Linear combinations of composition operators on H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{\infty }$\end{document} spaces over the unit ball and polydisk

In this paper, we characterize completely the compactness of linear combinations of composition operators acting on the space H∞(BN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{\infty }(\mathbb{B}_{N})$\end{document} of bounded holomorphic functions over the unit ball BN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{B}_{N}$\end{document} from two different aspects. The same problems are also investigated on the space H∞(DN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{\infty }(\mathbb{D}^{N})$\end{document} over the unit polydisk DN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{D}^{N}$\end{document}.


Introduction
Let H be a Banach space of holomorphic functions on a domain G of C N and ϕ be a holomorphic map of G into itself. The composition operator C ϕ is a linear operator defined by Such operators have been investigated mainly in various Banach spaces of holomorphic functions to characterize the operator theoretic behavior of C ϕ by the function theoretic properties of ϕ, see the books [3,15,22].
An area of considerable interest is the topological structure of the set of composition operators acting on a given function space. That work was originally investigated by Berkson [2] in the setting of Hardy-Hilbert space H 2 (D) on the open unit disk D, and then generalized by MacCluer [12] and Shapiro and Sundberg [16]. On the space of all bounded holomorphic functions on D, denoted by H ∞ (D), MacCluer, Ohno, and Zhao [13] studied the topological structure of the set of composition operators and characterized completely compact differences of composition operators. Hosokawa, Izuchi, and Zheng continued this investigation in [7], where they showed that a composition operator that is isolated in the norm topology is also isolated in the essential norm topology. Furthermore, Toews [19] generalized those results to the H ∞ space over the unit ball, and Wolf [21] characterized the boundedness and compactness of differences of composition operators between weighted Banach spaces of holomorphic functions on the unit polydisk.
After these works, many authors contributed to exploring norms and essential norms of differences of composition operators on H ∞ (D), see for example [1,5,6]. Moreover, Gorkin and Mortini [4] estimated norms and essential norms of linear sums of endomorphisms on uniform algebras. Following that, Izuchi and Ohno [9] characterized the compactness of linear combinations of composition operators on H ∞ (D) and computed norms and essential norms of them. In this paper, quite influenced by [9], we investigate to extend those results just mentioned to H ∞ spaces over the unit ball and polydisk. We want also to mention that some related results on difference of composition and weighted composition operators to weighted type spaces can be found in [11] and [18] and in the related references therein.
Recall that the unit ball B N resp. polydisk D N is defined as . . , z N ) ∈ C N : |z| := |z 1 | 2 + · · · + |z N | 2 1/2 < 1 resp., Let H ∞ (B N ) resp. H ∞ (D N ) be the Banach space of all bounded holomorphic functions with the supremum norms over the unit ball B N resp. the unit polydisk D N . Throughout this work, we use the same confusing notation · ∞ standing for the supremum norms over B N or D N , according to the context. This paper is organized as follows. Section 2 includes some background materials needed in the sequel. In Sect. 3 we determine conditions under which linear combinations of composition operators are compact on H ∞ (B N ) and H ∞ (D N ), respectively. One of the main difficult problems of our proof is how to construct suitable test functions.

Preliminaries and definitions
In order to handle linear combinations of composition operators, we need some auxiliary results. For z = (z 1 , . . . , z N ) and w = (w 1 , . . . , w N ) in C N , the inner product of z and w is defined by z, w := z 1 w 1 + · · · + z N w N , and then |z| = z, z 1/2 . For each z ∈ C N , denote by [z] the complex subspace spanned by z. The involutive automorphism of B N that interchanges a and 0 is given by where P a is the projection onto [a] (that is, P 0 = 0, P a (z) := z,a a,a a if a = 0), Q a (z) = (I -P a )(z) is the projection onto [z] ⊥ , and s a := (1a, a ) 1/2 . Clearly, P a (z), a = z, a . For z and w in B N , the pseudo-hyperbolic distance β(z, w) is defined by We also recall the following relations, for example, see [10,19].

Lemma 2.1 For any z and w in
an ellipsoid with center C zλ , where C zλ := (1λ 2 )z 1λ 2 |z| 2 and ρ zλ := 1 -|z| 2 1λ 2 |z| 2 (see [14, page 10] for details). As in [19], observe that is a disk centered at C zλ of radius λρ zλ as follows: The pseudo-hyperbolic distance between two points z and w in D N is defined by where we denote by ρ the pseudo-hyperbolic distance on D, i.e., ρ(a, b) = | a-b 1-ab | for a, b ∈ D. We also define the induced distance for any z and w in D N : The following relation can be obtained by using Schwarz's lemma for the polydisk and the argument in the proof of Lemma 2.2 in [20].
At the end of this section, we give compactness criterions for linear combinations of composition operators. Let X = H ∞ (B N ) (resp. H ∞ (D N )) and G = B N (resp. D N ). Let B(X) be the space of all bounded linear operators from X to X. Then an operator T ∈ B(X) is said to be compact if T(S) is compact in the norm topology in X, where S is the unit sphere of X.
Throughout this paper we use the notation X Y or Y X for nonnegative functions X and Y to mean that there exists C > 0 such that X ≤ CY , where C does not depend on the associated variables. Similarly, we use the notation X ≈ Y if both X Y and Y X hold.

