Volume 2009, Article ID 720217,13pages doi:10.1155/2009/720217
Research Article
On the Norm of Certain Weighted Composition Operators on the Hardy Space
M. Haji Shaabani and B. Khani Robati
Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran
Correspondence should be addressed to B. Khani Robati,[email protected] Received 22 January 2009; Revised 9 March 2009; Accepted 8 May 2009 Recommended by Stevo Stevic
We obtain a representation for the norm of certain compact weighted composition operatorCψ,ϕ
on the Hardy spaceH2, wheneverϕz azbandψz az−b. We also estimate the norm and essential norm of a class of noncompact weighted composition operators under certain conditions onϕandψ. Moreover, we characterize the norm and essential norm of such operators in a special case.
Copyrightq2009 M. Haji Shaabani and B. Khani Robati. 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
Let D denote the open unit disk in the complex plane. The Hardy space H2 is the space of analytic functions onD whose Taylor coefficients, in the expansion about the origin, are square summable. Also we recall that H∞ is the space of all bounded analytic function defined onD. Forα∈D, the reproducing kernel atαforH2is defined byKαz 1/1−αz.
An easy computation shows thatf, Kα fαwheneverf ∈ H2. For any analytic self- mapϕofD, the composition operatorCϕonH2is defined by the ruleCϕf f◦ϕ. Every composition operator is bounded, with
1
1−ϕ02 ≤Cϕ :H2 −→H2≤
1ϕ0
1−ϕ0 1.1
see1. We see from expression1.1that Cϕ 1 wheneverϕ0 0. There are few other cases for which the exact value of the norm has been known for many years. For example, the norm ofCϕwas obtained by Nordgren in2, wheneverϕis an inner function. In3this
norm was determined, whenϕz azb, with|a||b| ≤1,and if 0< s <1 and 0≤r ≤1 the norm was found in4forϕz rsz 1−s/r1−sz 1rs.
In 2003, Hammond5obtained exact values for the norms of composition operators Cϕ for certain linear fractional maps ϕ. In 6, Bourdon et al. determined the norm of Cϕ for a large class of linear-fractional maps, including those of the form ϕz b/d− z, where 0 < b < d − 1. The connection between the norm of certain composition operators Cϕ with linear-fractional symbol acting on the Hardy space and the roots of associated hypergeometric functions was first made by Basor and Retsek7. It was later refined by Hammond8. In9Effinger-Dean et al. computed the norms of composition operators with rational symbols that satisfy certain properties. Their work is based on the initial work of Hammond 5. Some other recent results regarding the calculation of the operator norm of some composition operators on the other spaces can be found in 10–14.
Ifψ is a bounded analytic function onDandϕis an analytic map fromDinto itself, the weighted composition operatorCψ,ϕis defined byCψ,ϕfz ψzfϕz. The mapϕ is called the composition map andψis called the weight. Ifψis a bounded analytic function on D, then the operator can be rewritten as Cψ,ϕ MψCϕ,where Mψ is a multiplication operator and Cϕ is a composition operator. Recall that if ϕ is an analytic self-map of D, then the composition operator Cϕ onH2 is bounded, hence in this case Cψ,ϕ is bounded, but in general every weighted composition operatorCψ,ϕ onH2 is not bounded. IfCψ,ϕ is bounded, thenCψ,ϕ1 ψbelongs toH2. These operators come up naturally. In 1964, Forelli 15showed that every isometry onHpfor 1 < p < ∞andp /2 is a weighted composition operator. Recently there has been a great interest in studying weighted composition operators in the unit disk, polydisk, or the unit ball; see12,16–27, and the references therein. In this paper we investigate the norm of certain bounded weighted composition operatorsCψ,ϕon H2.
2. Norm Calculation
In this section we obtain a representation for the norm of a class of compact weighted composition operatorsCψ,ϕon the Hardy spaceH2, wheneverϕz azb,ψz az−b,
|b|2 ≥ 1/2,and 2|a|2|b|2 ≤ 2/3. Also we give the norm and essential norm inequality for a class of noncompact weighted composition operatorsCψ,ϕonH2whenϕz aznb, for somen∈N,|a||b|1,andψis a bounded analytic map onDsuch that the radial limit of|ψ|
at one of thenth roots ofb|a|/a|b|is the supremum of|ψ|onD. Also, whenn1 we obtain the norm and essential norm of such operators.
