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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2020; 60(3): 519-534

Published online September 30, 2020

### Generalized Integration Operator between the Bloch-type Space and Weighted Dirichlet-type Spaces

Fariba Alighadr Ardebili, Hamid Vaezi*, Mostafa Hassanlou

Department of Mathematics, Sarab Branch, Islamic Azad University, Sarab, Iran
Department of Mathematics, Sarab Branch, Islamic Azad University, Sarab, Iran and Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
e-mail : hvaezi@tabrizu.ac.ir
Technical Faculty of Khoy, Urmia University, Urmia, Iran
e-mail : M.hassanlou@urmia.ac.ir

Received: September 29, 2019; Accepted: March 20, 2020

Let be the space of all holomorphic functions on the open unit disc in the complex plane ℂ. In this paper, we investigate the boundedness and compactness of the generalized integration operator Ig,ϕ(n)(f)(z)=0zf(n)(ϕ(ξ))g(ξ)dξ,         zD,

between Bloch-type and weighted Dirichlet-type spaces, where ϕ is a holomorphic self-map of , n ∈ ℕ and .

Keywords: integration operator, Bloch-type space, Dirichlet-type space

Let be the open unit disc in the complex plane ℂ and be the space of all holomorphic functions on . For α ∈ (0, ∞), the α-Bloch space ℬα is the space of all satisfying

fBα=supzD(1-z2)αf(z)<.

These are collectively referred to as Bloch-type spaces. The little Bloch-type space B0α consists of those functions f ∈ ℬα for which

limz1(1-z2)αf(z)=0.

The space ℬα is a Banach space with the norm

f=f(0)+fBα,

and B0α is a closed subspace of ℬα.

For p ∈ (0, ∞) and β > −1, Aβp denotes the space of all for which

fAβpp=Df(z)p(log1z)βdA(z)<,

where dA denotes the normalized Lebesgue area measure on . The space Aβp is called the weighted Bergman space. The weighted Bergman space Aβp is a Banach space for p ≥ 1 and a Hilbert space for p = 2. It is well-known that fAβp if and only if

Df(z)p(1-z2)βdA(z)<.

For p ∈ (0, ∞) and β > −1, the weighted Dirichlet-type space Dβp is the space of all functions for which

fDβpp=Df(z)p(log1z)βdA(z)<.

We note that fDβp if and only if fAβp.

Let u be a holomorphic function on and ϕ a nonconstant holomorphic self-map of . The weighted composition operator uCϕ induced by u and ϕ is defined on as follows:

uCϕ(f)=ufoϕ.

Putting u = 1, uCϕ reduces to the composition operator Cϕ. For general background on composition operators, we refer to [3, 14] and for weighted composition operators acting on Bloch-type spaces and Dirichlet-type spaces we refer for example to [2, 5, 13, 18].

In this paper, we consider an integration operator Ig,ϕ(n) which is defined on by

Ig,ϕ(n)(f)(z)=0zf(n)(ϕ(ξ))g(ξ)dξ,         zD,

where ϕ is a holomorphic self-map of , n ∈ ℕ and .

This operator, which was introduced in [15], is called the generalized integration operator. It is a generalization of the Riemann-Stieltjes operator Ig induced by g, defined by

Igf(z)=0zf(ζ)g(ζ)dζ,         zD.

Y. Yu and Y. Liu in [20] characterized the boundedness and compactness of Riemann-Stieltjes operator Ig from weighted Bloch spaces into Bergman-type spaces. The essential norm of the integral operator Ig on some spaces of holomorphic functions was studied by L. Liu, Z. Lou and C. Xiong in [10].

The operator Ig,ϕ(n) induces some known operators. For example, when n = 1, Ig,ϕ(n) reduces to an integration operator recently studied by S. Li and S. Stevic in [6, 7, 8]. Taking n = 1 and g(z) = ϕ′ (z), we obtain the composition operator Cϕ defined by Cϕf = f(ϕ) − f(ϕ (0)), fH(D).

Recently, S. D. Sharma and A. Sharma in [15] characterized the boundedness and compactness of generalized integration operator Ig,ϕ(n) from Bloch-type spaces to weighted BMOA.

