Product of differentiation and composition operators on the logarithmic Bloch space

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R E S E A R C H

Open Access

Product of differentiation and composition

operators on the logarithmic Bloch space

Jizhen Zhou

1

_{and Xiangling Zhu}

2*

*_{Correspondence: [email protected]} 2_{Faculty of Information Technology,}

Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau, China

Full list of author information is available at the end of the article

Abstract

We obtain a criterion for the boundedness and compactness of the products of diﬀerentiation and composition operatorsCϕDmon the logarithmic Bloch space in

terms of the sequence{zn}. An estimate for the essential norm ofCϕDmis given. MSC: 47B38; 30H30

1 Introduction

Denote byH(D) the space of all analytic functions on the unit diskD={z:|z|< }in the complex plane. LetH∞=H∞(D) denote the space of bounded analytic functions onD. Anf∈H(D) is said to belong to the Bloch spaceBif

fB=sup z∈D

f(z) –|z|<∞.

The logarithmic-Bloch space, denoted byLB, consists of allf ∈H(D) satisfying

flog=sup z∈D

 –|z|f(z)log e  –|z| <∞.

LBis a Banach space with the normfLB=|f()|+flog. It is well known thatLB∩H∞

is the space of multipliers of the Bloch spaceB(see [, ]). For some results on logarithmic-type spaces and operators on them, see, for example, [–].

Letϕbe an analytic self-map ofD. The composition operatorCϕis deﬁned by

Cϕ(f) =f ◦ϕ, f ∈H(D).

The diﬀerentiation operatorDis deﬁned byDf=f,f ∈H(D). For a nonnegative integer

m∈N, we deﬁne

Dmf =f(m), f ∈H(D).

The product of diﬀerentiation and composition operatorsCϕDmis deﬁned as follows:

CϕDmf =f(m)◦ϕ, f ∈H(D).

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A basic problem concerning concrete operators on various Banach spaces is to relate the operator theoretic properties of the operators to the function theoretic properties of their symbols, which attracted a lot of attention recently, the reader can refer to [–].

It is a well-known consequence of the Schwarz-Pick lemma that the composition oper-ator is bounded onB. See [–, , –, ] for the study of composition operators and weighted composition operators on the Bloch space. The product-type operators on or into Bloch type spaces have been studied in many papers recently; see [–, , – , , ] for example.

LetXandYbe two Banach spaces. Recall that a linear operatorT:X→Yis said to be compact if it takes bounded sets inXto sets inYwhich have compact closure. The essen-tial norm of an operatorTbetweenXandY is the distance to the compact operatorsK, that is,TX→Y

e =inf{T–K:Kis compact}, where · is the operator norm. It is easy to see thatTX→Y

e =  if and only ifTis compact. For some results in the topic, see, for example, [, , , , , , ].

In [], Wu and Wulan obtained a characterization for the compactness of the product of diﬀerentiation and composition operators acting on the Bloch space as follows:

Theorem A Letϕbe an analytic self-map ofD,m∈N.Then CϕDm:B→Bis compact if

and only if

lim n→∞CϕD

m_zn

B= .

The purpose of the paper is to extend Theorem A to the case ofLB. We will character-ize the boundedness and compactness ofCϕDmin terms of the sequence{zn}. Moreover, an estimate for the essential norm ofCϕDmwill be given. The main results are given in Sections  and .

In the paper, we say that a real sequence{an}n∈Nis asymptotic to another real sequence of{bn}n∈Nand write ‘an∼bn’ if and only if

lim n→∞

an

bn = .

In addition, we say thatABif there exists a constantCsuch thatA≤CB. The symbol

A≈Bmeans thatABA.

2 Auxiliary lemmas

In this section, we state and prove some auxiliary results which will be used to prove the main results in this paper.

Lemma . For m,n∈N,deﬁne the function Hm,n: [, )→[,∞)by

Hm,n(x) =

n! (n–m– )!x

n–m–_{( –}_x₎m+_log e

 –x. (.)

Then the following statements hold:

(i) Forn,m∈Nandn≥m+ ,there is a uniquexm,n∈[, )such thatHm,n(xm,n)is the

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(ii)

lim

n→∞xm,n=  (.)

and

lim n→∞

n( –xm,n)

=m+ . (.)

