Potential Anal (2014) 40:81–102 DOI 10.1007/s11118-013-9343-z
Compact Differences of Composition Operators on the Bergman Spaces Over the Ball Boo Rim Choe · Hyungwoon Koo · Inyoung Park
Received: 24 December 2012 / Accepted: 4 April 2013 / Published online: 21 April 2013 © Springer Science+Business Media Dordrecht 2013
Abstract The compact differences of composition operators acting on the weighted L2 -Bergman space over the unit disk is characterized by the angular derivative cancellation property and due to Moorhouse. In this paper we extend Moorhouse’s characterization, as well as some related results, to the ball and, at the same time, to the weighted L p -Bergman space for the full range of p. Keywords Composition operator · Compact difference · Compact combination · Weighted Bergman space · Ball Mathematics Subject Classifications (2010) Primary 47B33; Secondary 32A35 · 32A36
1 Introduction For a fixed positive integer n, let B = Bn be the open unit ball of the complex n-space Cn . Let ϕ be a holomorphic self-map of B. Such a holomorphic map ϕ induces the composition operator Cϕ : H(B) → H(B) defined by Cϕ f := f ◦ ϕ
H. Koo was supported by KRF of Korea (2012R1A1A2000705). B. R. Choe · H. Koo Department of Mathematics, Korea University, Seoul 136-701, Korea B. R. Choe e-mail:
[email protected] H. Koo e-mail:
[email protected] I. Park (B) BK21-Mathematical Sciences Division, Seoul National University, Seoul 151-747, Korea e-mail:
[email protected]
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where H(B) is the class of all holomorphic functions on B. We refer to [3] and [15] for various aspects on the theory of composition operators acting on subclasses of H(B). While quite an extensive study on the one-variable theory of composition operators has been established during the past four decades, the several-variable theory has been relatively less known. In this paper we study compact differences of composition operators and related properties on the weighted Bergman spaces over the ball. We first recall our function spaces to work on. For α > −1, let dv be the normalized volume measure on B and put α dvα (z) := cα 1 − |z|2 dv(z) where the constant cα > 0 is chosen so that vα (B) = 1. Now, given 0 < p < ∞, p the α-weighted Bergman space Aα (B) is the space of all f ∈ H(B) such that the “norm” f
p Aα
:=
1/ p | f (z)| dvα (z) p
B p
is finite. As is well known, the space Aα (B) equipped with the norm above is a Banach space for 1 ≤ p < ∞. On the other hand, it is a complete metric space for p 0 < p < 1 with respect the translation-invariant metric ( f, g) → f − g A p . α On the unit disk D := B1 , as a simple consequence of Littlewood’s Subordination p Principle, Cϕ is bounded on each of the weighted Bergman spaces Aα (D). Also, the compactness characterization for Cϕ on those spaces has been long known by MacCluer and Shapiro [11] by the non-existence of angular derivatives of the inducing function. Later Zhu [17] extended such angular-derivative characterization to the balls; see Lemma 4.2(2). Zhu’s result, however, applies only when composition operators under consideration are known to be already bounded on a smaller weighted Bergman space; the difficulties on the weighted Bergman spaces over the multi-dimensional balls are that, unlike the one-variable case, composition operators are not always bounded (see [9]) and that, other than the Carleson measure characterization, there is not a satisfactory characterization for the boundedness of a composition operator. Recently, Moorhouse [12] characterized compact differences of composition operators acting on A2α (D) by the angular derivative cancellation property. More precisely, she showed that, for α > −1 and holomorphic self-maps ϕ and ψ of D, Cϕ − Cψ is compact on A2α (D) if and only if
1 − |z|2 1 − |z|2 lim d(z) + |z|→1 1 − |ϕ(z)|2 1 − |ψ(z)|2
=0
(1.1)
where the cancellation factor d(z) is the pseudo-hyperbolic distance (see Section 2.2) between ϕ(z) and ψ(z). Roughly speaking, Eq. 1.1 means that appropriate cancellation should occur at boundary points where angular derivatives of the inducing functions exist. More recently, Kriete and Moorhouse [10] extended their
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study to general linear combinations. Also, the characterization (1.1), as well as some of related results in [10] and [12], has been extended by the current authors [1] to the polydisks. All of these studies were restricted to the case p = 2 so that the proofs extensively depended on Hilbert space methods. The purpose of this paper is to extend Moorhouse’s characterization (as well as some related properties) to the ball and, more significantly, to the full range of p. While the main scheme of our proofs is adapted from [10] and [12], our work with the full range of p (especially for the range 0 < p < 1) requires certain new approach and substantial amount of extra works. Our approach is to estimate directly by employing suitable test functions for the necessity and by utilizing subharmonicity for the sufficiency. Our first result (Theorem 3.5) is a lower estimate for the essential norms of general linear combinations of composition operators; such an estimate immediately yields the coefficient relation (Corollary 3.6) necessary for a compact combination. In case p = 2 a similar lower estimate was first noticed over the disk by Kriete and Moorhouse [10] and recently extended to the ball by Jiang and Ouyang [7]. While our estimate for general p is a bit weaker than those in [10] and [7], it yields the same coefficient relation. Our second result (Theorem 4.6) is a characterization for compact combinations of two composition operators. As a consequence we obtain the ball analogue (Theorem 4.7) of Moorhouse’s characterization (1.1) for compact differences. We remark that the Hardy space analogue for the compact difference has been also known over the disk, but with the restriction 1 ≤ p < ∞; see [13]. The following remark is in order in conjunction with our characterization for compact differences. Remark In [8] the authors prove Theorem 4.7 but we believe that their proofs contain some serious problems. In fact they worked only with p = 2 by claiming that the property of being a compact difference is independent of p. The reason they provided was that the compact difference could be characterized in terms of Carleson measure by the change of variables. But such an argument based on the change of variables works only for a single composition operator. Additionally, in their proof even with p = 2, we notice that the sufficiency part contains a non-fixable error. More precisely, the uniform convergence needed for the assertion (2.8) in the proof of [8, Theorem 1.3] is not guaranteed. Our final result (Theorem 5.3) is a characterization for compact combinations of arbitrarily many composition operators, but with certain additional assumptions on the angular derivative sets. As a consequence we obtain a characterization (Theorem 5.4) for a composition operator to be equal modulo compact operators to a sum of composition operators. Constants In the rest of the paper we use the same letter C to denote various positive constants which may change at each occurrence. Variables (other than n) indicating the dependency of constants C will be often specified in the parenthesis. We use the notation X Y or Y X for nonnegative quantities X and Y to mean X ≤ CY for some inessential constant C > 0. Similarly, we use the notation X ≈ Y if both X Y and Y X hold.
