1≤𝑘≤𝑛}sup_{𝑧∈𝐸𝑘𝛿}max_{1≤𝑗≤𝑛}𝛽(𝜑_𝑗(𝑧),𝜓_𝑗.(𝑧)) (3.4)(3.5)By direct calculation, it is easy to check thatlim_𝛿→0max_{1≤𝑘≤𝑛}sup_{𝑧∈𝐸𝑘𝛿}max_{1≤𝑗≤𝑛}𝛽(𝜑_𝑗(𝑧),𝜓_𝑗(𝑧))=max_{1≤𝑘≤𝑛}lim_𝛿→0sup_{𝑧∈𝐸𝑘𝛿}max_{1≤𝑗≤𝑛}𝛽(𝜑_𝑗(𝑧),𝜓_𝑗(𝑧)). (3.6)From which, and letting 𝛿→0 in (3.5), the upper estimate in (3.1) follows.
Now, we turn to the lower estimate.
Let𝑎_𝑘=lim_𝛿→0sup_{𝑧∈𝐸𝑘𝛿}max_{1≤𝑗≤𝑛}𝛽(𝜑_𝑗(𝑧),𝜓_𝑗(𝑧)),𝑘=1,…,𝑛. (3.7)If we set 𝛿_𝑚=1/𝑚, then 𝛿_𝑚→0 as 𝑚→∞, and there exists 𝑧_𝑚∈𝐸^𝑗_𝛿𝑚 and some 𝑗 such thatlim_𝑚→∞𝛽(𝜑_𝑗(𝑧_𝑚),𝜓_𝑗(𝑧_𝑚))=𝑎_𝑘. (3.8)
Since 𝑧_𝑚∈𝐸^𝑗_𝛿𝑚 and 𝛿_𝑚→0, we have |𝜑_𝑗(𝑧_𝑚)|→1 or |𝜙_𝑗(𝑧_𝑚)|→1. Without loss of generality, we can assume |𝜑_𝑗(𝑧_𝑚)|→1. Let 𝑤_𝑚=𝜑_𝑗(𝑧_𝑚), by Lemma 2.3, we have that there is a subsequence of 𝑤_𝑚 (we may denote it again by 𝑤_𝑚), a positive number 𝑀, and a sequence of functions 𝑓_𝑚∈𝐻^∞(𝔻) such that(i)𝑓_𝑚(𝑤_𝑘)=𝛿^𝑘_𝑚,(ii)∑_𝑚|𝑓_𝑚(𝑧)|≤𝑀<∞forany𝑧∈𝔻.
Now, for any 𝑧∈𝔻^𝑛, we define ^0𝑥0𝑓^02𝑑𝑐_𝑚(𝑧)∶=𝑓_𝑚(𝑧^𝑗), where 𝑧^𝑗 is the 𝑗th component of 𝑧, then ∑_𝑚|𝑓^{0𝑥002𝑑𝑐}_𝑚(𝑧)|≤𝑀<∞.
Next we claim that ^0𝑥0𝑓^02𝑑𝑐_𝑚 converge weakly to 0. Let 𝜆∈𝐻^∞(𝔻^𝑛)^∗. For any natural 𝑁, there exist some unimodular sequences 𝛼_𝑚 such that^𝑁_𝑚=0|𝜆(𝑓^{0𝑥002𝑑𝑐}_𝑚)|=^𝑁_𝑚=0𝛼_𝑚𝜆(𝑓^{0𝑥002𝑑𝑐}_𝑚)=𝜆(^𝑁_𝑚=0𝛼𝑓_𝑚^{0𝑥002𝑑𝑐}_𝑚)≤‖𝜆‖‖^𝑁_𝑚=0𝛼𝑓_𝑚^{0𝑥002𝑑𝑐}_𝑚‖_∞≤‖𝜆‖𝑀. (3.9)Thus, 𝜆(𝑓^{0𝑥002𝑑𝑐}_𝑚)→0 as 𝑚→∞, that is, ^0𝑥0𝑓^02𝑑𝑐_𝑚 converge weakly to 0.
Set𝑔_𝑚(𝑧)=^{0𝑥002𝑑𝑐}𝑓_𝑚(𝑧)𝑀𝑧^𝑗−𝜓_𝑗(𝑧_𝑚)1−𝜓_𝑗(𝑧_𝑚)𝑧^𝑗,𝑚∈ℕ. (3.10)Then, ‖𝑔_𝑚‖_∞≤1 and similarly to ^0𝑥0𝑓^02𝑑𝑐_𝑚, it is easy to see that 𝑔_𝑚 converge weakly to 0. Thus, for any compact operator 𝐾, we have ‖𝐾𝑔_𝑚‖_∞→0 as 𝑚→∞.
Now, we have𝐽=‖𝐶_𝜑−𝐶_𝜓−𝐾‖≥limsup_𝑚→∞‖(𝐶_𝜑−𝐶_𝜓−𝐾)𝑔_𝑚‖_∞≥limsup_𝑚→∞(‖(𝐶_𝜑−𝐶_𝜓)𝑔_𝑚‖_∞−‖𝐾𝑔_𝑚‖_∞)=limsup_𝑚→∞sup_{𝑧∈𝔻𝑛}|𝑔_𝑚(𝜑(𝑧))−𝑔_𝑚≥1(𝜓(𝑧))|𝑀limsup_𝑚→∞sup_{𝑧∈𝔻𝑛}|𝜑_𝑗(𝑧)−𝜓_𝑗(𝑧_𝑚)1−𝜓_𝑗(𝑧_𝑚)𝜑_𝑗(𝑧)^{0𝑥002𝑑𝑐}𝑓_𝑚𝜓(𝜑(𝑧))−^𝑗(𝑧)−𝜓_𝑗(𝑧_𝑚)1−𝜓_𝑗(𝑧_𝑚)𝜓_𝑗(𝑧)^{0𝑥002𝑑𝑐}𝑓_𝑚≥1(𝜓(𝑧))|𝑀limsup_𝑚→∞|𝜑_𝑗(𝑧_𝑚)−𝜓_𝑗(𝑧_𝑚)||1−𝜓_𝑗(𝑧_𝑚)𝜑_𝑗(𝑧_𝑚=1)|𝑀lim_𝑚→∞𝛽(𝜑_𝑗(𝑧_𝑚),𝜓_𝑗(𝑧_𝑚=1))𝑀𝑎_𝑘. (3.11)
Then, we have‖𝐶_𝜑−𝐶_𝜓‖_𝑒≥1𝑀max_{1≤𝑘≤𝑛}lim_𝛿→0sup_{𝑧∈𝐸𝑘𝛿}max_{1≤𝑗≤𝑛}𝛽(𝜑_𝑗(𝑧),𝜓_𝑗(𝑧)), (3.12)finishing the proof of the theorem.

