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R E S E A R C H

Open Access

Essential norm of generalized weighted

composition operators from the Bloch space

to the Zygmund space

Qinghua Hu

1

, Yafeng Shi

2

, Yecheng Shi

3

and Xiangling Zhu

4*

*Correspondence: jyuzxl@163.com 4Department of Mathematics, Jiaying University, Meizhou, Guangdong 515063, China Full list of author information is available at the end of the article

Abstract

In this paper, we give some estimates of the essential norm for generalized weighted composition operators from the Bloch space to the Zygmund space. Moreover, we give a new characterization for the boundedness and compactness of the operator.

MSC: 30H30; 47B38

Keywords: Bloch space; Zygmund space; essential norm; generalized weighted composition operator

1 Introduction

LetXandYbe Banach spaces. The essential norm of a bounded linear operatorT:XY is its distance to the set of compact operatorsKmappingXintoY, that is,

Te,XY=inf

T–KXY :Kis compact

,

where · XY is the operator norm.

LetDbe the open unit disk in the complex planeCandH(D) the space of analytic func-tions onD. Letϕbe a nonconstant analytic self-map ofD,uH(D), andnbe a nonneg-ative integer. Thegeneralized weighted composition operator, denoted byDn

ϕ,u, is defined

onH(D) by

Dnϕ,uf

(z) =u(z)f(n)ϕ(z), z∈D.

Whenn= , the generalized weighted composition operatorDn

ϕ,uis the weighted

compo-sition operator, denoted byuCϕ. In particular, whenn=  andu= , we get the compo-sition operatorCϕ. Ifn=  andu(z) =ϕ(z), thenDn

ϕ,u=DCϕ, which was widely studied,

for example, in [–]. Ifu(z) = , thenDn

ϕ,u=CϕDn, which was studied, for example, in

[, , , ]. For the study of the generalized weighted composition operator on various function spaces see, for example, [–]. Recently there has been a huge interest in the study of various related product-type operators containing composition operators; see, e.g., [–] and the references therein.

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The Bloch space, denoted byB, is defined to be the set of allfH(D) such that

fB=f()+sup z∈D

 –|z|f(z)<∞.

Bis a Banach space with the above norm. AnfBis said to belong to the little Bloch space Biflim|z|→|f(z)|( –|z|) = . See [] for more information of Bloch spaces.

Composi-tion operators, as well as weighted composiComposi-tion operators mapping into Bloch-type spaces were studied a lot see, for example, [, , , –].

The Zygmund space, denoted byZ, is the space consisting of allfH(D) such that

fZ=f()+f()+sup z∈D

 –|z|f(z)<∞.

It is easy to see thatZis a Banach space with the above norm · Z. See [, , , , , , , –] for some results of the Zygmund space and related operators mapping into the Zygmund space or into some of its generalizations.

In , Madigan and Matheson proved that:BBis compact if and only if (see [])

lim |ϕ(z)|→

( –|z|) ( –|ϕ(z)|)ϕ

(z)= .

In , Montes-Rodrieguez in [] obtained the exact value for the essential norm of the operator:BB,i.e.,

Cϕe,BB=lims sup

|ϕ(z)|>s

( –|z|)|ϕ(z)|

( –|ϕ(z)|) .

Tjani in [] proved that :BB is compact if and only if lim|a|→CϕσaB= , whereσa=–aza–z. Wulanet al.in [] showed that:BBis compact if and only if limj→∞ϕjB= . Ohnoet al.studied the boundedness and compactness of the operator

uCϕon the Bloch space in []. The estimate for the essential norm of the operatoruCϕ on the Bloch space was given in []. Some new estimates for the essential norm ofuCϕ on the Bloch space were given in [, ]. In [], Zhu has obtained some estimates for the essential norm ofDn

ϕ,uon the Bloch space whennis a positive integer.

Stević studied the boundedness and compactness ofDn

ϕ,u:BZin [] (see also []).

In [], Li and Fu obtained a new characterization for the boundedness, as well as the compactness forDn

ϕ,u:BZby using three families of functions. We combine the results

in [] and [] as follows.

