on convexity of composition and multiplication operators

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2009, Article ID 931020, 9 pagesdoi:10.1155/2009/931020

Research ArticleOn Convexity of Composition and MultiplicationOperators on Weighted Hardy Spaces

Karim Hedayatian and Lotfollah Karimi

Department of Mathematics, Shiraz University, Shiraz 71454, Iran

Correspondence should be addressed to Karim Hedayatian, [email protected]

Received 30 September 2009; Revised 2 November 2009; Accepted 4 November 2009

Recommended by Stevo Stevic

A bounded linear operator T on a Hilbert space H, satisfying ‖T2h‖2 + ‖h‖2 ≥ 2‖Th‖2 for everyh ∈ H, is called a convex operator. In this paper, we give necessary and sufficient conditions underwhich a convex composition operator on a large class of weighted Hardy spaces is an isometry.Also, we discuss convexity of multiplication operators.

Copyright q 2009 K. Hedayatian and L. Karimi. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

1. Introduction and Preliminaries

We denote by B(H) the space of all bounded linear operators on a Hilbert space H. Anoperator T ∈ B(H) is said to be convex, if

∥∥∥T2h

∥∥∥

2+ ‖h‖2 ≥ 2‖Th‖2 (1.1)

for each h ∈ H. Note that if T is a convex operator then the sequence (‖Tnh‖2)n∈N forms aconvex sequence for every h ∈ H. Taking ΔT = T ∗T − I, it is easily seen that T is a convexoperator if and only if T ∗ΔTT ≥ ΔT .

A weighted Hardy space is a Hilbert space of analytic functions on the open unit discD for which the sequence (zj)∞j=0 forms a complete orthogonal set of nonzero vectors. It isusually assumed that ‖1‖ = 1. Writing β(j) = ‖zj‖, this space is denoted by H2(β) and itsnorm is given by

∥∥∥∥∥∥

∞∑

j=0

ajzj

∥∥∥∥∥∥

2

=∞∑

j=0

∣∣aj

∣∣2β(

j)2. (1.2)

2 Abstract and Applied Analysis

Let ϕ be an analytic map of the open unit disc D into itself, and define Cϕ(f) = f ◦ ϕwhenever f is analytic onD. The function ϕ is called the symbol of the composition operator.For a positive integer n, the nth iterate of ϕ, denoted by ϕn, is the function obtained bycomposing ϕ with itself n times; also ϕ0 is defined to be the identity function. Denote thereproducing kernel at z ∈ D, for the space H2(β), by Kz. Then 〈f,Kz〉 = f(z) for everyf ∈ H2(β). It is known that C∗

ϕ(Kz) = Kϕ(z) for all z in D. The generating function forH2(β)is the function given by

k(z) =∞∑

j=0

zj

β(

j)2. (1.3)

This function is analytic on D. Moreover, if w ∈ D then Kw(z) = k(wz) and ‖Kw‖2 = k(|w|2)(see [1]).

Recently, there has been a great interest in studying operator theoretic propertiesof composition and weighted composition operators, see, for example, monographs [1, 2],papers [3–16], as well as the reference therein.

Isometric operators on weighted Hardy spaces, especially those that are compositionoperators are discussed by many authors. Isometries of the Hardy space H2 amongcomposition operators are characterized in [17, page 444], [18] and [12, page 66]. Indeed, itis shown that the only composition operators onH2 that are isometries are the ones inducedby inner functions vanishing at the origin. Bayart [5] generalized this result and showed thatevery composition operator on H2 which is similar to an isometry is induced by an innerfunction with a fixed point in the unit disc. The surjective isometries of Hp, 1 ≤ p < ∞that are weighted composition operators have been described by Forelli [19]. Carswell andHammond [6] proved that the isometric composition operators of the weighted Bergmanspace A2

α are the rotations. Cima and Wogen [20] have characterized all surjective isometriesof the Bloch space. Furthermore, the identification of all isometric composition operators onthe Bloch space is due to Colonna [8]. Some related results can be found also in [3, 4, 6, 21–25].

