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} 4_{Department 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:X→Y is its distance to the set of compact operatorsKmappingXintoY, that is,
Te,X→Y=inf
T–KX→Y :Kis compact
,
where · X→Y 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,u∈H(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.
The Bloch space, denoted byB, is deﬁned to be the set of allf∈H(D) such that
fB=f()+sup z∈D
–|z|f(z)<∞.
Bis a Banach space with the above norm. Anf ∈Bis said to belong to the little Bloch space Biflim|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 allf ∈H(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 thatCϕ:B→Bis 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 operatorCϕ:B→B,i.e.,
Cϕe,B→B=lim_{s}_{→}_{} sup
|ϕ(z)|>s
( –|z|_{)|}_{ϕ}_{(z)|}
( –|ϕ(z)|_{)} .
Tjani in [] proved that Cϕ :B→B is compact if and only if lim_{|}a|→CϕσaB= , whereσa=_{–az}a–z. Wulanet al.in [] showed thatCϕ:B→Bis 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:B→Zin [] (see also []).
In [], Li and Fu obtained a new characterization for the boundedness, as well as the compactness forDn
ϕ,u:B→Zby using three families of functions. We combine the results
in [] and [] as follows.
Theorem A Let n be a positive integer, u∈H(D), andϕ be an analytic self-map of D. Suppose that Dn
ϕ,u:B→Zis bounded,then the following statements are equivalent: (a) The operatorDn
ϕ,u:B→Zis compact. (b)
lim |ϕ(w)|→
Dnϕ,ufϕ(w)_{Z}= lim |ϕ(w)|→
Dnϕ,ugϕ(w)_{Z}= lim |ϕ(w)|→
where
fϕ(w)(z) =
–|ϕ(w)|
–ϕ(w)z , gϕ(w)(z) =
( –|ϕ(w)|_{)}
( –ϕ(w)z) ,
hϕ(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:B→Z. Moreover, we give a new
characteriza-tion for the boundedness, compactness, and essential norm of the operatorDn
ϕ,u:B→Z.
Throughout this paper, we say thatPQif there exists a constantCsuch thatP≤CQ. The symbolP≈Qmeans thatPQP.
2 Essential norm ofDn_{ϕ}_{,}_{u}:B→Z
In this section, we give two estimates of the essential norm for the operatorDn
ϕ,u:B→Z.
Theorem . Let n be a positive integer,u∈H(D),andϕbe an analytic self-map ofDsuch that Dn
ϕ,u:B→Zis bounded.Then
Dnϕ,ue,B→Z≈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,B→Z. Leta∈D. Deﬁne
fa(z) =
–|a|
( –az), ga(z) =
( –|a|_{)}
( –az) , ha(z) =
( –|a|_{)}
( –az) , z∈D.
It is easy to check thatfa,ga,ha∈BandfaB,gaB,haB for alla∈D
contained inBwhich 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–K_{B}→Zlim sup_{|}
a|→
Dnϕ,u–K
fa_{Z}
≥lim sup |a|→
Dn_{ϕ}_{,u}fa_{Z}–lim sup |a|→
KfaZ=A,
Dnϕ,u–K_{B}→Zlim sup_{|}
a|→
Dnϕ,u–K
ga_{Z}
≥lim sup |a|→
Dn_{ϕ}_{,u}ga_{Z}–lim sup |a|→
KgaZ=B,
and
Dnϕ,u–K_{B}→Zlim sup_{|}_{a}_{|→}_{} Dnϕ,u–K
ha_{Z}
≥lim sup |a|→
Dnϕ,uha_{Z}–lim sup |a|→ KhaZ
=C.
Therefore, from the deﬁnition of the essential norm, we obtain
Dnϕ,ue,B→Z=inf_{K}D n
ϕ,u–K_{B}→Zmax{A,B,C}. Next, we prove thatDn
ϕ,ue,B→Zmax{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 toBand converge to weakly
inB. Moreover,
k_{j}(n)ϕ(zj)
= , k_{j}(n+)ϕ(zj)
= , k_{j}(n+)ϕ(zj)=
n! n+
|ϕ(zj)|n+
( –|ϕ(zj)|)n+
,
l_{j}(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+
Then for any compact operatorK:B→Z, we obtain
Dnϕ,u–K_{B}→Zlim 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–K_{B}→Zlim sup_{j}_{→∞} Dnϕ,u(lj)_{Z}–lim sup j→∞
K(lj)_{Z}
lim sup j→∞
( –|zj|)|u(zj)||ϕ(zj)|n
( –|ϕ(zj)|)n
,
and
Dn_{ϕ}_{,u}–K_{B}_{→}_{Z}lim sup j→∞
Dn_{ϕ}_{,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,B→Z=inf_{K}D n
ϕ,u–K_{B}→Z 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,
Dn_{ϕ}_{,u}_{e,}_{B}_{→}_{Z}=inf K
Dn_{ϕ}_{,u}–K_{B}_{→}_{Z}
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,B→Z =inf_{K}D n
ϕ,u–K_{B}→Z
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
Now, we prove that
Dn_{ϕ}_{,u}_{e,}_{B}_{→}_{Z}max{A,B,C} and Dn_{ϕ}_{,u}_{e,}_{B}_{→}_{Z}max{E,F,G}.
