Essential norm of weighted composition operator between α Bloch space and β Bloch space in polydiscs

PII. S0161171204403548 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ESSENTIAL NORM OF WEIGHTED COMPOSITION

OPERATOR BETWEEN

α

-BLOCH SPACE AND

β

_{-BLOCH SPACE IN POLYDISCS}

LI SONGXIAO and ZHU XIANGLING

Received 29 March 2004

Letϕ(z)=(ϕ1(z),...,ϕn(z))be a holomorphic self-map ofDn_{andψ(z)a holomorphic} function onDn_{, whereD}n_{is the unit polydiscs ofC}n_{. Let 0< α,β <1, we compute the} essential norm of a weighted composition operatorψCϕ betweenα-Bloch spaceᏮα_(Dn₎ andβ-Bloch spaceᏮβ(Dn_).

2000 Mathematics Subject Classification: 47B35, 30H05.

1. Introduction. Let Dn _{be the unit polydiscs of C}n_{, the class of all holomorphic}

functions with domainDn_{will be denoted byH(D}n_{). Letϕbe a holomorphic self-map}

ofDn_{, the composition operatorC}

ϕ induced byϕis defined by(Cϕf )(z)=f (ϕ(z))

forzinDn_andf∈H(Dn_{). If, in addition,ψis a holomorphic function defined onD}n_,

the weighted composition operatorsψCϕinduced byψandϕis defined byψCϕ(z)= ψ(z)f (ϕ(z))forzinDn_andf∈H(Dn_).

Let 0< α <1, a functionfholomorphic inDn_{is said to belong to theα-Bloch space}

Ꮾα_(Dn_)if

fα=f (0)+sup z∈Dn

n

k=1

_∂z∂f_k(z)1−zk2 α

<+∞. (1.1)

It is easy to show thatᏮα(Dn_{)is a Banach space with the norm·}

α. These spaces are

called Lipschitz space Lipα(Dn)by Zhou (see [6,8]). It is easy to show that the usual

norm on Lip1−α(Dn)defined by

fLip=f (0)+ sup z,w∈Dn

f (z)−f (w)

|z−w|1−α (1.2)

induces a Banach space structure on Lip1−α(Dn). Clahane in [1] has shown that this

norm is equivalent to·α.

Essential norm formulas for composition operators are known in various settings. Shapiro has given a formula for Cϕe when Cϕ acting on the Hardy space H2(D)

in [5]; Montes-Rodríguez [4] has given the essential norm of composition operator on the Bloch space in the unit disc; Donaway [2] has given upper and lower estimates forCϕewhenCϕmaps the Bloch space, the Dirichlet space, or the Besov-pspace to

composition operator between the Bloch-type spaces in the unit disc, namely,

uCϕe=lim s→1

sup |ϕ(z)|>s

u(z)ϕ_(z)

1−|z|2β

1−ϕ(z)2α. (1.3)

Here,ϕis an analytic self-map ofDanduis a fixed analytic function onDand 0< α <

1,0< β <∞; Zhou and Shi [7] have given the essential norm of composition operator on the Bloch space in polydiscs, that is,

1

n2lim δ→0

sup

dist(ϕ(z),∂Dn_)<δ

n

k,l=1 ∂ϕl

∂zk(z)

1−zk2

1−ϕl(z)2

≤Cϕe≤2n2lim δ→0

sup

dist(ϕ(z),∂Dn_)<δ

n

k,l=1 ∂ϕl

∂zk(z)

1−zk2

1−ϕl(z)2 .

(1.4)

Recently, Zhou [6] studied weighted composition operators betweenα-Bloch space andβ-Bloch space in polydiscs. He proved the following theorems.

Theorem _1.1. _Letϕ=(ϕ₁,...,ϕ_n)_{be a holomorphic self-map of}Dn _andψ(z)_a

holomorphic function ofDn_{,0< α,β <1. Then,ψC}

ϕ:Ꮾα(Dn)→Ꮾβ(Dn)is bounded if

and only ifψ∈Ꮾβ_(Dn_)and

_ψ(z) n k,l=1

∂ϕl

∂zk(z)

1−zk2 β

1−ϕl(z)2

α =O(1) |z| →1−. (1.5)

