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p Aα(D). In this paper, we assume that ϕ satisfies the following conditions − 1 (∆ϕ(z)) 2 =: τ(z) ց 0, and τ ′(z) → 0 when |z| → 1− so, we easily check that we exclude the standard weights since τ(r) = 1 − r when ϕ(r) = log(1 − r). In fact, these weights described above decrease faster than the standard weight (1 − |z|)α, α > 0. We can refer to Lemma 2.3 of [12] for the proof. Furthermore, we assume that τ(z)(1− −C ′ 1 |z|) increases for some C > 0 or lim|z|→1− τ (z) log τ(z) = 0. Now, we say that the weight ω is in the class W if ϕ satisfies all conditions above. If ω belongs to the class W then the associated function τ(z) has the following properties (L):

(i) There exists a constant C1 > 0 such that

τ(z) ≤ C1(1 − |z|), for z ∈ D.

(ii) There exists a constant C2 > 0 such that

|τ(z) − τ(ξ)|≤ C2|z − ξ|, for z,ξ ∈ D. The typical weight in the class W is the type of −c ω(r) = (1 − r)β exp , β ≥ 0, α,c> 0, (1 − r)α  1+ α and its associated function τ(z) = (1 − |z|) 2 . We can find more examples of our weight functions and their τ functions in [12]. Additionally, we assume that fast weights ω ∈ W have the regularity imposing the rate of decreasing condition in [8], ω(1 − δt) lim = 0, 0 <δ< 1. (1.1) t→0 ω(1 − t) In [8], Kriete and Maccluer gave the following sufficient condition for the bound- 2 edness of Cφ and the equivalent conditions for the compactness of Cφ on A (ω). Theorem 1.1. Let ω be a regular weight in the class W. If ω(z) lim sup < ∞, (1.2) |z|→1 ω(φ(z)) 2 then Cφ is bounded on A (ω). Moreover, the following conditions are equivalent: 2 (1) Cφ is compact on A (ω). ω(z) (2) lim|z|→1 ω(φ(z)) = 0. D 1−|φ(z)| (3) dφ(ζ) > 1, ∀ζ ∈ ∂ where dφ(ζ) = lim infz→ζ 1−|z| . In the same paper, they show that (1.2) is even the equivalent condition for the 1 2 − z α boundedness of Cφ on A (ω) when ω(z)= e (1−| |) , α > 0. Therefore, we can expect that the condition (1.2) is still the equivalent condition for the boundedness of Cφ on the Bergman spaces with weights ω in the class W. In fact, there is almost the converse of Theorem 1.1 in [5, Corollary 5.10] as follows: THEWEIGHTEDCOMPOSITIONOPERATORS 3

Theorem 1.2. If ω(r) lim inf = ∞ r→1 ω(Mφ(r)) 2 iθ then Cφ is unbounded on A (ω). Here, Mφ(r) = supθ |φ(re )|. In Section 3, we show that the condition (1.2) is a necessary condition of the 2 boundedness of Cφ on A (ω), thus we have the following result for general p with the help of the Carleson measure theorem. Theorem 1.3. Let ω be a regular weight in the class W and φ be an analytic self- p map of D. Then the composition operator Cφ is bounded on A (ω), 0

2 In Section 4, we studied the weighted composition operator uCφ on A (ω). In [6], we can find the characterizations of the boundedness, compactness and mem- bership in the Schatten classes of uCφ defined on the standard weighted Bergman spaces in terms of the generalized Berezin transform. Using the same technique, we extended their results in [6] to the case of large weighted Bergman spaces A2(ω). Those our results are given by Theorem 4.3 and Theorem 4.5 in Section 4. Finally, we studied the membership of uCφ in the Schatten classes Sp, p > 0 in Section 5.

Constants. In the rest of this paper, 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.

2. SOME PRELIMINARIES In this section, we recall some well-known notions and collect related facts to be used in our proofs in later section.

