AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation27969110.1155/2008/279691279691Research ArticleEssential Norms of Weighted Composition Operators from the α-Bloch Space to a Weighted-Type Space on the Unit BallStevićStevoReichSimeonMathematical Institute of the Serbian Academy of Sciences and ArtsKnez Mihailova 36/III 11000 BeogradSerbiami.sanu.ac.yu200809092008200808052008050920082008Copyright © 2008This 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.

This paper finds some lower and upper bounds for the essential norm of the weighted composition operator from α-Bloch spaces to the weighted-type space Hμ on the unit ball for the case α1.

1. Introduction

Let 𝔹={zn:|z|<1} be the open unit ball in n, H(𝔹) be the class of all holomorphic functions on the unit ball, and let H(𝔹) be the class of all bounded holomorphic functions on 𝔹 with the normf=supz𝔹|f(z)|.Let z=(z1,,zn) and w=(w1,,wn) be points in n and z,w=k=1nzkw¯k. For a holomorphic function f, we denotef=(fz1,,fzn).

A positive continuous function ϕ on the interval [0,1) is called normal (see ) if there is δ[0,1) and a and b, 0<a<b such thatϕ(r)(1r)aisdecreasingon[δ,1)andlimr1ϕ(r)(1r)a=0,ϕ(r)(1r)bisincreasingon[δ,1)andlimr1ϕ(r)(1r)b=.From now on, if we say that a function ϕ:𝔹[0,) is normal, we will also assume that it is radial, that is, ϕ(z)=ϕ(|z|),z𝔹.

The weighted space Hμ=Hμ(𝔹) consists of all fH(𝔹) such thatsupz𝔹μ(z)|f(z)|<,where μ is normal on the interval [0,1). For μ(z)=(1|z|2)β, β>0, we obtain the weighted space Hβ=Hβ(𝔹) (see, e.g., [2, 3]).

The α-Bloch space α=α(𝔹),α>0, is the space of all fH(𝔹) such thatbα(f)=supz𝔹(1|z|2)α|f(z)|<.With the normfα=|f(0)|+bα(f),the space α is a Banach space ().

The little α-Bloch space 0α=0α(𝔹) is the subspace of α consisting of all fH(𝔹) such thatlim|z|1(1|z|2)α|f(z)|=0.

Let uH(𝔹) and φ be a holomorphic self-map of the unit ball. Weighted composition operator on H(𝔹), induced by u and φ is defined by(uCφf)(z)=u(z)f(φ(z)),z𝔹.This operator can be regarded as a generalization of a multiplication operator and a composition operator. It is interesting to provide a function theoretic characterization when u and φ induce a bounded or compact weighted composition operator between some spaces of holomorphic functions on 𝔹. (For some classical results in the topic see, e.g., . For some recent results on this and related operators, see, e.g., [24, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] and the references therein.)

In , Ohno has characterized the boundedness and compactness of weighted composition operators between H and the Bloch space on the unit disk. In the setting of the unit polydisk 𝔻n, we have given some necessary and sufficient conditions for a weighted composition operator to be bounded or compact from H(𝔻n) to the Bloch space (𝔻n) in  (see, also ). Corresponding results for the case of the unit ball are given in . Among other results, in , we have given some necessary and sufficient conditions for the compactness of the operator uCφ:α(𝔹)H(𝔹), which we incorporate in the following theorem.

Theorem 1.

Let φ=(φ1,,φn) be a holomorphic self-map of 𝔹 and uH(𝔹).

If α>1, then the following statements are equivalent:

uCφ:0αH is a compact operator,

uCφ:αH is a compact operator,

uH, andlim|φ(z)|1|u(z)|(1|φ(z)|2)α1=0.

If α=1, then the following statements are equivalent:

uCφ:0H is a compact operator,

uCφ:H is a compact operator,

uH, and lim|φ(z)|1|u(z)|ln21|φ(z)|2=0.

We would also like to point out that if α1, then the boundedness of uCφ:αHμ and uCφ:0αHμ are equivalent (see  for the case α=1, and the proof of Theorem 3 in ).

The essential norm of an operator is its distance in the operator norm from the compact operators. More precisely, assume that X1 and X2 are Banach spaces and A:X1X2 is a bounded linear operator, then the essential norm of A, denoted by Ae,X1X2, is defined as follows:Ae,X1X2=inf{A+LX1X2:L:X1X2,Liscompact},where X1X2 denotes the operator norm. If X1=X2, it is simply denoted by e (see, e.g., [5, page 132]). If A:X1X2 is an unbounded linear operator, then clearly Ae,X1X2=.