Forr∈[, ), setKr:H(D)→H(D) by (Krf)(z) =fr(z) =f(rz),f ∈H(D). It is obvious that
fr→f 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:B→Zis compact. By the deﬁnition
of the essential norm, we get
Dnϕ,ue,B→Z≤lim sup j→∞
Dnϕ,u–Dnϕ,uKrj_{B}_{→}_{Z}. (.)
Therefore, we only need to prove that
lim sup j→∞
Dnϕ,u–Dnϕ,uKrj_{B}_{→}_{Z}max{A,B,C}
and
lim sup j→∞
Dnϕ,u–Dnϕ,uKrjB→Zmax{E,F,G}.
For anyf ∈Bsuch thatfB≤, we consider
Dn_{ϕ}_{,u}–Dn_{ϕ}_{,u}Krj
f_{Z} =u()f(n)ϕ()–rn
ju()f (n)_{r}
jϕ()
+u()(f–frj)
(n)_{ϕ()}_{+}_{u()(f}_{–}_{f} 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()–r_{j}nu()f(n)rjϕ()= (.)
and
lim j→∞
u()(f–frj)
(n)_{ϕ()}_{+}_{u()(f}_{–}_{f} 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)
+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 allj≥N,
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:B→Zis bounded, by Theorem of [], we see thatu∈Z,
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+)j →f
(n+)_{, as well as}_{r}n+ j f
(n+) rj →f
(n+)_{uniformly on compact subsets of}_{D}
asj→ ∞, we have
Q≤Klim sup j→∞
sup |w|≤rN
f(n+)(w) –rn+_{j} f(n+)(rjw)= (.)
and
Q≤Klim sup j→∞ |wsup|≤rN
Similarly, from the fact thatu∈Zwe have
Q≤ uZlim sup
j→∞ sup |w|≤rN
f(n)(w) –rn_{j}f(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|r_{j}n+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
Dn_{ϕ}_{,u}
fa–
(n+ )ga
(n+ )(n+ )+
ha
(n+ )(n+ ) _{Z} sup
|a|>rN
Dn_{ϕ}_{,u}(fa)_{Z}+
n+ (n+ )(n+ )|asup|>rN
Dn_{ϕ}_{,u}(ga)_{Z}
+
(n+ )(n+ )|asup|>rN
Dnϕ,u(ha)_{Z}. (.)
Taking the limit asN→ ∞we obtain
lim sup j→∞
Slim 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 sup_{j}_{→∞}SA+B+C,i.e., we get
QA+B+Cmax{A,B,C}. (.)
From (.), we see that
lim sup
j→∞ Slim sup|ϕ(z)|→
( –|z|_{)}_{|}_{u}_{(z)ϕ}_{(z) +}_{u(z)ϕ}_{(z)}_{|}
Similarly we havelim sup_{j}_{→∞}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|rn_{j}f(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( + (n_{ϕ(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→∞
Slim 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 sup_{j}_{→∞}SA+B+C,i.e., we get
QA+B+Cmax{A,B,C}. (.)
From (.), we see that
lim sup
j→∞ Slim sup|ϕ(z)|→
( –|z|_{)}β_{|}_{u}_{(z)}_{|}
Similarly we havelim sup_{j}_{→∞}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+_{}_{f}_{B}
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
Dn_{ϕ}_{,u}(fa)_{Z}+
n+ |asup|>rN
Dn_{ϕ}_{,u}(ga)_{Z}+
(n+ )(n+ )|asup|>rN
Dn_{ϕ}_{,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|→
Dn_{ϕ}_{,u}(fa)_{Z}+lim sup |a|→
Dn_{ϕ}_{,u}(ga)_{Z}
+lim sup |a|→
Dnϕ,u(ha)_{Z}
=A+B+C.
