Jasbir Singh MANHAS

Department of Mathematics and Statistics,College of Science,Sultan Qaboos University,P.O.Box 36,P.C.123,Al-Khod,Oman

E-mail:manhas@squ.edu.om

Ruhan ZHAO

Department of Mathematics,SUNY Brockport,Brockport,NY 14420,USA

Department of Mathematics,Shantou University,Shantou 515063,China

E-mail:rzhao@brockport.edu

Abstract We characterize boundedness and compactness of products of differentiation operators and weighted composition operators between weighted Banach spaces of analytic functions and weighted Zygmund spaces or weighted Bloch spaces with general weights.

Key words differentiation operators;weighted composition operators;weighted Banach space of analytic functions;Bloch-type spaces

1 Introduction

Let D be the unit disk in the complex plane C,and let H(D)denote the space of analytic functions on D.For an analytic function ψ and an analytic self map ? on D,the weighted composition operator Wψ,?on H(D)is defined as

As a combination of composition operators and pointwise multiplication operators,weighted composition operators arise naturally.For example,surjective isometries on Hardy spaces Hp,and Bergman spaces Ap,1≤p<∞,p 6=2,are given by weighted composition operators.See[5,7].For the information on composition and weighted composition operators see,for example,books[4,15,16].

Let D be the differentiation operator,defined by Df=f′.It is usually unbounded on many analytic function spaces.In this paper,our goal is to study the product of weighted composition operators and differentiation operators,which are defined respectively by

for every f∈H(D).

For the special case ψ(z)=1,the above operators reduce to the products of composition operators and differentiation operators DC?and C?D,defined respectively by DC?f=f′(?)?′and C?Df=f′(?).Note that DC?=W?′,?D.These operators were first studied by Hibschweiler and Portnoy in[6]and then by Ohno in[14],where boundedness and compactness of DC?between Hardy spaces and Bergman spaces were investigated.Later many other authors also studied these operators between various function spaces.In a series of papers,[17–20],Stevi? studied operators DWψ,?from various spaces into weighted Banach spaces or nth weighted Banach spaces,either on the unit disk or on the unit ball.Li[8]studied DWψ,?and Wψ,?D on H∞,and Li,Wang and Zhang[9]studied DWψ,?between weighted Bergman spaces and H∞.

Recently,in[12,13],we studied boundedness,compactness and essential norms of the operators DWψ,?and Wψ,?D between weighted Banach space of analytic functions with general weights.In this paper we continue this line of research,to characterize boundedness and compactness of DWψ,?and Wψ,?D from weighted Banach space of analytic functions to weighted Zygmund spaces and weighted Bloch spaces on D,with general weights.

2 Preliminaries

Let v be a weight function that is strictly positive,continuous and bounded on D.The weighted Banach space of analytic functionsconsists of analytic functions f on D satisfying

For a weight function v,we also define the general weighted Bloch space Bvas follows

and the general weighted Zygmund space as follows

If we identify functions that differ by a constant,then Bvis a Banach space under the norm k·kBv,and the space Bvis isometric toby the differentiation operator D:f→f′.Similarly,if we identify functions that differ by a linear function,then Zvis a Banach space under the norm k·kZv,and the space Zvis isometric toby the second differentiation operator D2:f → f′′,and isometric to Bvby the differentiation operator D:f → f′.If we take the standard weight vα(z)=(1?|z|2)αin Bv,α >0,then we have the α-Bloch spaces.In particular,if α=1,the space Bvis the classical Bloch space.For more information on these spaces,we refer to[22].

For a given weight v its associated weight v is defined as follows

where δz:→C is the point evaluation linear functional.The associated weight plays an important role for the setting of general weighted spaces.From[2]we know that the following relations between the weights v andv hold

(v1)0

(v2)kfkv≤1 if and only if kfkv≤1;

(v3)for each z∈D,there exists fzin the closed unit ball ofsuch that|fz(z)|=1/(z).

We say that a weight v is radial if v(z)=v(|z|)for every z∈D,and a radial non-increasing weight is typical ifv(z)=0.We also say that a weight v is essential if there is a constant c>0 such that v(z)≤v(z)≤cv(z)for every z∈D.The following condition(L1)introduced by Lusky in[10]is crucial to our study.