The boundedness and compactness of Riemann-Stieltjes operators from mixed norm spaces to Zygmund-type spaces on the unit ball was studied by Y. Liu and Y. Yu in [11]. X. Zhu in [24] investigated the boundedness and compactness of generalized integration operators from H to Zygmund-type spaces. Z. He and G. Cao in [4] investigated the boundedness and compactness of generalized integration operators between Bloch-type spaces and F(p, q, s) spaces. For related integral-type operators on unit disc and also in ℂn, see for example [1, 9, 16]. Motivated by the above results, in this article we give an equivalent conditions for the boundedness and compactness of the generalized integration operator Ig,ϕ(n) between the Bloch-type and weighted Dirichlet-type spaces.

The notation ab means that there exists a positive constant C such that aCb. If both ab and ba occur, then a ~ b.

### 2. Boundednes and Compactness of the Operator Ig,ϕ(n):Bα→Dβp

In this section we characterize the boundedness and compactness of the generalized integration operator Ig,ϕ(n) from the Bloch-type space ℬα into Dirichlet-type space Dβp.

Let α > 0. From [12] it follows that there are two holomorphic functions f1, f2 ∈ ℬα such that

C(1-z2)αf1(z)+f2(z),         zD,

where C ia a positive constant.

If we define h1(z)=f1(z)-zf1(0) and h2(z)=f2(z)-zf2(0), using the following relation from [22],

(1-z2)α+1f(z)+f(0)~(1-z2)α+1f(z),

it can be shown that h1, h2 ∈ ℬα and

C(1-z2)α+1h1(z)+h2(z),         zD.

By repeating the above method, we have the following:

### Lemma 2.1.([4, 23])

Let α > 0 and n ∈ ℕ. There exist two holomorphic functions h1, h2 ∈ ℬαsuch that

C(1-z2)α+n-1h1(n)(z)+h2(n)(z),         zD,

where C is a positive constant.

For achieving the boundedness of Ig,ϕ(n):BαDβp we need to the following result from [23]:

### Lemma 2.2

Let α > 0 and n ∈ ℕ. If f ∈ ℬα, then

f(z)C{fBα0<α<1fBαln21-z2α=1fBα(1-z2)α-1α>1

and

f(n)(z)CfBα(1-z2)α+n-1,

where C is a positive constant.

### Theorem 2.3

Let, n ∈ ℕ, ϕ be a holomorphic self-map of, 0 < α,p < ∞ and β > −1. Then the following statements are equivalent:

• (i) Ig,ϕ(n):BαDβpis bounded.

• (ii) Ig,ϕ(n):B0αDβpis bounded.

• (iii) M=Dg(z)p(1-ϕ(z)2)p(α+n-1)(log1z)βdA(z)<.

Proof

(i) ⇒ (ii) is trivial, since B0αBα.

(ii) ⇒ (iii). First, note that if h ∈ ℬα, then by defining hs as hs(z) = h(sz) for every and s ∈ (0, 1), hsB0α and hsB0αhBα. By Lemma 2.1, there are two holomorphic functions h1, h2 ∈ ℬα such that the following inequality holds:

C(1-z2)α+n-1h1(n)(z)+h2(n)(z),         zD.

So, by (2.1),

Dsg(z)p(1-sϕ(z)2)p(α+n-1)(log1z)βdA(z)CD|h1s(n)(ϕ(z))|psg(z)p(log1z)βdA(z)+CD|h1s(n)(ϕ(z))|psg(z)p(log1z)βdA(z)C(Ig,ϕ(n)(h1s)Dβpp+Ig,ϕ(n)(h2s)Dβpp),

for every s ∈ (0, 1). Since boundedness of Ig,ϕ(n):B0αDβp implies that Ig,ϕ(n)(h1s)Dβpp< and Ig,ϕ(n)(h2s)Dβpp<, so, by an application of Fatou’s Lemma,

M=Dg(z)p(1-ϕ(z)2)p(α+n-1)(log1z)βdA(z)<.

(iii) ⇒ (i). From Lemma 2.2 we have

|f(n)(z)|CfBα(1-z2)α+n-1,

for every f ∈ ℬα. This implies that

Ig,ϕ(n)fDβpp=D|f(n)(ϕ(z))|pg(z)p(log1z)βdA(z)CfBαpDg(z)p(1-ϕ(z)2)p(α+n-1)p(log1z)βdA(z)=CMfBαp<.

Therefore, Ig,ϕ(n):BαDβpis bounded.

Now, we investigate the compactness of Ig,ϕ(n):BαDβp. For this investigation we need to the following Lemma which can be found for example in [17].

