International Journal of Analysis and Applications ISSN 2291-8639 Volume 12, Number 2 (2016), 87-97 http://www.etamaths.com PROPERTIES OF WEIGHTED COMPOSITION OPERATORS ON SOME WEIGHTED HOLOMORPHIC FUNCTION CLASSES IN THE UNIT BALL A. E. SHAMMAKY AND M. A. BAKHIT∗ Abstract. In this paper, we introduce NK-type spaces of holomorphic functions in the unit ball of Cn by the help of a non-decreasing function K : (0, ∞) → [0, ∞). Several important properties of these spaces in the unit ball are provided. The results are applied to characterize boundedness and compactness of weighted composition operators Wu,φ from NK(B) spaces into Beurling-type classes. We also find the essential norm estimates for Wu,φ from NK(B) spaces into Beurling-type classes. 1. Introduction Through this paper, B is the unit ball of the n-dimensional complex Euclidean space Cn, S is the boundary of B. We denote the class of all holomorphic functions, with the compact-open topology on the unit ball B by H(B). For any z = (z1,z2, . . . ,zn),w = (w1,w2, . . . ,wn) ∈ Cn, the inner product is defined by 〈z,w〉 = z1w1 + . . . + znwn, and write |z| = √ 〈z,z〉. Two quantities Af and Bf, both depending on a function f ∈ H(B), are said to be equivalent, written as Af ≈ Bf, if there exists a finite positive constant M not depending on f, such that 1 M Bf ≤ Af ≤ MBf for every f ∈H(B). If the quantities Af and Bf, are equivalent, then in particular we have Af < ∞ if and only if Bf < ∞. As usual, the letter M will denote a positive constant, possibly different on each occurrence. Given a point a ∈ B, we can associate wit it the following automorphism Φa(z) ∈ Aut(B) : Φa(z) = a−Pa(z) −SaQa(z) 1 −〈z,a〉 , z ∈ B,(1.1) where Sa = √ 1 −|a|2,Pa(z) is the orthogonal projection of Cn on a subspace [a] generated by a, that is Pa(z) = { 0, if a = 0; a〈z,a〉 |a|2 , if a 6= 0, and Qa = I −Pa the projection on orthogonal complement [a] (see, for example,[8] or [10]). The map Φa has the following properties that Φa(0) = a, Φa(a) = 0, Φa = Φ −1 a and 1 −〈Φa(z), Φa(w)〉 = (1 −|a|2)(1 −〈z,w〉) (1 −〈z,a〉)(1 −〈a,w〉) ,(1.2) where z and w are arbitrary points in B. In particular, 1 −|Φa(z)|2 = (1 −|a|2)(1 −|z|2) |1 −〈z,a〉|2 .(1.3) 2010 Mathematics Subject Classification. Primary 30D45, 47B33; Secondary 30B10. Key words and phrases. weighted composition operators; NK spaces; Beurling-type spaces. c©2016 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 87 88 SHAMMAKY AND BAKHIT Let V be the Lebesgue volume measure on Cn, normalized so that V (B) ≡ 1 and σ be the normalized surface measure on S, so that σ(B) ≡ 1. Let dτ(z) = dV (z) (1 −|z|2)n+1 , which is Möbius invariant, that is for any ψ ∈ Aut(B),f ∈ L1(B), we have∫ B f(z)dτ(z) = ∫ B f ◦ψ(z)dτ(z). For a ∈ B, the Möbius invariant Green function in B denoted by G(z,a) = g(Φa(z)) where g(z) is defined by: g(z) = n + 1 2n ∫ 1 |z| (1 − t2)n−1t1−2ndt.