(iii)

lim n→∞

max<t<Hm,n(t) log(n+ ) =

m+ 

e

m+

. (.)

Proof Directly computing we have

H_m,n(x) =

n! (n–m– )!x

n–m–_{( –}_x₎m

(n–m–  –nx)log e  –x+x .

Deﬁne

gm,n(x) = (n–m–  –nx)log

e

 –x+x, x∈[, ). (.)

It is easy to see thatgm,nis continuous on [,) andgm,n() =n–m– ≥,limx→–g_m_,_n(x) =

–∞. Furthermore,

g_m_,_n(x) = –nlog e  –x+n–

m+ 

 –x +  < , x∈[, ).

Thengm,nis decreasing on [, ). Whenn=m+ , we getmax≤x<Hm,n(x) =Hm,n(). When

n>m+ , the intermediate value theorem of continuous function gives the result that there exists a uniquexm,n∈(, ) such thatgm,n(xm,n) = . So we have

max

<t<Hm,n(x) =Hm,n(xm,n).

(i) has been proved. By (.), we havegm,n(xm,n) = . Thus

n–m– 

n –xm,n log e

 –xm,n

= –xm,n

n .

It follows fromlimn→∞xmn,n=  andlog e

–xm,n≥ that (.) holds. Also,gm,n(xm,n) =  gives the result that

n–m– 

n –xm,n= – xm,n

nlog_–_xme

,n .

So we have

n( –xm,n) –m–  = –

xm,n log e

–xm,n .

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Note that

nlogxm,n∼nlog

 + (xm,n– )

∼n(xm,n– )→–m–  asn→ ∞.

This and (.) give

lim n→∞x

n–m–

m,n =e–m–. (.)

By (.) and (.) we obtain

lim n→∞

Hm,n(xm,n) log(n+ ) =nlim→∞

n!xn–m–

m,n ( –xm,n)m+log_–_xme_,_n (n–m– )!log(n+ ) =e–m– lim

n→∞

n!((m+ )/n)m+_log en m+ (n–m– )!log(n+ ) =

m+ 

e

m+ ,

which shows that (iii) hold. The proof is complete.

Lemma . Let m,n∈Nand n–m–  > .Let rm,n= (n–m– )/n.Then Hm,nis increasing

on[rm,n–m,rm,n]and

min rm,n–m≤x≤rm,n

Hm,n(x) =Hm,n(rm,n–m)∼

m+ 

e

m+

log(n+ ) as n→ ∞. (.)

Consequently,

Hm,n(x)

zn_LB =

Hm,n(rm,n–m)

zn_LB ∼

(m+ )m+

em as n→ ∞. (.)

Proof Sincen–m–  > , we have

H_m_,_n(rm,n) =

n! (n–m– )!

n–m– 

n

n–m–

m+ 

n

m

n–m– 

n > .

By Lemma ., we haverm,n<xm,n, wherexm,nis given as in Lemma .. SinceHm,n(x) >  forx∈(,xm,n), we see thatHm,nis increasing on [rm,n–m,rm,n]. Thus

Hm,n(x) =Hm,n(rm,n–m)

= n!

(n–m– )!

n– m– 

n–m

n–m–

m+ 

n–m

m+

loge(n–m)

m+  . Applying the important limitlimn→∞( +n)n=ewe obtain the result that (.) holds.

By Lemma . we have

zn_LB=sup

|z|<

n|z|n– –|z|log e

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wherex,nis given in Lemma .. By Lemma . we have

lim n→∞

Hm,n(rm,n–m)

zn_LB =_nlim_→∞

Hm,n(rm,n–m) log(n+ )

log(n+ )

zn_LB = lim

n→∞

Hm,n(rm,n–m) log(n+ ) nlim→∞

log(n+ )

znLB =

(m+ )m+

em .

This gives (.). The proof is complete.

Lemma .[] For m∈N.Then f ∈LBif and only if

sup z∈D

 –|z|mf(m)(z)log e  –|z|<∞. Moreover,

fLB≈

m–

j=

f(j)()+sup z∈D

 –|z|mf(m)(z)log e  –|z|.