Theorem _1.2. _Letϕ=(ϕ₁,...,ϕ_n) _{be a holomorphic function of}Dn _andψ(z)_a

holomorphic function ofDn_{,0< α,β <1. Then,ψC}

ϕ:Ꮾα(Dn)→Ꮾβ(Dn)is compact if

and only if

(i) ψCϕis bounded,

(ii)

n

k=1

_∂z∂ψ_k(z)1−zk2 β

=o(1), ϕ(z) →∂Dn_{, (1.6)}

(iii)

ψ(z) n

k,l=1 ∂ϕl

∂zk(z)

1−zk2 β

1−ϕl(z)2

α=o(1), ϕ(z) →∂Dn. (1.7)

It is reasonable to expect that the essential norm ofψCϕ:Ꮾα(Dn)→Ꮾβ(Dn)should

ESSENTIAL NORM OF WEIGHTED COMPOSITION 3943

Theorem 1.3. Letϕ=(ϕ₁,...,ϕ_n)be a holomorphic self-map ofDn andψ(z)a

holomorphic function ofDn_{,0< α,β <1. Suppose the weighted composition operator} ψCϕ:Ꮾα(Dn)→Ꮾβ(Dn)is bounded, then,

1

nlim_δ→0

sup

dist(ϕ(z),∂Dn_)<δ

_ψ(z) n k,l=1

∂ϕl

∂zk(z)

1−zk2 β

1−ϕl(z)2 α

≤ψCϕe≤2nlim δ→0

sup

dist(ϕ(z),∂Dn_)<δ ψ(z)

n

k,l=1 ∂ϕl

∂zk(z)

1−zk2 β

1−ϕl(z)2 α.

(1.8)

2. The proof ofTheorem 1.3. In this section, we mainly give the proof of the main theorem of this paper. We divided our proof into two parts.

The lower estimates. _SinceψC_ϕ:Ꮾα(Dn)→Ꮾβ(Dn)_{is bounded, we have}ψ∈ Ꮾβ_(Dn_{)byTheorem 1.1.}

Note that form≥2,

zm

1α= sup z∈Dn

1−z12 α

mzm−1

1 =m

_2α

m−1+2α

α _m−1

m−1+2α

(m−1)/2 , (2.1)

where the maximum is attained at any point on the circle with radius

rm=

_m−1

m−1+2α 1/2

. (2.2)

Hence, the sequence{zm

1}m≥2is bounded inᏮα(Dn). Letfm=zm1/zm1α, thenfmα=

1 andfmis bounded sequence and converges weakly to 0 inᏮα(Dn). This follows since

a bounded sequence contained inᏮα_(Dn_{)which tends to 0 uniformly on compact}

sub-sets ofDn_{converges weakly to 0 inᏮ}α_(Dn_{). In particular, ifKis any compact operator}

fromᏮα(Dn)toᏮβ(Dn), then limm→∞Kfmβ=0.

Form≥2, let

Am=z=z1,z2,...,zn∈Dn, rm≤z1≤rm+1

, (2.3)

here,rm=((m−1)/(m−1+2α))1/2. Letg(x)=m(1−x2)αxm−1, then

g_{(x)= −mx}m−2_1−x2α−1_2αx2_{−(m−1)1−x}2_{≤0 (2.4)}

forx∈[((m−1)/(m−1+2α))1/2_{,1), that is,g(x)is a decreasing function forx∈} [((m−1)/(m−1+2α))1/2_{,1). Therefore,}

min

Am ∂fm

∂z1

1−z1 2α

=

1−r2

m+1mrmm+−11 zm

1α

=

_m−1+2α

m+2α

α _m2_+(2α−1)m

m2_{+(2α−1)m−2α}

(m−1)/2 =Cm.

Note that Cm tends to 1 as m→ ∞. Take any compact operator K fromᏮα(Dn)to

Ꮾβ_(Dn_{), we have}

ψCϕ−K

≥lim sup

m→∞

ψCϕ−Kfmβ

≥lim sup

m→∞

ψCϕfmβ−Kfmβ

=lim sup

m→∞

ψCϕfmβ

≥lim sup

m→∞ zsup∈Dn

n

k=1

∂ψfm◦ϕ ∂zk (z)

1−zk2 β

≥lim sup

m→∞ ϕ(z)sup∈Am

n

k=1

∂ψfm◦ϕ ∂zk (z)

