AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 608578 10.1155/2013/608578 608578 Research Article Differences of Composition Operators Followed by Differentiation between Weighted Banach Spaces of Holomorphic Functions Chen Cui Chen Ren-Yu http://orcid.org/0000-0001-9943-9010 Zhou Ze-Hua Jeribi Aref 1 Department of Mathematics Tianjin University Tianjin 300072 China tju.edu.cn 2013 12 11 2013 2013 25 05 2013 02 10 2013 2013 Copyright © 2013 Cui Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We characterize the boundedness and compactness of differences of the composition operators followed by differentiation between weighted Banach spaces of holomorphic functions in the unit disk. As their corollaries, some related results on the differences of composition operators acting from weighted Banach spaces to weighted Bloch type spaces are also obtained.

1. Introduction

Let H(𝔻) and S(𝔻) denote the class of holomorphic functions and analytic self-maps on the unit disk 𝔻 of the complex plane of , respectively. Let v be a strictly positive continuous and bounded function (weight) on 𝔻.

The weighted Bloch space v is defined to be the collection of all fH(𝔻) that satisfy (1)fv:=supz𝔻v(z)|f(z)|<. Provided we identify the functions that differ by a constant, ·v becomes a norm and v a Banach space.

The Hv={fH(𝔻):fv:=supz𝔻v(z)|f(z)|<} endowed with the weighted sup-norm ·v is referred to as the weighted Banach space. In setting the so-called associated weight plays an important role.

For a weight v, its associated weight v~ is defined as follows: (2)v~(z)=1sup{|f(z)|:fHv,fv1}=1δz(Hv), where δz denotes the point evaluation at z. By  the associated weight v~ is continuous, v~v>0, and for every z𝔻 we can find gzHv with gzv1 such that gz(z)=1/v~(z).

We say that a weight v is radial if v(z)=v(|z|) for every z𝔻. A positive continuous function v is called normal if there exist three positive numbers δ,t,s and t>s, such that for every z𝔻 with |z|[δ,1), (3)v(|z|)(1-|z|)s0,v(|z|)(1-|z|)t,as  |z|1. A radial, nonincreasing weight is called typical if lim|z|1v(z)=0. When studying the structure and isomorphism classes of the space Hv, Lusky [2, 3] introduced the following condition (L1) (renamed after the author) for radial weights: (L1)infnv(1-2-n-1)v(1-2-n)>0, which will play a great role in this paper. In case v is a radial weight, if it is also normal, then it satisfies the condition (L1). Moreover, the radial weights with (L1) are essential (e.g., see ); that is, we can find a constant C>0 such that (4)v(z)v~(z)Cv(z)for  everyz𝔻.

Let φS(𝔻); the composition operator Cφ induced by φ is defined by (5)(Cφf)(z)=f(φ(z)),fH(𝔻),z𝔻. This operator has been studied for many years. Readers interested in this topic are referred to the books , which are excellent sources for the development of the theory of composition operators, and to the recent papers [8, 9] and the references therein.

By differentiation we are led to the linear operator DCφ:H(𝔻)H(𝔻), f(fφ)φ, which is regarded as the product of the composition operator and the differentiation operator denoted by Df=f, fH(𝔻). The product operators have been studied, for example, in  and the references therein.

Recently, there has been an increasing interest in studying the compact difference of composition operators acting on different spaces of holomorphic functions. Some related results on differences of the composition operators or weighted composition operators on weighted Banach spaces of analytic functions, Bloch-type spaces, and weighted Bergman spaces can be found, for example . More recently, Wolf [28, 29] characterized the boundedness and compactness of differences of composition operators between weighted Bergman spaces or weighted Bloch spaces and weighted Banach spaces of holomorphic functions in the unit disk. The same problems between standard weighted Bergman spaces were discussed by Saukko .

For each φ and ψ in S(𝔻), we are interested in the operators DCφ-DCψ, and we characterize boundedness and compactness of the operators DCφ-DCψ between weighted Banach spaces of holomorphic functions in terms of the involved weights as well as the inducing maps. As a corollary we get a characterization of boundedness and compactness about the differences of composition operators Cφ-Cψ acting from weighted Banach spaces to weighted Bloch type spaces.