3 The boundedness ofCϕDmonLB

In this section, we will state the boundedness criterion for the operatorCϕDm onLB. Since the boundedness ofCϕDmonLBgivesϕ∈LB, we may always assume thatϕ∈LB. The main result of this section is stated as follows.

Theorem . Let m∈Nandϕbe an analytic self-map ofDsuch thatϕ∈LB.Then CϕDm

is bounded onLBif and only if

sup n∈N

CϕDm(zn)LB

zn_LB <∞. (.)

Proof ⇒) Assume thatCϕDm is bounded onLB, that is,CϕDmLB→LB<∞. Since the sequence{zn_/_zn_LB}_{is bounded in the logarithmic Bloch space}_LB_{, we have}

CϕDm(zn)LB

znLB ≤CϕD m

LB→LB

_znzLBn

LB≤

CϕDm_LB_→_LB<∞,

for anyn∈N, from which the implication follows.

⇐) We now assume that the condition (.) holds. On the one hand, for the case sup_z_∈_D|ϕ(z)|< , there is anr∈(, ) such that|ϕ(z)|<r. By (.), for any givenf ∈LB, we have

CϕDmf_LB =sup z∈D

 –|z|log e  –|z|f

(m+)_ϕ(_z₎_ϕ₍_z₎

≤sup z∈DϕLB

|f(m+)_(ϕ(_z₎₎_|_{( –}_|_ϕ(_z₎_|₎m+_log e –|ϕ(z)|

( –|ϕ(z)|)m+_log e –|ϕ(z)|

sup z∈D

ϕLBfLB

( –r)m+_ln e –r

<∞.

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On the other hand, for the casesup_z_∈_D|ϕ(z)|= . LetNbe the smallest positive integer such thatDN is not empty, where

Dn=

z∈D:rm,n–m≤ϕ(z)≤rm,n

andrm,nis given in Lemma .. Note thatHm,n(|ϕ(z)|) > , whenz∈Dn,n≥N, by (.) we obtain

:= inf z∈Dn

Hm,n(|ϕ(z)|)

zn_LB > .

For any givenf ∈LB, by Lemma . we have CϕDmf_LB=sup

z∈D

 –|z|log e  –|z|f

(m+)_ϕ(_z₎_ϕ₍_z₎

=sup n≥N

sup z∈Dn

 –|z|log e  –|z|f

(m+)_ϕ(_z₎_ϕ₍_z₎

=sup n≥N

sup z∈Dn

 –|z|log e  –|z|f

(m+)_ϕ(_z₎_ϕ₍_z₎ znLB

Hm,n(|ϕ(z)|)

zn_LB

fLB

nsup≥N sup z∈Dn

n! (n–m– )!

 –|z|log e  –|z|ϕ

₍_z₎|ϕ(z)|n–m– zn

LB

≤fLB

nsup≥N

CϕDm(zn)LB

zn_LB .

The proof is complete.

4 The essential norm ofCϕDmonLB

DenoteKrf(z) =f(rz) forr∈(, ). ThenKris a compact operator on the spaceLB. It is easy to see thatKr ≤. We denote byIthe identity operator.

In order to give the lower and upper estimate for the essential norm ofCϕDmonLB, we need the following result.

Lemma . There is a sequence{rk},with <rk< tending to,such that the compact

operator

Ln= 

n

k=

Krk

onLBsatisﬁes:

(i) for anyt∈(, ),limn→∞supf_LB≤tsup|z|≤t|((I–Ln)f)(z)|= , (iia) limn→∞supf_LB≤sup|z|<|(I–Ln)f(z)| ≤,

(iib) lim_n_→∞sup_f

LB≤sup|z|<s|(I–Ln)f(z)|= ,for anys∈(, ),

(iii) lim sup_n_→∞I–Ln ≤.

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any  <s< , we choose an increasing sequencerktending to  such thatrk≥ ––_ks. For

any givenz∈Dandrk,k= , , , . . . , there exists ansk∈(rk, ) such that

f(z) –frk(z)=zf(skz)(z–rkz). (.)