Corollary 3.2. The operator 𝐶_𝜑−𝐶_𝜓 is compact if and only if max_{1≤𝑘≤𝑛}lim_𝛿→0sup_{𝑧∈𝐸𝑘𝛿}max_{1≤𝑗≤𝑛}𝛽(𝜑_𝑗(𝑧),𝜓_𝑗(𝑧))=0. (3.13)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. Let 𝑛=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 thatmax_1≤𝑘≤2lim_𝛿→0sup_{𝑧∈𝐸𝑘𝛿}max_1≤𝑗≤2𝛽(𝜑_𝑗(𝑧),𝜓_𝑗1(𝑧))=5>0; (3.14)so by Corollary 3.2, 𝐶_𝜑−𝐶_𝜓 is not compact.

Example 3.4. Let 𝑛=2, 𝑝>1, 0<𝑐≤1, 𝜑(𝑧)=((𝑧₁+1)/2,(1/2)𝑧₂), and𝑧𝜓(𝑧)=(¹+12+𝑐(1−𝑧₁2)^𝑝,12𝑧₂), (3.15)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, max_1≤𝑗≤2𝛽(𝜑_𝑗(𝑧),𝜓_𝑗(𝑧))=𝛽(𝜑₁(𝑧),𝜓₁(𝑧)). By Corollary 3.2 and the proof of Example 1 of [25], we have, for these 𝑐,(1) if 1<𝑝≤2, then 𝐶_𝜑−𝐶_𝜓 is noncompact;(2) if 2<𝑝<∞, then 𝐶_𝜑−𝐶_𝜓 is compact.

Acknowledgments

The authors would like to express their sincere thanks to the editor S. Stević and referees for their helpful comments and suggestions which improved the manuscript considerably. Moreover, this work was supported in part by the National Natural Science Foundation of China (Grant nos. 10671141 and 10371091).