Composition Operator on Bergman Orlicz Space

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Volume 2009, Article ID 832686,14pages doi:10.1155/2009/832686

Research Article

Composition Operator on Bergman-Orlicz Space

Zhijie Jiang

1, 2

_{and Guangfu Cao}

1

1_{College of Mathematics and Information Science, Guangzhou University,}

Guangzhou, Guangdong, 510006, China

2_{Department of Mathematics, Sichuan University of Science and Engineering,}

Zigong, Sichuan 643000, China

Correspondence should be addressed to Zhijie Jiang,[email protected]

Received 19 May 2009; Accepted 22 October 2009

Recommended by Shusen Ding

LetDdenote the open unit disk in the complex plane and letdAzdenote the normalized area measure onD. Forα >−1 andΦa twice diﬀerentiable, nonconstant, nondecreasing, nonnegative, and convex function on0,∞, the Bergman-Orlicz spaceLΦα is defined as followsLΦα {f ∈

HD:_DΦlog|fz|1− |z|2αdAz<∞}.Letϕbe an analytic self-map ofD. The composition operatorCϕ induced by ϕ is defined byCϕf f◦ϕ forf analytic inD. We prove that the

composition operatorCϕis compact onLΦα if and only ifCϕis compact onA2α, andCϕhas closed

range onLΦα if and only ifCϕhas closed range onA2α.

Copyrightq2009 Z. Jiang and G. Cao. 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.

1. Introduction

LetDbe the open unit disk in the complex plane and letϕbe an analytic self-map ofD. The

composition operatorCϕinduced byϕis defined byCϕff◦ϕforfanalytic inD. The idea

of studying the general properties of composition operators originated from Nordgren1 .

As a sequence of Littlewood’s subordinate theorem, eachϕinduces a bounded composition

operator on the Hardy spacesHp_D_{for all}_p₀_{< p <}_∞_{and the weighted Bergman spaces}

ApαDfor allp0 < p < ∞and for allα−1 < α <∞. Thus, boundedness of composition

operators on these spaces becomes very clear. Nextly, a natural problem is how to characterize the compactness of composition operators on these spaces, which once was a central problem for mathematicians who were interested in the theory of composition operators. The study of compact composition operators was started by Schwartz, who obtained the first compactness

theorem in his thesis2 , showing that the integrability of1− |ϕ|−1 over∂D implied the

compactness ofCϕ onHp. The work was continued by Shapiro and Taylor3 , who showed

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∂D. Moreover, MacCluer and Shapiro4 pointed out that nonexistence of the finite angular

derivatives ofϕwas a suﬃcient condition for the compactness ofCϕ onApαbut it failed on

Hp_{. So looking for an appropriate tool of characterizing the compactness of}_C

ϕ onHpwas

diﬃcult at that time. Fortunately, Shapiro5 developed relations between the essential norm

ofCϕonH2and the Nevanlinna counting function ofϕ, and he obtained a nice essential norm

formula ofCϕin 1987. As a result, he completely gave a characterization of the compactness

ofCϕin terms of the function properties ofϕ.

Another solution to the compactness of Cϕ on H2 was done by the Aleksandrov

measures which was introduced by Cima and Matheson 6 . It is well known that the

harmonic functionRλϕz/λ−ϕzcan be expressed by the Possion integral

Rλϕz

λ−ϕz

∂D

Pz, ζdmλζ 1.1

for eachλ∈∂D. Cima and Matheson appliedσλthe singular part ofmλto give the following

expression:

_C_ϕ2

esup

λ∈∂Dσλ. 1.2

They showed thatCϕwas compact onH2if and only if all the measuresmλwere absolutely

continuous.