2.1. Carleson type measures. Let τ be the positive function on D satisfying the properties (L) introduced in Section 1. We define the Euclidean disk D(δτ(z)) having a center at z with radius δτ(z). Let min(1,C−1,C−1) m := 1 2 , (2.1) τ 4 where C1,C2 are in the properties (i), (ii) of (L). In [12], they show that for 0 <δ

Lemma 2.1. [12, Lemma 2.2] Let ω = e−ϕ, where ϕ is a subharmonic function. Suppose the function τ satisfies properties (L) and τ(z)2∆ϕ(z) ≤ C for some constant C > 0. For β ∈ R and 0

p β M p β |f(z)| ω(z) ≤ 2 2 |f| ω dA δ τ(z) ZD(δτ(z)) for a sufficiently small δ > 0, and f ∈ H(D). A positive Borel measure µ in D is called a (vanishing) Carleson measure for Ap(ω) if the embedding Ap(ω) ⊂ Lp(ωdµ) is (compact) continuous where

Lp(ωdµ) := f ∈ M(D) |f(z)|pω(z)dµ(z) < ∞ , (2.3)  ZD 

and M(D) is the set of µ-measurable functions on D. Now, we introduce the Carleson measure theorem on Ap(ω) given by [11]. Theorem 2.2 (Carleson measure theorem). Let ω ∈ W and µ be a positive Borel measure on D. Then for 0

∞ 2.2. The Reproducing Kernel. Let the system of functions {ek(z)}k=0 be an orthonormal basis of A2(ω). It is well known that the reproducing kernel for the Bergman space A2(ω) is defined by ∞ K(z,ξ)= Kz(ξ)= ek(z)ek(ξ). Xk=0 Unlike the standard weighted Bergman spaces, the explicit form of K(z,ξ) of A2(ω) has been unknown. However, we have the precise estimate near the diagonal given by Lemma 3.6 in [9], ω(z)−1ω(ξ)−1 |K(z,ξ)|2 ≈ K(z, z)K(ξ,ξ) ≈ , ξ ∈ D(δτ(z)), (2.4) τ(z)2τ(ξ)2 THEWEIGHTEDCOMPOSITIONOPERATORS 5 where δ ∈ (0,mτ /2). Recently, in [1], they introduced the upper estimate for the reproducing kernel as follows: for z,ξ ∈ D there exist constants C,σ > 0 such that 1 1 |γ′(t)| |K(z,ξ)|ω(z)1/2ω(ξ)1/2 ≤ C exp −σ inf dt , (2.5) τ(z)τ(ξ)  γ Z0 τ(γ(t))  where γ is a piecewise C1 curves γ : [0, 1] → D with γ(0) = z and γ(1) = ξ. In the same paper, they remarked that the distance βϕ denoted by 1 |γ′(t)| βϕ(z,ξ) = inf dt γ Z0 τ(γ(t)) is a complete distance because of the property (i) of (L). By the completeness of βϕ and the kernel estimate (2.5), we obtain the following lemma.

Lemma 2.3. Let ω ∈ W. Then the normalized kernel function kz(ξ) uniformly converges to 0 on every compact subsets of D when |z|→ 1−. Proof. Given a compact subset K of D and z ∈ D, (2.4) and (2.5) follow that −1/2 |Kz(ξ)| ω(ξ) −σβϕ(z,ξ) −σβϕ(z,ξ) |kz(ξ)| = . e ≤ CKe , ∀ξ ∈ K. kKzk2,ω τ(ξ)

By the completeness of βϕ, we conclude that kz(ξ) converges to 0 uniformly on compact subsets of D when z approaches to the boundary.  2.3. The Julia-Caratheodory Theorem. For a boundary point ζ and α > 1, we define the nontangential approach region at ζ by Γ(ζ, α)= {z ∈ D : |z − ζ| < α(1 − |z|)}. A function f is said to have a nontangential limit at ζ if ∠ lim f(z) < ∞, for each α> 1. z→ζ z∈Γ(ζ,α) Definition 2.4. We say φ has a finite angular derivative at a boundary point ζ if there is a point η on the circle such that φ(z) − η φ′(ζ) := ∠ lim < ∞, for each α> 1. z→ζ z − ζ z∈Γ(ζ,α) Theorem 2.5 (Julia-Caratheodory Theorem). For φ : D → D analytic and ζ ∈ ∂D, the following is equivalent: 1−|φ(z)| (1) dφ(ζ) = lim infz→ζ 1−|z| < ∞. (2) φ has a finite angular derivative φ′(ζ) at ζ. (3) Both φ and φ′ have (finite) nontangential limits at ζ, with |η| = 1 for η = limr→1 φ(rζ). ′ Moreover, when these conditions hold, we have φ (ζ) = dφ(ζ)ζη¯ and dφ(ζ) = 1−|φ(z)| ∠ limz→ζ 1−|z| . 6 I. PARK

