AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation93102010.1155/2009/931020931020Research ArticleOn Convexity of Composition and Multiplication Operators on Weighted Hardy SpacesHedayatianKarimKarimiLotfollahStevićStevoDepartment of MathematicsShiraz UniversityShiraz 71454Iranshirazu.ac.ir20092511200920093009200902112009041120092009Copyright © 2009This 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.

A bounded linear operator T on a Hilbert space , satisfying T2h2+h22Th2 for every h, is called a convex operator. In this paper, we give necessary and sufficient conditions under which a convex composition operator on a large class of weighted Hardy spaces is an isometry. Also, we discuss convexity of multiplication operators.

1. Introduction and Preliminaries

We denote by B() the space of all bounded linear operators on a Hilbert space . An operator TB() is said to be convex, if

T2h2+h22Th2 for each h. Note that if T is a convex operator then the sequence (Tnh2)nN forms a convex sequence for every h. Taking ΔT=T*T-I, it is easily seen that T is a convex operator if and only if T*ΔTTΔT.

A weighted Hardy space is a Hilbert space of analytic functions on the open unit disc D for which the sequence (zj)j=0 forms a complete orthogonal set of nonzero vectors. It is usually assumed that 1=1. Writing β(j)=zj, this space is denoted by H2(β) and its norm is given by

j=0ajzj2=j=0|aj|2β(j)2.

Let φ be an analytic map of the open unit disc D into itself, and define Cφ(f)=fφ whenever f is analytic on D. The function φ is called the symbol of the composition operator. For a positive integer n, the nth iterate of φ, denoted by φn, is the function obtained by composing φ with itself n times; also φ0 is defined to be the identity function. Denote the reproducing kernel at zD, for the space H2(β), by Kz. Then f,Kz=f(z) for every fH2(β). It is known that Cφ*(Kz)=Kφ(z) for all z in D. The generating function for H2(β) is the function given by

k(z)=j=0zjβ(j)2. This function is analytic on D. Moreover, if wD then Kw(z)=k(w̅z) and Kw2=k(|w|2) (see ).

Recently, there has been a great interest in studying operator theoretic properties of composition and weighted composition operators, see, for example, monographs [1, 2], papers , as well as the reference therein.

Isometric operators on weighted Hardy spaces, especially those that are composition operators are discussed by many authors. Isometries of the Hardy space H2 among composition operators are characterized in [17, page 444],  and [12, page 66]. Indeed, it is shown that the only composition operators on H2 that are isometries are the ones induced by inner functions vanishing at the origin. Bayart  generalized this result and showed that every composition operator on H2 which is similar to an isometry is induced by an inner function with a fixed point in the unit disc. The surjective isometries of Hp, 1p< that are weighted composition operators have been described by Forelli . Carswell and Hammond  proved that the isometric composition operators of the weighted Bergman space Aα2 are the rotations. Cima and Wogen  have characterized all surjective isometries of the Bloch space. Furthermore, the identification of all isometric composition operators on the Bloch space is due to Colonna . Some related results can be found also in [3, 4, 6, 2125].

Herein, we are interested in studying the convexity of composition and multiplication operators acting on a weighted Hardy space H2(β). First, we give some preliminary facts on convex operators. Next, we will offer necessary and sufficient conditions under which a convex composition operator may be isometry on a large class of weighted Hardy spaces containing Hardy, Bergman, and Dirichlet spaces. We also discuss on convexity of the adjoint of a composition operator. Finally, we will obtain similar results for multiplication operators and their adjoints. For a good reference on isometric multiplication operators the reader can see .

Throughout this paper, T is assumed to be a bounded linear operator on a Hilbert space . It is easy to see that for every convex operator T, the sequence (T*nΔTTn)n forms an increasing sequence. We use this fact to prove the following theorem.

Theorem 1.1.

If T is a convex operator then so is every nonnegative integer power of T.

Proof.