It is known that radial weights satisfying(L1)are always essential(see[3]).It is easy to see that the standard weights vα(z)=(1 ? |z|2)α,where α >0,and the logarithmic weight vβ(z)=(1?log(1?|z|2))β,where β <0,satisfy condition(L1).The weighted Banach spaces of analytic functions have important applications in functional analysis,complex analysis,partial differential equations,convolution equations and distribution theory.For more details on these spaces we refer to[1,2,10,11].

3 Boundedness and Compactness of the Operators DWψ,? and Wψ,?D from into Zw

In this section we give necessary and sufficient conditions for the boundedness and compactness of the products of weighted composition operators and differentiation operators frominto Zw.Recall that an operator T between two Banach spaces is said to be compact if T maps every bounded set into a relatively compact set.In order to prove boundedness and compactness of the operator DWψ,?(or Wψ,?D),we need the following two results.The first one is Lemma 5 in[21].

Proposition 3.1 Let v be a radial weight satisfying condition(L1).Then there exists a constant cv>0,depending only on the weight v,such that for every function f∈,

for every z∈D and every non-negative integer n.

The proof of the following result is similar to the one for Proposition 3.11 in[4].

Proposition 3.2 Let v and w be arbitrary weights on D,let ? be an analytic self-map of D,and let ψ ∈H(D).Then the operator DWψ,?(or Wψ,?D):→Zw(or Bw)is compact if and only if it is bounded and for every bounded sequence{fn}in H∞vsuch that fn→0 uniformly on compact subsets of D,DWψ,?fn→ 0(or Wψ,?Dfn→ 0)in Zw(or Bw).

We begin with proving the boundedness criterion for DWψ,?from H∞vto Zw.

Theorem 3.3 Let v be a radial weight satisfying condition(L1)and w be an arbitrary weight.Let ψ ∈H(D)and ? be an analytic self-map of D.Then the operator DWψ,?:H∞v→Zwis bounded if and only if the following conditions are satisfied

Proof In the following proof,we will simply denote kDWψ,?kHv∞→Zwby kDWψ,?k.First,suppose that the operator DWψ,?:→Zwis bounded.Fix a∈D,and let ?a(z)=(z??(a))/(1?)for every z∈ D.It is easy to compute that the derivative of ?ais given by

Also,by(v3),there exists a function fain the closed unit ball ofsuch that|fa(?(a))|=1/v(?(a)).Since v satisfies(L1),v is essential,and so without loss of generality we may replace v by v.Now,consider the function

Clearly kgakv≤1.It is easy to see that ga(?(a))=0,(?(a))=0,(?(a))=0,and

From this,it follows that This proves(iv).

Now,to prove condition(iii),again fix a ∈ D and define ha(z)=(?a(z))2fa(z)for every z∈ D.Clearly khakv≤ 1.It is easy to see that ha(?(a))=0,(?(a))=0,and

Further,using Proposition 3.1 in the above inequality,it follows that

Here we have used(3.2)in the last inequality.Thus

This proves(iii).

To prove condition(ii), fix a∈D and consider ka(z)=?a(z)fa(z)for every z∈D.Again we have kkakv≤ 1,ka(?(a))=0 and

Further,using Proposition 3.1 in the above inequality,it follows that

Here we have used(3.2)and(3.3)in the last inequality.Thus

This proves(ii).

Finally,to prove condition(i),we have

Further,using(3.2),(3.3),(3.4)and Proposition 3.1 the above inequality implies that

This proves(i).Hence conditions(i)–(iv)are all proved.

Conversely,suppose that conditions(i)–(iv)are satisfied.We shall show that DWψ,?is bounded.Let f∈.Then using Proposition 3.1,we have From this inequality and conditions(i)–(iv)we conclude that DWψ,?:→Zwis bounded.Also,by(3.2),(3.3),(3.4)and(3.5),there exists some constant C>0 such that

From(3.6)and(3.6),we also obtain the asymptotic relation(3.1).The proof is completed.?

In the next theorem we characterize the compactness of the operator DWψ,?:→Zw.