Theorem A Let n be a positive integer, uH(D), andϕ be an analytic self-map of D. Suppose that Dn

ϕ,u:BZis bounded,then the following statements are equivalent: (a) The operatorDn

ϕ,u:BZis compact. (b)

lim |ϕ(w)|→

Dnϕ,u(w)Z= lim |ϕ(w)|→

Dnϕ,u(w)Z= lim |ϕ(w)|→

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where

(w)(z) =

 –|ϕ(w)|

 –ϕ(w)z , (w)(z) =

( –|ϕ(w)|)

( –ϕ(w)z) ,

(w)(z) =

( –|ϕ(w)|)

( –ϕ(w)z) , z∈D.

(c)

lim |ϕ(z)|→

( –|z|)|u(z)|

( –|ϕ(z)|)n =|ϕ(z)lim|→

( –|z|)|u(z)||ϕ(z)| ( –|ϕ(z)|)n+

= lim |ϕ(z)|→

( –|z|)|u(z)ϕ(z) +u(z)ϕ(z)| ( –|ϕ(z)|)+n = .

Motivated by these observations, the purpose of this paper is to give some estimates of the essential norm for the operatorDn

ϕ,u:BZ. Moreover, we give a new

characteriza-tion for the boundedness, compactness, and essential norm of the operatorDn

ϕ,u:BZ.

Throughout this paper, we say thatPQif there exists a constantCsuch thatPCQ. The symbolPQmeans thatPQP.

2 Essential norm ofDnϕ,u:BZ

In this section, we give two estimates of the essential norm for the operatorDn

ϕ,u:BZ.

Theorem . Let n be a positive integer,uH(D),andϕbe an analytic self-map ofDsuch that Dn

ϕ,u:BZis bounded.Then

Dnϕ,ue,BZ≈max{A,B,C} ≈max{E,F,G},

where

A:=lim sup |a|→

Dnϕ,u

 –|a|

 –az Z, B:=lim sup|a|→

Dnϕ,u

( –|a|)

( –az)

Z,

C:=lim sup |a|→

Dnϕ,u

( –|a|)

( –az)

Z

, F:=lim sup

|ϕ(z)|→

( –|z|)|u(z)|

( –|ϕ(z)|)n ,

E:=lim sup |ϕ(z)|→

( –|z|)|u(z)ϕ(z) +u(z)ϕ(z)|

( –|ϕ(z)|)n+ ,

and

G:=lim sup |ϕ(z)|→

( –|z|)|u(z)||ϕ(z)| ( –|ϕ(z)|)n+ .

Proof First we prove thatmax{A,B,C} ≤ Dn

ϕ,ue,BZ. Leta∈D. Define

fa(z) =

 –|a|

( –az), ga(z) =

( –|a|)

( –az) , ha(z) =

( –|a|)

( –az) , z∈D.

It is easy to check thatfa,ga,haBandfaB,gaB,haB for alla∈D

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contained inBwhich converges uniformly to  on compact subsets ofDconverges weakly

to  inB(see [, ]). Thus, for any compact operatorK:B→Z, we have

lim

|a|→KfaZ= , |alim|→KgaZ= , |alim|→KhaZ= .

Hence

Dnϕ,u–KBZlim sup|

a|→

Dnϕ,u–K

faZ

≥lim sup |a|→

Dnϕ,ufaZ–lim sup |a|→

KfaZ=A,

Dnϕ,u–KBZlim sup|

a|→

Dnϕ,u–K

gaZ

≥lim sup |a|→

Dnϕ,ugaZ–lim sup |a|→

KgaZ=B,

and

Dnϕ,u–KBZlim sup|a|→ Dnϕ,u–K

haZ

≥lim sup |a|→

Dnϕ,uhaZ–lim sup |a|→ KhaZ

=C.

Therefore, from the definition of the essential norm, we obtain

Dnϕ,ue,BZ=infKD n

ϕ,u–KBZmax{A,B,C}. Next, we prove thatDn

ϕ,ue,BZmax{E,F,G}. Let{zj}j∈Nbe a sequence inDsuch that |ϕ(zj)| → asj→ ∞. Define

kj(z) =

 –|ϕ(zj)|

 –ϕ(zj)z

– n+  (n+ )(n+ )

( –|ϕ(zj)|)

( –ϕ(zj)z)

+ 

(n+ )(n+ )

( –|ϕ(zj)|)

( –ϕ(zj)z)

,

lj(z) =

 –|ϕ(zj)|

 –ϕ(zj)z

– (n+ )  + (n+ )(n+ )

( –|ϕ(zj)|)

( –ϕ(zj)z)

+ 

 + (n+ )(n+ )

( –|ϕ(zj)|)

( –ϕ(zj)z)

,

and

mj(z) =

 –|ϕ(zj)|

 –ϕ(zj)z

–  n+ 

( –|ϕ(zj)|)

( –ϕ(zj)z)

+ 

(n+ )(n+ )

( –|ϕ(zj)|)

( –ϕ(zj)z)

.