For anyf ∈LBwithfLB≤, we have (I–Ln)f(z)≤ 

n

k=

f(z) –frk(z)≤ 

n

k=

f(skz)( –rk)

≤ 

n

k=

 –rk ( –|rkz|)log_–_|e_r

kz|

≤ 

n

k=  = .

Thus

lim sup n→∞

sup

f_LB≤ sup

|z|<

(I–Ln)f(z)≤.

This shows that (iia) holds.

If|z| ≤s, by the equality (.), we have ₍_I_–_L_n)_f₍_z₎_≤ 

n

k=

 –rk ( –|sz|)log_–e_|_sz_|

≤ 

n

k=  –rk ( –s)=



n

k= 

k ≤ π n.

The above estimate gives (iib). The proof is complete.

The following lemma can be proved in a standard way; see, for example Proposition . in [].

Lemma . Let m∈Nandϕbe an analytic self-map ofD.Then CϕDmis compact onLB

if and only if CϕDm is bounded onLB and for any bounded sequence{fn}inLBwhich

converges to zero uniformly on compact subsets ofD,thenCϕDmfnLB→as n→ ∞.

Theorem . Let m∈N andϕ be an analytic self-map of D. Suppose that CϕDm is

bounded onLB.Then the estimate for the essential norm of CϕDmonLBis _C_ϕ_DmLB→LB

e ≈lim sup

n→∞

CϕDm(zn)LB

zn

LB . (.)

Proof We ﬁrst give the lower estimate for the essential norm. Without loss of generality, we assume thatn≥m+ . Choose the sequence of functionfn(z) =zn_/_zn_LB_,_n_∈_N_{. Then}

fnLB= , and{fn}converges to zero weakly onLBasn→ ∞. Thus we have

lim

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for any given compact operatorKonLB. The basic inequality gives CϕDm–KLB

→LB_≥

CϕDm–K

fn_LB≥CϕDmfn_LB–KfnLB.

Thus we obtain

CϕDm–KLB→LB≥lim sup n→∞

CϕDmfn_LB≥lim sup n→∞

CϕDmfn_LB.

So we have

CϕDmLB

→LB

e =inf_KCϕD

m_–_K_≥_{lim sup} n→∞

CϕDm(zn)LB

zn_LB .

Now we give the upper estimate for the essential norm. For the case ofsup_z_∈_D|ϕ(z)|< , there is a numberδ∈(, ) such thatsup_z_∈_D|ϕ(z)|<δ. In this case, the operatorCϕDmis compact onLB. In fact, choose a bounded sequence{fn}n∈NinLBwhich converges to zero uniformly on compact subset ofD. From Cauchy’s integral formula,{fn(m+)}converges to zero on compact subsets ofDasn→ ∞. It follows that

lim n→∞CϕD

m_f

n_LB = lim n→∞f

(m) n

ϕ()+CϕDmfn_log

= lim

n→∞sup_z_∈_D

 –|z|log e  –|z|f

(m+) n

ϕ(z)ϕ(z)

≤ ϕLB lim n→∞sup_z_∈_Df

(m+) n

ϕ(z)

=ϕLB lim

n→∞_|sup_w_|≤δ _f(m+)

n (w)= .

Then the operatorCϕDmis compact onLBby Lemma .. This gives

CϕDmLB_e →LB= . (.)

On the other hand, by Lemma . and (.) we obtain

zn_LB=H,n(x,n)≥H,n(r,n)≥  log(en), which implies that

lim sup n→∞

CϕDm(zn)LB

zn_LB

≤elim sup n→∞

sup z∈D

 –|z|log e  –|z|

n!

(n–m– )!ϕ(z) n–m–ϕ

(z)

≤eϕLB lim n→∞n

m_δn–m–_{= .}

Combining the last inequality with (.), we get the desired result.

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compact onLB. Hence

CϕDmLB_e →LB≤lim sup n→∞

CϕDm–CϕDmLn_LB_→_LB

=lim sup n→∞

CϕDm(I–Ln)_LB_→_LB

=lim sup n→∞

sup

f_LB≤

CϕDm(I–Ln)f_LB

=lim sup n→∞ fsupLB≤

(I–Ln)f(m)◦ϕ_LB

≤I+I,

where

I=lim sup n→∞

sup

f_LB≤

(I–Ln)f(m)ϕ()

and

I=lim sup n→∞ fsupLB≤

sup z∈D

(I–Ln)f(m+)ϕ(z)ϕ(z) –|z|log e  –|z|.