Compactness on H ∞ (B N )
As is well known, every holomorphic self-map ϕ of B N induces a bounded composition We may exclude such trivial ones from our linear sums and assume that sup z∈B N |ϕ i (z)| = 1 for each i throughout this subsection. Denote by Z := Z(ϕ 1 , . . . , ϕ M ) the family of sequences {z (j) } in B N satisfying the following conditions: } is a convergent sequence for every i, k. By our hypothesis, there is {z (j) } ∈ Z, and then we write Then it is easy to see that Under these notations, we can characterize completely the compactness of linear sums of composition operators on H ∞ (B N ) as follows.
Replacing ε by positive numbers ε j tending to zero, we have . Then there exists a sequence {g j } ⊆ H ∞ (B N ) with g j ∞ ≤ 1 such that it converges uniformly to zero on every compact subset of B N , and Then, for some constant ε 0 > 0, we can take z (j) ∈ B N such that |z (j) | → 1, and Considering subsequences of {z (j) }, we may assume that |ϕ i (z (j) )| → α i with α i ≥ 0, as j → ∞ for every i. Also {g j } converges uniformly to zero on every compact subset of B N , so α i = 1 for some i. Now we can assume that {z (j) } ∈ Z. And we get Note that {g j (ϕ i (z (j) ))} is bounded, and then considering a subsequence of {z (j) }, we may assume that g j (ϕ i (z (j) )) → ξ i as j → ∞ for every i. Recall that there is a subset {t 1 , . . . , t } ⊆ I({z (j) }) such that , due to the part (a) of Lemma 2.1 and lim t→0 by the hypothesis i∈I 0 ({z (j) },t) a i = 0. This contradicts (3.2). Hence the proof is complete.
The following corollaries can be obtained immediately from Theorem 3.1.
tends to zero, as L → ∞.
To prove the sufficiency, we continue to use the same notation in the proof of Theorem 3.1. A brief retrospective analysis has proved that the following statements are equivalent: (1) M i=1 a i C ϕ i is compact; (2) lim j→∞ Tf j ∞ = 0 for the functions f j given by (3.1); (3) i∈I 0 ({z (j) },t) a i = 0 for every {z (j) } ∈ Z and t ∈ I({z (j) }).
To end the proof, we first compute Note that goes to 0, because of |ϕ t (z (j) )| → 1, as j → ∞. Clearly, there exists a positive constant δ < 1 such that |ϕ k m (z (j) )| ≤ δ for each k m ∈ {1, . . . , M} \ I 0 ({z (j) }, t), so for j large enough, we have So T is compact, which completes the proof.

Compactness on H ∞ (D N )
Our for each i ∈ I (λ) 0 ({z (j) }, t). Now we get and then i∈I (λ) 0 ({z (j) },t) a i = 0 due to the compactness of M i=1 a i C ϕ i . Thus statement (1) is obtained.
(2) Here we argue by contradiction, and suppose that M i=1 a i C ϕ i is not compact on H ∞ (D N ). Then there exists a sequence {g j } ⊆ H ∞ (D N ) with g j ∞ ≤ 1 such that it converges uniformly to zero on each compact subset of D N , whereas For some constant ε 0 > 0, then we can take z (j) ∈ D N with |z (j) | max → 1 and M i=1 a i g j ϕ i z (j) > ε 0 .
Considering subsequences of {z (j) }, we may assume that ϕ i (z (j) ) → α i as j → ∞ for every i. Also {g j } converges uniformly to zero on each compact subset of D N , so |α i | max = 1 for some i. Now we may say that {z (j) } ∈ N λ=1 Z λ . And we get Note that {g j (ϕ i (z (j) ))} ∞ j=1 is bounded, and then considering a subsequence of {z (j) }, we may assume that g j (ϕ i (z (j) )) → ξ (i) as j → ∞ for every i. Without loss of generality, we may assume that {z (j) } ∈ Z 1 . Recall that there is a subset (z (j) )) → 0 as j → ∞, by the . Then it follows from Lemma 2.2 d) that by the hypothesis i∈I (λ) 0 ({z (j) },t) a i = 0. This contradicts (3.3), which completes the proof.
As an application, the following characterizes compact differences of composition operators.