1−zk2 β

≥lim sup

m→∞ sup

ϕ(z)∈Am

n

k=1

∂ψ(z) ∂zk fm

_ϕ(z) +ψ(z)

n

l=1 ∂fm ∂wl

_ϕ(z)∂ϕl ∂zk(z)

1−zk2 β

=lim sup

m→∞ ϕ(z)sup∈Am

n

k=1

∂ψ(z)_∂zk fmϕ(z)+ψ(z)∂f_∂wm 1

ϕ(z)∂ϕ1 ∂zk(z)

1−zk2 β

≥lim sup

m→∞ sup

ϕ(z)∈Am

n

k=1

ψ(z)∂fm ∂w1

ϕ(z)∂ϕ1 ∂zk(z)

×

1−zk2 β

1−ϕ1(z) 2α

1−ϕ1(z) 2α

− n

k=1

_∂z∂ψ_k(z)1−zk2 β

fmϕ(z) .

(2.6)

When 0< α <1, we know thatψ∈Ꮾβ_(Dn_{), so that}

lim sup

m→∞ sup

ϕ(z)∈Am

n

k=1

_∂z∂ψ_k(z)1−zk2 β

fmϕ(z)

≤ ψβlim sup m→∞

sup

ϕ(z)∈Am

fmϕ(z)

= ψβlim sup m→∞

_m/(m

+2α)m/2

_zm 1α

= ψβlim sup m→∞

_m/(m

+2α)m/2

m2α/(m−1+2α)α(m−1)/(m−1+2α)(m−1)/2

=0.

ESSENTIAL NORM OF WEIGHTED COMPOSITION 3945

Therefore,

ψCϕe

≥lim sup

m→∞ sup

ϕ(z)∈Am

n

k=1

ψ(z)∂fm ∂w1

_ϕ(z)∂ϕ1 ∂zk(z)

×

1−zk2 β

1−ϕ1(z) 2α

1−ϕ1(z) 2α

≥lim sup

m→∞ sup

ϕ(z)∈Am

n

k=1

ψ(z)∂ϕ1 ∂zk(z)

1−zk2 β

1−ϕ1(z) 2α

×lim inf

m ϕ(z)min∈Am ∂fm

∂w1

ϕ(z)1−ϕ1(z)2 α

≥lim sup

m→∞ ϕ(z)sup∈Am

n

k=1

ψ(z)∂ϕ1 ∂zk(z)

1−zk2 β

1−ϕ1(z)2

αlim inf_m Cm

≥lim sup

m→∞ ϕ(z)sup∈Am

n

k=1

ψ(z)∂ϕ1 ∂zk(z)

1−zk2 β

1−ϕ1(z) 2α.

(2.8)

So,

ψCϕe≥lim sup m→∞

sup

ϕ(z)∈Am

n

k=1

ψ(z)∂ϕ1 ∂zk(z)

1−zk2 β

1−ϕ1(z)2

α. (2.9)

Forl=1,2,...,n, define

al=lim δ→0

sup

dist(ϕ(z),∂Dn_)<δ

n

k=1

ψ(z)∂ϕl ∂zk(z)

1−zk2 β

1−ϕl(z)2

α. (2.10)

For any >0, (2.10) shows that there existsδ0, 0< δ0<1, if dist(ϕ(z),∂Dn) < δ0, then

n

k=1

ψ(z)∂ϕ1 ∂zk(z)

1−zk2 β

1−ϕ1(z)2

α > a1−. (2.11)

Becauserm→1(m→ ∞),rm>1−δ0formbeing large enough. Ifϕ(z)∈Am, then rm≤ |ϕ1(z)| ≤rm+1, that is, 1−rm+1≤1−|ϕ1(z)| ≤1−rm, that is,

distϕ(z),∂Dn_≤distϕ 1(z),∂D

By (2.9) and (2.11), we have

ψCϕe≥a1−. (2.13)

If we choosegm(z)=zml /zlm, repeating similar argument as in the casel=1, we have

ψCϕe≥al− (2.14)

forl=2,...,n. Hence,

ψCϕe≥

1

n n

l=1

al−

=_n1 n

l=1   lim

δ→0

sup

dist(ϕ(z),∂Dn_)<δ

n

k=1

ψ(z)∂ϕl ∂zk(z)

1−zk2 β

1−ϕl(z)2 α−

  

=_n1lim

δ→0

sup

dist(ϕ(z),∂Dn_)<δ

n

k,l=1

ψ(z)∂ϕl ∂zk(z)

1−zk2 β

1−ϕl(z)2 α−.