We argue by using mathematical induction. The convexity of T implies that the result holds for k=1. Suppose that T*nΔTnTnΔTn, then T*n+1ΔTn+1Tn+1-ΔTn+1=T*n+1(T*ΔTnT+ΔT)Tn+1-ΔTn+1=T*2(T*nΔTnTn)T2+T*n+1ΔTTn+1-ΔTn+1T*2ΔTnT2+T*nΔTTn-ΔTn+1=T*2(T*nTn-I)T2+T*nΔTTn-T*(T*nTn-I)T-ΔT=T*n(T*2T2)Tn-T*2T2+T*nΔTTn-T*n(T*T)Tn+T*T-ΔT=T*n(T*2T2-I)Tn-T*2T2+I2T*nΔTTn-T*2T2+I2T*ΔTT-T*2T2+I=T*ΔTT-ΔT0. So the result holds for k=n+1.

Proposition 1.2.

If T is a convex operator, then for every nonnegative integer n, T*nTnnΔT+I.

Proof.

We give the assertion by using mathematical induction on n. The result is clearly true for n=1. Suppose that T*nTnnΔT+I. Thus, T*n+1Tn+1=T*(T*nTn)TT*(nΔT+I)T=nT*ΔTT+T*T=n(T*2T2-2T*T+I)+nT*T+T*T-nI(n+1)T*T-nI=(n+1)ΔT+I. So the result holds for k=n+1.

Proposition 1.3.

Let T() be a convex operator and let h be such that supk0Tkh<. If ΔT0, then Th=h.

Proof.

By applying Proposition 1.2, we observe that for every nonnegative integer n, nΔTh,h+h2Tnh2supk0Tkh2<. Letting n, the positivity of ΔT implies that ΔTh=0; hence, Th=h.

Proposition 1.4.

Let {en}n=0 be an orthonormal basis for and let T() be a convex operator satisfying ΔT0. Suppose that there is a nonnegative integer i and a scalar αi with 0<|αi|1 so that Tei=αiei, then =ni{en} is an invariant subspace for T.

Proof.

Using Proposition 1.2, we see that ei2αinei2=Tnei2=T*nTnei,einΔTei,ei+ei2 for every n0. Let n. Since ΔT is a positive operator, we conclude that ΔTei=0. Consequently, T*ei=(1/αi)T*Tei=(1/αi)ei. Now, if f then Tf,ei=0; hence, Tf.

2. Composition Operators

Our purpose in this section is to discuss on convex composition operators on a weighted Hardy space. Recall that an operator T in B() is an isometry, if ΔT=0. At first, we give an example of a nonisometric composition operator T on a weighted Hardy space such that T*ΔTTΔT0. For simplicity of notation, ΔCφ is denoted by Δφ.

Example 2.1.

Consider the weighted Hardy space H2(β) with weight sequence (β(n))n given by β(n)=n+1. Define φ:DD by φ(z)=z2. It is easily seen that Cφ(H2(β))H2(β), and an application of the closed graph theorem shows that Cφ is bounded. Now, a simple calculation shows that (Cφ*ΔφCφ-Δφ)(zk),zk=Cφ2zk2-2Cφzk2+zk2>0 for all k0; besides Δφzk,zk=Cφzk2-zk2 which is positive for all k1, and zero whenever k=0. It follows that Cφ*ΔφCφΔφ0, but Cφ is not an isometry.

Proposition 2.2.

Suppose that T:H2(β)H2(β) is a convex operator satisfying T1=1 and ΔT0, then M={fH2(β):f(0)=0} is a nontrivial invariant subspace of T.

Proof.

Clearly M is a nontrivial closed subspace of T. To show that M is invariant for T, apply Proposition 1.4 for the Hilbert space =H2(β), the orthonormal basis {en}n given by en=zn/β(n),  i=0 and α0=1.

Example 2.3.