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2 Background In this section we recall some basic facts which will be used in later sections. 2.1 Compact Operator It seems better to clarify the notion of compact operators, since when 0 < p < 1 the spaces under consideration are not Banach spaces. Suppose X and Y are topological vector spaces whose topologies are induced by complete metrics. A continuous linear operator T : X → Y is said to be compact if T maps every bounded set in X onto a relatively compact set in Y. Due to the metric topology of Y, T will be compact if and only if the image of every bounded sequence in X has a subsequence that converges in Y. We have the following convenient compactness criterion for a linear combination of composition operators acting on the weighted Bergman spaces. Lemma 2.1 Let α > −1 and 0 < p < ∞. Let T be a linear combination of compop p sition operators. Then T is compact on Aα (B) if and only if T fk → 0 in Aα (B) p for any bounded sequence { fk } in Aα (B) such that fk → 0 uniformly on compact subsets of B. A proof can be found in [3, Proposition 3.11] for a single composition operator and it can be easily modified for a linear combination. 2.2 Pseudo-hyperbolic Distance The pseudo-hyperbolic distance between z, w ∈ B is given by ρ(z, w) := |σz (w)| where σz is the involutive automorphism of B that exchanges 0 and z. More explicitly, we have 1 − |z|2 1 − |w|2 2 1 − ρ(z, w) = . (2.1) |1 − z, w |2 Here, and in what follows, z, w denotes the Hermitian inner product of z, w ∈ Cn . The pseudo-hyperbolic ball with center z ∈ B and radius r ∈ (0, 1) is defined by Er (z) := {w ∈ B : ρ(z, w) < r}. Clearly, Er (0) = rB. It is well known that ρ is invariant under automorphisms on B. So, Er (z) = σz (rB). Also, it is well known that, given 0 < r < 1 and α > −1, we have 1 − |z|2 ≈ 1 − |w|2 ,
w ∈ Er (z)
(2.2)
and n+1+α ; vα [Er (z)] ≈ 1 − |z|2 the constants suppressed in these estimates depend only on r, α and n.
(2.3)
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Given 0 < r < 1 and α > −1, we will use the submean value type inequality C | f | p dvα , z∈B (2.4) | f (z)| p ≤ n+α+1 Er (z) 1 − |z|2 for all f ∈ H(B), 0 < p < ∞ and for some constant C = C(α, r) > 0. All the details for the statements above can be found in [17, Sections 1.4 and 2.4]. In most cases we will work with r = 1/2. So, we put E(z) := E1/2 (z) for brevity. 2.3 Carleson Measure Let α > −1 and μ be a positive finite Borel measure on B. For 0 < r < 1 and 0 < p < ∞, it is well known (see [16, Section 2.4]) that the embedding Aαp (B) ⊂ L p (dμ) is bounded ⇐⇒ sup
μ[Er (z)] −1 and 0 < p < ∞, we will use the norm equivalence
p
∇ f (z) dvα (z) ≈ | f (z) − f (0)| p dvα (z) (2.10) B
B
for f ∈ H(B). All the details for the statements above can be found in [16, Section 2.3]. 2.5 The Julia-Carathéodory Theorem The well-known Julia-Carathéodory Theorem on the disk has a natural extension to the ball. To state it we first recall some terminology. Let S be the unit sphere, the boundary of B. Given ζ ∈ S, a continuous function γ : [0, 1) → B with limt→1 γ (t) = ζ is said to be a restricted ζ -curve if |γ (t) − γ (t), ζ ζ |2 =0 t→1 1 − | γ (t), ζ |2
lim
and
|ζ − γ (t), ζ ζ | < ∞. 0≤t 0.
Theorem 2.2 For a holomorphic self-map ϕ of B and ζ ∈ S, the following three statements are equivalent: (1) ϕ has finite angular derivative at ζ ; (2) dϕ (ζ ) < ∞; (3) ϕ has restricted limit η ∈ S at ζ and Dζ φη (z) = ϕ (z)ζ, η has f inite restricted limit at ζ . Moreover, when these conditions hold, the following statements hold: (4) (5) (6)
Dζ φη has restricted limit at ζ with Dζ φη (ζ ) = dϕ (ζ ). Aϕ (ζ ) = dϕ (ζ ). ϕη⊥ (z) 1− z,ζ
has restricted limit 0 at ζ for any η⊥ ∈ S orthogonal to η.
For a proof of the equivalences of (1)–(3) and property (4), see [3, Theorem 2.81]. Properties (5) and (6) are proved in [14, Theorem 8.5.6] under the inessential restriction ϕ(0) = 0 and one may check that the proof goes through without such a restriction. We remark that some additional properties can be found in [14, Theorem 8.5.6].
3 Essential Norm Estimates p
For a linear operator T : Aα (B) → H(B), its essential norm |||T||| Aαp (possibly ∞) is defined by |||T||| Aαp := inf T − K p
where the infimum is taken over all compact operators K on Aα . Of course, the p notation · above stands for the operator norm on Aα (B) and is to be understood as ∞ for an unbounded operator. We introduce some notation. Let ϕ be a holomorphic self-map of B. We denote by F(ϕ) the set of all boundary points at which ϕ has finite angular derivatives. Note from the Julia-Carathéodory Theorem that F(ϕ) = ζ ∈ S : dϕ (ζ ) < ∞ . We put
D(ϕ, ζ ) := (ϕ(ζ ), dϕ (ζ )) ∈ S × R+ ,
ζ ∈ F(ϕ)
(3.1)
where R+ denotes the set of all positive real numbers. Let ϕ1 , . . . , ϕ N be finitely many holomorphic self-maps of B. For ζ ∈ F(ϕi ), we denote by Jζ (i) the set of all indices j for which ζ ∈ F(ϕ j) and D(ϕ j, ζ ) = D(ϕi , ζ ). In case p = 2, it is known that
2
2
N
N
1
a jCϕ j
≥ a j
(3.2)
n+1+α d (ζ )
j=1
ϕi i=1
A2α
j∈Jζ (i)
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for all a1 , . . . , a N ∈ C and ζ ∈ S. Here, the term inside the sum above is to be understood as 0 if ζ ∈ / F(ϕi ). The inequality (3.2) was first proved on the disk by Kriete and Moorhouse [10] and then extended to the balls by Jiang and Ouyang [7]. Note that Eq. 3.2 immediately yields a coefficient relation necessary for compactness. In this section we obtain a similar type of lower estimate for the essential norm for the full range of p. Working with general p, we directly estimate by applying our operator to suitable test functions. Unfortunately, our direct approach results in a slightly weaker lower bound. However, such a weaker lower bound is enough for our purpose, because it provides the same necessary coefficient relation for compactness. We begin with a simple consequence of the Julia-Carathéodory Theorem. In the following lemma ∠ lim denotes the nontangential limit on D; we refer to [3] for the notion of nontangential limits on D. Lemma 3.1 Let ϕ be a holomorphic self-map of B. Then 1 − |ϕ(λζ )|2 = dϕ (ζ ) λ→1 1 − |λ|2
∠ lim for ζ ∈ F(ϕ).