Herein, we are interested in studying the convexity of composition and multiplicationoperators acting on a weighted Hardy space H2(β). First, we give some preliminary factson convex operators. Next, we will offer necessary and sufficient conditions under which aconvex composition operator may be isometry on a large class of weighted Hardy spacescontaining Hardy, Bergman, and Dirichlet spaces. We also discuss on convexity of the adjointof a composition operator. Finally, we will obtain similar results for multiplication operatorsand their adjoints. For a good reference on isometric multiplication operators the reader cansee [3].

Throughout this paper, T is assumed to be a bounded linear operator on a HilbertspaceH. It is easy to see that for every convex operator T , the sequence (T ∗nΔTT

n)n forms anincreasing sequence. We use this fact to prove the following theorem.

Theorem 1.1. If T is a convex operator then so is every nonnegative integer power of T .

Proof. We argue by using mathematical induction. The convexity of T implies that the resultholds for k = 1. Suppose that T ∗nΔTnT

n ≥ ΔTn , then

T ∗n+1ΔTn+1Tn+1 −ΔTn+1 = T ∗n+1(T ∗ΔTnT + ΔT )Tn+1 −ΔTn+1

= T ∗2(

T ∗nΔTnTn)

T2 + T ∗n+1ΔTTn+1 −ΔTn+1

Abstract and Applied Analysis 3

≥ T ∗2ΔTnT2 + T ∗nΔTT

n −ΔTn+1

= T ∗2(

T ∗nTn − I)

T2 + T ∗nΔTTn − T ∗

(

T ∗nTn − I)

T −ΔT

= T ∗n(

T ∗2T2)

Tn − T ∗2T2 + T ∗nΔTTn − T ∗n(T ∗T)Tn + T ∗T −ΔT

= T ∗n(

T ∗2T2 − I)

Tn − T ∗2T2 + I

≥ 2T ∗nΔTTn − T ∗2T2 + I

≥ 2T ∗ΔTT − T ∗2T2 + I

= T ∗ΔTT −ΔT ≥ 0.

(1.4)

So the result holds for k = n + 1.

Proposition 1.2. If T is a convex operator, then for every nonnegative integer n,

T ∗nTn ≥ nΔT + I. (1.5)

Proof. We give the assertion by using mathematical induction on n. The result is clearly truefor n = 1. Suppose that T ∗nTn ≥ nΔT + I. Thus,

T ∗n+1Tn+1 = T ∗(

T ∗nTn)

T

≥ T ∗(nΔT + I)T

= nT ∗ΔTT + T ∗T

= n(

T ∗2T2 − 2T ∗T + I)

+ nT ∗T + T ∗T − nI

≥ (n + 1)T ∗T − nI= (n + 1)ΔT + I.

(1.6)

So the result holds for k = n + 1.

Proposition 1.3. Let T ∈ B(H) be a convex operator and let h ∈ H be such that supk≥0‖Tkh‖ <∞.If ΔT ≥ 0, then ‖Th‖ = ‖h‖.

Proof. By applying Proposition 1.2, we observe that for every nonnegative integer n,

n〈ΔTh, h〉 + ‖h‖2 ≤ ‖Tnh‖2 ≤ supk≥0

∥∥∥Tkh

∥∥∥

2<∞. (1.7)

Letting n → ∞, the positivity of ΔT implies that ΔTh = 0; hence, ‖Th‖ = ‖h‖.


Proposition 1.4. Let {en}∞n=0 be an orthonormal basis for H and let T ∈ B(H) be a convex operatorsatisfying ΔT ≥ 0. Suppose that there is a nonnegative integer i and a scalar αi with 0 < |αi| ≤ 1 sothat Tei = αiei, thenM = ∨n/= i{en} is an invariant subspace for T .

Proof. Using Proposition 1.2, we see that

‖ei‖2 ≥∥∥αni ei

∥∥2 = ‖Tnei‖2 = 〈T ∗nTnei, ei〉 ≥ n〈ΔTei, ei〉 + ‖ei‖2 (1.8)

for every n ≥ 0. Let n → ∞. Since ΔT is a positive operator, we conclude that ΔTei = 0.Consequently, T ∗ei = (1/αi)T ∗Tei = (1/αi)ei. Now, if f ∈ M then 〈Tf, ei〉 = 0; hence, Tf ∈M.