Since the set of all compact operators is a closed subset of the set of bounded operators, it follows that an operator A is compact if and only if Ae,X1X2=0.

Motivated by Theorem A, in this paper, we find some lower and upper bounds for the essential norm of the weighted composition operator uCφ:α(𝔹)(or0α(𝔹))Hμ(𝔹), when α1.

Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to another. The notation ab means that there is a positive constant C such that aCb. If both ab and ba hold, then we say that ab.

2. Auxiliary Results

In this section, we quote several auxiliary results which we need in the proofs of the main results in this paper. The following lemma should be folklore.

Lemma 2.1.

Let fα(𝔹),0<α<. Then, |f(z)|{Cfα,α(0,1),|f(0)|+b1(f)12ln1+|z|1|z|,α=1,Cfα(1|z|2)α1,α>1, for some C>0 independent of f.

The proof of the lemma for the case α1 can be found, for example, in . The formulation of the corresponding estimate in , for the case α=1, is slightly different. In this case, Lemma 2.1 follows from the following estimate:|f(z)f(0)|=|01f(tz),z¯|b1(f)01|z|dt1|z|2t2=b1(f)12ln1+|z|1|z|.

The next lemma can be proved in a standard way (see, e.g., the proofs of the corresponding results in [5, 2729]).

Lemma 2.2.

Assume α>0, gH(𝔹), μ is normal, and φ is an analytic self-map of 𝔹. Then, uCφ:α(or0α)Hμ is compact if and only if uCφ:α(or0α)Hμ is bounded and for any bounded sequence (fk)k in α(or0α) converging to zero uniformly on compacts of 𝔹 as k, one hasuCφfkHμ0 as k.

Lemma 2.3.

Let fw(z)=(1|w|2)ε(1z,w)α+ε1,w𝔹,where α>1 and ε(0,1]. Then,fwα=(1|w|2)ε+(α+ε1)(2α)α|w|(α2+|w|2ε2|w|2α2+ε)ε(α+ε)ε(α2+|w|2ε2|w|2α2+α)α.

Proof.

We have (1|z|2)α|fw(z)|=(α+ε1)(1|z|2)α(1|w|2)ε|w||1z,w|α+ε, from which it easily follows that bα(fw)(α+ε1)2α+ε. Set gs(x)=(α+ε1)s(1s2)ε(1x2)α(1sx)α+ε,x[0,1],s[0,1). Then, gs(x)=(α+ε1)s(1s2)ε(1x2)α1s(αε)x22αx+s(α+ε)(1sx)α+ε+1. Hence, the points xM,m=α±α2+s2ε2s2α2s(αε) are stationary for the function gs(x). Since xM>1, it follows that gs(x) attains its maximum on the interval [0,1] at the point xm=αα2+s2ε2s2α2s(αε)=s(α+ε)α+α2+s2ε2s2α2(0,1). By some long but elementary calculations, it follows that gs(xm)=(α+ε1)(2α)αs(α2+s2ε2s2α2+ε)ε(α+ε)ε(α2+s2ε2s2α2+α)α.From this and since fw(0)=(1|w|2)ε, (2.4) follows.

Remark 2.4.

Note thatlim|w|1fwα=lims10[(1s2)ε+gs(xm)]=(α+ε1)2α+εεεαα(α+ε)α+ε,limε0+(α+ε1)2α+εεεαα(α+ε)α+ε=(α1)2α.

3. Estimates of the Essential Norm of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M138"><mml:mrow><mml:mi>u</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>:</mml:mo><mml:msup><mml:mrow><mml:mi>ℬ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mtext>or</mml:mtext><mml:mtext> </mml:mtext><mml:msubsup><mml:mrow><mml:mi>ℬ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">)</mml:mo><mml:mo>→</mml:mo><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>

In this section, we prove the main results in this paper. Before we formulate and prove these results, we prove another auxiliary result.

Lemma 3.1.

Assume α(0,), uH(𝔹),μ is normal, φ is a holomorphic self-map of 𝔹 such that φ<1, and the operator uCφ:α(or0α)Hμ is bounded. Then, uCφ:α(or0α)Hμ is compact.

Proof.

First note that since uCφ:α(or0α)Hμ is bounded and f0(z)10αα, it follows that uCφ(f0)=uHμ. Now, assume that (fk)k is a bounded sequence in α(or0α) converging to zero on compacts of 𝔹 as k. Then, we have uCφ(fk)HμuHμsupwφ(𝔹)|fk(w)|0, as k, since φ(𝔹) is contained in the ball |w|φ which is a compact subset of 𝔹, according to the assumption, φ<1. Hence, by Lemma 2.2 , the operator uCφ:α(or0α)Hμ is compact.