In addition to the Julia-Caratheodory theorem, we use the Julia’s lemma which gives a useful geometric result. For k > 0 and ζ ∈ ∂D, let

E(ζ, k)= {z ∈ D : |ζ − z|2 ≤ k(1 − |z|2)}.

D 1 k A computation shows that E(ζ, k)= z ∈ : k+1 ζ − z ≤ 1+k . n o

Theorem 2.6 (Julia’s Lemma). Let ζ be a boundary point and φ : D → D be analytic. If dφ(ζ) < ∞ then |φ(ζ)| = 1 where limn→∞ φ(an) =: φ(ζ) and {an} is a sequence along which this lower limit is achieved. Moreover, φ(E(ζ, k)) ⊆ E(φ(ζ), kdφ(ζ)) for every k> 0, that is, |φ(ζ) − φ(z)|2 |ζ − z|2 ≤ d (ζ) for all z ∈ D. (2.6) 1 − |φ(z)|2 φ 1 − |z|2

Note that (2.6) shows that dφ(ζ) = 0 if and only if φ is a unimodular con- stant. Moreover, when dφ(ζ) ≤ 1, φ(E(ζ, k)) ⊆ E(φ(ζ), k) thus, the set E(ζ, k) contains its image of φ when φ(ζ)= ζ.

2.4. Schatten Class. For a positive compact operator T on a separable H, there exist orthonormal sets {ek} in H such that

Tx = λkhx, ekiek, x ∈ H, Xk where the points {λk} are nonnegative eigenvalues of T . This is referred to as the canonical form of a positive compact operator T . For 0

p p kT kSp = |λk| < ∞. Xk

When 1 ≤ p< ∞, Sp is the Banach space with the above and Sp is a metric space when 0

∗ p/2 ∗ (T T ) ∈ S1 ⇐⇒ T T ∈ Sp/2. (2.7)

In particular, when T ∈ S2, we say that T is a Hilbert-Schmidt integral operator. It is well known that every Hilbert-Schmidt operator on L2(X,µ) is an integral operator induced by a function K ∈ L2(X × X,µ × µ) such that

Tf(z)= K(x,y)f(y) dµ(y), ∀f ∈ L2(X,dµ). ZX The converse is also true. To study all of the basic properties of Schatten class operators above, we can refer to §1.4 and §3.1 in [16]. THEWEIGHTEDCOMPOSITIONOPERATORS 7

p 3. COMPOSITION OPERATORS Cφ ON A (ω)

In [8], they also gave the necessary condition for the boundedness of Cφ on large Bergman spaces in terms of the angular derivative, 1 − |φ(z)| dφ(ζ) = lim inf ≥ 1, ∀ζ ∈ ∂D. (3.1) z→ζ 1 − |z|

The following Lemma enables us to consider only the boundary points dφ(ζ) = 1 in the proof of Theorem 3.2.

Lemma 3.1. Let ω be a regular fast weight. Then if dφ(ζ) > 1 for ζ ∈ ∂D then ω(z) lim = 0. z→ζ ω(φ(z))

1−|φ(z)| Proof. We first assume that lim infz→ζ 1−|z| = dφ(ζ)=1+2ǫ< ∞ for some ζ ∈ ∂D. Then for ǫ> 0, there exists an open disk Dr(ζ) centered at ζ with radius r such that 1 − |φ(z)| > d (ζ) − ǫ, ∀z ∈ D (ζ) ∩ D. 1 − |z| φ r Thus, using the relation xA ≥ 1 − A(1 − x) when A> 1 and 0