Similarly, we havelim sup_{j}_{→∞}SA+B+C,i.e., we get
QA+B+Cmax{A,B,C}. (.)
From (.), we see that
lim sup j→∞
Slim sup |ϕ(z)|→
( –|z|_{)|}_{ϕ}_{(z)|}_{|u(z)|}
( –|ϕ(z)|_{)}n+ =G.
Similarly we havelim sup_{j}_{→∞}SG. Therefore
Hence, by (.), (.), (.), (.), (.), (.), (.), (.), (.), and (.) we get
lim sup j→∞
Dnϕ,u–Dϕn,uKrj_{B}_{→}_{Z}
=lim sup j→∞
sup fB≤
Dnϕ,u–Dnϕ,uKrj
f_{Z} =lim sup
j→∞ sup fB≤
u·(f –frj) (n)_{◦}_{ϕ}
∗max{A,B,C}. (.)
Similarly, by (.), (.), (.), (.), (.), (.), (.), (.), (.), and (.) we get
lim sup j→∞
Dn_{ϕ}_{,u}–Dn_{ϕ}_{,u}KrjB→Zmax{E,F,G}. (.)
Therefore, by (.), (.), and (.), we obtain
Dnϕ,ue,B→Zmax{A,B,C} and D n
ϕ,ue,B→Zmax{E,F,G}.
This completes the proof of Theorem ..
3 New characterization ofDn
ϕ,u:B→Z
In this section, we give a new characterization for the boundedness, compactness, and essential norm of the operatorDnϕ,u:B→Z. For this purpose, we present some deﬁnitions
and some lemmas which will be used later.
The weighted space, denoted byH_{v}∞, consists of allf ∈H(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):f ∈H_{v}∞,fv≤
–
, z∈D.
Whenv=vα(z) = ( –|z|_{)}α_{( <}_{α}_{<}_{∞), it is well known that}_{vα(z) =}_{˜} _{vα(z). In this case, we} denoteH_{v}∞byH_{v}∞_{α}.
Lemma .[] Forα> ,we havelim_{k}_{→∞}kα_{}_{z}k–_{} vα= (
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:H_{v}∞→H_{w}∞is bounded if and only if
sup_{z}_{∈}_{D}_{v(}_{˜}w(z)_{ϕ}_{(z))}|u(z)|<∞.Moreover,the following holds:
uCϕHv∞→Hw∞=sup z∈D
w(z) ˜
(b) SupposeuCϕ:H_{v}∞→H_{w}∞is bounded.Then
uCϕe,H∞v →Hw∞= 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:H_{v}∞→H_{w}∞is bounded if and only ifsup_{k}_{≥}_{}u_{}ϕkw
zk_{v} <∞,with the norm
comparable to the above supremum. (b) SupposeuCϕ:H_{v}∞→H_{w}∞is bounded.Then
uCϕe,H∞v →Hw∞=lim sup k→∞
uϕk_{} w
zk v
.
Theorem . Let n be a positive integer,u∈H(D),andϕ be an analytic self-map ofD. Then the operator Dnϕ,u:B→Zis bounded if and only if
⎧ ⎪ ⎨ ⎪ ⎩
sup_{j}_{≥}_{}jn+_{}_{(u}_{ϕ}_{+}_{uϕ}_{)ϕ}j–_{} v<∞,
sup_{j}_{≥}_{}jn_{}_{u}_{ϕ}j–_{} v<∞,
sup_{j}_{≥}_{}jn+uϕϕj–v<∞.
(.)
Proof By [],Dn
ϕ,u:B→Zis bounded if and only if
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
sup_{z}_{∈}_{D}(–|z|)|u(z)ϕ(z)+u(z)ϕ(z)| (–|ϕ(z)|_{)}n+ <∞,
sup_{z}_{∈}_{D}(–|z|)|u(z)| (–|ϕ(z)|_{)}n <∞, sup_{z}_{∈}_{D}(–|z|)|u(z)||ϕ(z)|
(–|ϕ(z)|_{)}n+ <∞.
(.)
By Lemma ., the ﬁrst inequality in (.) is equivalent to the weighted composition op-erator (uϕ+uϕ)Cϕ:H_{v}∞_{n}_{+}→H_{v}∞_{} is bounded. By Lemma ., this is equivalent to
sup j≥
(uϕ+uϕ)ϕj–_{} v zj–
v+n
<∞.