Theorem 3.2.

Assume α>1, μ is normal, uH(𝔹), φ=(φ1,,φn) is a holomorphic self-map of 𝔹, and uCφ:αHμ is bounded. Then, 2αα1limsup|φ(z)|1μ(z)|u(z)|(1|φ(z)|2)α1uCφe,0αHμuCφe,αHμClimsup|φ(z)|1μ(z)|u(z)|(1|φ(z)|2)α1, for some positive constant C.

Proof.

Since uCφ:αHμ is bounded, recall that uHμ. If φ<1, then, from Lemma 3.1, it follows that uCφ:αHμ is compact which is equivalent with uCφe,αHμ=0, and, consequently, uCφe,0αHμ=0. On the other hand, it is clear that in this case the condition |φ(z)|1 is vacuous, so that inequalities in (3.2) are vacuously satisfied.

Hence, assume φ=1. Let (zk)k be a sequence in 𝔹 such that limk|φ(zk)|=1 and ε(0,1) be fixed. Set fkε(z)=(1|φ(zk)|2)ε(1z,φ(zk))α+ε1,k. By Lemma 2.3, it follows that supkfkεα<, moreover, it is easy to see that fkε0α for every k and fkε0 uniformly on compacts of 𝔹 as k. Then, by [6, Theorem 7.5], it follows that fkε converges to zero weakly as k. Hence, for every compact operator L:0αHμ, we have LfkεHμ0 as k.

We have fkεαuCφ+L0αHμ(uCφ+L)(fkε)HμuCφfkεHμLfkεHμ for every compact operator L:0αHμ.

Letting k in (3.4) and using the definition of fkε, we obtain C(ε,α)uCφ+L0αHμ=limsupkuCφfkεHμlimsupkμ(zk)|u(zk)|(1|φ(zk)|2)α1,where C(ε,α) is the quantity in (2.12).

Taking in (3.5) the infimum over the set of all compact operators L:0αHμ, then letting ε0+ in such obtained inequality, and using (2.13), we obtain 2α(α1)uCφe,0αHμlimsupkμ(zk)|u(zk)|(1|φ(zk)|2)α1, from which the first inequality in (3.2) follows.

Since the second inequality in (3.2) is obvious, we only have to prove the third one. By Lemma 3.1, we have that for each fixed ρ(0,1) the operator uCρφ:αHμ is compact.

Let δ(0,1) be fixed, and let (ρm)m be a sequence of positive numbers which increasingly converges to 1, then for each fixed m, we have uCφe,αHμuCφuCρmφαHμ=supfα1(uCφuCρmφ)(f)Hμ=supz𝔹supfα1μ(z)|u(z)(f(φ(z))f(ρmφ(z)))|sup|φ(z)|δsupfα1μ(z)|u(z)||f(φ(z))f(ρmφ(z))|+sup|φ(z)|>δsupfα1μ(z)|u(z)||f(φ(z))f(ρmφ(z))|.

By the mean-value theorem, we have sup|φ(z)|δsupfα1μ(z)|u(z)(f(φ(z))f(ρmφ(z)))|(1ρm)sup|φ(z)|δsupfα1μ(z)|u(z)||φ(z)|sup|w|δ|f(w)|(1ρm)δ(1δ2)αuHμsupfα1sup|w|δ(1|w|2)α|f(w)|(1ρm)δ(1δ2)αuHμ0, as m.

Moreover, by Lemma 2.1 (case α>1), and known inequality frαfα, where fr(z)=f(rz), r[0,1), we have μ(z)|u(z)(f(φ(z))f(ρmφ(z)))|Cffρmαμ(z)|u(z)|(1|φ(z)|2)α12Cfαμ(z)|u(z)|(1|φ(z)|2)α1 for some positive constant C. Replacing (3.10) in (3.7), letting in such obtained inequality m, employing (3.8), and then letting δ1, the third inequality in (3.2) follows, finishing the proof of the theorem.

Corollary 3.3.

Assume α>1, μ is normal, uH(𝔹), φ=(φ1,,φn) is a holomorphic self-map of 𝔹, and the operator uCφ:α(or0α)Hμ is bounded. Then, uCφ:α(or0α)Hμ is compact if and only if limsup|φ(z)|1μ(z)|u(z)|(1|φ(z)|2)α1=0.

Theorem 3.4.