Proof Let ζ ∈ F(ϕ) and put η = ϕ(ζ ). We have by the Julia-Carathéodory Theorem 1 − ϕη (λζ ) 1 − ϕη (λζ ) = ∠ lim = dϕ (ζ ). λ→1 λ→1 1 − λζ, ζ 1−λ
∠ lim
So, the one-variable function λ ∈ D → ϕη (λζ ) has finite angular derivative dϕ (ζ ) at 1. Thus, by the Julia-Carathéodory Theorem on D (see [3, Theorem 2.44]) we have
1 − ϕη (λζ )
= dϕ (ζ ). λ→1 1 − |λ|
∠ lim
Now, for any nontangential approach region ⊂ D with vertex at 1, we have dϕ (ζ ) ≤ lim inf
λ→1,λ∈
1 − |ϕ(λζ )| 1 − |λ|
1 − |ϕ(λζ )| 1 − |λ| λ→1,λ∈
1 − ϕη (λζ )
≤ ∠ lim λ→1 1 − |λ| ≤ lim sup
= dϕ (ζ ). Thus we obtain
∠ lim
λ→1
1 − |ϕ(λζ )| = dϕ (ζ ), 1 − |λ|
which implies the lemma. The proof is complete.
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Also, for M > 1 and ζ ∈ S, we denote by M,ζ the ζ -curve consisting of points λζ such that λ ∈ D with Im λ ≥ 0 and |1 − λ| =
M 1 − |λ|2 . 2
Note that points on the curve M,ζ approaches to ζ non-tangentially over the copy of the unit disk in the complex line through ζ . Thus, M,ζ is a restricted ζ -curve. Proposition 3.2 below is one of the key estimate in [7] for the estimate (3.2). A proof for the second case is implicit in the proof of [7, Theorem 2.1]. But, the first case is outlined only. A detailed proof for the first case is included below for the convenience of the readers. Proposition 3.2 Let ϕ and ψ be holomorphic self-maps of B. Then the equality ⎧ 1 ⎪ ⎪ ⎨ d (ζ ) if ζ ∈ F(ϕ) ∩ F(ψ) and 2 1 − |z| ϕ lim lim = D(ϕ, ζ ) = D(ψ, ζ ) ⎪ M→∞ z→ζ 1 − ϕ(z), ψ(z) ⎪ ⎩ z∈ M,ζ 0 otherwise holds for ζ ∈ S. Proof Let ζ ∈ F(ϕ) ∩ F(ψ) and D(ϕ, ζ ) = D(ψ, ζ ). It is easily checked that dϕ (ζ ) = dV◦ϕ◦U (U ∗ ζ ) for any unitary operators U, V on Cn . Also, the left quantity is invariant under the unitary change of variables. Thus we may assume ζ = e and ϕ(e) = e where e := (1, 0, . . . , 0). Let ϕ j and ψ j be the j-th component functions of ϕ and ψ, respectively. Then a little manipulation yields 1− ψ(z), ϕ(z) 1−|ϕ(z)|2 1−z1 = + 1−|z|2 1−|z|2 1−|z|2 ⎡ ⎤ n 1 1 j 2 j j (z) 1−ϕ ψ |ϕ (z) (z) (z)| (z)ϕ 1−ψ ⎦ × ⎣ϕ 1 (z) − − + 1−z1 1 − z1 1−z 1−z 1 1 j=2 where z1 denote the first component of z. Note from the Julia-Carathéodory Theorem (Theorem 2.2(6)) that the second term in the right-hand side of the above has restricted limit 0 at e. Also, note from Lemma 3.1 that the first term converges to dϕ (e) as z → e along the curve M,e . So, we conclude that the left-hand side of the above converges to dϕ (e) as z → e along the curve M,e . The proof is complete. As an immediate consequence of Lemma 3.1 and Proposition 3.2, we have the following corollary. Corollary 3.3 Let ϕ and ψ be holomorphic self-maps of B. Then the equality 1 − |ϕ(z)|2 1 if ζ ∈ F(ψ) and D(ϕ, ζ ) = D(ψ, ζ ) = lim lim M→∞ z→ζ 1 − ϕ(z), ψ(z) 0 otherwise z∈ M,ζ
holds for ζ ∈ F(ϕ).
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In the following lemma · Lαp denotes the norm for the Lebesgue space L p (dvα ). Lemma 3.4 Let 0 < p < ∞ and α > −1. Fix δ > 0 and put δ/ p 1 − |w|2 gw (z) := , w ∈ B. (1 − z, w )(δ+α+n+1)/ p For 0 < r < 1, let χr be the characteristic function of the annulus B \ rB. Then there is a constant C = C(α, δ) > 0 such that p
p
|||T||| A p ≥ C lim sup χr Tgw L p α
α
|w|→1,r→1
p
for any linear operator T : Aα → H(B). p
Proof We only need to consider bounded linear operators T on Aα . Fix such an p p operator T and let K be a compact operator on Aα . As is well known, gw A p is α bounded by some constant independent of p; see, for example, [16, Theorem 1.12]. Thus we have p
T − K p ≥ C sup (T − K)gw A p
(3.3)
α
w∈B
for some constant C = C(α, δ) > 0. Meanwhile, since K is compact, we can extract a p sequence {w j} in B such that Kgw j converges to some g in Aα . So, when 0 < p < 1, writing g j = gw j for short, we have p
p
(T − K)g j A p ≥ χr (T − K)g j L p α
α
≥
p χr Tg j L p α
≥
p χr Tg j L p α
p
p
− χr (g − Kg j) L p − χr g L p α
− g −
p Kg j A p α
−
α
p χr g L p . α
Note the last two terms of the above tends to 0 as r → 1 and j → ∞. So, we obtain lim sup (T − K)g j Aαp ≥ lim sup χr Tg j Lαp . j→∞
(3.4)
j→∞,r→1
This remains valid for 1 ≤ p < ∞ by a similar argument. Now, we conclude the lemma by Eqs. 3.3 and 3.4. The proof is complete. We are now ready to establish a lower estimate for the essential norm of a general linear combination of composition operators acting on the weighted Bergman spaces. Theorem 3.5 Let 0 < p < ∞ and α > −1. Let ϕ1 , . . . , ϕ N be finitely many holomorphic self-maps of B. Then there is a constant C = C(α) > 0 such that
p
p ⎛
⎞
N
1
⎠ (3.5) a jCϕ j
≥ C max ⎝
a j
n+1+α 1≤i≤N
j=1
dϕi (ζ ) p
Aα
j∈Jζ (i)
/ F(ϕi ), the quantity inside the parenthesis for all ζ ∈ S and a1 , . . . , a N ∈ C. In case ζ ∈ above is to be understood as 0.