The study of compactness of composition operators is also an important subject on other analytic function spaces, and we have chosen two typical examples above, and for more

related materials one can consult7,8 . Another natural interesting subject is the composition

operator with closed range. Considering angular derivatives of ϕ, it is known that Cϕ is

compact onA2_{if and only if}_ϕ_{fails to have finite angular derivatives on}_∂_D_{, in this case,}_C

ϕ

does not have closed range sinceCϕis not a finite rank operator. And ifϕhas finite angular

derivatives on∂D, thenϕis necessarily a finite Blaschke product and hence one can easily

verify thatCϕhas closed range onA2. Zorboska has given a necessary and suﬃcient condition

forCϕwith closed range onH2, and she also has done onApα9 . Luecking10 considered

the same question on Dirichlet space after Zorboska’s work. Recently, Kumar and Partington

11 have studied the weighted composition operators with closed range on Hardy spaces

and Bergman spaces.

This paper will study the compactness of composition operator on Bergman-Orlicz space. We are mainly inspired by the following results.

iLiu et al. 12 showed that composition operator was bounded on Hardy-Orlicz

space. Lu and Cao13 also showed that composition operator was bounded on

Bergman-Orlicz space.

iiA composition operator was compact on the Nevanlinna classNif and only if it

was compact onH2₁₄ _.

iiiIf a composition operator was compact onHp_{for some}_{p >}_{0, then it was compact}

on Hp _{for all} _{p >} ₀ ₃ _{. Moreover, paper} ₁₅ _{compared the compactness of}

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CϕonA2αand on the Bergman-Orlicz space, and whether there is a equivalence for

the closed range of Cϕ onA2αand on the Bergman-Orlicz space. In this paper, we

are going to give aﬃrmative answers for the proceeding questions.

2. Preliminaries

LetHDdenote the space of all analytic functions onD. LetdAzdenote the normalized

area measure onD, that is,AD 1. LetS denote the class of strongly convex functions

Φ:0,∞ → 0,∞, which satisfies

i Φ0 Φ0 0,Φt/t → ∞ast → ∞,

ii Φexists on0,∞,

iii Φ2t≤CΦtfor some positive constantCand for allt >0.

ForΦ∈ Sandα >−1 the Bergman-Orlicz spaceLΦ_α is defined as follows:

LΦ_α

f∈HD:f_Φ

DΦ

logfz 1− |z|2αdAz<∞

, 2.1

where logx max{0,logx}. Although · _Φ does not define a norm in LΦ_α, it holds that

thedf, g f−g_Φdefines a metric on LΦ_α, and makesLΦ_α into a complete metric space.

Obviously, the inequalities

logx≤log1x≤1logx, x≥0,

2logx≤log 1x2≤12logx, x≥0,

2.2

and the fact thatΦis nondecreasing convex function imply that

Φlogx≤Φlog1x≤Φ1logx

≤ 1

2Φ2

1

2Φ

2 logx

≤ 1

2Φ2

1

2CΦ

logx,

Φlogx≤Φ2 logx≤Φ log 1x2

≤Φ12 logx

≤ 1

2Φ2

1

2Φ

4 logx

≤ 1

2Φ2

1

2CΦ

logx.

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Thenf ∈LΦ_α if and only if

DΦ

log1fz 1− |z|2αdAz<∞ 2.4

or if and only if

DΦ log 1 _f_z2

1− |z|2αdAz<∞. 2.5

Throughout this paper, constants are denoted byC, they are positive and may diﬀer

from one occurrence to the other. The notationabmeans that there is a positive constant

Csuch thata≤Cb. Moreover, if bothabandbahold, we writeaband say thatais

asymptotically equivalent tob.

In this section we will prove several auxiliary results which will be used in the proofs of the main results in this paper.

Lemma 2.1. Iff ∈LΦ_α, then

f_ΦΦ log 1f0|2

Df

Φ_z ₁_{− |}_z_|2α_dA_z_, _2.6

whereΔis Laplacian andfΦz ΔΦlog1|fz|2.

Proof. By the Green Theorem, ifu, v∈C2_Ω_{, where}_Ω_{is a domain in the plane with smooth}

boundary, then

ΩuΔv−vΔudx dy

∂Ω

u∂v ∂n−v

∂u ∂n

ds. 2.7

Let 0< ε < r <1,uz logr/|z|,vz Φlog1|fz|2, andΩ {z∈D:ε <|z|< r}.