Assume uH(𝔹),μ is normal, φ=(φ1,,φn) is a holomorphic self-map of 𝔹, and uCφ:Hμ is bounded. Then, Climsup|φ(z)|1μ(z)|u(z)|ln1+|φ(z)|1|φ(z)|uCφe,0HμuCφe,Hμlimsup|φ(z)|1μ(z)|u(z)|ln1+|φ(z)|1|φ(z)| for a positive constant C.

Proof.

Clearly, u=(uCφ)(1)Hμ. If φ<1, then the result follows as in Theorem 3.2.

Hence, assume φ=1. We use the following family of test functionshw(z)=(ln(1+|w|)21z,w)2(ln1+|w|1|w|)1,w𝔹.

We have supz𝔹(1|z|2)|hw(z)|=supz𝔹2(1|z|2)|w||1z,w||ln(1+|w|)21z,w|(ln1+|w|1|w|)14|w|(2ln1+|w|1|w|+2π)(ln1+|w|1|w|)14(2+2πln3), when |w|1/2.

From this and since |hw(0)|4(ln2)2/ln3, for |w|1/2, we have that hw4(2+2πln3+(ln2)2ln3)=C0.

Assume (φ(zk))k is a sequence in 𝔹 such that |φ(zk)|1 as k. Note that (hφ(zk))k is a bounded sequence in (moreover in 0) converging to zero uniformly on compacts of 𝔹. Then, by [6, Theorem 7.5], it follows that hφ(zk) converges to zero weakly as k. Hence, for every compact operator L:0Hμ, we have limkLhφ(zk)Hμ=0.

On the other hand, for every compact operator L:0Hμ, we havehφ(zk)uCφ+L0Hμ(uCφ+L)hφ(zk)HμuCφhφ(zk)HμLhφ(zk)Hμ. Using (3.15), letting k in (3.17), and applying (3.16), it follows that

C0uCφ+L0HμlimsupkuCφhφ(zk)Hμlimsupkμ(zk)|u(zk)hφ(zk)(φ(zk))|=limsupkμ(zk)|u(zk)|ln1+|φ(zk)|1|φ(zk)|.

Taking in (3.18) the infimum over the set of all compact operators L:0Hμ, we obtain uCφe,0Hμ1C0limsupkμ(zk)|u(zk)|ln1+|φ(zk)|1|φ(zk)|, from which the first inequality in (3.12) follows.

As in Theorem 3.2, we need only to prove the third inequality in (3.12).

Recall that for each ρ(0,1), the operator uCρφ:Hμ is compact. Let the sequence (ρm)m be as in Theorem 3.2. Note that inequality (3.7) and relationship (3.8) also hold for α=1. Hence, we should only estimate the quantity Imδ=sup|φ(z)|>δsupf1μ(z)|u(z)||f(φ(z))f(ρmφ(z))|. On the other hand, by Lemma 2.1 (case α=1) applied to the function ffρm, which belongs to the Bloch space for each m, and inequality (3.9) with α=1, we have μ(z)|u(z)(f(φ(z))f(ρmφ(z)))|12ffρmμ(z)|u(z)|ln1+|φ(z)|1|φ(z)|fμ(z)|u(z)|ln1+|φ(z)|1|φ(z)|.

From (3.21), by letting m in (3.7) and using (3.8) (with α=1), and letting δ1 in such obtained inequality, we obtainuCφe,Hμlimsup|φ(z)|1μ(z)|u(z)|ln1+|φ(z)|1|φ(z)|, as desired.

Corollary 3.5.

Assume uH(𝔹),μ is normal, φ=(φ1,,φn) is a holomorphic self-map of 𝔹, and the operator uCφ:(or0α)Hμ is bounded. Then, uCφ:(or0α)Hμ is compact if and only if limsup|φ(z)|1μ(z)|u(z)|ln1+|φ(z)|1|φ(z)|=0.