Consider the Bergman space A2(D) consisting of all analytic functions f on the open unit disc D, for which f2=D|f(z)|2dA(z)<, where dA(z) is the normalized Lebesgue area measure on D. If fA2(D) is represented by f(z)=n=0anzn, then f2=n=0|an|2n+1. Also, {zk}k forms an orthogonal basis for A2(D). Fix nonnegative integers k and n, and observe that Cφnzk2=φkn2=D|φkn(z)|2dA(z)DdA(z)=1. Thus, Proposition 1.3 implies that Cφ*ΔφCφΔφ0 if and only if Cφ is an isometry. In this case, taking T=Cφ and f(z)=z in Proposition 2.2, we conclude that φ(0)=0; thus, the Schwarz lemma implies that |φ(z)||z| for all zD. On the other hand, if f(z)=z then D|φ(z)|2dA(z)=Cφf2=f2=D|z|2dA(z), and so |φ(z)|=|z| almost everywhere with respect to the area measure. Hence, φ(z)=eiθz for some θ[0,2π).

Example 2.4.

Consider the Hardy space H2(D). If φ is an analytic self-map of the unit disc, then φ induces a bounded composition operator, and Cφnzk1 for all nonnegative integers n and k. Thus, by Proposition 1.3, Cφ*ΔφCφΔφ0 if and only if Cφ is an isometry.

Recall that the Dirichlet space 𝒟 is the set of all functions analytic on D whose derivatives lie in the Bergman space A2(D). The Dirichlet norm is defined by

f𝒟2=|f(0)|2+D|f(z)|2dA(z). If φ is a univalent self-map of D, then Cφ is bounded on 𝒟 [2, page 18]. Also, the area formula [1, page 36], shows that

Cφf𝒟2=|(foφ)(0)|2+D|f(z)|2nφ(z)dA(z), where nφ(z) is, as usual, the counting function defined as the cardinality of the set {wD:φ(w)=z}.

In the next theorem, we characterize all convex composition operators Cφ on 𝒟 satisfying Δφ0. Note that we cannot use Proposition 1.3 for the Dirichlet space, thanks to the fact that in general the positive powers of Cφ are not uniformly bounded on the zi's.

Theorem 2.5.

If Cφ is convex on the Dirichlet space 𝒟, then Δφ0 if and only if Cφ is an isometry.

Proof.

One implication is clear. Suppose that Δφ is a positive operator, and take T=Cφ in Proposition 2.2. Since the identity function is in the subspace M={f𝒟:f(0)=0}, we conclude that φ(0)=0. Thus, in light of (2.9), to show that Cφ is an isometry it is sufficient to prove that D|f(z)|(1-nφ)(z)dA(z)=0,f𝒟. Let f be any function in the Dirichlet space 𝒟. Then 0(Cφ*ΔφCφ-Δφ)(f),f=D|f(z)|2(nφ2-2nφ+1)(z)dA(z). Furthermore, 0Δφf,f=D|f(z)|2(nφ-1)(z)dA(z). By summing up these two relations we get D|f(z)|2(nφ2-nφ)(z)dA(z)0. But nφ2(z)nφ(z), and so D|f(z)|2(nφ2-nφ)(z)dA(z)=0,f𝒟. This, in turn, implies that nφ2(z)=nφ(z) almost everywhere. Substituting this in (2.11), and then considering (2.12) the assertion will be completed.

Observe that if φ(0)=0, nφ2-2nφ+10 almost everywhere, and Cφ is bounded on 𝒟 then it is convex. Indeed,

(Cφ*ΔφCφ-Δφ)f,f=D|f(z)|2(nφ2-2nφ+1)(z)dA(z)0.

In the next theorem, we turn to the adjoint of a composition operator and give necessary and sufficient conditions under which a convex operator Cφ* is an isometry.

Theorem 2.6.

Let φ be an analytic self-map of D with φ(0)=0. If Cφ* is a convex operator on H2(β), then it is an isometry if and only if ΔCφ*0.

Proof.