SinceΔuz 0, by2.7we have

ΩΔΦ log 1

_f_z2

log r

|z|dx dylog r ε

|z|ε

∂

∂nΦ log 1fz

2

ds

|z|r

Φ log 1fz2

r ds−

|z|ε

Φ log 1fz2

ε ds.

2.8

Since∂/∂nΦlog1|fz|2is bounded near to 0, we get

lim

ε→0log

r ε

|z|ε

∂

∂nΦ log 1fz

2

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Letε → 0 in2.8, we have

|z|<r

ΔΦ log 1fz2log r

|z|dx dy

₂_π

0 Φ

log

1f reiθ2

dθ−2πΦ log 1f02.

2.10

Integrating equality2.10with respect torfrom 0 to 1, we obtain

1

0

|z|<r

ΔΦ log 1fz2log r

|z| 1−r

2α_{rdr dA}_z_f

Φ−

2π

α1Φ log 1f0

2

.

2.11

Thus

_f

Φ

2π

α1Φ log 1f0

21

0

|z|<r

ΔΦ log 1fz2log r

|z| 1−r

2α_{rdr dA}_z

2π

α1Φ log 1f0

2

DΔΦ log 1

fz2dAz

1

|z|

log r

|z| 1−r

2α_rdr.

2.12

Since1_|_z_|logr/|z|1−r2α_rdr ₁₋_z_|22α_,

f_Φ 2π

α1Φ log 1f0

2

DΔΦ log 1

fz2 1− |z|2αdAz

Φ log 1f02

Df

Φ ₁_{− |}_z_|2α_dA_z_,

2.13

the proof is complete.

Letϕbe an analytic self-map ofD. The generalized Nevanlinna counting function ofϕ

is defined by

Nϕ,α2w

z∈ϕ−1_w

log 1

|z|

α2

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Lemma 2.2see9 . Ifϕis an analytic self-map ofDand g is a nonnegative measurable function in

D, then

Dg◦ϕz ϕz2

log 1

|z|

α2

dAz

DgzNϕ,α2zdAz.

2.15

Lemmas2.1and2.2see9 can lead to the following corollary.

Corollary 2.3. Letϕbe an analytic self-map ofDandf ∈HD, then

f◦ϕΦ 1logf◦ϕ02

Df

Φ_w_N

α2wdAw. 2.16

We will end this section with the following lemma, which illustrates that the counting

functionalδz:f→fzis continuous onLΦα.

Lemma 2.4. Letf ∈LΦ_α, then

fz≤exp ⎛ ⎜ ⎝Φ−1

⎛ ⎜

⎝ CfΦ

1− |z|2α2

⎞ ⎟ ⎠

⎞ ⎟

⎠ ∀ zinD. 2.17

Proof. By the subharmonicity of mapz→log1|fz|, we get

log1fz≤ 1

AαDz,1− |z|/2

Dz,1−|z|/2log

1fz 1−z|2αdAz. 2.18

SinceΦlog1|fz|is convex and increasing, we have

Φlog1fz≤ 1

AαDz,1− |z|/2

Dz,1−|z|/2Φ

log1fzdAαz

≤ C

1− |z|2α2

Dz,1−|z|/2Φ

log1fzdAαz

≤ C

1− |z|2α2

DΦ

log1fzdAαz

≤ CfΦ

1− |z|2α2 .

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SinceΦlogx≤Φlog1x, we get

Φlogfz≤ CfΦ

1− |z|2α2

, _2.20

that is, log|fz| ≤ Φ−1_C_f

Φ/1− |z|2α2. Thus|fz| ≤ expΦ−1CfΦ/1− |z|2α2.

3. Compactness

In this section, we are going to investigate the equivalence between compactness of

composition operator on the Bergman-Orlicz spaceLΦ_α and on the weighted Bergman space

A2

α. The following lemma characterizes the compactness ofCϕ onLΦα in terms of sequential

convergence, whose proof is similar to that in7, Proposition 3.11 .

