# Generalized Composition Operators fromℬμ Spaces to QK,ω(p,q) Spaces

Authors: Haiying Li; Tianshui Ma
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Let0<p<∞, let -2<q<∞, and let φ be an analytic self-map of 𝔻 and g∈H(𝔻). The boundedness and compactness of generalized composition operators (Cφgf)(z)=∫0z‍f'(φ(ξ))g(ξ)dξ,z∈𝔻,f∈H(𝔻), from ℬμ (ℬμ,0) spaces to QK,ω(p,q) spaces are investigated.


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## Body

## 1. Introduction and Preliminaries

Letφ be an analytic self-map of the open unit disc 𝔻 of the complex plane ℂ. Let H(𝔻) be the space of all analytic functions in 𝔻 and g∈H(𝔻). If X is a Banach space, then we denote the unit ball in X by BX. For 0<r<1, Ωr={z∈𝔻:|φ(z)|>r}.A positive continuous functionμ on the interval [0,1) is called normal if there exist three constants 0≤δ<1 and 0<a<b such that
(1)μ(r)(1-r)aisdecreasingon⁡[δ,1),lim⁡r→1μ(r)(1-r)a=0;μ(r)(1-r)bisincreasingon⁡[δ,1),lim⁡r→1μ(r)(1-r)b=∞.A functionf∈H(𝔻) belongs to the Bloch type space ℬμ if
(2)∥f∥ℬμ=|f(0)|+sup⁡z∈𝔻μ(z)|f′(z)|<∞,
where μ is normal and radial and μ(|z|)=μ(z). The space ℬμ is a Banach space with the norm ∥·∥ℬμ.The little Bloch type spaceℬμ,0 consists of all f∈ℬμ such that
(3)lim⁡|z|→1-μ(|z|)|f′(z)|=0.
For α>0, μ(|z|)=(1-|z|2)α, ℬμ is the α-Bloch space ℬα; for α=1, ℬα is the classical Bloch space; for example, see [1].For0<p<∞, -2<q<∞, a∈D, K:[0,∞)→[0,∞) is a nondecreasing function, and ω:(0,1]→(0,∞) is a given reasonable function. An analytic function f on D is said to belong to QK,ω(p,q) in [2] if
(4)∥f∥QK,ω(p,q)={sup⁡a∈D∫D‍|f′(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)}1/p<∞
and an analytic function f∈QK,ω,0(p,q) if
(5)lim⁡|a|→1-∫D‍|f′(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)=0,
where dA denotes the normalized Lebesgue area measure on D, g(z,a)=log⁡⁡(1/|ϕa(z)|) is a green function, and ϕa(z)=(a-z)/(1-a¯z).Q
K
,
ω
(
p
,
q
) classes are more general than many classes of analytic functions and have attracted a lot of attention in recent years. When ω≡1, QK,ω(p,q)=QK(p,q). When p=q=2, ω(t)=t, K(t)=tp, and QK,ω(p,q)=Qp. When ω≡1, K(t)=ts and QK,ω(p,q)=F(p,q,s). Moreover, the following results hold:(1)
QK,ω(p,q)⊂Bω(q+2)/p;(2)
QK,ω(p,q)=Bω(q+2)/p if and only if
(6)∫01‍K(log⁡1r)r(1-r2)2dr<∞,where(7)Bωα={(1-|z|)αω(1-|z|)f∈H(D):∥f∥Bωα=sup⁡z∈D(1-|z|)αω(1-|z|)|f′(z)|<∞,0<α<∞}.The composition operator is defined byCφf(z)=f(φ(z)), f∈H(𝔻). This operator has been studied for many years. The first setting was in the Hardy space H2, the space of functions analytic on 𝔻 (see [3]). Madigan and Matheson (see [1]) gave a characterization of the compact composition operators on the Bloch space ℬ. For more details, see [4–12]. In [13], Li and Stević defined the generalized composition operator as follows:
(8)(Cφgf)(z)=∫0z‍f′(φ(ξ))g(ξ)dξ,z∈𝔻,f∈H(𝔻).
The operator Cφg induces many known operators. When g=φ′, the operator Cφg is essentially (up to a constant) the composition operator Cφ. When φ(z)=z, the operator Cφg coincides with the operator Ig defined by
(9)(Igf)(z)=∫0z‍f′(ζ)g(ζ)dζ,ζ∈𝔻,f∈H(𝔻).
So the generalized composition operator Cφg can be considered as a generalization of the composition operator Cφ and the operator Ig.A fundamental problem in the study of generalized composition operatorsCφg is to investigate the relations between function theoretic properties of φ and g and operator theoretic properties of the restriction of Cφg to various Banach spaces of analytic functions. A lot of attentions have been attracted to study the problem on many Banach spaces of analytic functions in recent years. In [9], the authors studied composition operators from Bloch type spaces into QK(p,q) spaces. In [14], the authors characterized the boundedness and compactness of generalized composition operators on QK,ω(p,q) spaces. In [15], Rezaei and Mahyar studied generalized composition operators from logarithmic Bloch type spaces to QK type spaces. In [16], essential norms of generalized composition operators from Bloch type spaces to QK type spaces were given. In [17], generalized composition operators from F(p,q,s) spaces to Bloch-type spaces were characterized. In [18], Stević investigated generalized composition operators between mixed-norm space and some weighted spaces and from logarithmic Bloch spaces to mixed-norm spaces. In [3], Zhang and Liu studied generalized composition operators from Bloch type spaces to QK type spaces. In [19], generalized composition operator acting from Bloch-type spaces to mixed-norm space was studied. In [12], generalized composition operators from generalized weighted Bergman spaces to Bloch type spaces were investigated. In [20], generalized composition operators and Volterra composition operators on Bloch spaces on the unit ball were studied. This paper is devoted to investigating the boundedness and compactness of generalized composition operators Cφg from ℬμ (ℬμ,0) spaces to QK,ω(p,q) spaces. Throughout this paper, constants are denoted by C; they are positive and may differ from one occurrence to the other.