1+ǫ 1−|φ(z)| since ω(|φ(z)|) ≥ ω(|z| ) by (3.2). When dφ(ζ) = lim infz→ζ 1−|z| = ∞, we have the prompt result, |φ(z)| < |z|M , M > 1 for near ζ by the definition, since there exists M > 1 such that 1 1 log > M log , |φ(z)| |z| for |z| near 1. Thus, we complete the proof.  We now prove our first theorem. Theorem 3.2. Given a regular weight ω in W, let φ be an analytic self-map of D. 2 If the composition operator Cφ is bounded on A (ω), then ω(z) lim sup < ∞. z→ζ∈∂D ω(φ(z))

Proof. Since the composition operator is bounded, we have dφ(ζ) ≥ 1 for all boundary points ζ by (3.1). By Lemma 3.1, it suffices to check only the boundary 8 I. PARK point ζ satisfying dφ(ζ) = 1. For a boundary point ζ such that dφ(ζ) = 1, we have the following relation of inclusion by Theorem 2.6, 1 − r 1 − r φ E ζ, ⊆ E φ(ζ), , r ∈ (0, 1).   1+ r   1+ r  1−r Here, the set E(ζ, 1+r ) introduced in Section 2.3 is the closed disk centered at 1+r 1−r 1−r 2 ζ with radius 2 so, rζ is the point on the boundary of E(ζ, 1+r ) closest to 0. Thus, we have 1+ r 1 − r φ(ζ) − φ(rζ) ≤ , 2 2

that is, |φ(rζ)|≥ r for 0

τ(φ(|z|ζ)) ≤ τ(z) for r0 ≤ |z| < 1, (3.3) since τ(z) is a radial decreasing function when |z|→ 1−. Now, define the follow- ing radial function

Mφ(r)= sup |φ(rη)|, 0

τ(Mφ(|z|)) ≤ τ(φ(|z|ζ)) ≤ τ(z), for r0 < |z| < 1. (3.4) Now, we assume that there is a sequence {z } converges to ζ with ω(zn) → ∞ n ω(φ(zn)) when n → ∞. We can choose {ξn} in D such that |ξn| = |zn| and |φ(ξn)| = ∗ ∗ Mφ(|zn|). Using the relation CφKz(ξ)= hCφKz, Kξi = hKz,CφKξi = Kφ(z)(ξ), we have K (φ(z)) 2 ∗ 2 φ(z) τ(z) ω(z) D kCφkzk2,ω = 2 ≈ 2 , ∀z ∈ , kKzk2,ω τ(φ(z)) ω(φ(z)) thus by (3.4), we obtain 2 2 ∗ 2 ω(ξn) τ(ξn) kCφk & kCφkξn k2,ω & 2 ω(φ(ξn)) τ(φ(ξn)) 2 ω(zn) τ(zn) ω(zn) = 2 ≥ , ω(Mφ(|zn|)) τ(Mφ(|zn|)) ω(φ(zn)) for large n > N. Thus, we have its contradiction and complete our proof.  From Theorem 3.2 together with Theorem 1.1[KM], we conclude that (1.2) is 2 the equivalent condition for the boundedness of Cφ on A (ω). Furthermore, by the change of variables in the measure theory in Theorem C in §39 of [7], we have

p p p −1 kCφfkp,ω = |f ◦ φ(z)| ω(z) dA(z)= |f(z)| [ωdA] ◦ φ (z). ZD ZD Denote φ −1 −1 dµω(z) := ω(z) [ωdA] ◦ φ (z). THEWEIGHTEDCOMPOSITIONOPERATORS 9

p Then Cφ is (compact) bounded on A (ω) for 0

2 4. WEIGHTED COMPOSITION OPERATORS uCφ ON A (ω) For a positive Borel measure µ, we introduce the Berezin transform µ on D given by e 2 µ(z)= |kz(ξ)| ω(ξ)dµ(ξ), z ∈ D, ZD 2 where kz is the normalizede kernel of A (ω). For δ ∈ (0, 1), the averaging function µδ over the disks D(δτ(z)) is defined by µ(D(δτ(z))) b µ (z)= , z ∈ D. δ |D(δτ(z))| In [2], we already calculatedb the following relation between the Berezin transform and the averaging function,