### Lemma 2.4

Let X and Y be Banach spaces of holomorphic functions on. Suppose that

• (i) The point evaluation functions on X are continuous.

• (ii) The closed unit ball of X is a compact subset of X in the topology of uniform convergence on compact sets.

• (iii) T : XY is continuous when X and Y are given the topology of uniform convergence on compact sets.

Then, T is a compact operator if and only if given a bounded sequence {fn} in X such that fn → 0 uniformly on compact sets, then the sequence {Tfn} converges to zero in the norm of Y.

For X = ℬα and Y=Dβp, the above Lemma can be applied. So it follows that:

### Lemma 2.5

LetT:BαDβpbe a bounded operator. Then, T is compact if and only if given a bounded sequence {fn} inαsuch that fn → 0 uniformly on compact sets, then the sequence {Tfn} converges to zero in the norm ofDβp.

By the following result we characterize the compactness of Ig,ϕ(n):BαDβp.

### Theorem 2.6

Let, n ∈ ℕ, ϕ be a holomorphic self-map of, 0 < α,p < ∞ and β > −1. Then the following statements are equivalent:

• (i) Ig,ϕ(n):BαDβpis compact.

• (ii) Ig,ϕ(n):B0αDβpis compact.

• (iii) Ig,ϕ(n):B0αDβpis weakly compact.

• (iv)M=Dg(z)p(1-ϕ(z)2)p(α+n-1)(log1z)βdA(z)<,

and

limt1ϕ(z)>tg(z)p(1-ϕ(z)2)p(α+n-1)(log1z)βdA(z)=0.

Proof

(i) ⇒ (ii) is trivial.

(ii) ⇔ (iii). Clearly, Ig,ϕ(n):B0αDβp is weakly compact if and only if its adjoint, i.e. (Ig,ϕ(n))*:(Dβp)*(B0α)* is weakly compact. According to [21], (B0α)*=Aβ1. Since Aβ1 satisfies in the Schur property, (Ig,ϕ(n))*:(Dβp)*(B0α)* is compact. Thus Ig,ϕ(n):B0αDβp is compact.

(iii) ⇒ (iv). Assume that Ig,ϕ(n):B0αDβpis (weakly) compact. Then Theorem 2.3 implies that (2.3) holds. Let fk(z)=zkk1-α for k ∈ ℕ and . Then {fk}B0α is a norm bounded sequence and fk → 0 as k → ∞ for every k ∈ ℕ on any compact subset of . Thus, by Lemma 2.5, we have

limkIg,ϕ(n)fkDβp=0.

Hence, for every ɛ > 0 there is an N such that for every kN,

lim (kα(k-1)!(k-n)!)pDϕ(z)p(k-n)g(z)plog (1z)βdA(z)<ɛ.

Thus, for each r ∈ (0, 1),

(Nα(N-1)!(N-n)!)prp(N-n)ϕ(z)>rg(z)p(log1z)βdA(z)<ɛ.

If we choose r((N-n)!(N-1)!)1(N-n)N-αN-n, then we have

ϕ(z)>rg(z)p(log1z)βdA(z)<ɛ.

Let fBB0α, where BB0α is the unit ball of B0α. The compactness of Ig,ϕ(n):B0αDβp impliess that for every ɛ > 0, there exists r ∈ (0, 1) such that

D|(Ig,ϕ(n)(f-ft))(z)|p(log1z)βdA(z)<ɛ,

where ft(z) = f(tz), .

So, (2.8) and (2.9) imply that

ϕ(z)>r|(Ig,ϕ(n)f)(z)|p(log1z)βdA(z)Cϕ(z)>r|(Ig,ϕ(n)(f-ft))(z)|p(log1z)βdA(z)+Cϕ(z)>r|(Ig,ϕ(n)(ft))(z)|p(log1z)βdA(z)Cϕ(z)>r|(Ig,ϕ(n)(f-ft))(z)|p(log1z)βdA(z)+Cϕ(z)>rft(n)(ϕ(z))pg(z)p(log1z)βdA(z)Cɛ+CɛsupzDft(n)(z)p=Cɛ(1+supzDft(n)(z)p),

where C is a positive constant. Thus, for every fBB0α and every ɛ > 0, there is a δ = δ(f, ɛ) (depended on f and ɛ) such that for every r ∈ [δ, 1) we have

ϕ(z)>r|(Ig,ϕ(n)f)(z)|p(log1z)βdA(z)<ɛ.