## 2. Main Results and Their Proofs

To derive our results, we need the following lemmas.Lemma 1.
Assume that0<p<∞, -2<q<∞, K is a nonnegative nondecreasing function on [0,∞), and ω:(0,1]→(0,∞) is a given reasonable function. Assume that μ is a normal function, φ is an analytic self-map of 𝔻, and g∈H(𝔻). Then Cφg:ℬμ(ℬμ,0)→QK,ω(p,q) is compact if and only if, for every bounded sequence {fk} in ℬμ(ℬμ,0) which converges to 0 uniformly on compact subsets of 𝔻, lim⁡k→∞∥Cφgfk∥QK,ω(p,q)=0.Lemma1 can be proved in a standard way of Theorem  3.11 in [4].The following lemma is similar to Lemma  2.2 in [5, 7], using the results for the Hadamard gap series and following a technique used before in the Bloch space in [5, 7]. Specific details can be seen in [9].Lemma 2.
Letμ:[0,1)→[0,∞) be a nonincreasing radial weight function and normal on the interval [0,1). Then there exist two functions f1,f2∈ℬμ such that, for each z∈𝔻,
(10)|f1′(z)|+|f2′(z)|≥Cμ(|z|).Theorem 3.
Assume that0<p<∞, -2<q<∞, φ is an analytic self-map of 𝔻, μ is a normal function, K is nonnegative and nondecreasing in [0,∞), and ω:(0,1]→(0,∞) is a given reasonable function. Then the following statements are equivalent:(a)
Cφg:ℬμ→QK,ω(p,q) is bounded;(b)
Cφg:ℬμ,0→QK,ω(p,q) is bounded;(c)
(11)sup⁡a∈𝔻∫𝔻‍|g(z)|p(1-|z|2)qK(g(z,a))μp(|φ(z)|)ωp(1-|z|)dA(z)<∞.Proof.
(a)⇒ (b) Since ℬμ,0⊂ℬμ, then (a) implies (b).
(b)⇒ (c) Suppose (b) holds; then ∥Cφgf∥QK,ω(p,q)≤∥Cφg∥∥f∥ℬμ for all f∈ℬμ,0. For any given f∈ℬμ, the function ft(z)=f(tz), 0<t<1, belongs to ℬμ,0 and ∥ft∥ℬμ≤∥f∥ℬμ. Let f1, f2 be the functions from Lemma 2 and we get
(12)sup⁡a∈𝔻∫𝔻‍|g(z)|p(1-|z|2)qK(g(z,a))μp(t|φ(z)|)ωp(1-|z|)dA(z)≤2p∥Cφg∥p(∥f1t∥ℬμp+∥f2t∥ℬμp)≤2p∥Cφg∥p(∥f1∥ℬμp+∥f2∥ℬμp).
Then (11) holds with Fatou’s Lemma.
(c)⇒ (a) For f∈ℬμ,
(13)∥Cφgf∥QK,ω(p,q)p=sup⁡a∈𝔻∫𝔻‍|f′(φ(z))|p|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)≤∥f∥ℬμpsup⁡a∈𝔻∫𝔻‍|g(z)|p(1-|z|2)qK(g(z,a))μp(|φ(z)|)ωp(1-|z|)dA(z).Theorem 4.
Assume that0<p<∞, -2<q<∞, φ is an analytic self-map of 𝔻, μ is a normal function, K is nonnegative and nondecreasing in [0,∞), and ω:(0,1]→(0,∞) is a given reasonable function. Then the following statements are equivalent:(a)
Cφg:ℬμ→QK,ω(p,q) is compact;(b)
Cφg:ℬμ,0→QK,ω(p,q) is compact;(c)
(14)M=sup⁡a∈𝔻∫𝔻‍|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)<∞,(15)lim⁡r→1sup⁡a∈𝔻∫Ωr‍|g(z)|p(1-|z|2)qK(g(z,a))μp(|φ(z)|)ωp(1-|z|)dA(z)=0.Proof.
(a)⇒ (b) Since ℬμ,0⊂ℬμ, then (a) implies (b).
(b)⇒ (c) Assume that (b) holds; then we have (14), Let
(16)fn(z)=znnμ(1-(1/n)),z∈𝔻.