Lemma 3.1. Letϕbe an analytic self-map ofD, bounded operatorCϕis compact onLΦα if and only if

whenever{fn}is bounded inLΦ_α andfn → 0uniformly on compact subsets ofD, thenCϕfn_Φ → 0

asn → ∞.

In order to characterize the compactness ofCϕ, we need to introduce the notion of

Carleson measure. For|ξ| 1 andδ >0 we defineQδξ {z∈ D:|z−ξ|< δ}. A positive

Borel measureμonDis called a Carleson measure if sup_|_ξ_|₁μQδξ Oδα2. Moreover, if

μsatisfies the additional condition limδ→0μQδξ/δα20,μis called a vanishing Carleson

measuresee16 for the further information of Carleson measure. The following result for

the compactness ofCϕonA2αis useful in the proof ofTheorem 3.3.

Lemma 3.2see14,17 . Letϕ be an analytic self-map ofD. Then the following statements are equivalent:

iCϕis compact onA2α,iilim|z| →1Nϕ,α2z/1− |z|2 α2

0, andiiithe pull measure

μϕis a vanshing Carleson measure onD.

Theorem 3.3. Letϕ be an analytic self-map ofD, thenCϕ is compact onA2α if and only ifCϕ is

compact onLΦ_α.

Proof. First we assume thatCϕ is compact onA2α. Choose a sequence{fn}that is bounded

by a positive constantMinLΦ_α and converges to zero uniformly on compact subsets ofD.

ByLemma 3.1, it is enough to show thatfn◦ϕ_Φ → 0 asn → ∞. Letε > 0, we can find

0 < r < 1 such thatNϕ,α2z < ε1− |z|2α2 for all|z| > r.Since fn → 0 uniformly on

compact subsets ofDasn → ∞, so isf_n. Thus we can chooseN > 0 such that|fn| < εand

|f_n|< εonrD, whenevern > N. Hence for suchnwe have

_C_ϕ_f_n

Φ≤Φ log 1fn◦ϕ02

Df

Φ

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As|fn◦ϕ0| → 0 asn → ∞andΦlog1|fn◦ϕ0|2 → 0 asn → ∞, we only need to

verify that_Df_nΦNϕ,α2dAw → 0 asn → ∞. Now

Df

Φ

nNϕ,α2dAw

rDf

Φ

nNϕ,α2dAw

D\rDf

Φ

nNϕ,α2dAw III. 3.2

We first prove that the first term in previous equality is bounded by a constant multiple ofε

I

rDf

Φ

nNϕ,α2dAw

rD

Φ _log ₁_f nw2

fnw2 Φ log 1fnw2

× fnw

2

1fnw2

2Nϕ,α2wdAw

≤Φ_log₁_ε_ε_Φ_log₁_ε_ε rD

Nϕ,α2wdAw

≤Φ_log₁_ε_ε_Φ_log₁_ε_ε.

3.3

Now, we show that the previous second term above is also bounded by a constant multiple ofε

II

D\rDf

Φ

nNϕ,α2dAw≤εfnΦw 1− |w|2

α

≤εf_nΦ≤Mε. 3.4

Conversely, we assume thatCϕ is compact onLΦα. ByLemma 3.2, we need to verify

thatμϕ is a vanishing Carleson measure. For 0 < δ < 1 andξ ∈ ∂Dwe write a 1−δξ

andgaz 1− |a|2 α2

/1−az2α4. Then|ga| ∈L1_D_{, dA}

α. PutGz Φ|gaz|.Gis well

defined, beacuseGis nondecreasing on range of|ga|. SinceΦ−1is concave, there is a constant

C > 0 such that Φ−1_t _≤ _Ct_{for enough big}_t_{. Thus we get}_G_z _∈ _L1_D_{, dA}

α. Sethz

exp2₀πeit_z_/_eit₋_z_G_eit_dt_{. Since}_Φ_log_|_h_z_|_Φ_G_z _|_g

az| ∈ L1D, dAα, it

means thath ∈ LΦ_α. Let faz 1− |a|hz. Then clearlyfa → 0 uniformly on compact

subsets ofDas|a| → 1. Moreover,

_f_a

Φ

DΦ

logfazdAαz≤

DΦ

log|hz|dAαz

D

1− |a|2α2

|1−az|2α4 dAαz 1.