The compactness of Ig,ϕ(n):B0αDβp leads that Ig,ϕ(n)(BB0α)is a relatively compact subset of Dβp. Hence, for every ɛ > 0 there exists a finite family of functions f1,,fNBB0α such that for every fBB0α,Ig,ϕ(n)f-Ig,ϕ(n)fiDβp<ɛ for i ∈ {1, … , N}. i.e.,

D|(Ig,ϕ(n)(f-fi))(z)|p(log1z)βdA(z)<ɛ.

Hence, putting δ=max1iNδ(fi,ɛ), for any fBB0α we have

ϕ(z)>r|(Ig,ϕ(n)fi)(z)|p(log1z)βdA(z)<Cɛ,

if r ∈ [δ, 1).

Applying (2.12) to functions (fi)s(z) = fi(sz) for i = 1, 2 (the functions are as in Lemma 2.1), we obtain

ϕ(z)>rsg(z)p(1-sϕ(z)2)p(α+n-1)(log1z)βdA(z)Cϕ(z)>r|f1(n)(sϕ(z))|psg(z)p(log1z)βdA(z)+Cϕ(z)>r|f2(n)(sϕ(z))|psg(z)p(log1z)βdA(z)Cf1sB0αpϕ(z)>r|(Ig,ϕ(n)f1s)(z)|p(log1z)βdA(z)+Cf2sB0αpϕ(z)>r|(Ig,ϕ(n)f2s)(z)|p(log1z)βdA(z)<Cɛ,

for all r ∈ [δ, 1). By Fatou’s Lemma, this estimate implies (2.4).

(iv) ⇒ (i). Let {fk} be a bounded sequence in ℬα converges to zero on compact subsets of as k → ∞. Cauchy’s estimate implies that for any n ∈ ℕ, {fk(n) } also converges to zero on compact subset of as k → ∞. In particular

limksupwr|fk(n)(w)|=0.

By hypothesis, for every ɛ > 0 there is r ∈ (0, 1) such that,

ϕ(z)>rg(z)p(1-ϕ(z)2)p(α+n-1)(log1z)βdA(z)<ɛ.

Taking the function f(z) = zn, boundedness of Ig,ϕ(n) implies that

L=Dg(z)p(log1z)βdA(z)<.

So, using Lemma 2.2 and relations (2.14) and (2.15),

Ig,ϕ(n)fkDβpp=D|(Ig,ϕ(n)fk)(z)|p(log1z)βdA(z)=ϕ(z)r|fk(n)(ϕ(z))|pg(z)p(log1z)βdA(z)+r<ϕ(z)<1|fk(n)(ϕ(z))|pg(z)p(log1z)βdA(z)Lsupϕ(z)r|fk(n)(ϕ(z))|p+CɛfkBαp.

Letting k → ∞ and using (2.13), we conclude that Ig,ϕ(n)fkDβp0. Thus, by Lemma 2.5, Ig,ϕ(n):BαDβp is compact.

### 3. Boundednes and Compactness of the Operator Ig,ϕ(n):Dβp→Bα

In this section we study the boundedness and compactness of the generalized integration operator Ig,ϕ(n):DβpBα.

For every the holomorphic mapping from onto is defined by σa(z)=a-z1-a¯z.

### Lemma 3.1.([19])

Let β > −1, 0 < p < ∞ andfAβp. Then

f(z)(1-z2)2+βp((1+β)Df(z)p(1-z2)βdA(z))1pzD,

with equality if and only if f is a constant multiple of the functionfa(z)=(-σa(z))2+βp.

We recall the following fundamental lemma from [21]:

### Lemma 3.2.([21, Lemma 4.2.2])

Suppose, c is real, t > −1 and

Ic,t(z)=D(1-w2)t1-zw¯2+t+cdA(w).
• (a) If c < 0, then as a function of z, Ic,t(z) is bounded on.

• (b) If c > 0, then

Ic,t(z)~1(1-z2)c,         z1-.

• (c) If c = 0, then

I0,t(z)~log11-z2,         z1-.