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Proof We first introduce some temporary notation. Let a1 , . . . , a N ∈ C be given and put T := Nj=1 a jCϕ j . Fix any index i such that ζ ∈ F(ϕi ). For w ∈ B, put fw := gϕi (w) where g-functions are those introduced in Lemma 3.4 (with any fixed δ > 0). Also, denote by χw the characteristic function of the pseudohyperbolic ball E(w); recall E(w) = E1/2 (w). Recall that ϕi has restricted limit ϕi (ζ ) ∈ S. Thus, in particular, we have |ϕi (w)| → 1 as w → ζ along any M,ζ which is a restricted ζ -curve. So, we have by Lemma 3.4 ⎛
⎞
⎜ p p ⎟ |||T||| A p ≥ C sup ⎝lim sup χw T fw L p ⎠ α
M
w→ζ w∈ M,ζ
α
(3.6)
for some constant C = C(α) > 0. We now estimate the quantity in the parenthesis of Eq. 3.6. Given w ∈ B, we have by Eq. 2.4
p
χw T fw L p = α
|T fw (z)| p dvα (z) (1 − |w|)n+1+α |T fw (w)| p . E(w)
Meanwhile, setting t := n + α + 1 + δ, we have
p
N 2 δ/ p
1 − |ϕ (w)| i
|T fw (w)| p =
aj
t/ p
j=1 (1 − ϕ j(w), ϕi (w) )
t/ p
p
N
1 − |ϕi (w)|2 1
=
aj n+α+1 .
1 −
ϕ (w), ϕ (w) j i
j=1
1 − |ϕi (w)|2 It follows that p χw T fw L p α
N n+1+α t/ p p 2
1 − |ϕ 1 − |w|2 (w)| i
aj .
1 − |ϕ (w)|2 1 − ϕ j(w), ϕi (w) i
j=1
Note that we have by Corollary 3.3
t/ p p p 2
N
1 − |ϕ (w)| i
=
lim lim
aj a j
M→∞ w→ζ
1 −
ϕ (w), ϕ (w) j i
j∈Jζ (i)
w∈ M,ζ j=1 and by Lemma 3.1 lim
w→ζ w∈ M,ζ
1 1 − |w|2 = 2 1 − |ϕi (w)| dϕi (ζ )
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for each M > 1. Accordingly, we obtain p lim lim χw T fw L p α M→∞ w→ζ w∈ M,ζ
1 dϕi (ζ )n+1+α
p
a j
.
j∈Jζ (i)
One may check that the constant suppressed above depends only on n. Combining this estimate with Eq. 3.6, we conclude the theorem. Note that the maximum in Eq. 3.5 can be rephrased as
p ⎛
⎞
⎜
1 ⎟
a j
n+α+1 ⎟ max ⎜
⎝ ⎠. (η,s)∈S×R+
s ζ ∈F(ϕ ) j
D(ϕ ,ζ )=(η,s)
j
This immediately yields the following coefficient relation necessary for compactness. Corollary 3.6 With the notation as in Theorem 3.5, if p Aα (B), then
aj = 0
N j=1
a jCϕ j is compact on some
ζ ∈F(ϕ j ) D (ϕ j ,ζ )=(η,s)
for ζ ∈ S and (η, s) ∈ S × R+ . This coefficient relation yields further consequences for two special types of linear combinations. Proofs are the same as those of the polydisk analogues in [1]. Corollary 3.7 Let ϕ and ψ be holomorphic self-maps B such that F(ϕ) and F(ψ) p are not empty. Let a, b ∈ C \ {0}. If aCϕ + b Cψ is compact on some Aα (B), then the following statements hold: (1) a + b = 0; (2) F(ϕ) = F(ψ); (3) D(ϕ, ζ ) = D(ψ, ζ ) for each ζ ∈ F(ϕ). Corollary 3.8 Let ϕ, ϕ1 , . . . , ϕ N be holomorphic self-maps of B. If Cϕ − p compact on some Aα (B), then the following statements hold:
N j=1
Cϕ j is
(1) F(ϕ1 ), . . . , F(ϕ N ) are pairwise disjoint and F(ϕ) = ∪ Nj=1 F(ϕ j); (2) D(ϕ, ζ ) = D(ϕ j, ζ ) at each ζ ∈ F(ϕ j) for j = 1, . . . , N.