(1.4) Let H∞(B) denote the Banach space of bounded functions in H(B) with the norm ‖f‖∞ = sup z∈B |f(z)|. For α > 0, the Beurling-type space (sometimes also called the Bers-type space) H∞α (B) in the unit ball B consists of those functions f ∈H(B) for which ‖f‖H∞α (B) = sup z∈B |f(z)|(1 −|z|)α < ∞. The Bergman space A2(B) consists of those functions f ∈H(B) for which ‖f‖2A2(B) = ∫ B |f(z)|2dV (z) < ∞. Let K : (0,∞) → [0,∞) be a right-continuous, non-decreasing function and is not equal to zero identically. The NK(B) space consists of all functions f ∈H(B) such that ‖f‖2NK(B) = sup a∈B ∫ B |f(z)|2K(G(z,a))dτ(z) < ∞. Moreover, f ∈H(B) is said to belong to NK,0(B) if lim |a|→1 ∫ B |f(z)|2K(G(z,a))dτ(z) = 0. Clearly, if K(t) = tp, then NK(B) = Np(B); since G(z,a) ≈ (1 −|ϕa(z)|2) (see [6]). For K(t) = 1 it gives the Bergman space A2(B). If NK(B) consists of just the constant functions, we say that it is trivial. Several important properties of the NK(B) spaces in the unit disk in the complex plane have been characterized in [1], [4] and [9]. We assume from now that all K : (0,∞) → [0,∞) to appear in this paper are right-continuous, non-decreasing function and not equal to zero identically. Given u ∈ H(B) and φ a holomorphic self-map of B. The weighted composition operator Wu,φ : H(B) →H(B) is defined by Wu,φ(f)(z) = u(z)(f ◦φ)(z), z ∈ B. It is obvious that Wu,φ can be regarded as a generalization of the multiplication operator Muf = u ·f and composition operator Cφf = f◦φ. Weighted composition operators are a general class of operators and appear naturally in the study of isometries on most of the function spaces. Operators of this kind also appear in many branches of analysis; the theory of dynamical systems, semigroups, the theory of operator algebras, the theory of solubility of equations with deviating argument and so on. The behavior of those operators is studied extensively on various spaces of holomorphic functions (see for example [2], [5], [6] and [9]). Recall that the pseudohyperbolic metric in the ball is defined as: ρ(z,w) = |Φw(z)|, z ∈ B.(1.5) It is true metric (see [3]). Also it is easy to verify, in particular, that ρ(0,w) = |w| and ρ(Φa(z),w) = |z|. The following lemma was proved in [6]: PROPERTIES OF WEIGHTED COMPOSITION OPERATORS 89 Lemma 1.1. For z,w ∈ B, if ρ(z,w) ≤ 1 2 . Then (1.6) 1 6 ≤ 1 −|z|2 1 −|w|2 ≤ 6. The following proposition was proved as part of Lemma 2.3 in [6], and hence, we omit the details. Proposition 1.1. For z,w ∈ B, if ρ(z,w) ≤ 1 4 . Then |f(z) −f(w)| ≤ 2 √ n|f(Φa(z))|ρ(z,w). Recall that a linear operator T : X → Y is said to be bounded if there exists a constant M > 0 such that ‖T(f)‖Y ≤ M‖f‖X for all maps f ∈ X. Moreover, T : X → Y is said to be compact if it takes bounded sets in X to sets in Y which have compact closure. For a Complex Banach spaces X and Y of H(B), T : X → Y is compact (respectively weakly compact) if it maps the closed unit ball of X onto a relatively compact (respectively relatively weakly compact) set in Y. In this paper, we introduce NK(B) spaces, in terms of the right continuous and non-decreasing function K : (0,∞) → [0,∞) on the unit ball B. We prove that NK(B) contained in Beurling-type space H∞α (B),α = n+1 2 . A sufficient and necessary condition for NK(B) non-trivial is given. We discus the nesting property of NK(B). We obtain the complete characterizations of the boundedness and compactness of weigted composition operators from NK(B) spaces into Beurling-type classes. We also find the essential norm estimates for these operators. Our results contain the results in the unit disk as particular cases (for example [4], [6] and [9]). 