µδ(z) ≤ Cµ(z), z ∈ D. (4.1) Moreover, we can see that (2),(3), and (4) in the following Proposition are equiv- b e alent in [2]. Before proving Proposition 4.2, we introduce the covering lemma which plays an essential role in the proof of many theorems including Carleson measure theorem. Lemma 4.1. [11] Let τ be a positive function satisfying properties (L) and δ ∈ (0,mτ ). Then there exists a sequence of points {ak}⊂ D such that (1) aj ∈/ D(δτ(ak)), for j 6= k. D (2) k D(δτ(ak)) = . (3) DS(δτ(a )) ⊂ D(3δτ(a )), where D(δτ(a )) = D(δτ(z)) k k k z∈D(δτ(ak )) (4) {D(3δτ(a ))} is a covering of D of finite multiplicity N. e k e S ∞ Here, we call the sequence {ak}k=1 δ-sequence. Now, we characterize Carleson measures on A2(ω) in terms of averaging func- tion and Berezin transform.

Proposition 4.2. Let ω ∈ W and δ ∈ (0,mτ /2). For µ ≥ 0, the following conditions are equivalent: (1) µ is a Carleson measure on A2(ω). (2) µ is bounded on D. (3) µδ is bounded on D. (4) Thee sequence {µδ(ak)} is bounded for every δ- sequence {ak}. b b 10 I. PARK

Proof. It is clear that (1) ⇒ (2). Thus, we remain to show that (4) implies (1). By Lemma 2.1, (3) of Lemma 4.1 and (2.2), we have

2 1 2 sup |f(ξ)| ω(ξ) . sup 2 |f| ωdA ξ∈D(δτ(ak)) ξ∈D(δτ(ak )) τ(ξ) ZD(δτ(ξ))

1 2 . 2 |f| ωdA. τ(ak) ZD(3δτ(ak )) Therefore, by Lemma 4.1 and the inequality above, we have the desired result, ∞ |f(ξ)|2ω(ξ)dµ(ξ) ≤ |f(ξ)|2ω(ξ)dµ(ξ) ZD Z Xk=1 D(δτ(ak )) ∞ 2 ≤ µ(D(δτ(ak))) sup |f(ξ)| ω(ξ) Xk=1 ξ∈D(δτ(ak )) ∞ µ(D(δτ(a ))) . k |f|2ωdA |D(δτ(ak))| Z Xk=1 D(3δτ(ak )) 2 . sup µδ(a)Nkfk2,ω. (4.2) a∈D b  For an analytic self-map φ of D and a function u ∈ L1(D), we define the φ- Berezin transform of u by

2 Bφu(z)= |kz(φ(ξ))| u(ξ)ω(ξ)dA(ξ). ZD Theorem 4.3. Let ω ∈W and u be an analytic function on D. Then the weighted 2 2 ∞ composition operator uCφ is bounded on A (ω) if and only if Bφ(|u| ) ∈ L (D). 2 Moreover, kuCφk≈ supz∈D Bφ(|u| )(z). 2 Proof. Given an analytic function u, we let dµ|u|2 (z) = |u(z)| ω(z)dA(z). Now, we define the positive measure φ −1 −1 dµω,u(z) := ω(z) dµ|u|2 ◦ φ (z). (4.3) By the change of variables in the measure theory, we have

2 2 |u(z)(f ◦ φ)(z)| ω(z) dA(z)= |(f ◦ φ)(z)| dµ|u|2 (z) ZD ZD 2 φ = |f(z)| ω(z) dµω,u(z) ZD φ Thus, the fact that the measure dµω,u is a Carleson measure is equivalent to that φ the Berezin transform µω,u is bounded on D by Proposition 4.2. On the other hand,

φ g 2 φ µω,u(z)= |kz(ξ)| ω(ξ) dµω,u(ξ) ZD g 2 2 2 = |kz(φ(ξ))| |u(ξ)| ω(ξ) dA(ξ)= Bφ(|u| )(z), (4.4) ZD THE WEIGHTED COMPOSITION OPERATORS 11 for all z ∈ D. Thus, the proof is complete. Furthermore, from (4.2) and (4.1), we have 2 kuCφk . sup Bφ(|u| )(z). z∈D 2 2 Finally, we obtain kuCφk ≈ supz∈D Bφ(|u| )(z) since Bφ(|u| )(z) ≤ kuCφk for all z ∈ D. 