4 Compact Difference In this section we characterize compact combinations of two composition operators and then use it to characterize compact differences of two composition operators. Throughout the section we reserve ϕ and ψ to denote a pair of arbitrary holomorphic self-maps of B. As mentioned in the Introduction, the ball analogue of
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the angular derivative cancellation property (1.1) will be the key condition in our characterization. So, we separate it as 1 − |z|2 1 − |z|2 = 0. + (AC) lim ρ(ϕ(z), ψ(z)) |z|→1 1 − |ϕ(z)|2 1 − |ψ(z)|2 We start with the necessary condition for a combination of two composition operators to be compact. Proposition 4.1 Let α > −1 and 0 < p < ∞. Let a, b ∈ C \ {0}. If aCϕ + b Cψ is p compact on Aα (B), then either one of the following two conditions hold: p
(i) Both Cϕ and Cψ are compact on Aα (B); (ii) a + b = 0 and (AC) holds. p
Proof Suppose that aCϕ + b Cψ is compact on Aα (B). Note that if (i) fails, then both p Cϕ and Cψ must be not compact on Aα (B). So, we assume that both Cϕ and Cψ are p not compact on Aα (B) and show (ii). By Corollary 3.7 we have a + b = 0 and hence we may assume a = −b = 1. Assume (AC) fails. We will derive a contradiction. Since (AC) fails, there exists a sequence {wk } with |wk | → 1 such that 1 − |wk |2 1 − |wk |2 + > 0. lim ρ(ϕ(wk ), ψ(wk )) k→∞ 1 − |ϕ(wk )|2 1 − |ψ(wk )|2 By passing to a subsequence if necessary, we may assume lim ρ(ϕ(wk ), ψ(wk ))
k→∞
1 − |wk |2 >0 1 − |ϕ(wk )|2
with 1 − |ψ(wk )|2 ≥ 1 − |ϕ(wk )|2
(4.1)
1−|wk | for all k. Note that the sequence { 1−|ϕ(w 2 } is bounded by Eq. 2.12. Thus, by passing k )| to another subsequence if necessary, we may further assume 2
lim ρ(ϕ(wk ), ψ(wk )) = c1
k→∞
and
1 − |wk |2 = c2 k→∞ 1 − |ϕ(wk )|2 lim
(4.2)
for some constants c1 , c2 > 0 with c1 ≤ 1. Also, note from Eqs. 4.1 and 2.1 that 1 − ρ(ϕ(wk ), ψ(wk ))2 ≥
1 − |ϕ(wk )|2 |1 − ϕ(wk ), ϕ(wk ) |
2 (4.3)
for all k. Fix δ > 0 and put fk := gϕ(wk ) for each k where g-functions are those introduced in Lemma 3.4. Note that { fk } is p bounded in Aα (B). Using these test functions, we will show (Cϕ − Cψ ) fk → 0 in p Aα (B), which is a contradiction by Lemma 2.1.
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Put t := n + α + 1 + δ for brevity. We have by Eq. 2.4 p | fk ◦ ϕ − fk ◦ ψ| p dvα (Cϕ − Cψ ) fk A p = α
≥
B
E(wk )
| fk ◦ ϕ − fk ◦ ψ| p dvα
n+α+1 1 − |wk |2 | fk (ϕ(wk )) − fk (ψ(wk ))| p n+1+α
t/ p
p 1 − |wk |2 1 − |ϕ(wk )|2
=
1 −
.
1 − |ϕ(wk )|2 1 − ψ(wk ), ϕ(wk ) for all k. It follows from Eq. 4.3 that n+1+α
! p 1 − |wk |2
p 2 t/2 p
− 1 − ρ(ϕ(w ), ψ(w )) (Cϕ − Cψ ) fk A p
1
k k α 1 − |ϕ(wk )|2 for all k. Consequently, we have by Eq. 4.2 " t/2 p # p p 1 − 1 − c21 >0 lim inf (Cϕ − Cψ ) fk A p cn+1+α 2 α
k→∞
p
and thus conclude, as asserted, that (Cϕ − Cψ ) fk → 0 in Aα (B). The proof is complete. We now proceed to the proof of the converse of Proposition 4.1. We first recall the following result taken from [2, Theorem 3.3] and [16, Theorem 11], respectively. Lemma 4.1 Let α > β > −1 and 0 < p, q < ∞. Suppose that Cϕ is bounded on q Aβ (B). Then the following statements hold: p
(1) Cϕ is bounded on Aα (B); p (2) Cϕ is compact on Aα (B) if and only if F(ϕ) = ∅. Given α > −1 and a nonnegative bounded Borel function W on B, put dWα = W dvα . Associated with this measure is the weighted pullback measure d(Wα ◦ ϕ −1 ) defined by ! Wα ◦ ϕ −1 (E) = Wα ϕ −1 (E) for Borel sets E ⊂ B. In case of the disk the following lemma is [12, Lemma 1]. The proof for the ball is similar and included below for the sake of completeness. Lemma 4.3 Let α > β > −1 and 0 < p, q < ∞. Suppose that Cϕ is bounded on q Aβ (B). Let W be a nonnegative bounded Borel function on B. If lim W(z)
|z|→1 p
1 − |z|2 = 0, 1 − |ϕ(z)|2
then Cϕ : Aα (B) → L p (dWα ) is compact.
(4.4)
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Proof As in Eq. 2.7, we have p Cϕ f L p (dWα )
| f ◦ ϕ| dWα =
=
p
B
| f | p d Wα ◦ ϕ −1
B
p p f ∈ Aα (B). We see from the above that Cϕ : Aα (B) only if Wα ◦ ϕ −1 is a compact α-Carleson measure. −1
for all → L p (B, dWα ) is compact if and So, it suffices to show that Wα ◦ ϕ is a compact α-Carleson measure. Note from Lemma 4.2(1) that we may assume 0 < α − β < 1. Also, we may assume that W is bounded by 1. Given z ∈ B, put $ %α−β 1 − |w|2 . W(w) (z) := sup 1 − |ϕ(w)|2 w∈ϕ −1 [E(z)] Note vβ ◦ ϕ −1 β < ∞, because Cϕ is bounded on Aβ (B). Also, note from Eq. 2.2 that q
1 − |ϕ(w)|2 ≈ 1 − |z|2 We thus have
whenever w ∈ ϕ −1 [E(z)].
(4.5)
−1
(Wα ◦ ϕ )[E(z)] = =
ϕ −1 [E(z)]
ϕ −1 [E(z)]
W(w) dvα (w) !α−β W(w)1−α+β W(w) 1 − |w|2 dvβ (w)
≤ (z)
ϕ −1 [E(z)]
1 − |ϕ(w)|2
α−β
dvβ (w).