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On the other hand, if|1−zζ|/1− |a|< γfor some fixed 0< γ <1/4, whereζa/|a|, that is,

z∈Qγδζ, we have

1− |a|2

|1−az|2

1− |a|2

|1−az|2

1− |a|2

⎛ ⎜

⎝1|a| 1−zζ

1− |a|

⎞ ⎟ ⎠ −2

≥ 1

1γ2

1− |a|2

1− |a|2 >

1− |a|2

41− |a|2 ≥

1 4δ.

3.6

Hence, forz∈Qγδζwe have

Φ−1

1− |a|2

|1−az|2

α2

≥Φ−1

1 4δ

α2

. 3.7

Thus, forz∈Qγδζwe obtain

Φlogfaz Φ

log1− |a|hz Φlog1− |a| log|hz|

≥Φlog|hz|gaz

1− |a|2

|1−az|2

α2

≥

1 4δ

α2

.

3.8

So, for allξ∈∂Dand 0< δ <1 we get

1 4δ

α2

μϕ

Qγδξ

≤

QγδξΦ

logfazdμϕ ≤

DΦ

logfazdμϕ

DΦ

logfa◦ϕzdAαz Cϕfa_Φ.

3.9

For the compactness of Cϕ, we know that Cϕfa_Φ → 0 as |a| → 1, which means that

limδ→1μϕQγδξ/δα2 0 uniformly for ξ ∈ ∂D. This means that μϕ is a vanishing

Carleson measure. ByLemma 3.2,Cϕis compact onA2α.

For special caseΦt tp_{p >} ₁_{, the Bergman-Orlicz space}_LΦ

α is called the area-type

Nevanlinna class and we writeNpα.

Corollary 3.4. Letϕ be an analytic self-map ofD, thenCϕ is compact onA2α if and only if Cϕ is

compact onNpα.

Remark 3.5. Theorem 3.3may be not true ifΦdoes not satisfy the given conditions in this

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Φis nondecreasing butΦx>0 for somex /0. Then the compactness ofCϕon the Bergman

spaceA2_i.e.,_α₀_{is di}_ﬀ_{erent from that on}_LΦ

α. HereLΦα is defined as follows:

LΦ_α

f∈HD:∃t >0, s.t.

DΦ

logtfzdAz<∞

. 3.10

If we takeΦx 0 forx ≤ 1, and Φx ∞forx > 1, thenLΦ_α isH∞D. We know that

Cϕ is compact onH∞Dif and only ifϕ∞ < 1 consult2 . But MacCluer and Shapiro

constructed an inner functionϕin4 such thatCϕwas compact onA2.

4. Closed Range

In this section we will develop a relatively tractable if and only if condition for the

composition operator onLΦ_α with closed range. Considering that any analytic automorphism

ofDhas the formϕaz cz−a/1−az, where|c| 1 anda∈D. By13 , we have the

following lemma.

Lemma 4.1. If one ofCϕ,Cϕ◦ϕa,Cϕa◦ϕhas closed range onLΦα, so have the other two.

Now thatLΦ_α,₀{f ∈LΦ_α :f0 0}is a closed subspace ofLΦ_α and dimLΦ_α/LΦ_α,₀ 1,

the following lemma is easily proved.

Lemma 4.2. Letϕbe an analytic self-map ofD, thenCϕhas closed range onLΦα if and only ifCϕhas

closed range onLΦ_α,₀.