Throughout this paper, we will use the symbol C to denote a finite positive number, and it may differ from one occurrence to another. And for each ω𝔻, gω denotes a function in Hu with gωu1 such that |gω(ω)|=1/u~(ω). The existence of this function is a consequence of Montel's theorem as can be seen in .

2. Background and Some Lemmas

Now let us state a couple of lemmas, which are used in the proofs of the main results in the next sections. The first lemma is taken from .

Lemma 1.

Let v be a radial weight satisfying condition (L1). There is a constant C>0 (depending only on the weight v) such that for all fHv, (6)|f(z)|Cfvv(z)(1-|z|2), for every z𝔻.

In order to handle the differences, we need the pseudohyperbolic metric. Recall that for any point a𝔻, let φa(z)=(z-a)/(1-a¯z), z𝔻. It is well known that each φa is a homeomorphism of the closed unit disk 𝔻¯ onto itself. The pseudohyperbolic metric on 𝔻 is defined by (7)ρ(a,z)=|φa(z)|. We know that ρ(a,z) is invariant under automorphisms (see, e.g., ).

Lemma 2.

Let v be a radial weight satisfying condition (L1). There is a constant C>0 such that for all fHv, (8)|v(z)(1-|z|2)f(z)-v(ω)(1-|ω|2)f(ω)|Cfvρ(z,ω), for all z,ω𝔻.

Proof.

For fHv, let u(z)=v(z)(1-|z|2), by Lemma 1, we obtain fHu, so by Lemma 3.2 in  and Lemma 1, there is a constant C>0 such that (9)|u(z)f(z)-u(ω)f(ω)|Cfuρ(z,ω)Cfvρ(z,ω) for each z,ω𝔻. This completes the proof.

Remark 3.

From Lemma 2, it is not hard to see that for any z,ωr𝔻={z𝔻:|z|2<r<1}, then (10)|v(z)(1-|z|2)f(z)-v(ω)(1-|ω|2)f(ω)|Cfrvρ(z,ω), for any fHv, where frv=supzr𝔻v(z)|f(z)|.

The following result is well known (see, e.g., ).

Lemma 4.

Assume that v is a normal weight. Then for every fH(𝔻) the following asymptotic relationship holds: (11)supz𝔻v(z)|f(z)||f(0)|+supz𝔻v(z)(1-|z|)|f(z)|.

Here and below we use the abbreviated notation AB to mean A/CBCA for some inessential constant C>0.

The following lemma is the crucial criterion for compactness, and its proof is an easy modification of that of Proposition 3.11 of .

Lemma 5.

Suppose that u,vH(𝔻) and φ,ψS(𝔻). Then the operator DCφ-DCψ:HuHv is compact if and only if whenever {fn} is a bounded sequence in Hu with fn0 uniformly on compact subsets of 𝔻, and then (DCφ-DCψ)fn0, as n.

3. The Boundedness of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M109"><mml:mi>D</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>φ</mml:mi></mml:mrow></mml:msub><mml:mo mathvariant="bold">-</mml:mo><mml:mi>D</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

In this section we will characterize the boundedness of DCφ-DCψ:HuHv. For this purpose, we consider the following three conditions: (12)supz𝔻v(z)|φ(z)|ρ(φ(z),ψ(z))u(φ(z))(1-|φ(z)|2)<;(13)supz𝔻v(z)|ψ(z)|ρ(φ(z),ψ(z))u(ψ(z))(1-|ψ(z)|2)<;(14)supz𝔻|v(z)φ(z)(1-|φ(z)|2)u(φ(z))-v(z)ψ(z)(1-|ψ(z)|2)u(ψ(z))|<.

Theorem 6.

Suppose that v is an arbitrary weight and that u is a normal and radial weight. Then the following statements are equivalent.

DCφ-DCψ:HuHv is bounded.

The conditions (12) and (14) hold.

The conditions (13) and (14) hold.

Proof.