Then {fn}n∈ℕ is bounded in ℬμ,0 and fn→0 uniformly on the compact subsets of 𝔻 as n→∞. Since Cφg:ℬμ,0→QK,ω(p,q) is compact, then by Lemma 1(17)lim⁡n→∞∥Cφgfn∥QK,ω(p,q)=0.
This means, for any given ɛ>0, there exists N∈ℕ such that n≥N implies
(18)sup⁡a∈𝔻∫𝔻‍|φn-1(z)|pμp(1-(1/n))ωp(1-|z|)×|g(z)|p(1-|z|2)qK(g(z,a))dA(z)<ɛ.
Hence, for 0<r<1,
(19)sup⁡a∈𝔻1μp(1-(1/N))×∫𝔻‍|φN-1(z)|p|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)≥sup⁡a∈𝔻1μp(1-(1/N))×∫Ωr‍|φN-1(z)|p|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)≥r(N-1)pμp(1-(1/N))×sup⁡a∈𝔻∫Ωr‍|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z).
Choosing r such that r(N-1)p/μp(1-(1/N))>1, then
(20)sup⁡a∈𝔻∫Ωr‍|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)<ɛ.
Forf∈ℬμ,0, let ft(z)=f(tz) for 0<t<1. Then ft∈ℬμ,0 and ft→f uniformly on compact subsets of 𝔻 as t→1. Since Cφg is compact, then ∥Cφgft-Cφgf∥QK,ω(p,q)→0 as t→1. Then for every ɛ>0 there exists t0∈(0,1) such that
(21)∫𝔻‍((|ft0′(φ(z))-f′(φ(z))|p|g(z)|p×(1-|z|2)qK(g(z,a))|ft0′(φ(z))-f′(φ(z))|p|g(z)|p)×(ωp(1-|z|))-1)dA(z)<ɛ.
By the triangle inequality, then
(22)sup⁡a∈𝔻∫Ωr‍|f′(φ(z))|p|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)≤2psup⁡a∈𝔻∫Ωr‍((|ft0′(φ(z))-f′(φ(z))|p|g(z)|p×(1-|z|2)qK(g(z,a))|ft0′(φ(z))-f′(φ(z))|p|g(z)|p)×(ωp(1-|z|))-1)dA(z)+2psup⁡a∈𝔻∫Ωr‍((|ft0′(φ(z))|p|g(z)|p×(1-|z|2)qK(g(z,a))|ft0′(φ(z))|p|g(z)|p)×(ωp(1-|z|))-1)dA(z)<2pɛ+2p∥ft0′∥H∞p×∫Ωr‍|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)<2p(1+∥ft0′∥H∞p)ɛ,
which means, for any ɛ>0 and f∈Bℬμ,0, there exists δ=δ(ɛ,f)>0 such that for r∈[δ,1)(23)sup⁡a∈𝔻∫Ωr‍|f′(φ(z))|p|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)<ɛ.
SinceCφg is compact, Cφg(Bℬμ,0) is relatively compact in QK,ω(p,q); then there are finite functions f1,f2,…,fm∈Bℬμ,0 such that, for any ɛ>0 and f∈Bℬμ,0, we can find fk(1≤k≤m) satisfying
(24)sup⁡a∈𝔻∫𝔻‍((|f′(φ(z))-fk′(φ(z))|p|g(z)|p×(1-|z|2)qK(g(z,a))|f′(φ(z))-fk′(φ(z))|p|g(z)|p)×(ωp(1-|z|))-1)dA(z)<ɛ.
Takeδ=max⁡1≤j≤mδ(ɛ,fj). Then for r∈[δ,1)(25)sup⁡a∈𝔻∫Ωr‍|fk′(φ(z))|p|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)<ε.
Then(26)sup⁡a∈𝔻∫Ωr‍|f′(φ(z))|p|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)<2ɛ.
Hence, we have shown that for anyɛ>0 there exists δ∈[0,1) such that for all f∈Bℬμ,0(27)sup⁡a∈𝔻∫Ωr‍|f′(φ(z))|p|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)<2ɛ.
Letfj,j=1,2, be the functions in Lemma 2; then, for 0<t<1, the functions fjt(z)=fj(tz) are included in ℬμ,0. Thus by Lemma 2 and Fatou’s Lemma, we get (15).
(c)⇒ (a) Assume that (14) and (15) hold. Assume that {fn}n∈ℕ is a bounded sequence in ℬμ such that fn→0 uniformly on compact subsets of 𝔻. Assume ∥fn∥ℬμ≤1; by (15), for any given ɛ>0, there exists r∈[0,1) such that