2. NK(B) spaces in the unit ball The following results play an important role in the proof of our main result. They also have their own interest. Proposition 2.1. Let K : (0,∞) → [0,∞) be non-decreasing function. Then NK(B) ⊂ H∞n+1 2 (B). Proof. For a ∈ B, let B 1 2 = {z ∈ B : |z| < 1 2 }. Without loss of generality, assume that K( 3 4 ) > 0. If f ∈NK(B), then ‖f‖2NK(B) ≥ K(3/4) ∫ B 1 2 |f(z)|2dτ(z). By By the subharmonicity of |f(z)|2 and hence by ([7], Theorem 2.1.4), we have |f(0)|2 ≤ 1 V (B 1 2 ) ∫ B 1 2 |f(z)|2dτ(z) = 4n ∫ B 1 2 |f(z)|2dτ(z). Thus |f(0)|2 ≤ 4n K(3/4) ‖f‖2NK(B), f ∈NK(B). For every fixed z ∈ B, we put (2.1) F(w) = (1 −|z|2) n+1 2 |f(Φa(w))| (1 −〈w,z〉)n+1 , w ∈ B, which is clearly F ∈H(B). We can prove that ‖F‖2NK(B) ≤‖f‖ 2 NK(B), and so F ∈NK(B). Then, we have |f(a)|2(1 −|a|2)n+1 = |F(0)|2 ≤ 4n K(3/4) ‖f‖2NK(B), for all z ∈ B, which implies that: ‖f‖2H∞ n+1 2 (B) = sup a∈B |f(a)|2(1 −|a|2)n+1 ≤ 4n K(3/4) ‖f‖2NK(B). That is, NK(B) ⊂ H∞n+1 2 (B). 90 SHAMMAKY AND BAKHIT Proposition 2.2. For z,w ∈ B and f ∈NK(B), we have (2.2) |f(z) −f(w)| ≤ M‖f‖NK(B) max{(1 −|z| 2)− n+1 2 , (1 −|w|2)− n+1 2 }ρ(z,w). Here, M = 6 n+1 2 22(n+1) √ n(3+2 √ 3) K(3/4) . Proof. We consider two cases: Case 1: ρ(z,w) ≥ 1 4 . Since |f(z) −f(w)| ≤ |f(z)| + |f(w)|, by Proposition 2.1, we have min{(1 −|z|2) n+1 2 , (1 −|w|2) n+1 2 }|f(z) −f(w)| ≤ (1 −|z|2) n+1 2 |f(z)| + (1 −|w|2) n+1 2 |f(w)| ≤ 2‖f‖H∞ n+1 2 (B) ≤ 22n+1 K(3/4) ‖f‖2NK(B) ≤ 22n+3ρ(z,w) K(3/4) ‖f‖2NK(B). Which implies that |f(z) −f(w)| ≤ 22n+3 K(3/4) ‖f‖NK(B) max{(1 −|z| 2)− n+1 2 , (1 −|w|2)− n+1 2 }ρ(z,w). Case 2: ρ(z,w) > 1 4 . Take and fix w ∈ B, from ρ(Φa(z),w) = |z| it follows that if z ∈ B 1 2 , (B = B∪S), then ρ(Φa(z),w) < 12. In this case, by Proposition 2.1 and Lemma 1.1, we have |f(Φw(z))| ≤ ‖f‖H∞ n+1 2 (B) (1 −|Φw(z)|2) n+1 2 ≤ 22n+1 K(3/4) ‖f‖2NK(B) (1 −|Φw(z)|2) n+1 2 = 22n+1 K(3/4) ‖f‖2NK(B) (1 −|w|2) n+1 2 [ 1 −|w|2 1 −|Φw(z)|2 ]n+1 2 ≤ 6 n+1 2 22n+1 K(3/4) ‖f‖2NK(B) (1 −|w|2) n+1 2 . By Proposition 1.1, we have |f(z) −f(w)| ≤ 6 n+1 2 22(n+1) √ n(3 + 2 √ 3) K(3/4) ‖f‖2NK(B) (1 −|w|2) n+1 2 ρ(z,w). Combining the results of the two cases yields |f(z) −f(w)| ≤ M‖f‖NK(B) max{(1 −|z| 2)− n+1 2 , (1 −|w|2)− n+1 2 }ρ(z,w), where M = 6 n+1 2 22(n+1) √ n(3+2 √ 3) K(3/4) . Lemma 2.1. For a ∈ B, 0 < δ < 1 and f ∈NK(B), we have (2.3) |f(a) −f(δa)| ≤ M (1 −|a|2) n+1 2 ‖f‖2NK(B). Consequently, for any 0 < r < 1, we have (2.4) sup |a|≤r |f(a) −f(δa)| ≤ M (1 −r2) n+1 2 ‖f‖2NK(B). Here, M is the constant from Proposition 2.2. PROPERTIES OF WEIGHTED COMPOSITION OPERATORS 91 Proof. Proposition 2.2 shows that |f(a) −f(δa)| ≤ M‖f‖NK(B) max{(1 −|a| 2)− n+1 2 , (1 −|δa|2)− n+1 2 }ρ(a,δa). The well-known formula 1 −|ρ(a,δa)|2 = 1 −|Φa(δa)|2 = (1 −|a|2)(1 −|δa|2) |1 −〈a,δa〉|2 , together with simple calculations gives ρ(a,δa) = (1 −|a|2)|a| 1 − δ|a|2 ≤ 1. On the other hand, (1 −|δa|2)− n+1 2 ≤ (1 −|a|2)− n+1 2 . The inequalities in (2.3) now follow. If |a| ≤ r, then 1 −δ|a|2 ≥ 1 −r2. Taking supremum of (2.3) in a yields (2.4). Theorem 2.1. If (2.5) ∫ 1 0 r2n−1 (1 −r2)n+1 K(g(r))dr < ∞, then NK(B) contains all polynomials; otherwise, NK(B) contains only constant functions. Proof. First assume that (2.5) holds. Let f(z) be a polynomial i.e. (there exists a M > 0 such that |f(z)|2 ≤ M for all z ∈ B. Then,∫ B |f(z)|2K(G(z,a))dτ(z) = ∫ B |f(Φa(z))|2K(g(z)) dV (z) (1 −|z|2)n+1 = 2n ∫ 1 0 r2n−1 (1 −r2)n+1 K(g(r))dr ∫ S |f ◦ϕa(rζ)|2dσ(ζ) ≤ 2nM ∫ 1 0 r2n−1 (1 −r2)n+1 K(g(r))dr. Since a is arbitrary, it follows that ‖f‖NK(B) ≤ 2nM ∫ 1 0 r2n−1 (1 −r2)n+1 K(g(r))dr < ∞. Thus, f ∈NK(B) and the first half of the theorem is proved. Now, we assume that the integral in (2.5) is divergent. Let α = (α1, · · · ,αn) is an n-tuple of non-negative integers, |α| = α1 + α2 + · · · + αn ≥ 1, f(z) = zα. Then, we have |f(rζ)|2 = r2|α||ζα|2, and ∫ S |(rζ)α|2dσ(rζ) ≥ r2|α| (n− 1)!α! (n− 1 + |α|)! ≥ Mr2|α|. Thus, ‖f‖NK(B) ≥ nM 22|α|−1 ∫ 1 1 2 r2n−1 (1 −r2)n+1 K(g(r))dr.(2.6) There exists a ∈ B such that f(a) 6= 0, by the subharmonicity of |f ◦ Φa(rζ)|2, ‖f‖NK(B) ≥ 3n 2 |f(a)|2 ∫ 1 2 0 r2n−1 (1 −r2)n+1 K(g(r))dr.(2.7) Combining (2.6) and (2.7), we see that (2.5) implies that ‖f‖NK(B) = ∞. It is proved that f /∈NK(B) and, since α is arbitrary, any non-constant polynomial is not contained in NK(B). We conclude that NK(B) contains only constant functions. The theorem is proved. 92 SHAMMAKY AND BAKHIT Lemma 2.2. For w ∈ B we define the probe function in NK(B) as hw(z) = (1 −|w|2)n+1 (1 −〈z,w〉) 3 2 (n+1) . Suppose that condition (2.5) is satisfied. Then hw ∈NK(B) and ‖hw‖NK(B) ≤ 1. Proof. Trivially hw ∈NK(B). It is also easy to see that ‖hw‖2NK(B) = sup a∈B ∫ B ∣∣∣∣ (1 −|w|2)n+1 (1 −〈z,w〉) 3 2 (n+1) ∣∣∣∣2K(G(z,a))dτ(z) ≤ 1, this by a change of variables and since condition (2.5) is satisfied. Such hw is a normalized reproducing kernel function in the Bergman space A 2(B). Also note that hw(w) = ( 1 1 −|w|2 )n+1 2 , ∀w ∈ B. In Section 4, we will discuss the estimation for the lower bounded of ‖Wu,φ‖e We will make use of weakly convergent sequences in the Bergman space A2(B). The following lemma plays an important role. Lemma 2.3. Suppose {fm}m≥1 ∈ A2(B) is a sequence that converges weakly to zero in A2(B). Then {fm}m≥1 converges weakly to zero in NK(B) as well. Proof. Let Γ ∈NK(B) be a bounded liner functional on NK(B). By the fact that ‖f‖NK(B) ≤‖f‖A2(B), then ‖Γ‖ A2(B) = sup f∈A2(B) |Γ(f)| ‖f‖A2(B) ≤ sup f∈A2(B) |Γ(f)| ‖f‖NK(B) ≤ sup f∈NK(B) |Γ(f)| ‖f‖NK(B) = ‖Γ‖NK(B), which implies Γ is also a bounded linear functional on A2(B). Since fm → 0 weakly in A2(B), we conclude that Γ(fm) → 0. Therefore, fm → 0 weakly in NK(B) as well. Corollary 2.1. Let {wm}m∈N ⊂ B and |wm| → 1 as m → ∞, then {hwm} converges weakly to zero in NK(B). Proof. It is well known that hwm → 0 weakly in A2(B) as m → ∞. Indeed, for any f ∈ A2(B), using the reproducing property, we have 〈f,hwm〉 = (1 −|wm| 2) n+1 2 f(wm), which converges to zero as m → ∞, because the set of polynomials is dense in A2(B) (see [10], Proposition 2.6). The conclusion of the corollary follows immediately from Lemma 2.3. 