First, we prove the implication (i) (ii). Assume that DCφ-DCψ:HuHv is bounded. Fixing w𝔻, we consider the function fw defined by (15)fw(z)=0zφψ(w)(t)gφ(w)(t)1-|φ(w)|2(1-φ(w)¯t)2dt,z𝔻.

Next prove that fwHu. In fact, (16)fw(z)=φψ(w)(z)gφ(w)(z)1-|φ(w)|2(1-φ(w)¯z)2. By Lemma 4 we have (17)supz𝔻u(z)|fw(z)||fw(0)|+supz𝔻u(z)(1-|z|)|fw(z)|=supz𝔻u(z)(1-|z|)|φψ(w)(z)|×|gφ(w)(z)|1-|φ(w)|2|1-φ(w)¯z|2Csupz𝔻u(z)|gφ(w)(z)|, thus fwHu, and fwuC. Note that fw(φ(w))=ρ(φ(w),ψ(w))/u~(φ(w))(1-|φ(w)|2), and fw(ψ(w))=0. So by the boundedness of DCφ-DCψ:HuHv, it then follows that (18)>(DCφ-DCψ)fwv=supz𝔻v(z)|fw(φ(z))φ(z)-fw(ψ(z))ψ(z)|v(w)|fw(φ(w))φ(w)-fw(ψ(w))ψ(w)|=v(w)|φ(w)|u~(φ(w))(1-|φ(w)|2)ρ(φ(w),ψ(w)), for any w𝔻. Since w𝔻 is an arbitrary element, then from (18) and (4), we can obtain (12).

Next we prove (14). For given w𝔻, we consider the function (19)hw(z)=0zgψ(w)(t)1-|ψ(w)|2(1-ψ(w)¯t)2dt,z𝔻. Like for fw above, we can show that hwHu with hw(z)=gψ(w)(z)((1-|ψ(w)|2)/(1-ψ(w)¯z)2). One sees that hw(ψ(w))=1/u~(ψ(w))(1-|ψ(w)|2). Then (20)>(DCφ-DCψ)hwvv(w)|hw(φ(w))φ(w)-hw(ψ(w))ψ(w)|=|I(w)+J(w)|, where (21)I(w)=v(w)φ(w)u(φ(w))(1-|φ(w)|2)×[u(φ(w))(1-|φ(w)|2)hw(φ(w))-u(ψ(w))(1-|ψ(w)|2)hw(ψ(w))],J(w)=u(ψ(w))(1-|ψ(w)|2)hw(ψ(w))×[v(w)φ(w)u(φ(w))(1-|φ(w)|2)-v(w)ψ(w)u(ψ(w))(1-|ψ(w)|2)]. By Lemma 2 and (12), we conclude that |I(w)|<, which combines with (20), and we obtain that (22)|J(w)|=C|v(w)φ(w)u(φ(w))(1-|φ(w)|2)-v(w)ψ(w)u(ψ(w))(1-|ψ(w)|2)|< for all w𝔻; thus (14) holds.

(ii) (iii). Assume that (12) and (14) hold, we need only to show that (13) holds. In fact, (23)v(z)|ψ(z)|u(ψ(z))(1-|ψ(z)|2)ρ(φ(z),ψ(z))v(z)|φ(z)|u(φ(z))(1-|φ(z)|2)ρ(φ(z),ψ(z))+|v(z)φ(z)u(φ(z))(1-|φ(z)|2)-v(z)ψ(z)u(ψ(z))(1-|ψ(z)|2)  |×ρ(φ(z),ψ(z)), from which, using (12) and (14), the desired condition (13) holds.

(iii) (i). Assume that (13) and (14) hold. By Lemmas 1 and 2, for any fHu, we have (24)v(z)|(DCφ-DCψ)f(z)|=v(z)|f(φ(z))φ(z)-f(ψ(z))ψ(z)|=v(z)|ψ(z)|u(ψ(z))(1-|ψ(z)|2)×|u(ψ(z))(1-|ψ(z)|2)f(ψ(z))-u(φ(z))(1-|φ(z)|2)f(φ(z))|+u(φ(z))(1-|φ(z)|2)|f(φ(z))|×|v(z)ψ(z)u(ψ(z))(1-|ψ(z)|2)-v(z)φ(z)u(φ(z))(1-|φ(z)|2)|Cfuv(z)|ψ(z)|u(ψ(z))(1-|ψ(z)|2)ρ(φ(z),ψ(z))+CfuCfu, from which it follows that DCφ-DCψ:HuHv is bounded. The whole proof is complete.