The following lemma was inspired by a similar result for unweighted composition operators28, Theorem 1.4. See29for a similar proof.
Lemma 2.1. LetKwbe the reproducing kernel atw. Then
C∗ψ,ϕKwψwKϕw. 2.1
In the next proposition we generalize the lower bound in1.1.
Proposition 2.2. Letϕbe a nonconstant analytic self-map ofD, and letψbe a nonzero analytic map onD. Ifnis the smallest nonnegative integer such thatψn0/0, then
Cψ,ϕ≥
ψn0 n!
1
1−ϕ02. 2.2
Proof. We note that iffis inH2, then for everyn∈N∪ {0}we have|fn0/n!| ≤ f 2. Hence we have
Cψ,ϕ≥ Cψ,ϕKϕ0 Kϕ0 ψ.
Kϕ0◦ϕ Kϕ0
≥ ψn0/n! Kϕ0◦ϕ 0 Kϕ0
ψn0 n!
1
1−ϕ02.
2.3
LetT be a bounded operator on a Hilbert spaceH. We recall that T e, the essential norm ofT, is the norm of its equivalence class in the Calkin algebra. Since the spectral radius of the operator T∗T equals T∗T T 2, we study the spectrum of T∗T when trying to determine T . We say that the operator T is norm-attaining if there is a nonzeroh ∈ H such that Th T h .We know that Th T h if and only if T∗Th T 2h.
Moreover, if T e< T , then the operatorTis norm-attaining and so the quantity T 2equals the largest eigenvalue ofT∗T; see5for more details. Ifϕz azb,ψz az−b,|b|2≥1/2, and 2|a|2|b|2 ≤ 2/3, then the operatorCψ,ϕ is compactsee the proof ofProposition 2.5.
Hence 0 Cψ,ϕ e< Cψ,ϕ and soCψ,ϕis norm-attaining.
Now our goal is to find a functional equation that relates an eigenvalue ofC∗ψ,ϕCψ,ϕto the values of its eigenfunctions at particular points in the disk. In what follows we use the techniques used in5,6,30and present some results that help us to obtain the norm ofCψ,ϕ.
Letϕbe an analytic self-map ofDand letψbe a bounded analytic map onD. Then Cψ,ϕ ∗
MψCϕ ∗C∗ϕM∗ψ C∗ϕTψ∗. 2.4
But ifϕz azbsuch that|a||b| ≤1, then by3or28 Cψ,ϕ ∗TgCσTh∗Tψ∗TgCσ
Tψh ∗, 2.5
wherehz 1,gz 1/−bz1,andσz az/−bz1.
From now on, unless otherwise stated, we assume thatψz czd, ϕz azb, and|a||b| ≤1. SinceTz∗is the backward shift onH2, we see that
C∗ψ,ϕCψ,ϕfz TgCσTψ∗TψCϕfz TgCσTczd∗
ψ·f ϕz TgCσ
c
ψ·f
ϕz −ψ·f ϕ0 z
TgCσ
dψz·f ϕz Tg
c
ψσz·f
ϕσz −ψ0·f ϕ0 σz
dgzψσz·f
ϕσz gz
c
ψσz·f
ϕσz −ψ0·f ϕ0 σz
dgzψσz·f
ϕσz γzfτz χzf
ϕ0
2.6
for allzinDnot equal to 0, where
γz
c 1−bz
daz d
1−bz acz az
1−bz2 ,
τz
|a|2− |b|2 zb
−bz1 , χz −cd
az.
2.7
In particular, ifg is an eigenfunction forC∗ψ,ϕCψ,ϕcorresponding to an eigenvalue λ, then
λgz γzgτz χzg
ϕ0 . 2.8
Formula2.8is essentially identical to5, Formula3.3. Using2.8we can find a set of conditions under which we determine C∗ψ,ϕCψ,ϕ . In the trivial casea0 we have Cψ,ϕ ψ 21/
1− |b|2.Also ifd 0, then Cψ,ϕ |c| Cϕ and ifc 0, then Cψ,ϕ |d| Cϕ . Therefore we assume thata, b, c, dare nonzero.