As an immediate consequence of Theorem 4.3 we obtain more useful necessary condition for the boundedness of uCφ.

Corollary 4.4. Let ω ∈W and u be an analytic function D. If uCφ is bounded on A2(ω), then τ(z) ω(z)1/2 sup |u(z)| < ∞. (4.5) 1/2 z∈D τ(φ(z)) ω(φ(z)) Proof. For any z ∈ D, Theorem 4.3 follows that

2 2 2 ∞ >Bφ(|u| )(φ(z)) = |kφ(z)(φ(ξ))| |u(ξ)| ω(ξ) dA(ξ) ZD 2 2 ≥ |kφ(z)(φ(ξ))| |u(ξ)| ω(ξ) dA(ξ) ZD(δτ(z)) 2 2 2 & τ(z) |kφ(z)(φ(z))| |u(z)| ω(z) τ(z)2 ω(z) & |u(z)|2. τ(φ(z))2 ω(φ(z)) 

Before characterizing the compactness of uCφ, we recall the definition of the essential norm kT ke for a bounded operator T on a X as follows:

kT ke = inf{kT − Kk : K is any compact operator on X}. 2 For a bounded operator uCφ on A (ω), we use the following formula introduced in [14] to estimate the essential norm of uCφ,

kuCϕke = lim kuCϕRnk, (4.6) n→∞ 2 n 2 where Rn is the orthogonal projection of A (ω) onto z A (ω) defined by ∞ ∞ k k Rnf(z)= akz for f(z)= akz . kX=n Xk=0 The formula (4.6) can be obtained by the similar proof of Proposition 5.1 in [14]. In fact, we can find the proof of (4.6) adjusting to large weighted Bergman spaces in [8, p.775]. Now, we prove the following estimate for the essential norm of a 2 bounded weighted composition operator uCφ on A (ω). 12 I. PARK

Theorem 4.5. Let ω ∈W and u be an analytic function on D. If uCφ is bounded on A2(ω), then there is an absolute constant C ≥ 1 such that

2 2 lim sup Bφ(|u| )(z) ≤ kuCφke ≤ C lim sup Bφ(|u| )(z). |z|→1 |z|→1

Proof. For kfk2,ω ≤ 1 and a fixed 0

2 2 φ k(uCφRn)fk2,ω = |Rnf(ξ)| ω(ξ) dµω,u(ξ) ZD 2 φ 2 φ = |Rnf(ξ)| ω(ξ) dµω,u(ξ)+ |Rnf(ξ)| ω(ξ) dµω,u(ξ), ZD\rD ZrD where rD = {z ∈ D : |z|≤ r}. By the orthogonality of Rn, we have |Rnf(ξ)|≤ kfk2,ωkRnKξk2,ω and the following series is uniformly bounded on |ξ|≤ r< 1,

∞ 1 2 1 2k 2k+1 kRnKξk2,ω . r < ∞, where pk = s ω(s)ds. pk Z kX=n 0

Thus, kRnKξk→ 0 as n →∞ so that |Rnf(ξ)| also uniformly converges to 0 on φ rD as n → ∞. Therefore the second integral vanishes as n → ∞ since µω,u is a φ φ Carleson measure. For the first integral, we denote by µω,u,r := µω,u|D\rD. For a fixed r > 0, we easily calculate (D \ rD) ∩ D(δτ(z)) 6= ∅ for |z| + δτ(z) > r. Thus by (4.1) and (4.2), the first integral is dominated by

2 φ \φ 2 φ |Rnf| ω dµω,u,r . sup µω,u,rδ(z)kRnfk2,ω ≤ sup µω,uδ(z) ZD z∈D |z|+δτ(z)>r d φ . sup µω,u(z), |z|+δτ(z)>r g for all n> 0. Here, letting r → 1, then |z|→ 1− since τ(z) → 0. Thus we obtain the following upper estimate by (4.6) and the above inequality,

2 2 φ kuCφke = lim sup k(uCφRn)fk2,ω . lim sup µω,u(z) n→∞ kfk ≤1 |z|→1 2,ω g 2 = limsup Bφ(|u| )(z). |z|→1