The second inequality in (.) is equivalent to the operatoruCϕ:H_{v}∞_{n} →H_{v}∞_{} is bounded. By Lemma ., this is equivalent to
sup j≥
u_{ϕ}j–_{} v zj–_{}
vn
<∞.
The third inequality in (.) is equivalent to the operatoruϕ_{Cϕ}_{:}_{H}∞ vn+→H
∞
v is bounded. By Lemma ., this is equivalent to
sup j≥
uϕ_{ϕ}j–_{} v zj–
vn+ <∞.
By Lemma ., we see thatDn
ϕ,u:B→Zis bounded if and only if
sup j≥
jn+uϕ+uϕϕj–_{v} ≈sup
j≥
jn+_{}_{(u}_{ϕ}_{+}_{uϕ}_{)ϕ}j–_{} v jn+zj–
v+n
sup j≥
jnuϕj–_{v} ≈sup
j≥
jn_{u}_{ϕ}j–_{} v jnzj–
vn
<∞,
and
sup j≥
jn+uϕϕj–_{v} ≈sup
j≥
jn+uϕϕj–v jn+_{}_{z}j–_{}
vn+ <∞.
The proof is completed.
Theorem . Let n be a positive integer,u∈H(D),andϕ be an analytic self-map ofD such that Dn
ϕ,u:B→Zis bounded.Then
Dn_{ϕ}_{,u}_{e,}_{B}_{→}_{Z}≈max{M,M,M},
where
M:=lim sup j→∞ j
+n_{u}_{ϕ}_{+}_{uϕ}_{ϕ}j– 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:B→Z
is equivalent to the boundedness of the operators (uϕ+uϕ)Cϕ:H_{v}∞_{+}_{n}→H_{v}∞_{},uCϕ: H_{v}∞_{n} →H_{v}∞_{}, anduϕCϕ:H_{v}∞_{n}_{+}→H_{v}∞_{}.
The upper estimate. By Lemmas . and ., we get
uϕ+uϕCϕ_{e,H}∞
v+n→Hv∞ =
lim sup j→∞
(u_{ϕ}_{+}_{uϕ}_{)ϕ}j–_{} v zj–_{}
v+n
=lim sup j→∞
j+n_{}_{(u}_{ϕ}_{+}_{uϕ}_{)ϕ}j–_{} v j+nzj–
v+n
≈lim sup j→∞ j
+n_{u}_{ϕ}_{+}_{uϕ}_{ϕ}j– v,
uCϕ_{e,H}∞
vn→H∞v =
lim sup j→∞
u_{ϕ}j–_{} v zj–_{}
vn
=lim sup j→∞
jα+n–_{u}_{ϕ}j–_{} v jα+n–_{z}j–_{}
vn
≈lim sup j→∞
jnuϕj–
v,
and
_{uϕ}_{Cϕ}
e,H_{vn}∞_{+}→H∞v_{} =lim sup j→∞
uϕ_{ϕ}j–_{} v zj–
vn+
=lim sup j→∞
jn+_{}_{uϕ}_{ϕ}j–_{} v jn+zj–
vn+
≈lim sup j→∞ j
n+_{uϕ}_{ϕ}j– v.
It follows that
_{D}n
ϕ,ue,B→Zuϕ+uϕ
Cϕ_{e,H}∞
v_{+}_{n}→Hv∞_{} +u _{Cϕ}
e,H_{vn}∞→Hv∞_{}
+uϕCϕ_{e,H}∞
vn+→Hv∞
The lower estimate. From Theorem ., and Lemmas . and ., we have
Dnϕ,ue,B→ZE=uϕ+uϕ
Cϕ_{e,H}∞
v+n→H∞v
=lim sup j→∞
(u_{ϕ}_{+}_{uϕ}_{)ϕ}j–_{} v zj–_{}
v+n
≈lim sup j→∞ j
α+n_{u}_{ϕ}_{+}_{uϕ}_{ϕ}j– v,
Dn_{ϕ}_{,u}_{e,}_{B}_{→}_{Z}F=uCϕ_{e,H}∞
vn→Hv∞_{} =lim sup j→∞
u_{ϕ}j–_{} v zj–_{}
vn
≈lim sup j→∞
jα+n–_{u}_{ϕ}j– v,
and
Dn_{ϕ}_{,u}_{e,}_{B}_{→}_{Z}G=uϕCϕ_{e,H}∞
vn+→Hv∞
=lim sup j→∞
uϕ_{ϕ}j–_{} v zj–
vn+
≈lim sup j→∞
jn+uϕϕj–_{v} .