Throughout this paper, we writeτj to denote thejth iterate ofτ,that is, τ0 is the identity map onDandτj1τ◦τj.
By a similar argument as in the proof of5, Proposition 5.1, we have the following lemma.
Lemma 2.3. Letgbe an eigenfunction forCψ,ϕ∗ Cψ,ϕcorresponding to an eigenvalueλ,z∈Dand for each nonnegative integerj,τjz/0. Then one has
λj1gz g
τj1zj
k0
γ
τkz
j
k0
g
ϕ0 χ
τkzk−1
m0
γ
τmz λj−k,
2.9
where one takes−1
m0· 1.
Lemma 2.4. For eachn ∈ N,τn0 αnb, where{αn}is strictly increasing sequence such that αn≥1 for eachn∈N. Alsoαn11αn|a|2/1−αn|b|2.
Proof. By inductionSinceτ0 band τ20 1|a|2/1− |b|2b, the claim holds for n1. Assume the claim holds forn−1. We will prove it forn. We have
τn0 τ
τn−10
ταn−1b
1 αn−1|a|2 1−αn−1|b|2
b. 2.10
Now if we setαn1αn−1|a|2/1−αn−1|b|2, thenτn0 αnb. But by hypothesisαn−1< αn, so
1 αn−1|a|2
1−αn−1|b|2 <1 αn|a|2
1−αn|b|2, 2.11
which implies thatαn < αn1 alsoτn10 ταnb 1αn|a|2/1−αn|b|2b.Hence the proof is complete.
Proposition 2.5. Letac,b−dand letλ Cψ,ϕ 2. If|b|2 ≥1/2,and 2|a|2|b|2≤2/3, then for eachz∈Dwith the property thatτjz/0 for every nonnegative integerj, one has
gz ∞
k0
g
ϕ0 χ
τkzk−1
m0
γ
τmz 1
λk1. 2.12
Proof. Since 2|a|2|b|2 ≤ 2/3, it is easy to see that|a||b| 1 if and only if|a| 1/3 and
|b|2/3. By assumption|b|2 ≥1/2, so|a||b|<1. ThereforeCϕis compact and, sinceCψ,ϕ MψCϕ, the operatorCψ,ϕis compact. Now according to the paragraph afterProposition 2.2,
there is functiong in H2 such thatC∗ψ,ϕCψ,ϕg λg. Letz ∈ D and for each integer j ≥ 0, τjz/0. ByLemma 2.3, we have
λj1gz g
τj1zj
k0
γ
τkz
j k0
g
ϕ0 χ
τkzk−1
m0
γ
τmz λj−k.
2.13
Hence
gz g
τj1zj
k0
γ τkz
λ
j
k0
g
ϕ0 χ
τkzk−1
m0
γ
τmz 1 λk1.
2.14
Now ifw0is the Denjoy-Wolffpoint ofτ,it suffices to show that
γw0 λ
<1. 2.15
Suppose the above inequality holds. Then we conclude that there is 0< β <1 andN∈Nsuch that fork > Nwe have|γτkz/λ|< β <1. Now we break the proof into two parts.
1The Denjoy-Wolffpointw0 ofτ lies insideD, thengτjzconverges togw0. Hence
jlim→ ∞
g
τj1zj
k0
γ τkz
λ
≤ lim
j→ ∞g
τj1z βj−N
N
k0
γ τkz
λ
0. 2.16
2The Denjoy-Wolffpointw0 ofτ lies on∂D, then by31, Lemma 5.1τ must be parabolic and by6, Lemma 3.3there is a constantCsuch that
1
1−τjz≤Cj. 2.17
Thus it follows that
g
τjzg, Kτjz
≤g·Kτjz g·
1
1−τjz2
≤g· jC.
2.18
Hence
jlim→ ∞
g
τj1zj
k0
γ τkz
λ
≤ lim
j→ ∞g
τj1z βj−N
N
k0
γ τkz
λ
≤ lim
j→ ∞g·
j1 C·βj−N
N k0
γ τkz
λ 0.