(2.15)

Let→0, we obtain the result.

The upper estimates. _{For this purpose, we define operator}K_m(m≥₂)_{as follows:}

Kmf (z)=f _m−1

m z . (2.16)

Using the method of [7], we can show thatKmhas the following properties:

(a) Kmis compact operator fromᏮα(Dn)toᏮα(Dn),

(b) limm→∞supfα≤1supz∈Dn|(I−K_m)f (z)| =0,

(c) for anyf∈Ꮾα_(Dn_{),(I−Km)f→0 uniformly on compact subsets ofD}n_{, hence,}

forl=1,2,...,n,∂(I−Km)f /∂wl→0 uniformly on compact subsets ofDn,

(d) I−Km ≤2.

The details of parts (b) and (c) are left to the reader, we will show the details for part (a) and (d).

First, we show the details of part (a). In fact, letE= {((m−1)/m)z, z∈Dn}, then

E is a compact subset ofDn_{. For any sequence {f}

j} ⊂Ꮾα(Dn), there exists a

sub-sequence{fjs}of{fj}converging uniformly to f∈Ꮾα(Dn)on compact subsets of Dn_andf

α≤M. It is obvious that{∂fjs/∂zi},i=1,2,...,n, converges uniformly to {∂f /∂zi}on compact subsets ofDn. So, for large enoughs,w∈Eandl=1,2,...,n,

∂fjs−f

∂wl (w)

ESSENTIAL NORM OF WEIGHTED COMPOSITION 3947

Hence,

_{Kmfjs−Kmfα}

=fjs _m−1

m z −f

_m−1

m z α

≤sup

z∈Dn

n

k=1

∂fjs−f

(m−1)/mz ∂zk

1−zk2 α

≤sup

z∈Dn

n

k=1 n

l=1

∂fjs−f

(m−1)/mz ∂wl

m_m−11−zk2 α

≤sup

z∈Dn

n

k=1 n

l=1

∂fjs−f

(m−1)/mz ∂wl

m_m−1

≤sup

w∈E m−1

m n

k=1 n

l=1

∂fjs−f

∂wl (w) →0

(2.18)

fors→ ∞. These show thatKmis a compact operator.

Next, we show the details of part (d). In fact, for anyf∈Ꮾα_(Dn_{), we have}

_I−Km

f_α≤sup

z∈Dn

n

k=1

∂I−Kmf ∂zk (z)

1−zk2 α

=sup

z∈Dn

n

k=1

_∂z∂f_k(z)−1− 1

m ∂f ∂zk

1− 1

m z

1−zk2 α

≤sup

z∈Dn

n

k=1

_∂z∂f_k(z)1−zk2 α

+1−_m1 sup

z∈Dn

n

k=1

_∂z∂f_k1−_m1 z 1−1−_m1 zk 2 α

≤ fα+fα=2fα.

(2.19)

Since eachKmis compact operator fromᏮα(Dn)toᏮα(Dn), so isψCϕKm. Hence,

ψCϕe≤ψCϕ−ψCϕKm=ψCϕI−Km= sup

fα≤1

ψCϕI−Kmfβ. (2.20)

We bound the last expression from above by

sup fα≤1

ψ(0)I−Kmfϕ(0) (2.21)

+ sup fα≤1

sup

z∈Dn

n

k,l=1

ψ(z)∂

I−Kmf ∂wl

ϕ(z)∂ϕl ∂zk(z)

1−zk2 β

(2.22)

+ sup fα≤1

sup

z∈Dn

n

k=1

_∂z∂ψ_k(z)I−Kmfϕ(z)1−zk2 β

By the property (b), we know that the supremum in (2.21) can be made arbitrarily small asm→ ∞. Sinceψ(z)∈Ꮾβ_(Dn_),

sup

z∈Dn

n

k=1

_∂z∂ψ_k(z)1−zk2 β

<∞. (2.24)

Property (b) also ensures that the supremum in (2.23) tends to 0 asm→ ∞. Now, we need only consider the term

sup fα≤1

sup

z∈Dn

n

k,l=1

ψ(z)∂

I−Kmf ∂wl

ϕ(z)∂ϕl ∂zk(z)

1−zk2 β

. (2.25)