2. Composition Operators

Our purpose in this section is to discuss on convex composition operators on a weightedHardy space. Recall that an operator T in B(H) is an isometry, if ΔT = 0. At first, we givean example of a nonisometric composition operator T on a weighted Hardy space such thatT ∗ΔTT ≥ ΔT ≥ 0. For simplicity of notation, ΔCϕ is denoted by Δϕ.

Example 2.1. Consider the weighted Hardy spaceH2(β) with weight sequence (β(n))n givenby β(n) = n + 1. Define ϕ : D → D by ϕ(z) = z2. It is easily seen that Cϕ(H2(β)) ⊆ H2(β),and an application of the closed graph theorem shows that Cϕ is bounded. Now, a simplecalculation shows that

⟨(

C∗ϕΔϕCϕ −Δϕ

)(

zk)

, zk⟩

=∥∥∥Cϕ2z

k∥∥∥

2 − 2∥∥∥Cϕz

k∥∥∥

2+∥∥∥zk

∥∥∥

2> 0 (2.1)

for all k ≥ 0; besides

⟨

Δϕzk, zk

⟩

=∥∥∥Cϕz

k∥∥∥

2 −∥∥∥zk

∥∥∥

2(2.2)

which is positive for all k ≥ 1, and zero whenever k = 0. It follows that C∗ϕΔϕCϕ ≥ Δϕ ≥ 0, but

Cϕ is not an isometry.

Proposition 2.2. Suppose that T : H2(β) → H2(β) is a convex operator satisfying T1 = 1 andΔT ≥ 0, then

M ={

f ∈ H2(β)

: f(0) = 0}

(2.3)

is a nontrivial invariant subspace of T .

Proof. Clearly M is a nontrivial closed subspace of T . To show that M is invariant for T ,apply Proposition 1.4 for the Hilbert space H = H2(β), the orthonormal basis {en}n given byen = zn/β(n), i = 0 and α0 = 1.


Example 2.3. Consider the Bergman space A2(D) consisting of all analytic functions f on theopen unit disc D, for which

∥∥f

∥∥2 =

∫

D

∣∣f(z)

∣∣2dA(z) <∞, (2.4)

where dA(z) is the normalized Lebesgue area measure on D. If f ∈ A2(D) is represented byf(z) =

∑∞n=0 anz

n, then

∥∥f

∥∥2 =

∞∑

n=0

|an|2n + 1

. (2.5)

Also, {zk}k forms an orthogonal basis for A2(D). Fix nonnegative integers k and n, andobserve that

∥∥∥Cn

ϕzk∥∥∥

2=∥∥ϕnk

∥∥2 =

∫

D

∣∣ϕnk(z)

∣∣2dA(z) ≤

∫

DdA(z) = 1. (2.6)

Thus, Proposition 1.3 implies that C∗ϕΔϕCϕ ≥ Δϕ ≥ 0 if and only if Cϕ is an isometry. In this

case, taking T = Cϕ and f(z) = z in Proposition 2.2, we conclude that ϕ(0) = 0; thus, theSchwarz lemma implies that |ϕ(z)| ≤ |z| for all z ∈ D. On the other hand, if f(z) = z then

∫

D

∣∣ϕ(z)

∣∣2dA(z) =

∥∥Cϕf

∥∥2 =

∥∥f

∥∥2 =

∫

D|z|2dA(z), (2.7)

and so |ϕ(z)| = |z| almost everywhere with respect to the area measure. Hence, ϕ(z) = eiθzfor some θ ∈ [0, 2π).

Example 2.4. Consider the Hardy space H2(D). If ϕ is an analytic self-map of the unit disc,then ϕ induces a bounded composition operator, and ‖Cn

ϕzk‖ ≤ 1 for all nonnegative integers

n and k. Thus, by Proposition 1.3, C∗ϕΔϕCϕ ≥ Δϕ ≥ 0 if and only if Cϕ is an isometry.