This, together with Eqs. 2.2 and 2.3, yields α−β & & Wα ◦ ϕ −1 [E(z)] 1 − |z|2 vβ ◦ ϕ −1 [(E(z)] ≤ (z) &vβ ◦ ϕ −1 &β (4.6) (z) vα [E(z)] vα [E(z)] for all z ∈ B. Note that the constant suppressed above is independent of z. Also, note from Eqs. 2.12 and 4.5 that |w| → 1 uniformly in w ∈ ϕ −1 [E(z)] as |z| → 1. Now, assuming that Eq. 4.4 holds, we have (z) → 0 as |z| → 1 and hence conclude from Eq. 4.6 that Wα ◦ ϕ −1 is a compact α-Carleson measure, as required. The proof is complete. In the case of p = 2 on the disk, the proof in [12] of the converse of Proposition 4.1 utilizes the Fundamental Theorem of Calculus to estimate the size of the left-hand side of the inequality below. On the other hand, in the case of p = 2 on the polydisk, the proof in [1] is carried out via kernel-norm estimates. Such approaches cause certain difficulties for general p. The following estimate is a substitute to overcome those difficulties. Lemma 4.4 Let α > −1, 0 < p < ∞ and 0 < r1 < r2 < 1. Then there is a constant C = C(α, p, r1 , r2 ) > 0 such that ρ p (a, b ) f | p dvα | f (a) − f (b )| p ≤ C |∇ n+α+1 2 Er2 (a) 1 − |a|
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for all f ∈ H(B) and a, b ∈ B with a ∈ Er1 (b ). Proof Let f ∈ H(B). Put dτ (w) := (1 − |w|2 )−(n+1) dv(w), which is the automorphism invariant measure on B; see [16, p. 17]. We first show
p
| f (ξ ) − f (0)| p ≤ C|ξ | p ξ ∈ r1 B (4.7) ∇ f (w) dτ (w), r2 B
for some constant C = C( p, r1 , r2 ) > 0. Let ξ ∈ r1 B. We have | f (ξ ) − f (0)| ≤ |ξ | sup |∇ f (z)|. z∈r1 B
Let r = r2 − r1 . Note Er (z) ⊂ r2 B for z ∈ r1 B. We thus have by Eqs. 2.4 and 2.3 |∇ f (w)| p dv(w) sup |∇ f (z)| p ≤ C1 z∈r1 B
r2 B
for some constant C1 = C1 ( p, r1 , r2 ) > 0. Meanwhile, we have by Eq. 2.8 p 1 − |w|2 |∇ f (w)| p dτ (w) |∇ f (w)| p dv(w) ≤ C2 r2 B
r2 B
≤ C2
p
∇ f (w) dτ (w)
r2 B
for some constantC2 = C2 ( p, r2 ) > 0. Combining these observations, we have Eq. 4.7, as asserted. Now, let b ∈ Er1 (a). Then b = σa (ξ ) where ξ := σa (b ) ∈ r1 B. Thus, applying Eq. 4.7 to f ◦ σa in place of f (recall that σa denotes the automorphism of B that exchanges 0 and a), we obtain
p p
∇( f ◦ σa )(w) dτ (w) | f (b ) − f (a)| ρ (a, b ) r2 B
p
∇ f (w) dτ (w)
= ρ p (a, b ) Er2 (a)
ρ p (a, b ) ≈ n+α+1 1 − |a|2
by Eq. 2.9
p
∇ f (w) dvα (w)
by Eq. 2.2;
Er2 (a)
the constants suppressed above depend only on the allowed parameters. The proof is complete. We are now ready to prove the converse of Proposition 4.1. Note that it is enough to consider differences only. Proposition 4.5 Let α > β > −1 and 0 < p, q < ∞. Suppose that Cϕ and Cψ are q p bounded on Aβ (B). If (AC) holds, then Cϕ − Cψ is compact on Aα (B). p
Proof Assume (AC). Let { fk } be an arbitrary bounded sequence in Aα (B) such that fk → 0 uniformly on compact subsets of B. By Lemma 2.1 it suffices to show that (Cϕ − Cψ ) fk Aαp → 0
(4.8)
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as k → ∞. In order to prove this, given 0 < < 1/3, we put Q := {z ∈ B : ρ(ϕ(z), ψ(z)) < } and
Q := B \ Q.
p
Using these sets, we decompose (Cϕ − Cψ ) fk A p into a sum of two integrals α p | fk ◦ ϕ − fk ◦ ψ| p dvα = + (Cϕ − Cψ ) fk A p = α
B
Q
Q
(4.9)
for each k. We first estimate the second term in the right-hand side of Eq. 4.9. Let χ Q be the characteristic function of Q . Since χ Q ≤ ρ(ϕ, ψ), we have by (AC) 1 − |z|2 1 − |z|2 lim χ Q (z) = 0. + |z|→1 1 − |ϕ(z)|2 1 − |ψ(z)|2 This, together with Lemma 4.3, yields | fk ◦ ϕ| p χ Q dvα + | fk ◦ ψ| p χ Q dvα → 0 Q
B
(4.10)
B
as k → ∞ (with fixed). Next, we estimate the first term in the right-hand side of Eq. 4.9. By Lemma 4.4 (with r1 = 1/3 and r2 = 1/2) and Fubini’s Theorem we have
p
p
∇ fk (w) dvα (w) dvα (z) n+α+1 Q B 1 − |ϕ(z)|2 E(ϕ(z)) (4.11) '
p dvα (z) p
∇ fk (w) = n+α+1 dvα (w). B ϕ −1 [E(w)] 1 − |ϕ(z)|2 Meanwhile, since 1 − |ϕ(z)|2 ≈ 1 − |w|2 for z ∈ ϕ −1 [E(w)] by Eq. 2.2, we see from Eq. 2.3 that the inner integral above is comparable to vα ◦ ϕ −1 [E(w)] , vα [E(w)] which is bounded by vα ◦ ϕ −1 α < ∞ by Lemma 4.2. So, we obtain from Eqs. 4.11 and 2.10 that p vα ◦ ϕ −1 α | fk (w) − fk (0)| p dvα (w) p ; (4.12) Q
B p
the last estimate comes from the boundedness of { fk } in Aα (B). Note that the constants suppressed above are independent of and k. Now, having estimates (4.10) and (4.12), we obtain p
lim sup (Cϕ − Cψ ) fk A p p , k→∞
α
which implies Eq. 4.8, because 0 < < 1/3 is arbitrary. The proof is complete.
Combining Propositions 4.1 and 4.5, we finally have the following characterization for compact linear combinations of two composition operators.