Corollary 7.

Suppose that v is an arbitrary weight and that u is a normal and radial weight satisfying condition (L1). Then the following statements are equivalent.

Cφ-Cψ:Huv is bounded.

The conditions (12) and (14) hold.

The conditions (13) and (14) hold.

4. The Compactness of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M152"><mml:mi>D</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>φ</mml:mi></mml:mrow></mml:msub><mml:mo mathvariant="bold">-</mml:mo><mml:mi>D</mml:mi><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

In this section, we turn our attention to the question of compact difference. Here we consider the following conditions: (25)v(z)φ(z)u(φ(z))(1-|φ(z)|2)ρ(φ(z),ψ(z))0as|φ(z)|1;(26)v(z)ψ(z)u(ψ(z))(1-|ψ(z)|2)ρ(φ(z),ψ(z))0as|ψ(z)|1;(27)v(z)φ(z)u(φ(z))(1-|φ(z)|2)-v(z)ψ(z)u(ψ(z))(1-|ψ(z)|2)0as|φ(z)|1,|ψ(z)|1.

Theorem 8.

Suppose that v is an arbitrary weight and that u is a normal and radial weight. Then DCφ-DCψ:HuHv is compact if and only if DCφ-DCψ:HuHv is bounded and the conditions (25)–(27) hold.

Proof.

First we suppose that DCφ-DCψ:HuHv is bounded and the conditions (25)–(27) hold. Then the conditions (12)–(14) hold by Theorem 6. From (25)–(27), it follows that for any ε>0, there exists 0<r<1 such that (28)v(z)|φ(z)|u(φ(z))(1-|φ(z)|2)ρ(φ(z),ψ(z))εfor  |φ(z)|>r,(29)v(z)|ψ(z)|u(ψ(z))(1-|ψ(z)|2)ρ(φ(z),ψ(z))εfor  |ψ(z)|>r,(30)|v(z)φ(z)u(φ(z))(1-|φ(z)|2)-v(z)ψ(z)u(ψ(z))(1-|ψ(z)|2)|εfor  |φ(z)|>r,|ψ(z)|>r.

Now, let {fn} be a sequence in Hu such that fnuL (constant) and {fn}0 uniformly on compact subsets of 𝔻. By Lemma 5 we need only to show that (DCφ-DCψ)fnv0 as n. A direct calculation shows that (31)v(z)|fn(φ(z))φ(z)-fn(ψ(z))ψ(z)|=|In(z)+Jn(z)|, where (32)In(z)=v(z)φ(z)u(φ(z))(1-|φ(z)|2)×[u(φ(z))(1-|φ(z)|2)fn(φ(z))-u(ψ(z))(1-|ψ(z)|2)fn(ψ(z))],Jn(z)=u(ψ(z))(1-|ψ(z)|2)fn(ψ(z))×[v(z)φ(z)u(φ(z))(1-|φ(z)|2)-v(z)ψ(z)u(ψ(z))(1-|ψ(z)|2)].

We divide the argument into a few cases.

Case 1 ( | φ ( z ) | r and | ψ ( z ) | r ). By the assumption, note that {fn} converges to zero uniformly on E={w:|w|r} as n; using (14) and Cauchy's integral formula, it is easy to check that Jn(z)0,  n uniformly for all z with |ψ(z)|r.

On the other hand, it follows from Remark 3 after Lemma 2 and (12) that(33)|In(z)|Cv(z)|φ(z)|u(φ(z))(1-|φ(z)|2)ρ(φ(z),ψ(z))×sup|w|ru(w)|fn(w)|Cε.

Case 2 ( | φ ( z ) | > r and | ψ ( z ) | r ). As in the proof of Case 1, Jn(z)0 uniformly as n. On the other hand, using Lemma 2 and (28) we obtain |In(z)|CLε.