Suppose that ΔCφ*0, and assume that φ is not the identity or an elliptic automorphism. By the Denjoy-Wolff theorem φn converges uniformly to zero on compact subsets of D , and so for every zD, limnKφn(z)=K0. Proposition 1.2 coupled with the fact that Cφ*nKz=Kφn(z) implies that for all zD and all nonnegative integers n, Kφn(z)2n(Kφ(z)2-Kz2)+Kz2. Furthermore, the positivity of ΔT shows that Kφ(z)Kz. Thus, in light of (2.16) and (2.17) we conclude that Kz=Kφ(z) for all zD, and so Kz=Kφn(z) for every positive integer n. Consequently, Kz=K0 for all zD. It follows that 1=K02=Kz2=k(|z|2)=1+j=1(|z|2)jβ(j)2,forzD. This contradiction shows that φ is the identity or an elliptic automorphism. Thus, there is a θ[0,2π) so that φ(z)=eiθz for all zD. Now, if ωD then Cφ*Kω(z)=Kφ(ω)(z)=k(φ(ω)¯z)=Kω(e-iθz)=Kω(φ-1(z))=Cφ-1Kω(z).

It follows that Cφ*=Cφ-1. But it is easily seen that Cφ-1f=f for every fH2(β). Hence, Cφ* is an isometry. The converse is obvious.

3. Multiplication Operators

This section deals with convex multiplication operators on a weighted Hardy space. Recall that a multiplier of H2(β) is an analytic function φ on D such that φH2(β)H2(β). The set of all multipliers of H2(β) is denoted by M(H2(β)). It is known that M(H2(β))H. In fact, if φM(H2(β)) and f is the constant function 1 then for every positive integer n and for every zD we have

|φ(z)|=|Mφnf,Kz|1/nMφnf1/nKz1/nMφKz1/n. Now, letting n, we conclude that φ is bounded. This coupled with the fact that φH2(β) implies that φH. If φ is a multiplier, then the multiplication operator Mφ, defined by Mφf=φf, is bounded on H2(β). Also note that for each λD, Mφ*Kλ=φ(λ)¯Kλ.

In what follows, the operator Mφ is assumed to be convex. First, we present an example of a nonisometric convex multiplication operator T with ΔT0.

Example 3.1.

Consider the weighted Hardy space H2(β) with weight sequence (β(n))n given by β(n)=n+1. Define the mapping φ on D by φ(z)=z2. Obviously, Mφ is bounded. Furthermore, it is easy to see that for every nonnegative integer k, Mφ2zk2-2Mφzk2+zk2>0,Mφzk>zk. Consequently, Mφ is convex but not an isometry. Besides, ΔMφ is a positive operator.

Theorem 3.2.

Let H consist of all multipliers of H2(β), and let φH be such that φ1. If T=Mφ or T=Mφ* then T*ΔTTΔT0 if and only if T is an isometry.

Proof.

Suppose that T is Mφ or Mφ* and T*ΔTTΔT0. Define the linear mapping S:H(H2(β)) by S(ψ)=Mψ. An application of the closed graph theorem implies that S is bounded. Therefore, there is c>0 such that for all ψH, Mψcψ. It follows that for every fH2(β) and every nonnegative integer n, Mφnfcφnfcf. Thus, supn0Mφnf< for every fH2(β). Since Mψ*=Mψ for all ψH, by a similar method one can show that supn0Mφ*nf< for all fH2(β). Therefore, the result follows from Proposition 1.3.

Example 3.3.

Let be the Bergman space or the Hardy space and let T be Mφ or its adjoint on . It is well known that M()=H. So if φ is a multiplier with φ1, then by applying the preceding theorem, we observe that T*ΔTTΔT0 if and only if T is an isometry.

We remark herein that if φ(z)=z and T=Mφ on the Dirichlet space 𝒟, then it is easily seen that T*ΔTTΔT0 but T is not an isometry.

Acknowledgments

The authors would like to thank Dr. Faghih Ahmadi for her assistance and the referee for a number of helpful comments and suggestions. This research was in part supported by a grant no. (88-GR-SC-27) from Shiraz University Research Council.

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