ShieldsA. L.WilliamsD. L.Bonded projections, duality, and multipliers in spaces of analytic functionsTransactions of the American Mathematical Society1971162287302MR028355910.2307/1995754ZBL0227.46034StevićS.Weighted composition operators between mixed norm spaces and Hα spaces in the unit ballJournal of Inequalities and Applications2007200792862910.1155/2007/28629MR2377529ZBL1138.47019TangX.Weighted composition operators between Bers-type spaces and Bergman spacesApplied Mathematics. A Journal of Chinese Universities B20072216168MR229871210.1007/s11766-007-0008-9ZBL1125.47017ClahaneD. D.StevićS.Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ballJournal of Inequalities and Applications20062006116101810.1155/JIA/2006/61018MR2253412ZBL1131.47018CowenC. C.MacCluerB. D.Composition Operators on Spaces of Analytic Functions1995Boca Raton, Fla, USACRC Pressxii+388Studies in Advanced MathematicsMR1397026ZBL0873.47017ZhuK.Spaces of Holomorphic Functions in the Unit Ball2005226New York, NY, USASpringerx+271Graduate Texts in MathematicsMR2115155ZBL1067.32005FuX.ZhuX.Weighted composition operators on some weighted spaces in the unit ballAbstract and Applied Analysis20082008860580710.1155/2008/605807MR2417227GorkinP.MacCluerB. D.Essential norms of composition operatorsIntegral Equations and Operator Theory20044812740MR2029942ZBL1065.47027LiS.StevićS.Composition followed by differentiation between Bloch type spacesJournal of Computational Analysis and Applications200792195205MR2292805ZBL1132.47026LiS.StevićS.Weighted composition operators from Bergman-type spaces into Bloch spacesProceedings of the Indian Academy of Sciences. Mathematical Sciences20071173371385MR2352056ZBL1130.47016LiS.StevićS.Weighted composition operators from α-Bloch space to H on the polydiscNumerical Functional Analysis and Optimization2007287-8911925MR2347688ZBL1130.4701510.1080/01630560701493222LiS.StevićS.Weighted composition operators from H to the Bloch space on the polydiscAbstract and Applied Analysis20072007134847810.1155/2007/48478MR2320803LiS.StevićS.Generalized composition operators on Zygmund spaces and Bloch type spacesJournal of Mathematical Analysis and Applications2008338212821295MR238649610.1016/j.jmaa.2007.06.013ZBL1135.47021LiS.StevićS.Weighted composition operators between H and α-Bloch space in the unit ballto appear in Taiwanese Journal of MathematicsLuoL.UekiS.-I.Weighted composition operators between weighted Bergman spaces and Hardy spaces on the unit ball of nJournal of Mathematical Analysis and Applications2007326188100MR227776810.1016/j.jmaa.2006.02.038ZBL1114.32003MacCluerB. D.ZhaoR.Essential norms of weighted composition operators between Bloch-type spacesThe Rocky Mountain Journal of Mathematics200333414371458MR205249810.1216/rmjm/1181075473ZBL1061.30023Montes-RodríguezA.Weighted composition operators on weighted Banach spaces of analytic functionsJournal of the London Mathematical Society2000613872884MR1766111ZBL0959.47016OhnoS.Weighted composition operators between H and the Bloch spaceTaiwanese Journal of Mathematics200153555563MR1849777ZBL0997.47025OhnoS.StroethoffK.ZhaoR.Weighted composition operators between Bloch-type spacesThe Rocky Mountain Journal of Mathematics2003331191215MR199448710.1216/rmjm/1181069993ZBL1042.47018ShiJ.LuoL.Composition operators on the Bloch space of several complex variablesActa Mathematica Sinica20001618598MR176052510.1007/s101149900028ZBL0967.32007StevićS.Composition operators between H and α-Bloch spaces on the polydiscZeitschrift für Analysis und ihre Anwendungen2006254457466MR2285095ZBL1118.47015StevićS.Norm of weighted composition operators from Bloch space to Hμ on the unit ballArs Combinatoria200888125127UekiS.-I.LuoL.Compact weighted composition operators and multiplication operators between Hardy spacesAbstract and Applied Analysis200820081219649810.1155/2008/196498MR2393121ZhuX.Generalized weighted composition operators from Bloch type spaces to weighted Bergman spacesIndian Journal of Mathematics2007492139150MR2341712ZBL1130.47017ZhuX.Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spacesIntegral Transforms and Special Functions2007183223231MR231958410.1080/10652460701210250ZBL1119.47035StevićS.On an integral operator on the unit ball in nJournal of Inequalities and Applications2005200518188MR214571810.1155/JIA.2005.81ZBL1074.47013LiS.StevićS.Riemann-Stieltjes-type integral operators on the unit ball in nComplex Variables and Elliptic Equations2007526495517MR2326188ZBL1124.4702210.1080/17476930701235225StevićS.Boundedness and compactness of an integral operator on a weighted space on the polydiscIndian Journal of Pure and Applied Mathematics2006376343355MR2285372ZBL1121.47032StevićS.Boundedness and compactness of an integral operator in a mixed norm space on the polydiskSibirskiĭ Matematicheskiĭ Zhurnal2007483694706MR2347917