Similarly to the above we see that allkj,lj, andmjbelong toBand converge to  weakly

inB. Moreover,

kj(n)ϕ(zj)

= , kj(n+)ϕ(zj)

= , kj(n+)ϕ(zj)=

n! n+ 

|ϕ(zj)|n+

( –|ϕ(zj)|)n+

,

lj(n+)ϕ(zj)

= , l(n+)j ϕ(zj)

= , l(n)j ϕ(zj)=

n!  + (n+ )(n+ )

|ϕ(zj)|n

( –|ϕ(zj)|)n

,

m(n)j ϕ(zj)

= , m(n+)j ϕ(zj)

= , m(n+)j ϕ(zj)= n! |

ϕ(zj)|n+

( –|ϕ(zj)|)n+

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Then for any compact operatorK:BZ, we obtain

Dnϕ,u–KBZlim sup

j→∞

Dnϕ,u(kj)Z–lim sup j→∞

K(kj)Z

lim sup j→∞

( –|zj|)|u(zj)ϕ(zj) +u(zj)ϕ(zj)||ϕ(zj)|n+

( –|ϕ(zj)|)n+

,

Dnϕ,u–KBZlim supj→∞ Dnϕ,u(lj)Z–lim sup j→∞

K(lj)Z

lim sup j→∞

( –|zj|)|u(zj)||ϕ(zj)|n

( –|ϕ(zj)|)n

,

and

Dnϕ,uKBZlim sup j→∞

Dnϕ,u(mj)Z–lim sup j→∞

K(mj)Z

lim sup

j→∞

( –|zj|)|u(zj)||ϕ(zj)||ϕ(zj)|n+

( –|ϕ(zj)|)n+

.

From the definition of the essential norm, we obtain

Dnϕ,ue,BZ=infKD n

ϕ,u–KBZ lim sup

j→∞

( –|zj|)|u(zj)ϕ(zj) +u(zj)ϕ(zj)||ϕ(zj)|n+

( –|ϕ(zj)|)n+

=lim sup |ϕ(z)|→

( –|z|)|u(z)ϕ(z) +u(z)ϕ(z)|

( –|ϕ(z)|)n+ =E,

Dnϕ,ue,BZ=inf K

Dnϕ,uKBZ

lim sup j→∞

( –|zj|)|u(zj)||ϕ(zj)|n

( –|ϕ(zj)|)n

=lim sup |ϕ(z)|→

( –|z|)|u(z)| ( –|ϕ(z)|)n =F,

and

Dnϕ,ue,BZ =infKD n

ϕ,u–KBZ

lim sup

j→∞

( –|zj|)|u(zj)||ϕ(zj)||ϕ(zj)|n+

( –|ϕ(zj)|)n+

=lim sup |ϕ(z)|→

( –|z|)|u(z)||ϕ(z)|

( –|ϕ(z)|)n+ =G.

Hence

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Now, we prove that

Dnϕ,ue,BZmax{A,B,C} and Dnϕ,ue,BZmax{E,F,G}.

Forr∈[, ), setKr:H(D)→H(D) by (Krf)(z) =fr(z) =f(rz),fH(D). It is obvious that

frf uniformly on compact subsets ofDasr→. Moreover, the operatorKris compact

onBandKrB→B≤ (see []). Let{rj} ⊂(, ) be a sequence such thatrj→ asj→ ∞.

Then for all positive integerj, the operatorDn

ϕ,uKrj:BZis compact. By the definition

of the essential norm, we get

Dnϕ,ue,BZ≤lim sup j→∞

Dnϕ,u–Dnϕ,uKrjBZ. (.)

Therefore, we only need to prove that

lim sup j→∞

Dnϕ,u–Dnϕ,uKrjBZmax{A,B,C}

and

lim sup j→∞

Dnϕ,u–Dnϕ,uKrjBZmax{E,F,G}.