Recall that the pseudohyperbolic metricρz, w,z, w∈Dis given by

ρz, w w−z

1−wz

. 4.1

For z ∈ D and 0 < r < 1 we define Dz, r {w ∈ D : ρz, w < r}. Forε > 0 we put

Ωε {z ∈ D : 1 − |z|2/1− |ϕz|2 ≥ ε}and Gε ϕΩε. We say that Gε satisfies the

Φ-reverse Carleson measure condition if there exists a positive constantηsuch that

Gε

Φ 1logfz2 1− |z|2α2dAz≥η

DΦ 1log

fz2 1− |z|2α2dAz,

4.2

wherefis analytic inDand_DΦ1log|fz|21− |z|2α2dAz<∞.

Theorem 4.3. Letϕbe an analytic self-map ofD. ThenCϕhas closed range onLΦα if and only if there

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Proof. We first assume that there existsε > 0 such thatGε satisfies theΦ-reverse Carleson

measure condition. Iff∈LΦ_α, then

_f_◦_ϕ

ΦΦ log 1f◦ϕ02

D

f◦ϕΦz 1− |z|2α2dAz

≥Φ log 1f◦ϕ02

Ωε

f◦ϕΦz 1− |z|2α2dAz

≥εα2

Φ log 1f◦ϕ02

Ωε

f◦ϕΦz 1−ϕz2α2dAz

εα2

n

Ωε∩Rn

f◦ϕΦz 1−ϕz2α2dAz εα2Φ log 1f◦ϕ02,

4.3

whereZis the zero point set ofϕand{Rn}is a partition ofD\Zinto at most countably many

semiclosed polar rectangles such thatϕis univalent on eachRn. LetSnϕRn∩Ωε. Then by

the change of variables involvingϕ, the last line above becomes

εα2

n

Gε

fΦ 1− |w|2α2χSnwdAz εα2Φ log 1f◦ϕ02

≥εα2

Gε

fΦ 1− |w|2α2dAz εα2Φ log 1f◦ϕ02

≥ηεα2

Df

Φ ₁_{− |}_w_|2α2_dA_z _ηεα2_Φ _log ₁_f_◦_ϕ₀2

.

4.4

So we show thatCϕhas closed range onLΦα.

Conversely, byLemma 4.2, we need to prove thatCϕhas closed range onLΦ_α,₀. Suppose

that there does not existε >0 such that GεsatisfiesΦ-reverse Carleson measure condition.

We can choose a sequence{fn}inLΦ_α,₀such that_DΦ1log|fnz|2dAα2z 1 for allnand

yet_GnfΦ1−w|2α2dA → 0 asn → ∞, whereGn ϕΩnandΩn {z∈D: 1− |ϕz|2 ≤

n1− |z|2}. Now

fn◦ϕ_ΦΦ log 1fn◦ϕ02

D

fn◦ϕ

Φ_z_dA α2z

Φ log 1fn◦ϕ02

Ωn

fn◦ϕ

Φ

zdAα2z

D\Ωn

fn◦ϕ

Φ_z_dA α2z.

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Sinceϕis an analytic self-map ofD. The Nevanlinna counting functionNϕ,α2satisfies

Nϕ,α2z O

log 1

|z|

α2

4.6

as|z| → 1. Using4.6and decompositions of the disk into polar rectangles8 , one can find

a positive constantc1such that

Ωn

fn◦ϕ

_Φ

dAα2z

Ωn

f_nΦNϕ,α2dAz≤c1

Gn

f_nΦz 1− |z|2α2dAz−→0 4.7

asn → ∞, and

D\Ωn

fn◦ϕ

_Φ

dAα2z≤ 1

nα2

D\Ωn

fn◦ϕ

_Φ

1−ϕz2α2dAz

≤ 1

nα2

Df

Φ

nNϕ,α2dAz

≤ 1

nα2fnΦ−→0

4.8

asn → ∞. Evidently,fn◦ϕ_Φ → 0 asn → ∞, thoughfn_Φ1 for alln. It follows thatCϕ

does not have closed range onLΦ_α.

We have oﬀered a criterion for the composition operator with closed range onLΦ_α, but

it seems that it is diﬃcult to check whether or notGεsatisfies theΦ-reverse Carleson measure

condition.

Theorem 4.4. The composition operatorCϕ has closed range onLΦα if and only if there are positive

constantsε, c, andrsuch thatAαGε∩Dz, r≥c|Dz, r|α2for allz∈D.