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Theorem 4.6 Let α > β > −1 and 0 < p, q < ∞. Suppose that Cϕ and Cψ are q p bounded on Aβ (B). Let a, b ∈ C \ {0}. Then aCϕ + b Cψ is compact on Aα (B) if and only if either one of the following two conditions holds: p
(i) Both Cϕ and Cψ are compact on Aα (B); (ii) a + b = 0 and (AC) holds. Note from Lemma 4.2(2) that condition (i) above also implies (AC). So, as a consequence of Theorem 4.6, we obtain the following characterization for compact p differences. In the theorem below, note that the compactness of Cϕ − Cψ on Aα (B) is independent of p, because (AC) is. Theorem 4.7 Let α > β > −1 and 0 < p, q < ∞. Suppose that Cϕ and Cψ are q p bounded on Aβ (B). Then Cϕ − Cψ is compact on Aα (B) if and only if (AC) holds. Combining Theorems 4.6 and 4.7, we have the following corollary. Corollary 4.8 Let α > β > −1 and 0 < p, q < ∞. Suppose that Cϕ and Cψ are q p bounded on Aβ (B), but not compact on Aα (B). Let a ∈ C \ {0}. Then Cϕ − aCψ is p p compact on Aα (B) if and only if a = 1 and Cϕ − Cψ is compact on Aα (B). We close the section with another consequence concerning linear fractional composition operators. By a linear fractional self-map on B, we mean a holomorphic self-map ϕ of the form ϕ(z) =
Az + B
z, C + D
where A is an n × n matrix , B, C are vectors in Cn , and D is a complex number. A composition operator induced by a linear fractional self-map of B is naturally referred to as a linear fractional composition operator. It is known that every linear p fractional composition operator is bounded on any Aα (B); see [4]. In case p = 2, the following characterization is independently proved by Heller, MacCluer and Weir [6] and Jiang and Ouyang [7]. So, it remains valid for general p by Theorem 4.7. Corollary 4.9 Let α > −1 and 0 < p < ∞. Let ϕ and ψ be distinct linear fractional p self-maps of B. Then Cϕ − Cψ is compact on Aα (B) if and only if both Cϕ and Cψ are p compact on Aα (B).
5 Compact Combination: A Special Case In this section we characterize compact combinations of a special type, which is motivated by the result of the previous section. As a consequence, we will characterize when a composition operator is equal modulo compact operators to a sum of finitely many composition operators. Throughout the section we reserve ϕ and ϕ1 , . . . , ϕ N to denote a class of finitely many arbitrary holomorphic self-maps of B. Note from Corollary 3.8 that there is a
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necessary condition the angular derivative sets for Cϕ to be equal modulo compact operators to the sum of Cϕ j ’s. We separate it as (M)
F(ϕ1 ), . . . , F(ϕ N ) are pairwise disjoint and F(ϕ) = ∪ Nj=1 F(ϕ j);
the letter M is after Moorhouse [12] who considered this condition for the first time. Under this additional condition, we will characterize compact combinations of the form Cϕ − Nj=1 a jCϕ j . In this setting the angular derivative cancellation property corresponding to (AC) turns out to be 1 − |z|2 1 − |z|2 (ACN ) lim ρ(ϕ(z), ϕ j(z)) = 0, ζ ∈ F(ϕ j), + z→ζ 1 − |ϕ(z)|2 1 − |ϕ j(z)|2 for each j = 1, . . . , N. As in the previous section, we start with the necessary condition. Proposition 5.1 Let α > β > −1 and 0 < p, q < ∞. Suppose that (M) holds and that q p Cϕ , Cϕ1 , . . . , Cϕ N are bounded on Aβ (B), but not compact on Aα (B). Let a1 , . . . , a N ∈ N p C \ {0}. If Cϕ − j=1 a jCϕ j is compact on Aα (B), then a1 = · · · = a N = 1 and (AC N ) holds. Proof For ease of notation put T := Nj=1 a jCϕ j . Assume that Cϕ − T is compact on p Aα (B). By Lemma 4.2(2) F(ϕ j) = ∅ for each j and thus F(ϕ) = ∅ by (M). Thus by (M) and Corollary 3.6 we have a1 = · · · = a N = 1. Assume that (ACN ) fails. We will derive a contradiction. Since (ACN ) fails, there exist ζ ∈ F(ϕi ) for some i and a sequence {wk } ⊂ B such that wk → ζ and 1 − |wk |2 1 − |wk |2 > 0. + lim ρ(ϕ(wk ), ϕi (wk )) k→∞ 1 − |ϕ(wk )|2 1 − |ϕi (wk )|2 Thus, taking a subsequence if necessary as in the proof of Proposition 4.1, we may assume that there are constants c1 , c2 > 0 with c1 ≤ 1 such that lim ρ(ϕ(wk ), ϕi (wk )) = c1
(5.1)
k→∞
and
1 − |wk |2 1 − |wk |2 , lim max k→∞ 1 − |ϕ(wk )|2 1 − |ϕi (wk )|2
= c2 .
(5.2)
In addition, we may assume that either one of the following two properties: (i) 1 − |ϕ(wk )|2 ≤ 1 − |ϕi (wk )|2 for all k; (ii) 1 − |ϕ(wk )|2 ≥ 1 − |ϕi (wk )|2 for all k. In case of (i) we take the same test functions as those in the proof of Proposition 4.1, i.e., fk := gϕ(wk ) for each k where g-functions are those introduced in Lemma 3.4 (with any fixed δ > 0). In case of (ii) our test functions will be the functions above with ϕ replaced by ϕi .
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We only consider the first case and the second case can be handled similarly. Considering the first case, note from Eq. 5.2 1 − |wk |2 = c2 . k→∞ 1 − |ϕ(wk )|2
(5.3)
lim
Also, note ζ ∈ F(ϕ j) for j = i by (M). Thus we have 1 − |wk |2 = 0, k→∞ 1 − |ϕ j (wk )|2 lim
j = i,
(5.4)
by the Julia-Carathéodory Theorem. Now, since a1 = · · · = a N = 1, the same argument as in the proof of Proposition 4.1 yields
t/ p p N 2 n+1+α
2
&
&p | )| 1 − |w 1 − |ϕ(w k k & Cϕ − T fk & p
1 −
Aα 2 1 − |ϕ(wk )| 1 −
ϕ (w ), ϕ(w ) j k k
j=1 where t = n + α + 1 + δ for all k. Meanwhile, we have 2 1 − |ϕ(wk )|2 1 − |ϕ(wk )|2 = [1 − ρ 2 (ϕ(wk ), ϕ j(wk ))] 2 |1 − ϕ j(wk ), ϕ(wk ) | 1 − |ϕ j(wk )|2 1 − |ϕ(wk )|2 1 − |ϕ j(wk )|2 1 − |ϕ(wk )|2 1 − |wk |2 = 1 − |wk |2 1 − |ϕ j(wk )|2 ≤
for all k and j. Note from Eqs. 5.3 and 5.4 that the last expression above tends to 0 as k → ∞ for j = i. So, as in the proof of Proposition 4.1 (with ϕi in place of ψ), we obtain " & &p t/2 p # p lim inf & Cϕ − T fk & Aαp c2n+1+α 1 − 1 − c21 > 0, k→∞
which is a desired contradiction. The proof is complete.