For anyfBsuch thatfB≤, we consider

Dnϕ,uDnϕ,uKrj

fZ =u()f(n)ϕ()rn

ju()f (n)r

jϕ()

+u()(f–frj)

(n)ϕ()+u()(ff rj)

(n+)ϕ()ϕ()

+u·(f –frj) (n)ϕ

∗, (.)

wheref∗=supz∈D( –|z|)|f(z)|.

It is obvious that

lim j→∞

u()f(n)ϕ()rjnu()f(n)rjϕ()=  (.)

and

lim j→∞

u()(f–frj)

(n)ϕ()+u()(ff rj)

(n+)ϕ()ϕ()= . (.)

Now, we consider

lim sup j→∞

u·(f–frj) (n)ϕ

≤lim sup j→∞

sup |ϕ(z)|≤rN

 –|z|(f–frj)

(n+)ϕ(z)u(z)ϕ(z) +u(z)ϕ(z)

+lim sup j→∞

sup |ϕ(z)|>rN

 –|z|(f –frj)

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+lim sup j→∞

sup |ϕ(z)|≤rN

 –|z|(f–frj)

(n)ϕ(z)u(z)

+lim sup j→∞

sup |ϕ(z)|>rN

 –|z|(f –frj)

(n)ϕ(z)u(z)

+lim sup j→∞

sup |ϕ(z)|≤rN

 –|z|(f–frj)

(n+)ϕ(z)ϕ(z)

u(z)

+lim sup j→∞ |ϕ(z)sup|>rN

 –|z|(f –frj)

(n+)ϕ(z)ϕ(z)u(z)

=Q+Q+Q+Q+Q+Q, (.)

whereN∈Nis large enough such thatrj≥for alljN,

Q:=lim sup j→∞

sup |ϕ(z)|≤rN

 –|z|(f–frj)

(n+)ϕ(z)u(z)ϕ(z) +u(z)ϕ(z),

Q:=lim sup j→∞ |ϕ(z)sup|>rN

 –|z|(f–frj)

(n+)ϕ(z)u(z)ϕ(z) +u(z)ϕ(z),

Q:=lim sup j→∞ |ϕ(z)sup|≤rN

 –|z|(f–frj)

(n)ϕ(z)u(z),

Q:=lim sup j→∞ |ϕ(z)sup|>rN

 –|z|(f –frj)

(n)ϕ(z)u(z),

Q:=lim sup j→∞

sup |ϕ(z)|≤rN

 –|z|(f–frj)

(n+)ϕ(z)ϕ(z)

u(z),

and

Q:=lim sup j→∞ |ϕ(z)sup|>rN

 –|z|(f –frj)

(n+)ϕ(z)ϕ(z)u(z).

SinceDnϕ,u:BZis bounded, by Theorem  of [], we see thatuZ,

K:=sup z∈D

 –|z|u(z)ϕ(z) +u(z)ϕ(z)<∞

and

K:=sup z∈D

 –|z|ϕ(z)u(z)<∞.

Sincern+j fr(n+)jf

(n+), as well asrn+ j f

(n+) rjf

(n+)uniformly on compact subsets ofD

asj→ ∞, we have

Q≤Klim sup j→∞

sup |w|≤rN

f(n+)(w) –rn+j f(n+)(rjw)=  (.)

and

Q≤Klim sup j→∞ |wsup|≤rN

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Similarly, from the fact thatuZwe have

Q≤ uZlim sup

j→∞ sup |w|≤rN

f(n)(w) –rnjf(n)(rjw)= . (.)

Next we considerQ. We haveQ≤lim supj→∞(S+S), where

S:= sup |ϕ(z)|>rN

 –|z|f(n+)ϕ(z)u(z)ϕ(z) +u(z)ϕ(z)

and

S:= sup |ϕ(z)|>rN

 –|z|rjn+f(n+)rjϕ(z)u(z)ϕ(z) +u(z)ϕ(z).

First we estimateS. Using the fact thatfB≤ and Theorem . in [], we have

S = sup |ϕ(z)|>rN

 –|z|f(n+)ϕ(z)u(z)ϕ(z) +u(z)ϕ(z)

×( –|ϕ(z)||ϕ(z)|)n+n+(n+ ) n!

|ϕ(z)|n+n!