Let 0 < p < ∞, β > −1 and fAβp. Then

f(z)fAβp(1-z2)2+βp,

for ([21]). Also, for fAβpand , we have

f(z)=(β+1)D(1-w2)βf(w)(1-zw¯)2+βdA(w),

See 4.2.1 of [21]. Differentiating under the integral sign n times, we obtain a constant Kn > 0 such that

f(n)(z)=KnD(1-w2)β(1-zw¯)n+2+βw¯nf(w)dA(w).

### Lemma 3.3

Let 0 < p < ∞, β > −1, m ∈ ℕ andfAβp. Then there exists a constant C > 0 such that

f(m)(z)CfAβp(1-z2)2+βp+m.
Proof

By (3.1), (3.3) and setting t=β-2+βp in Lemma 3.2, we have

f(m)(z)KmD(1-w2)βw¯m1-zw¯2+m+βf(w)dA(w)KmD(1-w2)β1-zw¯2+m+β·fAβp(1-w2)2+βpdA(w)=KmfAβpD(1-w2)t1-zw¯2+t+(m+2+βp)dA(w)~KmfAβp(1-z2)2+βp+m.

So, there exists a constant C such that

f(m)(z)CfAβp(1-z2)2+βp+m.

### Theorem 3.4

Let, n ∈ ℕ and ϕ be a holomorphic self-map of, 0 < α,p < ∞ and β > −1. Then the following statements are equivalent:

• Ig,ϕ(n):DβpBαis bounded.

• M=supzD(1-z2)αg(z)(1-ϕ(z)2)β+2p+n-1<.

Proof

(ii) ⇒ (i). Let fDβp. Then fAβp and from Lemma 3.3, there is a constant C > 0 such that

f(n)(z)=|(f(z))(n-1)|CfAβp(1-z2)2+βp+n-1CfDβp(1-z2)2+βp+n-1,

for . By (3.6),

supzD(1-z2)α|(Ig,ϕ(n)f)(z)|=supzD(1-z2)α|f(n)(ϕ(z))|g(z)CsupzD(1-z2)αg(z)(1-ϕ(z)2)β+2p+n-1fDβpCMfDβp.

Hence, Ig,ϕ(n):DβpBα is bounded.

(i) ⇒ (ii). Assume that (i) holds. Taking the function f(z)=znn!, the boundedness of Ig,ϕ(n) implies that

L=supzD(1-z2)αg(z)<.

Define the functions fa for every as follows:

fa(z)=0z(1-a2(1-a¯w)2)β+2pdw.

Then Lemma 3.1 implies that faDβp and faDβp~1. The boundedness of Ig,ϕ(n):DβpBα implies that there exists a constant C > 0 such that Ig,ϕ(n)faBαCfaDβpC. Also, it is easy to see that for any n ∈ ℕ and ,

fa(n+1)(z)=cn+1a¯n(1-a2)β+2p(1-a¯z)2(β+2)p+n,

where cn+1=Πm=0n-1(2(β+2)p+m). Since

fa(z)=(1-a2)β+2p(1-a¯z)2(β+2)p,

so, fa(z) follows from (3.8) by letting n = 0 and c1 = 1.

Since Ig,ϕ(n)fa(0)=0, letting a = ϕ(z) and using (3.8),

CIg,ϕ(n)fϕ(z)Ig,ϕ(n)fϕ(z)Bα=supzD(1-z2)α|fϕ(z)(n)(ϕ(z))|g(z)=supzD(1-z2)α|cnϕ(z)¯n-1(1-ϕ(z)2)β+2p(1-ϕ(z)2)2(β+2)p+n-1|g(z)cnϕ(z)n-1(1-z2)αg(z)(1-ϕ(z)2)β+2p+n-1

This shows that

supzDϕ(z)n-1(1-z2)αg(z)(1-ϕ(z)2)β+2p+n-1<.

For any δ, 0 < δ < 1, by (3.9),

supϕ(z)>δ(1-z2)αg(z)(1-ϕ(z)2)β+2p+n-1<.

For such that |ϕ(z)| ≤ δ, we have

(1-z2)αg(z)(1-ϕ(z)2)β+2p+n-1(1-z2)αg(z)(1-δ2)β+2p+n-1.

Hence, from (3.7) and (3.10), we have

supϕ(z)δ(1-z2)αg(z)(1-ϕ(z)2)β+2p+n-1<.

So,

supzD(1-z2)αg(z)(1-ϕ(z)2)β+2p+n-1<.

Thus, (3.5) holds and the proof of the theorem is completed.