For arbitrary 0< δ <1, we define

G1=z∈Dn: distϕ(z),∂Dn< δ, G2=

z∈Dn_{: distϕ(z),∂D}n_≥δ. (2.26)

Here,G2is compact subset ofCn. We consider

sup fα≤1

sup

z∈Dn

n

k,l=1

ψ(z)∂

_I −Kmf ∂wl

_ϕ(z)∂ϕl ∂zk(z)

1−zk2 β

≤ sup fα≤1

sup

z∈G1 n

k,l=1

ψ(z)∂

_I −Kmf ∂wl

_ϕ(z)∂ϕl ∂zk(z)

1−zk2 β

+ sup fα≤1

sup

z∈G2 n

k,l=1

_ψ(z)∂I−Kmf ∂wl

ϕ(z)∂ϕl ∂zk(z)

1−zk2 β

=I+II.

(2.27)

First, we writeIas follows:

I= sup fα≤1

sup

z∈G1 n

k,l=1

_ψ(z)∂ϕl ∂zk(z)

1−zk2 β

1−ϕl(z)2 α

×∂

I−Kmf ∂wl

ϕ(z)1−ϕl(z)2 α

(2.28)

and observe that this is bounded above by

nI−Kmsup z∈G1

n

k,l=1

ψ(z)∂ϕl ∂zk(z)

1−zk2 β

1−ϕl(z)2

ESSENTIAL NORM OF WEIGHTED COMPOSITION 3949

Then, by property (d), we know that

I≤2nsup

z∈G1 n

k,l=1

ψ(z)∂ϕl ∂zk(z)

1−zk2 β

1−ϕl(z)2

α. (2.30)

Next, we prove that limm→∞II=0. SinceψCϕis bounded fromᏮα(Dn)toᏮβ(Dn),

byTheorem 1.1we have

ψ(z) n

k,l=1 ∂ϕl

∂zk(z)

1−zk2 β

1−ϕl(z)2

α<∞ |z|→1−. (2.31)

Thus,

sup

z∈G2 ψ(z)

n

k,l=1 ∂ϕl

∂zk(z)

1−zk2 β

<∞. (2.32)

Then, using property (c), we have

lim

m→∞II=_mlim_{→∞ f}sup_α≤1zsup∈G2 n

k,l=1

ψ(z)∂

_I −Kmf ∂wl

_ϕ(z)∂ϕl ∂zk(z)

1−zk2 β

=0.

(2.33)

Combining the estimates for (2.21), (2.22), and (2.23) asm→ ∞, we get

ψCϕe≤2nlim δ→0

sup

dist(ϕ(z),∂Dn_)<δ

n

k,l=1

ψ(z)∂ϕl ∂zk(z)

1−zk2 β

1−ϕl(z)2

α. (2.34)

Acknowledgments. _{The authors thank the referee for several helpful comments}

and suggestions for improvements. The first author also thanks professor Zehua Zhou for many help. The first author is supported in part by the National Natural Science Foundation of China (no. 10371051) and the NSF of the Zhejiang province of China (no. 102025).

References

[1] D. D. Clahane,Bounded composition operators on Holomorphic Lipschitz and Bloch spaces of the polydisk, Integral Equations Operator Theory99(2004), 1–8.

[2] R. Donaway,Norm and essential norm estimates of composition operators on Besov type spaces, Ph.D. thesis, University of Virginia, Virginia, 1999.

[3] B. D. MacCluer and R. Zhao,Essential norms of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math.33(2003), no. 4, 1437–1458.

[4] A. Montes-Rodríguez,The essential norm of a composition operator on Bloch spaces, Pacific J. Math.188(1999), no. 2, 339–351.

[5] J. H. Shapiro,The essential norm of a composition operator, Ann. of Math. (2)125(1987), no. 2, 375–404.

[7] Z. H. Zhou and J. H. Shi,The essential norm of a composition operator on the Bloch space in polydiscs, Chinese Ann. Math. Ser. A24(2003), no. 2, 199–208.

[8] Z. H. Zhou and S. B. Zeng,Composition operators on the Lipschitz space in polydisk, Sci. China Ser. A32(2002), no. 5, 385–389.

Li Songxiao: Department of Mathematics, Jiaying University, Meizhou 514015, Guangdong, China

E-mail address:[email protected]

Zhu Xiangling: Department of Mathematics, Shantou University, Shantou 515063, Guangdong, China