Recall that the Dirichlet space D is the set of all functions analytic on D whosederivatives lie in the Bergman space A2(D). The Dirichlet norm is defined by

∥∥f

∥∥2D =

∣∣f(0)

∣∣2 +

∫

D

∣∣f ′(z)

∣∣2dA(z). (2.8)

If ϕ is a univalent self-map ofD, then Cϕ is bounded onD [2, page 18]. Also, the area formula[1, page 36], shows that

∥∥Cϕf

∥∥2D =

∣∣(

foϕ)

(0)∣∣2 +

∫

D

∣∣f ′(z)

∣∣2nϕ(z)dA(z), (2.9)

where nϕ(z) is, as usual, the counting function defined as the cardinality of the set {w ∈ D :ϕ(w) = z}.


In the next theorem, we characterize all convex composition operators Cϕ on Dsatisfying Δϕ ≥ 0. Note that we cannot use Proposition 1.3 for the Dirichlet space, thanksto the fact that in general the positive powers of Cϕ are not uniformly bounded on the zi’s.

Theorem 2.5. If Cϕ is convex on the Dirichlet space D, then Δϕ ≥ 0 if and only if Cϕ is an isometry.

Proof. One implication is clear. Suppose that Δϕ is a positive operator, and take T = Cϕ inProposition 2.2. Since the identity function is in the subspace M = {f ∈ D : f(0) = 0}, weconclude that ϕ(0) = 0. Thus, in light of (2.9), to show that Cϕ is an isometry it is sufficient toprove that

∫

D

∣∣f ′(z)

∣∣(

1 − nϕ)

(z)dA(z) = 0, ∀f ∈ D. (2.10)

Let f be any function in the Dirichlet space D. Then

0 ≤⟨(


)(

f)

, f⟩

=∫

D

∣∣f ′(z)

∣∣2(nϕ2 − 2nϕ + 1

)

(z)dA(z). (2.11)

Furthermore,

0 ≤ 〈Δϕf, f〉 =∫

D

∣∣f ′(z)

∣∣2(nϕ − 1

)

(z)dA(z). (2.12)

By summing up these two relations we get

∫

D

∣∣f ′(z)

∣∣2(nϕ2 − nϕ

)

(z)dA(z) ≥ 0. (2.13)

But nϕ2(z) ≤ nϕ(z), and so

∫

D

∣∣f ′(z)

∣∣2(nϕ2 − nϕ

)

(z)dA(z) = 0, ∀f ∈ D. (2.14)

This, in turn, implies that nϕ2(z) = nϕ(z) almost everywhere. Substituting this in (2.11), andthen considering (2.12) the assertion will be completed.

Observe that if ϕ(0) = 0, nϕ2 − 2nϕ + 1 ≥ 0 almost everywhere, and Cϕ is bounded on Dthen it is convex. Indeed,

⟨(


)

f, f⟩

=∫

D

∣∣f ′(z)

∣∣2(nϕ2 − 2nϕ + 1

)

(z)dA(z) ≥ 0. (2.15)

In the next theorem, we turn to the adjoint of a composition operator and givenecessary and sufficient conditions under which a convex operator C∗

ϕ is an isometry.


Theorem 2.6. Let ϕ be an analytic self-map ofD with ϕ(0) = 0. If C∗ϕ is a convex operator onH

2(β),then it is an isometry if and only if ΔC∗

ϕ≥ 0.

Proof. Suppose that ΔC∗ϕ

≥ 0, and assume that ϕ is not the identity or an ellipticautomorphism. By the Denjoy-Wolff theorem ϕn converges uniformly to zero on compactsubsets of D [1], and so for every z ∈ D,

limn→∞

∥∥Kϕn(z)

∥∥ = ‖K0‖. (2.16)

Proposition 1.2 coupled with the fact that C∗nϕ Kz = Kϕn(z) implies that for all z ∈ D and all

nonnegative integers n,

∥∥Kϕn(z)

∥∥2 ≥ n

(∥∥Kϕ(z)

∥∥2 − ‖Kz‖2

)

+ ‖Kz‖2. (2.17)

Furthermore, the positivity of ΔT shows that ‖Kϕ(z)‖ ≥ ‖Kz‖. Thus, in light of (2.16) and(2.17) we conclude that ‖Kz‖ = ‖Kϕ(z)‖ for all z ∈ D, and so ‖Kz‖ = ‖Kϕn(z)‖ for everypositive integer n. Consequently, ‖Kz‖ = ‖K0‖ for all z ∈ D. It follows that