(n+ )( –|ϕ(z)|)n+

(n+ )fB n!rn+

N

sup |ϕ(z)|>rN

 –|z|u(z)ϕ(z) +u(z)ϕ(z)

× n!|ϕ(z)|n+ (n+ )( –|ϕ(z)|)n+

sup |ϕ(z)|>rN

 –|z|u(z)ϕ(z) +u(z)ϕ(z) n!|ϕ(z)|

n+

(n+ )( –|ϕ(z)|)n+

sup |a|>rN

Dnϕ,u

fa

(n+ )ga

(n+ )(n+ )+

ha

(n+ )(n+ ) Z sup

|a|>rN

Dnϕ,u(fa)Z+

n+  (n+ )(n+ )|asup|>rN

Dnϕ,u(ga)Z

+ 

(n+ )(n+ )|asup|>rN

Dnϕ,u(ha)Z. (.)

Taking the limit asN→ ∞we obtain

lim sup j→∞

Slim sup |a|→

Dnϕ,u(fa)Z+lim sup |a|→

Dnϕ,u(ga)Z

+lim sup |a|→

Dnϕ,u(ha)Z

=A+B+C.

Similarly, we havelim supj→∞SA+B+C,i.e., we get

QA+B+Cmax{A,B,C}. (.)

From (.), we see that

lim sup

j→∞ Slim sup|ϕ(z)|→

( –|z|)|u(z)ϕ(z) +u(z)ϕ(z)|

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Similarly we havelim supj→∞SE. Therefore

QE. (.)

Next we considerQ. We haveQ≤lim supj→∞(S+S), where

S:= sup |ϕ(z)|>rN

 –|z|f(n)ϕ(z)u(z)

and

S:= sup |ϕ(z)|>rN

 –|z|rnjf(n)rjϕ(z)u(z).

After some calculation, we have

S = sup |ϕ(z)|>rN

 –|z|f(n)ϕ(z)u(z)

×( –|ϕ(z)|n!|)n( + (nϕ(z)|n+ )(n+ ))   + (n+ )(n+ )

n!|ϕ(z)|n

( –|ϕ(z)|)n

n( + (n+ )(n+ ))

n! fB|ϕ(z)sup|>rN

 –|z|u(z)

× 

 + (n+ )(n+ )

n!|ϕ(z)|n

( –|ϕ(z)|)n

sup |ϕ(z)|>rN

n!  + (n+ )(n+ )

( –|z|)|u(z)||ϕ(z)|n

( –|ϕ(z)|)n

sup |a|>rN

Dnϕ,u(fa)Z+

(n+ )

 + (n+ )(n+ )|asup|>rN

Dnϕ,u(ga)Z

+ 

 + (n+ )(n+ )|asup|>rN

Dnϕ,u(ha)Z

sup |a|>rN

Dnϕ,u(fa)Z+ sup |a|>rN

Dnϕ,u(ga)Z+ sup |a|>rN

Dnϕ,u(ha)Z. (.)

Taking the limit asN→ ∞we obtain

lim sup j→∞

Slim sup |a|→

Dnϕ,u(fa)Z+lim sup |a|→

Dnϕ,u(ga)Z

+lim sup |a|→

Dnϕ,u(ha)Z

=A+B+C.

Similarly, we havelim supj→∞SA+B+C,i.e., we get

QA+B+Cmax{A,B,C}. (.)

From (.), we see that

lim sup

j→∞ Slim sup|ϕ(z)|→

( –|z|)β|u(z)|

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Similarly we havelim supj→∞SF. Therefore

QF. (.)

Finally we considerQ. We haveQ≤lim supj→∞(S+S), where

S:= sup |ϕ(z)|>rN

 –|z|f(n+)ϕ(z)ϕ(z)u(z)

and

S:= sup |ϕ(z)|>rN

 –|z|rn+j f(n+)rjϕ(z)ϕ(z)u(z).

After some calculation, we have

S

n+fB

n! |ϕ(z)sup|>rN

 –|z|ϕ(z)u(z) n!|ϕ(z)|

n+

( –|ϕ(z)|)n+

n+ n! |ϕ(z)sup|>rN

 –|z|ϕ(z)u(z) n!|ϕ(z)|

n+

( –|ϕ(z)|)n+

sup |a|>rN

Dnϕ,u

fa

n+ ga+

(n+ )(n+ )ha Z sup

|a|>rN

Dnϕ,u(fa)Z+

n+ |asup|>rN

Dnϕ,u(ga)Z+

(n+ )(n+ )|asup|>rN

Dnϕ,u(ha)Z

≤ sup |a|>rN

Dnϕ,u(fa)Z+ sup |a|>rN

Dnϕ,u(ga)Z+ sup |a|>rN

Dnϕ,u(ha)Z. (.)