Theorem 3.4 Let v be a radial weight satisfying condition(L1)and w be an arbitrary weight.Let ψ ∈H(D)and ? be an analytic self-map of D such that DWψ,?:→Zwis bounded.Then DWψ,?:H∞v→Zwis compact if and only if the following conditions are satisfied

Proof Suppose that the operator DWψ,?:→Zwis bounded.Let f(z)=1 and g(z)=z for every z∈D.Then clearly f,g∈and hence

Using(3.7)and(3.8)and the boundedness of ?(z),it follows that

Now,let f(z)=z2and g(z)=z3for every z∈D.Then again f,g∈,and hence

Using boundedness of ?(z),(3.8)implies that

and

Using boundedness of ?(z),(3.9),(3.10)and(3.12)imply that

Also,inequalities(3.9),(3.11),(3.13),(3.14)and the boundedness of ?(z)imply that

Assume that the operator DWψ,?:→Zwis compact.To prove condition(iv),let{zn}be a sequence with|?(zn)|→ 1 such that

By choosing a subsequence we may assume that there exists n0∈ N such that|?(zn)|n≥ 1/2 for every n ≥ n0.For each ?(zn),we define the function

for every z∈D.Also,by(v3),there exists a function fn∈such that kfnkv≤1 and|fn(?(zn))|=1/(?(zn)).Since v satisfies(L1),v is essential,andv can be replaced by v.For each n∈N,we define the function

Clearly,kgnkv≤1.It is easy to see that gn(?(zn))=0,g′n(?(zn))=0,g′′n(?(zn))=0,and

for all n≥n0.Thus{gn}is a bounded sequence in H∞vthat tends to zero uniformly on compact subsets of D.Since the operator DWψ,?:H∞v→Zwis compact,by Proposition 3.2,kDWψ,?gnkZw→0 as n→ ∞.Now

From this,it follows that

which proves condition(iv).

To prove condition(iii),again let{zn} ? D be a sequence with|?(zn)|→ 1 such that

Again,using functions ?nand fnas obtained earlier,we define the function

Clearly,khnkv≤ 1.It is easy to see that hn(?(zn))=0,h′

n(?(zn))=0,and

for all n≥n0.Thus{hn}is a bounded sequence in H∞vthat tends to zero uniformly on compact subsets of D.Since the operator DWψ,?:H∞v→Zwis compact,again by Proposition 3.2,kDWψ,?hnkZw→0 as n→ ∞.Now

Using Proposition 3.1,the above inequality implies that

Using(3.16),the above inequality implies that

which proves condition(iii).

To prove condition(ii),again let{zn} ? D be a sequence with|?(zn)|→ 1 such that

Again,using functions ?nand fnas obtained earlier,we define the function

Clearly,kknkv≤ 1.It is easy to see that kn(?(zn))=0,and

for all n≥n0.Thus{kn}is a bounded sequence inthat tends to zero uniformly on compact subsets of D.Since the operator DWψ,?:→Zwis compact,by Proposition 3.2,kDWψ,?knkZw→0 as n→∞.Thus

Using Proposition 3.1,the above inequality implies that

Using(3.16)and(3.17),the above inequality implies that

which proves condition(ii).

Finally,to prove condition(i),again let{zn} ? D be a sequence with|?(zn)|→ 1 such that

For each n,we define the function

It is easy to see that kFnkv≤1,and

Thus{Fn}is a bounded sequence inthat tends to zero uniformly on compact subsets of D.Since the operator DWψ,?:→Zwis compact,by Proposition 3.2,kDWψ,?FnkZw→0 as n→∞.Thus

Using Proposition 3.1,the above inequality implies that

From(3.16)(3.17)and(3.18),the above inequality implies that

which proves condition(i).This completes the proof of necessary part.