2.19
Now we show that|γw0/λ|<1. Sinceacandb−d, we see that
γw0 λ
1−2bw0
−b
1−bw0
aaw0 λw0
1−bw0
2
. 2.20 By30, we have
w0 1− |a|2|b|2−
1− |a|2|b|22
−4|b|2
2b . 2.21
Applying the assumptions|b|2≥1/2 and 2|a|2|b|2≤2/3, an easy computation shows that
0≤2bw0−1≤1−bw0. 2.22
Also by usingProposition 2.2, 1/λ < 1− |b|2/|b|2,and byLemma 2.4, there isαn ≥ 1 such thatτn0 αnb. Therefore
γw0 λ
1−2bw0
−b 1−bw0
aaw0
λw0
1−bw02
2bw0−1−b
1−bw0
aaw0 λ|w0|
1−bw02
2bw0−1−b
1−blimn→ ∞αnb
aalimn→ ∞αnb λ|w0|
1−bw0
1−blimn→ ∞αnb
≤
2bw0−1
limn→ ∞|b|
1−αn
|b|2|a|2
λ|b|
1−bw0
limn→ ∞1−αn|b|2
<
1− |b|2
2bw0−1
limn→ ∞ 1−αn
|b|2|a|2
|b|2 1−bw0
limn→ ∞1−αn|b|2
≤ 1− |b|2
|b|2
≤1.
2.23
Proposition 2.6. Letac,b−d,|b|2 ≥1/2,and 2|a|2|b|2 ≤2/3. Thenλ Cψ,ϕ 2satisfies the equation
1∞
k0
χ
τk10k−1 m0
γ
τm10 1
λk1. 2.24
Proof. Since for every integerj ≥0,τkϕ0/0, inProposition 2.5we setzϕ0, then we have
g
ϕ0 ∞
k0
g
ϕ0 χ τk
ϕ0 k−1
m0
γ
τm ϕ0
1
λk1. 2.25
Sinceϕ0 τ0, we see that
g
ϕ0 ∞
k0
g
ϕ0 χ
τk10k−1
m0
γ
τm10 1
λk1. 2.26
But gϕ0/0, because otherwise Proposition 2.5 would dictate that the function gz is identically 0. Thus eigenfunctiongmust have the property thatgϕ0/0. Hence we have
1∞
k0
χ
τk10k−1
m0
γ
τm10 1
λk1. 2.27
We define
Fz ∞
k0
χ
τk10k−1
m0
γ
τm10
zk1. 2.28
Now we characterize the properties ofFand by using these properties we obtain a formula for the norm ofCψ,ϕ. The idea behindProposition 2.7is similar to the one found in30.
Proposition 2.7. Letac,b−d,|b|2 ≥1/2,and 2|a|2|b|2≤2/3. ThenFzhas the following properties.
aThe power series that definesFzhas radius of convergencer0larger than 1/λ.
bFxis non-negative real number for allxin the interval0, r0. cFx>0 for allxin the interval0, r0.
Proof. aByLemma 2.4, for each positive integernthere isαn ≥ 1 such thatτn0 αnb, thenχτm10 1/αm1 ≤1. Also in the proof ofProposition 2.5we have|γw0/λ|<1, hence there is 0< β <1 andN∈Nsuch that ifn > N, then
γ τn0
λ
< β <1. 2.29
Now letβ < β1 <1 and 0< < λβ1−β/β1. Then ifn > Nwe have
γ
τn0 λ
<
γ
τn0 λ−
< β1. 2.30
Therefore there is a constantCsuch that
∞ k0
χ
τk10k−1 m0
γ
τm10 1
λ−k1 ≤∞
k0
1 λ−
k−1 m0
γ
τm10 λ−
≤C ∞ k0
βk1
<∞.
2.31
By Lemma 2.4, there is strictly increasing sequenceαn ≥ 1 such thatτn0 αnb, and by hypothesis|b|>√
2/2, hence 1−2αn|b|2<1−2|b|2<0. Also we have|a|2|b|2≤ |b| ≤ |b/w0|<
1/αn, so we conclude that−1−αn|b|2 |a|2αn<0. Therefore
γ τm1
0 γαm1b
1−2αm1|b|2
−b
1−αm1|b|2
|a|2αm1b αm1b
1−αm1|b|22
1−2αm1|b|2
−
1−αm1|b|2
|a|2αm1 αm1
1−αm1|b|22
>0.