Taking the limit asN→ ∞we obtain

lim sup j→∞

S lim sup |a|→

Dnϕ,u(fa)Z+lim sup |a|→

Dnϕ,u(ga)Z

+lim sup |a|→

Dnϕ,u(ha)Z

=A+B+C.

Similarly, we havelim supj→∞SA+B+C,i.e., we get

QA+B+Cmax{A,B,C}. (.)

From (.), we see that

lim sup j→∞

Slim sup |ϕ(z)|→

( –|z|)|ϕ(z)||u(z)|

( –|ϕ(z)|)n+ =G.

Similarly we havelim supj→∞SG. Therefore

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Hence, by (.), (.), (.), (.), (.), (.), (.), (.), (.), and (.) we get

lim sup j→∞

Dnϕ,u–Dϕn,uKrjBZ

=lim sup j→∞

sup fB≤

Dnϕ,u–Dnϕ,uKrj

fZ =lim sup

j→∞ sup fB≤

u·(f –frj) (n)ϕ

∗max{A,B,C}. (.)

Similarly, by (.), (.), (.), (.), (.), (.), (.), (.), (.), and (.) we get

lim sup j→∞

Dnϕ,uDnϕ,uKrjBZmax{E,F,G}. (.)

Therefore, by (.), (.), and (.), we obtain

Dnϕ,ue,BZmax{A,B,C} and D n

ϕ,ue,BZmax{E,F,G}.

This completes the proof of Theorem ..

3 New characterization ofDn

ϕ,u:BZ

In this section, we give a new characterization for the boundedness, compactness, and essential norm of the operatorDnϕ,u:BZ. For this purpose, we present some definitions

and some lemmas which will be used later.

The weighted space, denoted byHv∞, consists of allfH(D) such that

fv=sup z∈D

v(z)f(z)<∞,

wherev:D→R+is a continuous, strictly positive, and bounded function.Hv∞is a Banach

space under the norm · v. The weightedvis called radial ifv(z) =v(|z|) for allz∈D. The

associated weightv˜ofvis as follows:

˜

v=supf(z):fHv∞,fv≤

–

, z∈D.

Whenv=vα(z) = ( –|z|)α( <α<∞), it is well known thatvα(z) =˜ vα(z). In this case, we denoteHv∞byHvα.

Lemma .[] Forα> ,we havelimk→∞zk– = (

α

e)

α.

Lemma .[] Let v and w be radial,non-increasing weights tending to zero at the bound-ary ofD.Then the following statements hold.

(a) The weighted composition operatoruCϕ:Hv∞→Hwis bounded if and only if

supzDv(˜w(z)ϕ(z))|u(z)|<∞.Moreover,the following holds:

uCϕHv∞→Hw∞=sup z∈D

w(z) ˜

(12)

(b) SupposeuCϕ:Hv∞→Hwis bounded.Then

uCϕe,HvHw∞= lim s→– sup

|ϕ(z)|>s

w(z) ˜

v(ϕ(z))u(z).

Lemma .[] Let v and w be radial,non-increasing weights tending to zero at the bound-ary ofD.Then the following statements hold.

(a) uCϕ:Hv∞→Hwis bounded if and only ifsupkuϕkw

zkv <∞,with the norm

comparable to the above supremum. (b) SupposeuCϕ:Hv∞→Hwis bounded.Then

uCϕe,HvHw∞=lim sup k→∞

uϕk w

zk v

.

Theorem . Let n be a positive integer,uH(D),andϕ be an analytic self-map ofD. Then the operator Dnϕ,u:BZis bounded if and only if

⎧ ⎪ ⎨ ⎪ ⎩

supjjn+(uϕ+j– v<∞,

supjjnuϕj– v<∞,

supjjn+uϕϕj–v<∞.

(.)

Proof By [],Dn

ϕ,u:BZis bounded if and only if

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

supzD(–|z|)|u(z)ϕ(z)+u(z)ϕ(z)| (–|ϕ(z)|)n+ <∞,

supzD(–|z|)|u(z)| (–|ϕ(z)|)n <∞, supzD(–|z|)|u(z)||ϕ(z)|

(–|ϕ(z)|)n+ <∞.

(.)