ThereforeDn
ϕ,ue,B→Zmax{M,M,M}. This completes the proof.
From Theorem ., we immediately get the following result.
Theorem . Let n be a positive integer,u∈H(D),andϕ be an analytic self-map ofD such that Dn
ϕ,u:B→Zis bounded.Then Dnϕ,u:B→Zis compact if and only if
lim sup j→∞
j+nuϕ+uϕϕj–
v= , lim sup
j→∞
jnuϕj–
v= ,
and
lim sup j→∞
jn+uϕϕj–
v= .
4 Conclusion
The boundedness and compactness ofDnϕ,u:B→Zwere characterized in [] and [].
In this paper, we give a new characterization for the boundedness and compactness of Dnϕ,u:B→Z. Moreover, using the method in [], we completely characterize the
essen-tial norm ofDnϕ,u:B→Z.
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
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
References
1. Hibschweiler, R, Portnoy, N: Composition followed by diﬀerentiation between Bergman and Hardy spaces. Rocky Mt. J. Math.35, 843-855 (2005)
2. Li, S, Stevi´c, S: Composition followed by diﬀerentiation between Bloch type spaces. J. Comput. Anal. Appl.9, 195-205 (2007)
3. Li, S, Stevi´c, S: Composition followed by diﬀerentiation betweenH∞andα-Bloch spaces. Houst. J. Math.35, 327-340 (2009)
4. Li, S, Stevi´c, S: Products of composition and diﬀerentiation operators from Zygmund spaces to Bloch spaces and Bers spaces. Appl. Math. Comput.217, 3144-3154 (2010)
5. Stevi´c, S: Norm and essential norm of composition followed by diﬀerentiation fromα-Bloch spaces toH∞μ. Appl.
Math. Comput.207, 225-229 (2009)
6. Stevi´c, S: Products of composition and diﬀerentiation operators on the weighted Bergman space. Bull. Belg. Math. Soc. Simon Stevin16, 623-635 (2009)
7. Stevi´c, S: Composition followed by diﬀerentiation fromH∞and the Bloch space tonth weighted-type spaces on the unit disk. Appl. Math. Comput.216, 3450-3458 (2010)
8. Stevi´c, S: Characterizations of composition followed by diﬀerentiation between Bloch-type spaces. Appl. Math. Comput.218, 4312-4316 (2011)
9. Yang, W: Products of composition diﬀerentiation operators fromQK(p,q) spaces to Bloch-type spaces. Abstr. Appl.
Anal.2009, Article ID 741920 (2009)
10. Stevi´c, S, Sharma, A: Iterated diﬀerentiation followed by composition from Bloch-type spaces to weighted BMOA spaces. Appl. Math. Comput.218, 3574-3580 (2011)
11. Wu, Y, Wulan, H: Products of diﬀerentiation and composition operators on the Bloch space. Collect. Math.63, 93-107 (2012)
12. Li, H, Fu, X: A new characterization of generalized weighted composition operators from the Bloch space into the Zygmund space. J. Funct. Spaces Appl.2013, Article ID 925901 (2013)
13. Li, S, Stevi´c, S: Generalized weighted composition operators fromα-Bloch spaces into weighted-type spaces. J. Inequal. Appl.2015, Article ID 265 (2015)
14. Stevi´c, S: Weighted diﬀerentiation composition operators from mixed-norm spaces to weighted-type spaces. Appl. Math. Comput.211, 222-233 (2009)
15. Stevi´c, S: Weighted diﬀerentiation composition operators from mixed-norm spaces to thenth weighted-type space on the unit disk. Abstr. Appl. Anal.2010, Article ID 246287 (2010)
16. Stevi´c, S: Weighted diﬀerentiation composition operators fromH∞and Bloch spaces tonth weighted-type spaces on the unit disk. Appl. Math. Comput.216, 3634-3641 (2010)
17. Yang, W, Zhu, X: Generalized weighted composition operators from area Nevanlinna spaces to Bloch-type spaces. Taiwan. J. Math.16, 869-883 (2012)
18. Zhu, X: Products of diﬀerentiation, composition and multiplication from Bergman type spaces to Bers type space. Integral Transforms Spec. Funct.18, 223-231 (2007)
19. Zhu, X: Generalized weighted composition operators on weighted Bergman spaces. Numer. Funct. Anal. Optim.30, 881-893 (2009)