Case 3 ( | φ ( z ) | > r and | ψ ( z ) | > r ). For n sufficiently large, by Lemma 2 and (28) we obtain that |In(z)|CLε. Meanwhile, |Jn(z)|CLε by Lemma 1 and (30).

Case 4 ( | φ ( z ) | r and | ψ ( z ) | > r ). We rewrite (34)v(z)|fn(φ(z))φ(z)-fn(ψ(z))ψ(z)|=|Pn(z)+Qn(z)|, where (35)Pn(z)=v(z)ψ(z)u(ψ(z))(1-|ψ(z)|2)×[u(φ(z))(1-|φ(z)|2)fn(φ(z))-u(ψ(z))(1-|ψ(z)|2)fn(ψ(z))],Qn(z)=u(φ(z))(1-|φ(z)|2)fn(φ(z))×[v(z)φ(z)u(φ(z))(1-|φ(z)|2)-v(z)ψ(z)u(ψ(z))(1-|ψ(z)|2)].

The desired result follows by an argument analogous to that in the proof of Case 2. Thus, together with the above cases, we conclude that (36)(DCφ-DCψ)fnv=supz𝔻v(z)|fn(φ(z))φ(z)-fn(ψ(z))ψ(z)|Cε, for sufficiently large n. Employing Lemma 5 we obtain the compactness of DCφ-DCψ:HuHv.

For the converse direction, we suppose that DCφ-DCψ:HuHv is compact. From which we can easily obtain the boundedness of DCφ-DCψ:HuHv. Next we only need to show that (25)–(27) hold.

Let {zn} be a sequence of points in 𝔻 such that |φ(zn)|1 as n. Define the functions (37)fn(z)=0zφψ(zn)(t)gφ(zn)(t)1-|φ(zn)|2(1-φ(zn)¯t)2dt. Clearly, {fn} converges to 0 uniformly on compact subsets of 𝔻 as n and fnHu with fnuL for all n. Moreover, (38)fn(φ(zn))=ρ(φ(zn),ψ(zn))u~(φ(zn))(1-|φ(zn)|2),fn(ψ(zn))=0.

By the compactness of DCφ-DCψ:HuHv and Lemma 5, it follows that (DCφ-DCψ)fnv0,  n. On the other hand, using (38) we have (39)(DCφ-DCψ)fnv=supz𝔻v(z)|fn(φ(z))φ(z)-fn(ψ(z))ψ(z)|v(zn)|fn(φ(zn))φ(zn)-fn(ψ(zn))ψ(zn)|=v(zn)|φ(zn)|u~(φ(zn))(1-|φ(zn)|2)ρ(φ(zn),ψ(zn)).

Letting n in (39), it follows that (25) holds. The condition (26) holds for the similar arguments.

Now we need only to show the condition (27) holds. Assume that {zn} is a sequence in 𝔻 such that |φ(zn)|1 and |ψ(zn)|1 as n. Define the function (40)hn(z)=0zgψ(zn)(t)1-|ψ(zn)|2(1-ψ(zn)¯t)2dt. It is easy to check that {hn} converges to 0 uniformly on compact subsets of 𝔻 as n and hnHu with hnuL for all nN. Note that hn(z)=gψ(zn)(z)((1-|ψ(zn)|2)/(1-ψ(zn)¯z)2), then hn(ψ(zn))=[u~(ψ(zn))(1-|ψ(zn)|2)]-1, and it follows from Lemma 5 that (DCφ-DCψ)hnv0, n. On the other hand we obtain that (41)(DCφ-DCψ)hnvv(zn)|hn(φ(zn))φ(zn)-hn(ψ(zn))ψ(zn)|=|I(zn)+J(zn)|, where (42)I(zn)=v(zn)φ(zn)u(φ(zn))(1-|φ(zn)|2)×[u(φ(zn))(1-|φ(zn)|2)hn(φ(zn))-u(ψ(zn))(1-|ψ(zn)|2)hn(ψ(zn))],J(zn)=u(ψ(zn))(1-|ψ(zn)|2)hn(ψ(zn))×[v(zn)φ(zn)u(φ(zn))(1-|φ(zn)|2)-v(zn)ψ(zn)u(ψ(zn))(1-|ψ(zn)|2)]=C[v(zn)φ(zn)u(φ(zn))(1-|φ(zn)|2)-v(zn)ψ(zn)u(ψ(zn))(1-|ψ(zn)|2)].