1 = ‖K0‖2 = ‖Kz‖2 = k(

|z|2)

= 1 +∞∑

j=1

(

|z|2)j

β(

j)2

, for z ∈ D. (2.18)

This contradiction shows that ϕ is the identity or an elliptic automorphism. Thus, there is aθ ∈ [0, 2π) so that ϕ(z) = eiθz for all z ∈ D. Now, if ω ∈ D then

C∗ϕKω(z) = Kϕ(ω)(z) = k

(

ϕ(ω)z)

= Kω

(

e−iθz)

= Kω

(

ϕ−1(z))

= Cϕ−1Kω(z). (2.19)

It follows that C∗ϕ = Cϕ−1 . But it is easily seen that ‖Cϕ−1f‖ = ‖f‖ for every f ∈ H2(β).

Hence, C∗ϕ is an isometry. The converse is obvious.

3. Multiplication Operators

This section deals with convex multiplication operators on a weighted Hardy space. Recallthat a multiplier ofH2(β) is an analytic function ϕ onD such that ϕH2(β) ⊆ H2(β). The set ofall multipliers ofH2(β) is denoted byM(H2(β)). It is known thatM(H2(β)) ⊆ H∞. In fact, ifϕ ∈M(H2(β)) and f is the constant function 1 then for every positive integer n and for everyz ∈ Dwe have

∣∣ϕ(z)

∣∣ =

∣∣∣

⟨

Mnϕf,Kz

⟩∣∣∣

1/n ≤∥∥∥Mn

ϕf∥∥∥

1/n‖Kz‖1/n ≤ ∥

∥Mϕ

∥∥‖Kz‖1/n. (3.1)

Now, letting n → ∞, we conclude that ϕ is bounded. This coupled with the fact that ϕ ∈H2(β) implies that ϕ ∈ H∞. If ϕ is a multiplier, then the multiplication operatorMϕ, definedbyMϕf = ϕf , is bounded onH2(β). Also note that for each λ ∈ D,M∗

ϕKλ = ϕ(λ)Kλ.


In what follows, the operator Mϕ is assumed to be convex. First, we present anexample of a nonisometric convex multiplication operator T with ΔT ≥ 0.

Example 3.1. Consider the weighted Hardy spaceH2(β) with weight sequence (β(n))n givenby β(n) = n + 1. Define the mapping ϕ on D by ϕ(z) = z2. Obviously, Mϕ is bounded.Furthermore, it is easy to see that for every nonnegative integer k,

∥∥∥M2

ϕzk∥∥∥

2 − 2∥∥∥Mϕz

k∥∥∥

2+∥∥∥zk

∥∥∥

2> 0,

∥∥∥Mϕz

k∥∥∥ >

∥∥∥zk

∥∥∥.

(3.2)

Consequently,Mϕ is convex but not an isometry. Besides, ΔMϕ is a positive operator.

Theorem 3.2. LetH∞ consist of all multipliers ofH2(β), and let ϕ ∈ H∞ be such that ‖ϕ‖∞ ≤ 1. IfT =Mϕ or T =M∗

ϕ then T ∗ΔTT ≥ ΔT ≥ 0 if and only if T is an isometry.

Proof. Suppose that T is Mϕ or M∗ϕ and T ∗ΔTT ≥ ΔT ≥ 0. Define the linear mapping S :

H∞ → B(H2(β)) by S(ψ) = Mψ . An application of the closed graph theorem implies that Sis bounded. Therefore, there is c > 0 such that for all ψ ∈ H∞,

∥∥Mψ

∥∥ ≤ c∥∥ψ∥∥∞. (3.3)

It follows that for every f ∈ H2(β) and every nonnegative integer n,

∥∥∥Mn

ϕf∥∥∥ ≤ c∥∥ϕn∥∥∞

∥∥f

∥∥ ≤ c∥∥f∥∥. (3.4)

Thus, supn≥0‖Mnϕf‖ < ∞ for every f ∈ H2(β). Since ‖M∗

ψ‖ = ‖Mψ‖ for all ψ ∈ H∞, by asimilar method one can show that supn≥0‖M∗n

ϕ f‖ < ∞ for all f ∈ H2(β). Therefore, the resultfollows from Proposition 1.3.