We now prove the converse of Proposition 5.1. Again, we only need to consider the case a1 = · · · = a N = 1. The scheme of our proof is the same as the one for p = 2 in [12] for the disk or [1] for the polydisk except that we use Lemma 4.4. Proposition 5.2 Let α > β > −1 and 0 < p, q < ∞. Suppose that (M) holds and that q Cϕ , Cϕ1 , . . . , Cϕ N are bounded on Aβ (B). If (AC N ) holds, then Cϕ − Nj=1 Cϕ j is p compact on Aα (B). Proof Assume (ACN ). We first introduce some notation. Given an index i = 1, . . . , N, let 1 − |z|2 1 − |z|2 ≥ for all j = i . Di := z ∈ B : 1 − |ϕi (z)|2 1 − |ϕ j(z)|2
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Fix 0 < < 1/3 and put Qi := {z ∈ Di : ρ(ϕ(z), ϕi (z)) < } and
Qi := Di \ Qi .
Finally, χ will denote the characteristic function of the set attached to its subscript. By a straightforward modification of the argument in the proof of [12, Theorem 5] or [1, Lemma 5.3], one can deduce from (M) and (ACN ) that 1 − |z|2 1 − |z|2 + = 0 for all i, j (5.5) lim χ Qi (z) |z|→1 1 − |ϕ(z)|2 1 − |ϕ j(z)|2 and lim χ Qi (z)
|z|→1
1 − |z|2 = 0 whenever i = j. 1 − |ϕ j(z)|2
(5.6)
p
Now, let { fk } be a bounded sequence in Aα (B) such that fk → 0 uniformly on N compact subsets of B. Since B = ∪i=1 Di , we have (Cϕ − T)
p fk A p α
p
N N
= fk ◦ ϕ − fk ◦ ϕ j
dvα ≤ B
j=1
i=1
+ Qi
N
i=1
Qi
.
Note, as in the proof of Proposition 4.5, that the second sum of the above tends to 0 as k → ∞ by Eq. 5.5 and Lemma 4.3. For the i-th term of the first sum, we have
| fk ◦ ϕ − fk ◦ ϕi | p dvα + | fk ◦ ϕ j| p dvα . Qi
Qi
j=i
Qi
Note from Eq. 5.6 and Lemma 4.3 that the second term of the above tends to 0 as k → ∞. So, combining these observations, we obtain p
lim sup (Cϕ − T) fk A p ≤ lim sup k→∞
α
k→∞
N
i=1
| fk ◦ ϕ − fk ◦ ϕi | p dvα . Qi
Finally, we have seen in the proof of Proposition 4.5 that the right-hand side of the above is dominated by some constant (independent of ) times p . So, we conclude (Cϕ − T) fk → 0 as k → ∞, which completes the proof by Lemma 2.1. Combining Propositions 5.1 and 5.2, we finally have the following characterization. Theorem 5.3 Let α > β > −1 and 0 < p, q < ∞. Suppose that (M) holds and that q p Cϕ , Cϕ1 , . . . , Cϕ N are bounded on Aβ (B), but not compact on Aα (B). Let a1 , . . . , a N ∈ N p C \ {0}. Then Cϕ − j=1 a jCϕ j is compact on Aα (B) if and only if a1 = · · · = a N = 1 and (AC N ) holds. As a consequence of Theorem 5.3 and Corollary 3.8, we obtain the following characterization for a composition operator to be equal modulo compact operators to a sum of composition operators. In the theorem below, note that the compactness p of Cϕ − Nj=1 Cϕ j on Aα (B) is independent of p, because (M) and (ACN ) are.
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Theorem 5.4 Let α > β > −1 and 0 < p, q < ∞. Suppose that Cϕ , Cϕ1 , . . . , Cϕ N are q p bounded on Aβ (B). Then Cϕ − Nj=1 Cϕ j is compact on Aα (B) if and only if (M) and (AC N ) hold. Finally, combining Theorems 5.3 and 5.4, we have the following corollary. Corollary 5.5 Let α > β > −1 and 0 < p, q < ∞. Suppose that (M) holds and that q p Cϕ , Cϕ1 , . . . , Cϕ N are bounded on Aβ (B), but not compact on Aα (B). Let a1 , . . . , a N ∈ N p C \ {0}. Then Cϕ − j=1 a jCϕ j is compact on Aα (B) if and only if a1 = · · · = a N = 1 N p and Cϕ − j=1 Cϕ j is compact on Aα (B). References 1. Choe, B.R., Koo, H., Park, I.: Compact differences of composition operators over polydisks. Integr. Equ. Oper. Theory 73, 57–91 (2012) 2. Clahane, D.: Compact composition operators on weighted Bergman spaces of the unit ball. J. Oper. Theory 45, 335–355 (2001) 3. Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions. CRC Press, New York (1995) 4. Cowen, C., MacCluer, B.: Linear frcational maps of the unit ball and their composition operators. Acta Sci. Math. (Szeged) 66, 351–376 (2000) 5. Halmos, P.: Measure Theory. Springer, New York (1974) 6. Heller, K., MacCluer, B., Weir, R.: Compact differences of composition operators in several variables. Integr. Equ. Oper. Theory 69, 247–268 (2011) 7. Jiang, L., Ouyang, C.: Compact differences of composition operators on holomorphic function spaces in the unit ball. Acta Math. Sci. Ser. B 31, 1679–1693 (2011) 8. Khoi, L., Yang, X.: Compact differences of composition operators on Bergman spaces in the unit ball. J. Aust. Math. Soc. 89, 407–418 (2010) 9. Koo, H., Smith, W.: Composition operators induced by smooth self-maps of the unit ball in C N . J. Math. Anal. Appl. 329, 617–633 (2007) 10. Kriete, T., Moorhouse, J.: Linear relations in the Calkin algebra for composition operators. Trans. Am. Math. Soc. 359, 2915–2944 (2007) 11. MacCluer, B., Shapiro, J.H.: Angular derivatives and compact composition operators on the Hardy and Bergman spaces. Canad. J. Math. 38, 878–906 (1986) 12. Moorhouse, J.: Compact differences of composition operators. J. Funct. Anal. 219, 70–92 (2005) 13. Nieminen, P., Saksman, E.: On compactness of the difference of composition operators. J. Math. Anal. Appl. 298, 501–522 (2004) 14. Rudin, W.: Function Theory in the Unit Ball of Cn . Springer, New York (1980) 15. Shapiro, J.H.: Composition Operators and Classical Function Theory. Springer, New York (1993) 16. Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Springer, New York (2005) 17. Zhu, K.: Compact composition operators on Bergman spaces of the unit ball. Houston J. Math. 33, 273–283 (2007)