3. Weighted composition operators from NK(B) into H∞α (B) In this section, we will consider the operator Wu,φ : NK(B) → H∞α (B). Theorem 3.1. Let φ : B → B be a holomorphic mapping and u ∈ H(B). For 0 < α < ∞, then Wu,φ : H ∞ α (B) →NK(B) is a bounded operator if and only if (3.1) sup a∈B |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 < ∞. Proof. First assume that condition (3.1) holds, by Proposition 1.1, we have ‖Wu,φ(f)‖H∞α (B) = sup z∈B |u(z)||f(φ(z))|(1 −|z|2)α ≤ ‖f‖H∞ n+1 2 sup z∈B |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ≤ C‖f‖NK(B). PROPERTIES OF WEIGHTED COMPOSITION OPERATORS 93 This implies that Wu,φ : H ∞ α (B) →NK(B) is a bounded operator. Conversely, assume that Wu,φ : NK(B) → H∞α (B) is bounded, then ‖Wu,φ(f)‖H∞α (B) ≤‖f‖NK(B). Let hw be the test function in Lemma 2.2 with w = φ(z), then we get hφ(z)(φ(z)) = ( 1 1 −|φ(z)|2 )n+1 2 . Hence, there exist a positive constant M such that: M ≥‖hw‖NK(B) ≥‖Wu,φ(hw)‖H∞α (B) ≥ |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . This completes the proof of the theorem. Using the standard arguments similar to those outlined in proposition 3.11 of [2], we have the following lemma: Lemma 3.1. Let φ : B → B be a holomorphic mapping and u ∈ H(B). For 0 < α < ∞, then Wu,φ : H ∞ α (B) →NK(B) is compact if and only if lim m→∞ ‖Wu,φ(fm)‖NK(B) = 0, for every bounded sequence {fm}⊂NK(B) which converges to 0 uniformly on any compact subsets of B as m →∞. Theorem 3.2. Let φ : B → B be a holomorphic mapping and u ∈ H(B). For 0 < α < ∞, then Wu,φ : NK(B) → H∞α (B) is compact if and only if (3.2) lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 = 0. Proof. First assume that Wu,φ : NK(B) → H∞α (B) is compact, then it is bounded. By Theorem theorem 3.1, we have L = sup a∈B |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 < ∞, Note that lim r→1− L(r) always exists, where: L(r) = sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . Now, we show that (3.4) holds. Assume on the contrary that there exists ε0 > 0 such that lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 = ε0. There exists an r0 ∈ (0, 1) such that r0 < r < 1, we have L(r) > ε02 . Then, by the standard diagonal process, we can construct a sequence {zm} ⊂ B such that |φ(z)| → 1 asm → ∞, and also for each m ∈ N, |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ≥ ε0 4 . Clearly, we can assume that wn = φ(zm) tends to w0 ∈ ∂B as m → ∞. Let hwm = (1−|wm|2)n+1 (1−〈zm,wm〉) 3 2 (n+1) be the function in Lemma 2.2 with wn = φ(zm). Then hwn → hw0 with respect to the compact-open topology. Define fm = hwm −hw0 . Then ‖fm‖NK(B) ≤ 1 and fm → 0 uniformly on compact subsets of B. Thus, fm ◦φ → 0 in H∞α (B) by assumption. But, for m big enough, ‖Wu,φ(fn)‖H∞α (B) ≥ |u(zm)|(1 −|zm|2)α (1 −|φ(zm)|2) n+1 2 ≥ ε0 4 , which is