Now, we investigate the compactness of Ig,ϕ(n):DβpBα. We use the following lemma which can be obtained from 2.4 by taking X=Dβpand Y =ℬα.

### Lemma 3.5

LetT:DβpBαbe a bounded operator. Then, T is compact if and only if given a bounded sequence {fn} inDβpsuch that fn → 0 uniformly on compact sets, then the sequence {Tfn} converges to zero in the norm ofα.

### Theorem 3.6

Let, n ∈ ℕ, ϕ be a holomorphic self-map of, 0 < α, p < ∞ and β > −1. If ||ϕ|| < 1 andIg,ϕ(n):DβpBαis bounded, ThenIg,ϕ(n)is compact.

Proof

Since Ig,ϕ(n):DβpBα is bounded, Theorem 3.4 implies that

M=supzD(1-z2)αg(z)(1-ϕ(z)2)β+2p+n-1<.

Let {fk} be a bounded sequence in the unit ball of Dβp that converges to 0 uniformly on compact subsets of as k → ∞. Then, Cauchy’s estimate implies that {fk(n) } for n ∈ ℕ also converges uniformly to 0 on compact subset of as k → ∞. This implies that,

limksupwϕ(D)fk(n)(w)=0.

So,

Ig,ϕ(n)fkBα=supzD(1-z2)α|(Ig,ϕ(n)fk)(z)|=supzD(1-z2)α|fk(n)(ϕ(z))|g(z)MsupzD|fk(n)(ϕ(z))|0         as k.

Hence, by Lemma 3.5, Ig,ϕ(n):DβpBα is compact.

### Theorem 3.7

Let, n ∈ ℕ, ϕ be a holomorphic self-map of, 0 < α,p < ∞ and β > −1. If ||ϕ|| = 1, then the following statements are equivalent:

• (i) Ig,ϕ(n):DβpBαis compact.

• (ii) Ig,ϕ(n):DβpBαis bounded and

limϕ(z)1(1-z2)αg(z)(1-ϕ(z)2)β+2p+n-1=0.

Proof

(i) ⇒ (ii). Suppose that Ig,ϕ(n):DβpBα is compact. Obviously, it is bounded. We consider the function fa for defined as in Theorem 3.4. This function converges to zero uniformly on compact subsets of as |a| → 1.

Now, pick the sequence such that |ϕ (zm)| → 1 as m → ∞. Using the test function fm(z) = fϕ(zm) (z), we obtain

Ig,ϕ(n)fmBα=supzD(1-z2)α|(Ig,ϕ(n)fm)(z)|=supzD(1-z2)α|fm(n)(ϕ(z))|g(z)=supzD(1-z2)α|fϕ(zm)(n)(ϕ(z))|g(z)(1-zm2)α|ϕ(zm)¯n(1-ϕ(zm)2)β+2p(1-ϕ(zm)¯ϕ(zm))2(β+2)p+n|g(z)(1-zm2)α|ϕ(zm)n-1(1-ϕ(zm)2)β+2p+n-1|g(zm).

As we mentioned above, since fm = fϕ(zm) converges to zero uniformly on compact subsets of as |ϕ (zm)| → 1, from Lemma 3.5 it follows that Ig,ϕ(n)fmBα0 as |ϕ (zm)| → 1 and so, (3.11) holds.

(ii) ⇒ (i). Let {fk} be a bounded sequence in the unit ball of Dβp that converges to 0 uniformly on compact subsets of as k → ∞. The relation (3.11) implies that for every ɛ > 0 there is a δ ∈ (0, 1) such that

sup{z:δ<ϕ(z)<1}(1-z2)αg(z)(1-ϕ(z)2)β+2p+n<ɛ.

Also, the uniform convergence of {fk} on compact subset of together with Cauchy’s estimate, implies that {fk(n) } for n ∈ ℕ converges to 0 on compact subset of as k → ∞. This implies that

limksupwδ|fk(n)(w)|=0.

Then, by (3.6), (3.7) and (3.12) we get the following:

Ig,ϕ(n)fkBα=supzD(1-z2)α|(Ig,ϕ(n)fk)(z)|=supzD(1-z2)α|fk(n)(ϕ(z))|g(z)=supϕ(z)δ(1-z2)α|fk(n)(ϕ(z))|g(z)+supδ<ϕ(z)<1(1-z2)α|fk(n)(ϕ(z))|g(z)Lsupwδ|fk(n)(w)|+CfkDβpsupδ<ϕ(z)<1(1-z2)αg(z)(1-ϕ(z)2)β+2p+n-1Lsupϕ(z)δ|fk(n)(ϕ(z))|+CɛfkDβp.