20. Zhu, X: Generalized weighted composition operators on Bloch-type spaces. J. Inequal. Appl.2015, Article ID 59 (2015)
21. Zhu, X: Essential norm of generalized weighted composition operators on Bloch-type spaces. Appl. Math. Comput.
274, 133-142 (2016)
22. Li, S, Stevi´c, S: Generalized composition operators on Zygmund spaces and Bloch type spaces. J. Math. Anal. Appl.
338, 1282-1295 (2008)
23. Li, S, Stevi´c, S: Products of composition and integral type operators fromH∞to the Bloch space. Complex Var. Elliptic Equ.53(5), 463-474 (2008)
24. Li, S, Stevi´c, S: Composition followed by diﬀerentiation from mixed-norm spaces toα-Bloch spaces. Sb. Math.
199(12), 1847-1857 (2008)
25. Li, S, Stevi´c, S: Products of integral-type operators and composition operators between Bloch-type spaces. J. Math. Anal. Appl.349, 596-610 (2009)
26. Stevi´c, S: On an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces. Nonlinear Anal. TMA71, 6323-6342 (2009)
27. Stevi´c, S: Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces. Sib. Math. J.50, 726-736 (2009)
28. Stevi´c, S: On some integral-type operators between a general space and Bloch-type spaces. Appl. Math. Comput.
218, 2600-2618 (2011)
29. Stevi´c, S, Sharma, A, Bhat, A: Products of multiplication composition and diﬀerentiation operators on weighted Bergman spaces. Appl. Math. Comput.217, 8115-8125 (2011)
30. Stevi´c, S, Sharma, A, Bhat, A: Essential norm of products of multiplication composition and diﬀerentiation operators on weighted Bergman spaces. Appl. Math. Comput.218, 2386-2397 (2011)
31. Zhu, K: Operator Theory in Function Spaces, 2nd edn. Am. Math. Soc., Providence (2007)
32. Cowen, C, Maccluer, B: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995) 33. Hyvärinen, O, Lindström, M: Estimates of essential norm of weighted composition operators between Bloch-type
spaces. J. Math. Anal. Appl.393, 38-44 (2012)
35. Li, S, Stevi´c, S: Weighted composition operators betweenH∞andα-Bloch spaces in the unit ball. Taiwan. J. Math.12, 1625-1639 (2008)
36. Li, S, Stevi´c, S: Weighted composition operators from Zygmund spaces into Bloch spaces. Appl. Math. Comput.206, 825-831 (2008)
37. MacCluer, B, Zhao, R: Essential norm of weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math.33, 1437-1458 (2003)
38. Madigan, K, Matheson, A: Compact composition operators on the Bloch space. Trans. Am. Math. Soc.347, 2679-2687 (1995)
39. Manhas, J, Zhao, R: New estimates of essential norms of weighted composition operators between Bloch type spaces. J. Math. Anal. Appl.389, 32-47 (2012)
40. Montes-Rodriguez, A: The essential norm of composition operators on the Bloch space. Pac. J. Math.188, 339-351 (1999)
41. Ohno, S, Stroethoﬀ, K, Zhao, R: Weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math.33, 191-215 (2003)
42. Stevi´c, S: Essential norms of weighted composition operators from theα-Bloch space to a weighted-type space on the unit ball. Abstr. Appl. Anal.2008, Article ID 279691 (2008)
43. Tjani, M: Compact composition operators on some Möbius invariant Banach space. PhD dissertation, Michigan State University (1996)
44. Wulan, H, Zheng, D, Zhu, K: Compact composition operators on BMOA and the Bloch space. Proc. Am. Math. Soc.
137, 3861-3868 (2009)
45. Zhao, R: Essential norms of composition operators between Bloch type spaces. Proc. Am. Math. Soc.138, 2537-2546 (2010)
46. Choe, B, Koo, H, Smith, W: Composition operators on small spaces. Integral Equ. Oper. Theory56, 357-380 (2006) 47. Duren, P: Theory ofHpSpaces. Academic Press, New York (1970)
48. Esmaeili, K, Lindström, M: Weighted composition operators between Zygmund type spaces and their essential norms. Integral Equ. Oper. Theory75, 473-490 (2013)
49. Li, S, Stevi´c, S: Volterra type operators on Zygmund spaces. J. Inequal. Appl.2007, Article ID 32124 (2007)
50. Yu, Y, Liu, Y: Weighted diﬀerentiation composition operators fromH∞to Zygmund spaces. Integral Transforms Spec. Funct.22, 507-520 (2011)
51. Montes-Rodriguez, A: Weighed composition operators on weighted Banach spaces of analytic functions. J. Lond. Math. Soc.61, 872-884 (2000)