Conversely,we assume that conditions(i)–(iv)holds.Let{fn}be a bounded sequence inwhich converges to zero uniformly on compact subsets of D.We may assume that kfnkv≤1 for every n ∈ N.To show that the operator DWψ,?is compact,according to Proposition 3.2,it is enough to show that the operator DWψ,?is bounded and kDWψ,?fnkZw→ 0 as n → ∞.In view of(3.8),(3.9),(3.14)and(3.15),let

Also,from the given conditions(i)–(iv),for every ε>0,there exists 0

where cvis the constant given in Proposition 3.1.Also,since fn→0 uniformly on compact subsets of D,Cauchy’s estimate gives that,andconverge to 0 uniformly on compact subsets of D.This implies that there exists n0∈N such that for every n≥n0,we have

Now,applying Proposition 3.1 and using(3.19)–(3.30),we have that for every n ≥ n0,

Using the facts that fn,andconverge to 0 uniformly on compact subsets of D as n→∞,it can be easily shown that|(ψ ·fn? ?)′(0)|→ 0 and|(ψ ·fn? ?)′′(0)|→ 0 as n → ∞.Thus we have shown that kDWψ,?fnkZw→ 0 as n → ∞.This proves that DWψ,?is a compact operator.The proof of the theorem is completed. ?

Similar to the proofs of Theorem 3.3 and Theorem 3.4,we get the following results related to the boundedness and compactness of the operator Wψ,?D:→Zw.We omit the proofs.

Theorem 3.5 Let v be a radial weight satisfying condition(L1)and w be an arbitrary weight.Let ψ ∈H(D)and ? be an analytic self-map of D.Then the operator Wψ,?D:→Zwis bounded if and only if the following conditions are satisfied

Moreover,if Wψ,?D:→Zwis bounded then

Theorem 3.6 Let v be a radial weight satisfying condition(L1)and w be an arbitrary weight.Let ψ ∈H(D)and ? be an analytic self-map of D such that Wψ,?D:→Zwis bounded.Then Wψ,?D:H∞v→Zwis compact if and only if the following conditions are satisfied

From Theorems 3.3–3.6,we get the following corollaries related to the product of composition operators and differentiation operators.

Corollary 3.7 Let v be a radial weight satisfying condition(L1)and w be an arbitrary weight.Let ? be an analytic self-map of D.Then the operator DC?:H∞v→Zwis bounded if and only if the following conditions are satisfied

Corollary 3.8 Let v be a radial weight satisfying condition(L1)and w be an arbitrary weight.Let ? be an analytic self-map of D such that DC?:→Zwis bounded.Then DC?:→Zwis compact if and only if the following conditions are satisfied

Corollary 3.9 Let v be a radial weight satisfying condition(L1)and w be an arbitrary weight.Let ? be an analytic self-map of D.Then the operator C?D:H∞v→Zwis bounded if and only if the following conditions are satisfied

Corollary 3.10 Let v be a radial weight satisfying condition(L1)and w be an arbitrary weight.Let ? be an analytic self-map of D such that C?D:H∞v→Zwis bounded.Then C?D:H∞v→Zwis compact if and only if the following conditions are satisfied

4 Boundedness and Compactness of the Operators DWψ,? and Wψ,?D frominto Bw

Using the techniques of Theorem 3.3 and Theorem 3.4,we can obtain the following similar results of the operator DWψ,?(or Wψ,?D):H∞v→Bw.We omit the similar proofs.

Theorem 4.1 Let v be a radial weight satisfying condition(L1)and w be an arbitrary weight.Let ψ ∈H(D)and ? be an analytic self-map of D.Then the operator DWψ,?:H∞v→Bwis bounded if and only if the following conditions are satisfied

Theorem 4.2 Let v be a radial weight satisfying condition(L1)and w be an arbitrary weight.Let ψ ∈H(D)and ? be an analytic self-map of D such that DWψ,?:H∞v→Bwis bounded.Then DWψ,?:H∞v→Bwis compact if and only if the following conditions are satisfied

Theorem 4.3 Let v be a radial weight satisfying condition(L1)and w be an arbitrary weight.Let ψ ∈H(D)and ? be an analytic self-map of D.Then the operator Wψ,?D:H∞v→Bwis bounded if and only if the following conditions are satisfied

Theorem 4.4 Let v be a radial weight satisfying condition(L1)and w be an arbitrary weight.Let ψ ∈H(D)and ? be an analytic self-map of D such that Wψ,?D:H∞v→Bwis bounded.Then Wψ,?D:H∞v→Bwis compact if and only if the following conditions are satisfied

AcknowledgementsThe second author would like to thank Sultan Qaboos University for the support and hospitality.