By Lemma 2 and the condition (25) that has been proved, we get I(zn)0, n. This combines with (41), and we obtain J(zn)0, n. This shows that (27) is true. The whole proof is complete.

Corollary 9.

Suppose that v is an arbitrary weight and that u is a normal and radial weight satisfying condition (L1). Then Cφ-Cψ:Huv is compact if and only if Cφ-Cψ:Huv is bounded and the conditions (25)–(27) hold.

5. Examples

In this final section we give an example of function u,v,φ,ψ for which the operator DCφ-DCψ between the weighted Banach spaces to show that the condition in Theorem 8 that DCφ-DCψ is bounded is necessary.

Example 1.

In this example we will show that there exist weight u (normal, radial) and v, analytic self-maps on the unit disk φ,ψ such that the conditions (25)–(27) in Theorem 8 are satisfied while DCφ-DCψ:HuHv is not compact.

Let (43)φ(z)=1M+1j=1zjj2, and ψ(z)=-φ(z), where M=j=1(1/j2).

Since for |z|<1, we have |φ(z)|<M/(M+1) so φ belongs to S(𝔻), as well as ψ. Moreover, |ψ(z)| and |ψ(z)| can never tend to 1 for any z𝔻, which means that conditions (25)–(27) hold trivially.

Now we will show that DCφ-DCψ:HuHv is not bounded, and then not compact. Let zk=1-1/k, and then it is easy to check that φ(zk)M/(M+1) and ψ(zk)-M/(M+1) as k. So (44)ρ(φ(zk),ψ(zk))2(M/(M+1))1+(M/(M+1))2>0. However, since φ(z)=(1/(M+1))j=1(zj-1/j), then |φ(zk)| as k. Thus choose u(z)=1-|z|2 and v(z)=u(φ(z)), and we can obtain (45)v(zk)|φ(zk)|ρ(φ(zk),ψ(zk))u(φ(zk))(1-|φ(zk)|2),ask. Hence DCφ-DCψ does not map Hu boundedly into Hv by Theorem 6.

Acknowledgment

This work was supported in part by the National Natural Science Foundation of China (Grant nos. 11371276, 11301373, and 11201331).