It follows from Lemma .(iib) and Cauchy’s integral formula thatI= . For each positive integern≥m+ , we deﬁne

Dn=

z∈D:rm,n–m≤ϕ(z)<rm,n

,

whererm,nis given in Lemma .. Letkbe the smallest positive integer such thatDk= . Sincesup_z_∈_D|ϕ(z)|= ,Dnis not empty for every integern≥kandD=

_∞

n=kDn, we have

sup

f_LB≤ sup z∈D

(I–Ln)f(m+)ϕ(z)ϕ(z) –|z|log e

 –|z|=I+I,

where

I= sup

fLB≤ sup k≤i≤N–

sup z∈Di

(I–Ln)f (m+)

ϕ(z)ϕ(z) –|z|log e  –|z|

and

I= sup

f_LB≤ sup N≤i

sup z∈Di

(I–Ln)f(m+)ϕ(z)ϕ(z) –|z|log e  –|z|.

HereNis a positive integer determined as follows. By (.),

lim i→∞

zi_LB

Hm,i(rm,i–m)

= e

m

(m+ )m+.

Hence, for any givenε> , there exists anNsuch that

zi_LB

Hm,i(rm,i–m)≤

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wheni≥N. For suchNit follows that

I= sup

f_LB≤ sup N≤i

sup z∈Di

(I–Ln)f(m+)ϕ(z)ϕ(z) –|z|log e  –|z|

= sup

fLB≤ sup N≤i

sup z∈Di

(I–Ln)f(m+)ϕ(z)ϕ(z)

· –|z|log e  –|z|

Hm,i(|ϕ(z)|)

zi_LB

ziLB Hm,i(|ϕ(z)|)

em

(m+ )m+ +ε sup fLB≤

(I–Ln)f_LBsup N≤i

sup z∈Di

ϕ(z)

· –|z|log e  –|z|

i! (i–m– )!

|ϕ(z)|i–m–

zi_LB

≤

em

(m+ )m+ +ε I–Lnsup N≤i

CϕDm(zi)LB

zi_LB . Thus lim sup n→∞ I em

(m+ )m+ +ε sup N≤i

CϕDm(zi)LB

zi

LB . (.)

By (ii) of Lemma . and Cauchy’s integral formula, we have

lim sup n→∞ I

=lim sup n→∞

sup

fLB≤ sup k≤i<N–

sup z∈Di

(I–Ln)f (m+)

ϕ(z)ϕ(z) –|z|log e  –|z|

≤ ϕLBlim sup n→∞

sup

f_LB≤ sup

|ϕ(z)|<rm,N–

(I–Ln)f(m+)ϕ(z)

= ,

which together with (.) implies that

I

em

CϕDm(zi)LB

zi_LB . (.)

From (.) we obtain

CϕDmLB

→LB

e ≤I+I

em

CϕDm(zi)LB

zi_LB . By the arbitrariness ofε, we get

CϕDmLB

→LB

e

em

(m+ )m+lim sup n→∞

CϕDm(zn)LB

zn_LB .

The proof is complete.

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Corollary . Let m∈Nandϕbe an analytic self-map ofDsuch that CϕDmis bounded

onLB.Then CϕDmis compact onLBif and only if

lim sup n→∞

CϕDm(zn)LB

zn_LB = .

Especially, whenm= , from the proof of Theorem ., we get the exact formula for essential norm of composition operator onLB.

Corollary . Letϕbe an analytic self-map ofD.Suppose that Cϕis bounded onLB;then

CϕLBe →LB=lim sup n→∞

ϕnLB

zn_LB.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Sciences, Anhui University of Science and Technology, Huainan, Anhui 232001, China.}2_{Faculty of Information}

Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau, China.

Acknowledgements

The project is supported by Department of education of Anhui Province of China (No. KJ2013A101), the Natural Science Foundation of Guangdong (No. S2013010011978) and NSF of China (No. 11471143).

Received: 22 September 2014 Accepted: 29 October 2014 Published:10 Nov 2014 References

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