Proof. We first assume thatCϕ has closed range onLΦα. Then there is a constant ε > 0 such

thatGε satisfies theΦ-reverse Carleson measure condition. Thus, applying the proceeding

constructed functionhzto theΦ-reverse Carleson condition gives

Gε

1− |a|2

|1−az|2

α2

dAαz≥η

D

1− |a|2

|1−az|2

α2

dAαz η. 4.9

Since_D1− |z|2αdAz<∞, it allows to choose a fixed constantr >0 such that

D\Db,r 1− |z|

2α_dA_z_≤ 1

2C

D 1− |z|

2α_dA_z_.

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Changingw b−a 1−abz/1−ba b−azin4.10gives

D\Da,r

1− |a|2

|1−az|2

α2

dAαz≤

1 2C

D

1− |a|2

|1−az|2

α2

dAαz. 4.11

Combing4.11gives

Gε∩Da,r

1− |a|2

|1−az|2

α2

dAαz≥ 1

2C

D 1− |z|

2α_dA_z_. _4.12

The integral in the left of4.12is dominated by

AαGε∩Da, rsup

⎧ ⎨ ⎩

1− |a|2

|1−az|2 α2

:z∈Da, r

⎫ ⎬

⎭. 4.13

Since1−a|2_/_|₁₋_az_|2₁_/_|_D_{a, r}_|_for_z_∈_D_{a, r}_{, we get}

AαGε∩Da, r |Da, r|α2 ≥

1 2C

D 1− |z|

2α_dA_z_.

4.14

The converse can be derived from modification of18 , so we omit it here.

Remark 4.5. From 18 , we find that the composition operator has closed range on the

weighted Bergman space Apαif and only if there are positive constants ε, cand r such that

AαGε∩Dz, r≥c|Dz, r|α2for allz∈D. Thus, we have the following fact.

The composition operatorCϕhas closed range onLΦα if and only ifCϕhas closed range

onApα.

Let us further investigate the Φ-reverse Carleson measure condition, which can be

formulated as follows.

The space{f|Gε :f∈LΦ_α}is a closed subspace ofLΦ_α if and only if there exists a constant

ε >0 such thatGεsatisfiesΦ-reverse Carleson measure condition.

From the perspective of closed subspace, we will see the following special setting. Let

F {zn : n 1,2, . . .} be aδ-sequence inD. That is, there isk withρz, zk < δfor every

z∈D. We also assume thatFisγseparated for some fixedγ >0, that is,ρzm, zn≥γfor all

m /n. Using the subharmonicity of log|fz|for analytic functionfz, it is easy to see that

logfzk≤ C

Aα

Dzk, γ/2

Dzk,γ/2

logfzdAαz. 4.15

SinceΦlog|fz|is convex and increasing, we have

Φlogfzk≤

C Aα

Dzk, γ/2

Dzk,γ/2

(14)

Moreover, the formulaAαDzk, γ/21− |zk|2 α2

allows us to write

Φlogfzk≤C 1− |zk|2

−α−2

Dzk,γ/2

ΦlogfzdAαz. 4.17

SinceDzk, γ/2are disjoint, we obtain

k

Φlogfzk 1− |zk|2

α2

≤

DΦ

logfzdAαz. 4.18

Hence, the mapσ:f →f|_FtakesLΦ_α intoLΦF, μ, whereμis a measure onFthat assignszk

to the mass1− |zk|2α2and spaceLΦF, μ {f:F → C|_FΦlog|f|dμ <∞}. Of course,

the mapσmay be one to one. If the mapσis one to one, the mapσhas closed range if and

only if

DΦ

logfzdAαz≤

FΦ

logfdμ

k

Φlogfzk 1− |zk|2

α2

. 4.19

Acknowledgments

The authors are extremely thankful to the editor for pointing out several errors. This work

was supported by the Science Foundation of Sichuan Province no. 20072A04 and the

Scientific Research Fund of School of Science SUSE.

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