By Lemma ., the first inequality in (.) is equivalent to the weighted composition op-erator (uϕ+)Cϕ:Hvn+Hv is bounded. By Lemma ., this is equivalent to

sup j≥

(uϕ+j– v zj–

v+n

<∞.

The second inequality in (.) is equivalent to the operatoruCϕ:HvnHv is bounded. By Lemma ., this is equivalent to

sup j≥

uϕj– vzj–

vn

<∞.

The third inequality in (.) is equivalent to the operator:Hvn+→H

v is bounded. By Lemma ., this is equivalent to

sup j≥

uϕϕj– v zj–

vn+ <∞.

By Lemma ., we see thatDn

ϕ,u:BZis bounded if and only if

sup j≥

jn+uϕ+uϕϕj–v ≈sup

j≥

jn+(uϕ+j– vjn+zj–

v+n

(13)

sup j≥

jnuϕj–v ≈sup

j≥

jnuϕj– vjnzj–

vn

<∞,

and

sup j≥

jn+uϕϕj–v ≈sup

j≥

jn+uϕϕj–vjn+zj–

vn+ <∞.

The proof is completed.

Theorem . Let n be a positive integer,uH(D),andϕ be an analytic self-map ofD such that Dn

ϕ,u:BZis bounded.Then

Dnϕ,ue,BZ≈max{M,M,M},

where

M:=lim sup j→∞ j

+nuϕ+ϕj– v,

M:=lim sup j→∞

jnuϕj–

v, M:=lim sup

j→∞

jn+uϕϕj–

v.

Proof From the proof of Theorem . we know that the boundedness ofDn

ϕ,u:BZ

is equivalent to the boundedness of the operators (uϕ+)Cϕ:Hv+nHv,uCϕ: HvnHv, and:Hvn+Hv.

The upper estimate. By Lemmas . and ., we get

uϕ+uϕCϕe,H

v+nHv∞ =

lim sup j→∞

(uϕ+j– v zj–

v+n

=lim sup j→∞

j+n(uϕ+j– vj+nzj–

v+n

≈lim sup j→∞ j

+nuϕ+ϕj– v,

uCϕe,H

vnHv =

lim sup j→∞

uϕj– v zj–

vn

=lim sup j→∞

+n–uϕj– v+n–zj–

vn

≈lim sup j→∞

jnuϕj–

v,

and

e,Hvn+Hv =lim sup j→∞

ϕj– v zj–

vn+

=lim sup j→∞

jn+ϕj– vjn+zj–

vn+

≈lim sup j→∞ j

n+ϕj– v.

It follows that

Dn

ϕ,ue,BZuϕ+

e,H

v+nHv +u

e,Hvn∞→Hv

+e,H

vn+→Hv∞

(14)

The lower estimate. From Theorem ., and Lemmas . and ., we have

Dnϕ,ue,BZE=uϕ+

e,H

v+nHv

=lim sup j→∞

(uϕ+j– vzj–

v+n

≈lim sup j→∞ j

α+nuϕ+ϕj– v,

Dnϕ,ue,BZF=uCϕe,H

vnHv =lim sup j→∞

uϕj– v zj–

vn

≈lim sup j→∞

+n–uϕj– v,

and

Dnϕ,ue,BZG=e,H

vn+→Hv∞

=lim sup j→∞

uϕϕj– v zj–

vn+

≈lim sup j→∞

jn+uϕϕj–v .

ThereforeDn

ϕ,ue,BZmax{M,M,M}. This completes the proof.

From Theorem ., we immediately get the following result.

Theorem . Let n be a positive integer,uH(D),andϕ be an analytic self-map ofD such that Dn

ϕ,u:BZis bounded.Then Dnϕ,u:BZis compact if and only if

lim sup j→∞

j+nuϕ+uϕϕj–

v= , lim sup

j→∞

jnuϕj–

v= ,

and

lim sup j→∞

jn+uϕϕj–

v= .

4 Conclusion

The boundedness and compactness ofDnϕ,u:BZwere characterized in [] and [].

In this paper, we give a new characterization for the boundedness and compactness of Dnϕ,u:BZ. Moreover, using the method in [], we completely characterize the

essen-tial norm ofDnϕ,u:BZ.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The four authors contributed equally to the writing of this paper. They read and approved the final version of the manuscript.

Author details

(15)

Acknowledgements

This project is partially supported by the Macao Science and Technology Development Fund (No. 083/2014/A2).

Received: 19 January 2016 Accepted: 1 April 2016

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