2.32
Also it is obvious that
χ
τm10
−cd aαm1b 1
αm1 >0. 2.33
Hence the proof of partbis complete.
cEvery coefficient ofFis positive and soFx>0 for allxin the interval0, r0. Now we find an equation that involves the norm ofCψ,ϕ.
Theorem 2.8. Letac,b−d,|b|2 ≥1/2 and 2|a|2|b|2≤2/3. Thenλ Cψ,ϕ 2is the unique positive real solution of the equation
1∞
k0
χ
τk10k−1 m0
γ
τm10 1
λk1. 2.34
Proof. By Propositions2.6and2.7, there is exactly one positive real numberλwhich satisfies equation2.34, and this number must be equal to Cψ,ϕ 2.
Corollary 2.9. InTheorem 2.8 if one replaces a0 with aand b0 with b such that |a| |a0|, and
|b||b0|, then norm ofCψ,ϕdoes not change.
Proof. We haveτn0 αnb. But byLemma 2.4,αn1αn−1|a|2/1−αn−1|b|2. Hence if one replacesa0withaandb0withbsuch that|a| |a0|and|b| |b0|, thenαn,γτm10and χτm10 1/αm1do not change. Hence by2.34, the norm ofCψ,ϕdoes not change.
Example 2.10. Letϕz azbandψz az−b, where|a|1/10 and|b|8/10. Then we have
χz 4
5z, τz 63z−80
80z−100, γz 5−8z−1613z
z10−8z2 . 2.35
For positive integerk0, letλk0denote the positive solution of
1k0
k0
χ
τk10k−1
m0
γ
τm10 1
λk1. 2.36
Now by using numerical methods, we have
λ10≈1.796745850919, λ20≈1.797084678603, λ30≈1.797084948747, λ50≈1.797084948963, λ70≈1.797084948963, λ100≈1.797084948963.
2.37
Hence we see that Cψ,ϕ 2≈1.797084948.
The hypotheses of Theorem 2.8 restrict us to considering the norms of compact operators. In the remainder of this section we investigate the norm and essential norm of a class of noncompact weighted composition operators.
Theorem 2.11. Letϕz aznb, for somen∈N, where|a||b|1,ψ ∈H∞,letαbe one of the nth roots ofb|a|/a|b|such thatψhas radial limit atα,and let|ψ|attains its supremum onD∪ {α}
atα. Then
1
n|a|ψα≤Cψ,ϕ
e≤Cψ,ϕ≤ 1
|a|ψα. 2.38
Proof. Let 0< r <1. Takingβrα, by a similar proof for unweighted composition operators 28, Proposition 3.13, we have
Cψ,ϕ2
e≥ lim
r→1−
C∗ψ,ϕKβ2 Kβ2 lim
r→1−
ψβ2· lim
r→1−
Kϕβ2 Kβ2 ψα2· lim
r→1−
1−r2 1−rn|a||b|2 1
n|a||a||b|ψα2 1
n|a|ψα2.
2.39
Therefore
Cψ,ϕ
e ≥ 1
n|a|ψα. 2.40
On the other hand, by3, we have Cψ,ϕ
e≤Cψ,ϕ≤MψCϕ≤ψ
∞ Cazb 1
|a|ψα. 2.41
Therefore
1
n|a|ψα≤Cψ,ϕ
e≤Cψ,ϕ≤ 1
|a|ψα. 2.42
Corollary 2.12. InTheorem 2.11ifn1, then Cψ,ϕCψ,ϕ
e 1
|a|ψα. 2.43 Example 2.13. 1Ifϕz 1/2z1/2 andψz z1/2, then Cψ,ϕ √
2.
2 Ifϕz 1/3z 2/3iandψz z5−2z3i, then Cψ,ϕ 4√ 3.
3 Ifϕz −1/4iz3/4 andψz 7z5−5z32i/z22, then Cψ,ϕ 28.
Acknowledgment
The authors would like to thank the referee for his valuable comments and suggestions.
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