Bierstedt K. D. Bonet J. Taskinen J. Associated weights and spaces of holomorphic functions Studia Mathematica 1998 127 2 137 168 MR1488148 Lusky W. On weighted spaces of harmonic and holomorphic functions Journal of the London Mathematical Society 1995 51 2 309 320 10.4064/sm175-1-2 MR1325574 Lusky W. On the isomorphism classes of weighted spaces of harmonic and holomorphic functions Studia Mathematica 2006 175 1 19 45 10.4064/sm175-1-2 MR2261698 Bonet J. Domański P. Lindström M. Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions Canadian Mathematical Bulletin 1999 42 2 139 148 10.4153/CMB-1999-016-x MR1692002 Cowen C. C. MacCluer B. D. Composition Operators on Spaces of Analytic Functions 1995 Boca Raton, Fla, USA CRC Press MR1397026 Shapiro J. H. Composition Operators and Classical Function Theory 1993 New York, NY, USA Springer MR1237406 Zhu K. H. Operator Theory in Function Spaces 1990 139 New York, NY, USA Marcel Dekker MR1074007 Zhou Z. H. Chen R. Y. Weighted composition operators from F(p, q, s) to Bloch type spaces on the unit ball International Journal of Mathematics 2008 19 8 899 926 10.1142/S0129167X08004984 MR2446507 Zhou Z. Shi J. Compactness of composition operators on the Bloch space in classical bounded symmetric domains Michigan Mathematical Journal 2002 50 2 381 405 10.1307/mmj/1028575740 MR1914071 Li S. Stević S. Composition followed by differentiation between H and α-Bloch spaces Houston Journal of Mathematics 2009 35 1 327 340 MR2491884 Li S. Stević S. Products of integral-type operators and composition operators between Bloch-type spaces Journal of Mathematical Analysis and Applications 2009 349 2 596 610 10.1016/j.jmaa.2008.09.014 MR2456215 Stević S. Norm and essential norm of composition followed by differentiation from α-Bloch spaces to H Applied Mathematics and Computation 2009 207 1 225 229 10.1016/j.amc.2008.10.032 MR2492736 Stević S. Weighted differentiation composition operators from H and Bloch spaces to nth weighted-type spaces on the unit disk Applied Mathematics and Computation 2010 216 12 3634 3641 10.1016/j.amc.2010.05.014 MR2661728 Wolf E. Composition followed by differentiation between weighted Banach spaces of holomorphic functions Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales A 2011 105 2 315 322 10.1007/s13398-011-0040-8 MR2826710 Wu Y. Wulan H. Products of differentiation and composition operators on the Bloch space Collectanea Mathematica 2012 63 1 93 107 10.1007/s13348-010-0030-8 MR2887113 Zhu X. Multiplication followed by differentiation on Bloch-type spaces Bulletin of the Allahabad Mathematical Society 2008 23 1 25 39 MR2405835 Bonet J. Lindström M. Wolf E. Differences of composition operators between weighted Banach spaces of holomorphic functions Journal of the Australian Mathematical Society 2008 84 1 9 20 10.1017/S144678870800013X MR2469264 Hosokawa T. Differences of weighted composition operators on the Bloch spaces Complex Analysis and Operator Theory 2009 3 4 847 866 10.1007/s11785-008-0062-1 MR2570115 Hosokawa T. Izuchi K. Essential norms of differences of composition operators on H Journal of the Mathematical Society of Japan 2005 57 3 669 690 MR2139727 Hosokawa T. Ohno S. Differences of composition operators on the Bloch spaces Journal of Operator Theory 2007 57 2 229 242 MR2328996 Lindström M. Wolf E. Essential norm of the difference of weighted composition operators Monatshefte für Mathematik 2008 153 2 133 143 10.1007/s00605-007-0493-1 MR2373366 Manhas J. S. Compact differences of weighted composition operators on weighted Banach spaces of analytic functions Integral Equations and Operator Theory 2008 62 3 419 428 10.1007/s00020-008-1630-5 MR2461128 Moorhouse J. Compact differences of composition operators Journal of Functional Analysis 2005 219 1 70 92 10.1016/j.jfa.2004.01.012 MR2108359 Ohno S. Differences of weighted composition operators on the disk algebra Bulletin of the Belgian Mathematical Society 2010 17 1 101 107 MR2656674 Song X. J. Zhou Z. H. Differences of weighted composition operators from Bloch space to H on the unit ball Journal of Mathematical Analysis and Applications 2013 401 1 447 457 10.1016/j.jmaa.2012.12.030 MR3011286 Wolf E. Compact differences of composition operators Bulletin of the Australian Mathematical Society 2008 77 1 161 165 10.1017/S0004972708000166 MR2411875 Zhou Z. H. Liang Y. X. Differences of weighted composition operators from Hardy space to weighted-type spaces on the unit ball Czechoslovak Mathematical Journal 2012 62 3 695 708 10.1007/s10587-012-0040-7 MR2984629 Wolf E. Differences of weighted composition operators between weighted Banach spaces of holomorphic functions and weighted Bloch type spaces Cubo 2010 12 2 19 27 MR2724877 Wolf E. Differences of composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions Glasgow Mathematical Journal 2010 52 2 325 332 10.1017/S0017089510000029 MR2610976 Saukko E. Difference of composition operators between standard weighted Bergman spaces Journal of Mathematical Analysis and Applications 2011 381 2 789 798 10.1016/j.jmaa.2011.03.058 MR2802114 Dai J. N. Ouyang C. H. Differences of weighted composition operators on Hα(BN) Journal of Inequalities and Applications 2009 2009 1 19 127431 Avetisyan K. Stević S. Extended Cesàro operators between different Hardy spaces Applied Mathematics and Computation 2009 207 2 346 350 10.1016/j.amc.2008.10.055 MR2489106