Letting k → ∞ and using (3.13), it folloes that Ig,ϕ(n)fkBα0. Thus, Lemma 3.5 implies that Ig,ϕ(n):DβpBα is compact.

1. DC. Chang, S. Li, and S. Stevic. On some integral operators on the unit polydisc and unit ball. Taiwanese J Math., 11(5)(2007), 1251-1285.
2. F. Colonna, and S. Li. Weighted composition operators from the Besov spaces into the Bloch spaces. Bull Malays Math Sci Soc., 36(4)(2013), 1027-1039.
3. CC. Cowen, and BD. MacCluer. Composition operators on spaces of analytic functions. Studies in Advanced Mathematics, , CRC Press, Boca Raton, FL, 1995.
4. ZH. He, and G. Cao. Generalized integration operators between Bloch-type spaces and F (p, q, s) spaces. Taiwanese J Math., 17(4)(2013), 1211-1225.
5. S. Kummar. Weighted composition operators between spaces of Dirichlet-type. Rev Math Complut., 22(2)(2009), 469-488.
6. S. Li, and S. Stevic. Generalized composition operators on Zygmund spaces and Bloch type spaces. J Math Anal Appl., 338(2)(2008), 1282-1295.
7. S. Li, and S. Stevic. Products of volterra type operator and composition operator from H and Bloch spaces to Zygmund spaces. J Math Anal Appl., 345(1)(2008), 40-52.
8. S. Li, and S. Stevic. Products of integral type operators and composition operators between Bloch-type spaces. J Math Anal Appl., 349(2)(2009), 596-610.
9. S. Li, and H. Wulan. Volterra type operators on Qk spaces. Taiwanese J Math., 14(1)(2010), 195-211.
10. L. Liu, Z. Lou, and C. Xiong. Essential norms of integral operators on spaces of analytic functions. Nonlinear Anal., 75(13)(2012), 5145-5156.
11. Y. Liu, and Y. Yu. Riemann-Stieltjes operator from mixed norm spaces to Zygmund-type spaces on the unit ball. Taiwanese J Math., 17(5)(2013), 1751-1764.
12. W. Ramey, and D. Ullrich. Bounded mean oscillation of Bloch pull-backs. Math Ann., 291(4)(1991), 591-606.
13. A. Salaryan, and H. Vaezi. Adjoints of rationally induced weighted composition operators on the Hardy, Bergman and Dirichlet spaces. Period Math Hungar., 72(2016), 76-89.
14. JH. Shapiro. Composition operators and classical function theory, , Springer-Verlag, New York, 1993.
15. SD. Sharma, and A. Sharma. Generalized integration operators from Bloch type spaces to weighted BMOA spaces. Demonstratio Math., 44(2)(2011), 373-390.
16. S. Stevic. Boundedness and Compactness of an integral type operator from Bloch-Type spaces with normal weights to F (p, q, s) space. Appl Math Comput., 218(9)(2012), 5414-5421.
17. Tjani. M. Compact composition operators on some moebius invariant Banach spaces, Thesis (Ph D), Michigan State University, 1996.
18. H. Vaeiz, and S. Houdfar. Composition and weighted composition operators from Bloch-type to Besov-type spaces. Math Reports., to appear.
19. D. Vukotic. A sharp estimate for functions in ℂn. Proc Amer Math Soc., 117(1993), 753-756.
20. Y. Yu, and Y. Liu. Integral-type operators from weighted Bloch spaces into Bergman-type spaces. Integral Transforms Spec Funct., 20(6)(2009), 419-428.
21. K. Zhu. Operator theory in function spaces. Monographs and Textbooks in Pure and Applied Mathematics, 139, Marcel Dekker, Inc, New York, 1990.
22. K. Zhu. Bloch type spaces of analytic functions. Rochy Mountain J Math., 23(1993), 1143-1177.
23. K. Zhu. Generalized weighted composition operators from Bloch type spaces to weighted Bergman spaces. Indian J Math., 49(2)(2007), 139-150.
24. X. Zhu. An integral type operator from H to Zygmund-type spaces. Bull Malays Math Sci Soc., 35(3)(2012), 679-686.