(28)sup⁡a∈𝔻∫Ωr‍|g(z)|p(1-|z|2)qK(g(z,a))μp(|φ(z)|)ωp(1-|z|)dA(z)<ɛ.
Sincefn→0 uniformly on compact subsets of 𝔻, then fn′→0 uniformly on compact subsets of 𝔻. Then for above ɛ, there exists N∈ℕ such that n>N implies |fn′|<ɛ for |z|≤r. Thus,
(29)∫𝔻‍|fn′(φ(z))|p|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)≤{∫Ωr‍+∫𝔻∖Ωr‍}((|fn′(φ(z))|p|g(z)|p×(1-|z|2)qK(g(z,a))|fn′(φ(z))|p|g(z)|p)×(ωp(1-|z|))-1)dA(z)≤∥fn∥ℬμp∫Ωr‍|g(z)|p(1-|z|2)qK(g(z,a))μp(|φ(z)|)ωp(1-|z|)dA(z)+ɛp∫𝔻‍|g(z)|p(1-|z|2)qK(g(z,a))ωp(1-|z|)dA(z)≤ɛ+ɛpM.
Hence,∥Cφgfn∥QK,ω(p,q)→0 as n→∞. Thus Cφg:ℬμ→QK,ω(p,q) is compact.Remark 5.
Forα>0, μ(|z|)=(1-|z|2)α, ℬμ is the α-Bloch space ℬα. Let μ(|z|)=(1-|z|2)α and ω≡1 in Theorems 3 and 4; we easily obtain the following results in [3].Corollary 6.
Assume that0<p<∞, -2<q<∞, α>0, φ is an analytic self-map of 𝔻, and K is a nonnegative nondecreasing function on [0,∞). Then the following statements are equivalent:(a)
Cφg:ℬα→QK(p,q) is bounded;(b)
Cφg:ℬ0α→QK(p,q) is bounded;(c)
(30)sup⁡a∈𝔻∫𝔻‍|g(z)|p(1-|z|2)qK(g(z,a))(1-|φ(z)|2)pαdA(z)<∞.Corollary 7.
Assume that0<p<∞, -2<q<∞, α>0, φ is an analytic self-map of 𝔻, and K is a nonnegative nondecreasing function on [0,∞). Then the following statements are equivalent:(a)
Cφg:ℬα→QK(p,q) is compact;(b)
Cφg:ℬ0α→QK(p,q) is compact;(c)
(31)sup⁡a∈𝔻∫𝔻‍|g(z)|p(1-|z|2)qK(g(z,a))dA(z)<∞,lim⁡r→1sup⁡a∈𝔻∫Ωr‍|g(z)|p(1-|z|2)qK(g(z,a))(1-|φ(z)|2)pαdA(z)=0.Remark 8.
Asg=φ′, the operator Cφg is essentially the composition operator Cφ, since the difference Cφg-Cφ is constant. Moreover, ω≡1; QK,ω(p,q)=QK(p,q). Let g=φ′and ω≡1 in Theorems 3 and 4; we easily obtain the following results in [9].Corollary 9.
Assume that0<p<∞, -2<q<∞, φ is an analytic self-map of 𝔻, μ is a normal function, and K is nonnegative and nondecreasing in [0,∞). Then the following statements are equivalent:(a)
Cφ:ℬμ→QK(p,q) is bounded;(b)
Cφ:ℬμ,0→QK(p,q) is bounded;(c)
(32)sup⁡a∈𝔻∫𝔻‍|φ′(z)|p(1-|z|2)qK(g(z,a))μp(|φ(z)|)dA(z)<∞.Corollary 10.
Assume that0<p<∞, -2<q<∞, φ is an analytic self-map of 𝔻, μ is a normal function, and K is nonnegative and nondecreasing in [0,∞). Then the following statements are equivalent:(a)
Cφ:ℬμ→QK(p,q) is compact;(b)
Cφ:ℬμ,0→QK(p,q) is compact;(c)
φ∈QK(p,q) and
(33)lim⁡r→1sup⁡a∈𝔻∫Ωr‍|φ′(z)|p(1-|z|2)qK(g(z,a))μp(|φ(z)|)dA(z)=0.Problem 11.
Can the boundedness and compactness of the generalized composition operatorCφg:QK,ω(p,q)→ℬμ be characterized by use of function theoretic properties of φ and g?



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