For the lower estimate, for any compact operator K on A2(ω), we have

2 2 2 2 kuCφke ≥ kuCφ − Kk ≥ lim sup k(uCφ)kzk2,ω = limsup Bφ(|u| )(z). |z|→1 |z|→1 

In addition to Theorem 4.5, we have the following useful necessary condition 2 for the compactness of uCφ on A (ω). THE WEIGHTED COMPOSITION OPERATORS 13

Corollary 4.6. Let ω ∈ W and u be an analytic function on D. If the weighted 2 composition operator uCφ is compact on A (ω) then τ(z)2 ω(z) lim |u(z)|2 = 0. (4.7) |z|→1− τ(φ(z))2 ω(φ(z))

Proof. By Lemma 2.3, we know that the normalized kernel sequence {kz} con- ∗ verges to 0 weakly when |z|→ 1. Since (uCφ) Kz = u(z)Kφ(z), we have

∗ 2 2 K(φ(z), φ(z)) 0 = lim k(uCφ) kzk2,ω = lim |u(z)| |z|→1 |z|→1 K(z, z) τ(z)2 ω(z) & lim |u(z)|2 . |z|→1 τ(φ(z))2 ω(φ(z)) 

5. SCHATTEN CLASS WEIGHTED COMPOSITION OPERATORS In order to study the Schatten class composition operator, we use some known results of Toeplitz operators acting on A2(ω). First of all, we recall that the defini- tion and some facts of Toeplitz operators defined on A2(ω). For a finite complex Borel measure µ on D, the Toeplitz operator Tµ is defined by

Tµf(z)= f(ξ)K(z,ξ)ω(ξ)dµ(ξ), ZD for f ∈ A2(ω). Here, it is not clear the integrals above will converge, so we give the following additional condition on µ,

|K(z,ξ)|2ω(ξ)d|µ|(ξ) < ∞, (5.1) ZD for all z ∈ D. The following lemma is from Lemma 2.2 in [2]. Lemma 5.1. Let ω ∈W, µ ≥ 0 and µ be a Carleson measure. Then we have

2 hTµf, giω = f(ξ)g(ξ)ω(ξ)dµ(ξ), f,g ∈ A (ω), ZD where hf, giω = D f(z)g(z)ω(z) dA(z). It is well knownR that composition operators are closely related to Toeplitz oper- ators on weighted Bergman spaces. From Lemma 5.1, we have the relation,

∗ 2 h(uCφ) (uCϕ)f, giω = f(φ(z))g(φ(z))|u(z)| ω(z)dA(z) ZD φ = f(z)g(z)ω(z)dµω,u(z)= hT φ f, giω, (5.2) ZD µω,u φ where dµω,u is defined by (4.3). Thus, we can use the results of T φ to see when µω,u the composition operators uCϕ belong to Sp. In other words, it suffices to show ∗ when T φ is in S since uCϕ ∈ Sp is equivalent to (uCϕ) (uCϕ) ∈ S as we µω,u p/2 p/2 studied in Section 2.4. 14 I. PARK

The following lemma is the characterization of the membership in the Schatten ideals of a Toeplitz operator acting on A2(ω).

Lemma 5.2 (Theorem 4.6 [2]). Let ω ∈ W, δ ∈ (0,mτ /4) and 0

2 |Bφ(|u| )(z)|∆ϕ(z)dA(z) ZD 2 2 1 = |kz(φ(ξ))| |u(ξ)| ω(ξ) dA(ξ) dA(z) ZD ZD τ(z)2 ≈ |K(z, φ(ξ))|2|u(ξ)|2ω(ξ) dA(ξ)ω(z)dA(z) ZD ZD = K(φ(ξ), φ(ξ))|u(ξ)|2ω(ξ) dA(ξ) ZD τ(ξ)2 ω(ξ) ≈ |u(ξ)|2∆ϕ(ξ)dA(ξ) < ∞. ZD τ(φ(ξ))2 ω(φ(ξ))  Acknowledgements. The author would like to thank the referee for indicating various mistakes and giving helpful comments.

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BK21-MATHEMATICAL SCIENCES DIVISION,PUSAN NATIONAL UNIVERSITY,BUSAN 46241, REPUBLIC OF KOREA E-mail address: [email protected]