a contradiction. Conversely, if (3.4) holds, we assume that {fm} is a bounded sequence in NK(B) norm which converges 94 SHAMMAKY AND BAKHIT to zero uniformly on every compact subset of mathbbB, then for all ε > 0 there exists δ ∈ (0, 1) and mε < m such that for δ < r < 1, we have ‖Wu,φ(fn)‖H∞α (B) ≤ sup |φ(z)|>r |u(z)||fm(φ(z))|(1 −|z|2)α + sup |φ(z)|≤r |fm(φ(z))|(1 −|z|2)α ≤ ε. From this ‖Wu,φ(fn)‖H∞α (B) → 0 as m → ∞, it follows that Wu,φ : NK(B) → H ∞ α (B) is a compact operator. This completes the proof of the theorem. As a corollary of Theorems 3.1 and 3.2, we have: Corollary 3.1. Let φ : B → B be a holomorphic mapping and 0 < α < ∞. Then composition operator Cφ : NK(B) → H∞α (B) • is bounded if and only if (3.3) sup z∈B (1 −|z|2)α (1 −|φ(z)|2) n+1 2 < ∞; • is compact if and only if (3.4) lim r→1− sup |φ(z)|>r (1 −|z|2)α (1 −|φ(z)|2) n+1 2 = 0. 4. Essential norms of weighted composition operators from NK(B) into H∞α (B) In this section, we study the essential norm of weighted composition operator Wu,φ : NK(B) → H∞α (B). Let us denote by C := C(NK(B),H∞α (B)) the set of all compact operators acting from NK(B) into H∞α (B). Then the essential norm of Wu,φ is defined as follows: (4.1) ‖Wu,φ‖e = inf O∈C {‖Wu,φ −O‖}. Obviously, the essential norm of a compact operator is zero. Note that by using the standard argument, it can be shown that a composition operator Cφ : NK(B) → NK(B) is compact if and only if for any bounded sequence {fm} ⊂ NK(B) converging to zero uniformly on every compact subset of B, the sequence {‖fm ◦φ‖} converges to zero as m →∞. Lemma 4.1. Suppose φ : B → B is a holomorphic mapping such that ‖φ‖∞ < 1, and u ∈ H(B). For 0 < α < ∞, then Wu,φ : H∞α (B) →NK(B) is compact if and only if lim m→∞ ‖Wu,φ(fm)‖NK(B) = 0, Proof. Let r = ‖φ‖∞ and take an arbitrary f ∈NK(B). Then we have ‖Wu,φ(f)‖NK(B) ≤‖f ◦φ‖∞‖u‖NK(B) ≤ ( sup {z:|z|≤r} |f(z)| ) ‖u‖NK(B) < ∞. This show that Wu,φ maps NK(B) into itself. Now suppose that {fm} is a bounded sequence in NK(B) that converges to zero uniformly on every compact subset of B. Applying the above estimate with f = fm, we have ‖Wu,φ(fm)‖NK(B) ≤ ( sup {z:|z|≤r} |fm(z)| ) ‖u‖NK(B) < ∞. Since the set z : |z| ≤ r is compact, the right-hand side of the last quantity converges to 0 as m →∞, hence so does the sequence {‖Wu,φ(fm)‖NK(B)}. This means that Wu,φ is compact. In the following theorem we formulate and prove an estimate for the upper bound of the essential norm of Wu,φ : NK(B) → H∞α (B). PROPERTIES OF WEIGHTED COMPOSITION OPERATORS 95 Theorem 4.1. Let φ : B → B be a holomorphic mapping and u ∈H(B). For 0 < α < ∞, suppose that Wu,φ : NK(B) → H∞α (B) is a bounded operator. Then (4.2) ‖Wu,φ‖e ≤ M lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 , where M = 6 n+1 2 22(n+1) √ n(3+2 √ 3) K(3/4) is the constant from Proposition 2.2. Proof. Since Wu,φ is bounded, we see that u ∈ H∞α (B), and Theorems 3.1, 3.2 shows that lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 exists and is a real number. First, we prove that for any r ∈ [0, 1), (4.3) ‖Wu,φ‖e ≤ M sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . For each k ∈ N, set φk(z) = kzk+1 for all z ∈ B. By Lemma 4.1, Cφk is compact on NK(B), and hence, O = Wu,φ ◦Cφk ∈C is compact acting from NK(B) into H ∞ α (B). Then for any k ∈ N, we have ‖Wu,φ‖e = inf O∈C {‖Wu,φ −O‖}≤‖Wu,φ −Wu,φ ◦Cφk‖e = sup ‖f‖NK(B)≤1 ‖(Wu,φ −Wu,φ ◦Cφk)(f)‖H∞α (B), which implies that (4.4) ‖Wu,φ‖e ≤ inf k∈N { sup ‖f‖NK(B)≤1 ‖(Wu,φ −Wu,φ ◦Cφk)(f)‖H∞α (B) } . For f ∈NK(B), we estimate ‖(Wu,φ −Wu,φ ◦Cφk)(f)‖H∞α (B) = sup z∈B { |u(z)| ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣(1 −|z|2)α } ≤ sup |φ(z)|>r { |u(z)| ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣(1 −|z|2)α } + sup |φ(z)|≤r { |u(z)| ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣(1 −|z|2)α } . On the one hand, by Lemma 2.1, equation (2.3), we have sup |φ(z)|>r { |u(z)| ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣(1 −|z|2)α } ≤ sup |φ(z)|>r ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣sup z∈B { |u(z)|(1 −|z|2)α } ≤ ( sup |φ(z)|>r M|u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ) ‖f‖NK(B). On the one hand, by Lemma 2.1, equation (2.4), we have sup |φ(z)|≤r { |u(z)| ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣(1 −|z|2)α } ≤ ( Mr‖u‖H∞α (B) (k + 1)(1 −r2)α ) ‖f‖NK(B). 96 SHAMMAKY AND BAKHIT Therefore, if ‖f‖NK(B) ≤ 1, then ‖(Wu,φ −Wu,φ ◦Cφk)(f)‖H∞α (B) ≤ sup z∈B { |u(z)| ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣(1 −|z|2)α } ≤ M {( sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ) + Mr‖u‖H∞α (B) (k + 1)(1 −r2)α } . It then follows that inf k∈N { sup ‖f‖NK(B)≤1 ‖(Wu,φ −Wu,φ ◦Cφk)(f)‖H∞α (B) } ≤ M inf k∈N {( sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ) + Mr‖u‖H∞α (B) (k + 1)(1 −r2)α } ≤ M ( sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ) .(4.5) Combining (4.4) and (4.5), we obtain (4.3). Now letting r → 1 in (4.3), we arrive at the desired inequality (4.6) ‖Wu,φ‖e ≤ M lim r→1 ( sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ) . This completes the proof of the theorem. We now discuss the estimation for the lower bound of the essential norm of Wu,φ : NK(B) → H∞α (B). Theorem 4.2. Let φ : B → B be a holomorphic mapping and u ∈H(B). For 0 < α < ∞, suppose that Wu,φ : NK(B) → H∞α (B) is a bounded operator. Then (4.7) ‖Wu,φ‖e ≥ lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . Proof. The case ‖φ‖∞ < 1 is obvious since the right hand side is zero. Now assume that ‖φ‖∞ = 1. For any r ∈ (0; 1), the set Sr := {z ∈ B : |φ(z)| > r is not empty. For each z ∈ B, consider the probe function hw in Lemma 2.2 with w = φ(z). Then for any compact operator O ∈C we have ‖Wu,φ −O‖ = sup ‖f‖NK(B)≤1 ‖(Wu,φ −O)(f)‖H∞α (B) ≥ ‖(Wu,φ −O)(hφ(z))‖H∞α (B) ≥ ‖Wu,φ(hφ(z))‖H∞α (B) −‖O(hφ(z))‖H∞α (B) ≥ |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 −‖O(hφ(z))‖H∞α (B), which is equivalent to (4.8) ‖Wu,φ −O‖ + ‖O(hφ(z))‖H∞α (B) ≥ |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . Taking the supremum on z over the set Sr on both sides of (4.8) yields ‖Wu,φ −O‖ + sup z∈Sr ‖O(hφ(z))‖H∞α (B) ≥ sup z∈Sr |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . which is (4.9) ‖Wu,φ −O‖ + sup |φ(z)|>r ‖O(hφ(z))‖H∞α (B) ≥ sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . Denote H(r) = sup |φ(z)|>r ‖O(hφ(z))‖H∞α (B). Since H(r) decreases as r increases, limr→1 H(r) exists. We claim that this limit is necessarily zero. For the purpose of obtaining a contradiction, assume that PROPERTIES OF WEIGHTED COMPOSITION OPERATORS 97 lim r→1 H(r) = L > 0. Then there is a sequence {zm}⊂ B satisfying |φ(zm)|→ 1 as m →∞, and for each m ∈ N, (4.10) ‖O(hφ(z))‖H∞α (B) ≥ 1 2 L. By Corollary 3.1, {hφ(zm)} converges weakly to zero in NK(B). Since O is compact, we have {‖O(hφ(z))‖H∞α (B)} converges to zero as m →∞, which contradicts (4.10). Therefore, lim r→1− sup |φ(z)|>r ‖O(hφ(z))‖H∞α (B) = 0. Letting r → 1− on both sides of (4.9), we conclude that for any compact operator O ∈C, ‖Wu,φ −O‖≥ lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . From this, it follows that ‖Wu,φ‖e = inf O∈C ‖Wu,φ −O‖≥ lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . This completes the proof of the theorem. In conclusion, combining Theorems 4.1 and 4.2, we obtain a full description of the essential norm of Wu,φ. Theorem 4.3. Let φ : B → B be a holomorphic mapping and u ∈H(B). For 0 < α < ∞, suppose that Wu,φ : NK(B) → H∞α (B) is a bounded operator. Then (4.11) ‖Wu,φ‖e ≈ lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . Acknowledgement. The authors would like to express their thanks to the Deanship of Scien- tific Research at Jazan University Saudi Arabia for funding the work through research project no. 207/SABIC 2/36. References [1] M.A. Bakhit, A.E. Shammaky, Predual Norms of Some Holomorphic Function Spaces, International J. Functional Analysis, Operator Theory and Applications, 6(3) (2014), 153-176. [2] C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, 1995. [3] P. Duren and R. Weir, The pseudohyperbolic metric and Bergman spaces in the ball, Trans. Amer. Math. Soc. 359 (2007), 63-76. [4] A. El-Sayed Ahmed and M.A. Bakhit, Holomorphic NK and Bergman-type spaces, Birkhuser Series on Operator Theory: AdVances and Applications (2009), BirkhuserVerlag Publisher BaselSwitzerland, 195 (2009), 121-138. [5] A. El-Sayed Ahmed and M.A. Bakhit, Composition operators acting betweensome weighted Möbius invariant spaces, J. Ann. Funct. Anal. 2 (2011), 138-152. [6] B. Hu and L.H. Khoi, Compact difference of weighted composition operators on Np-spaces in the ball, Romanian J. Pure Appl. Math. 60(2) (2015), 101-116. [7] S. Krantz, Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992. [8] W. Rudin, Function Theory in the Unit Ball of Cn, Springer-Verlag, New York, 1980. [9] A.E. Shammaky, Weighted composition operators acting between kind of weighted Bergman-type spaces and the Bers-type space, International Journal of Mathematical, Computational Science and Engineering, 8(3) (2014), 496- 499. [10] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer-Verlag, New York, 2004. Department of Mathematics, Faculty of Science, Jazan university, Jazan, Saudi Arabia ∗Corresponding author: mabakhit@jazanu.edu.sa