Example 3.3. LetH be the Bergman space or the Hardy space and let T beMϕ or its adjoint onH. It is well known thatM(H) = H∞. So if ϕ is a multiplier with ‖ϕ‖∞ ≤ 1, then by applyingthe preceding theorem, we observe that T ∗ΔTT ≥ ΔT ≥ 0 if and only if T is an isometry.

We remark herein that if ϕ(z) = z and T =Mϕ on the Dirichlet spaceD, then it is easilyseen that T ∗ΔTT ≥ ΔT ≥ 0 but T is not an isometry.

Acknowledgments

The authors would like to thank Dr. Faghih Ahmadi for her assistance and the referee for anumber of helpful comments and suggestions. This research was in part supported by a grantno. (88-GR-SC-27) from Shiraz University Research Council.


References

[1] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies inAdvanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995.

[2] J. H. Shapiro, Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics,Springer, New York, NY, USA, 1993.

[3] R. F. Allen and F. Colonna, “Isometries and spectra of multiplication operators on the Bloch space,”Bulletin of the Australian Mathematical Society, vol. 79, no. 1, pp. 147–160, 2009.

[4] R. F. Allen and F. Colonna, “On the isometric composition operators on the Bloch space in Cn,” Journalof Mathematical Analysis and Applications, vol. 355, no. 2, pp. 675–688, 2009.

[5] F. Bayart, “Similarity to an isometry of a composition operator,” Proceedings of the AmericanMathematical Society, vol. 131, no. 6, pp. 1789–1791, 2003.

[6] B. J. Carswell and C. Hammond, “Composition operators with maximal norm on weighted Bergmanspaces,” Proceedings of the American Mathematical Society, vol. 134, no. 9, pp. 2599–2605, 2006.

[7] B. A. Cload, “Composition operators: hyperinvariant subspaces, quasi-normals and isometries,”Proceedings of the American Mathematical Society, vol. 127, no. 6, pp. 1697–1703, 1999.

[8] F. Colonna, “Characterisation of the isometric composition operators on the Bloch space,” Bulletin ofthe Australian Mathematical Society, vol. 72, no. 2, pp. 283–290, 2005.

[9] S. Li and S. Stevic, “Weighted composition operators from H∞ to the Bloch space on the polydisc,”Abstract and Applied Analysis, vol. 2007, Article ID 48478, 13 pages, 2007.

[10] S. Li and S. Stevic, “Weighted composition operators from Zygmund spaces into Bloch spaces,”Applied Mathematics and Computation, vol. 206, no. 2, pp. 825–831, 2008.

[11] S. Ohno and R. Zhao, “Weighted composition operators on the Bloch space,” Bulletin of the AustralianMathematical Society, vol. 63, no. 2, pp. 177–185, 2001.

[12] J. H. Shapiro, “What do composition operators know about inner functions?” Monatshefte furMathematik, vol. 130, no. 1, pp. 57–70, 2000.

[13] S. Stevic, “Essential norms of weighted composition operators from the α-Bloch space to a weighted-type space on the unit ball,” Abstract and Applied Analysis, vol. 2008, Article ID 279691, 11 pages, 2008.

[14] S. Stevic, “Norms of some operators from Bergman spaces to weighted and Bloch-type spaces,”Utilitas Mathematica, vol. 76, pp. 59–64, 2008.

[15] S. Stevic, “Norm of weighted composition operators from Bloch space to H∞μ on the unit ball,” Ars

Combinatoria, vol. 88, pp. 125–127, 2008.[16] S. I. Ueki and L. Luo, “Compact weighted composition operators and multiplication operaors

between Hardy spaces,” Abstract and Applied Analysis, vol. 2008, Article ID 196498, 12 pages, 2008.[17] E. A. Nordgren, “Composition operators,” Canadian Journal of Mathematics, vol. 20, pp. 442–449, 1968.[18] H. Schwarz, Composition operators on Hp, Ph.D. thesis, University of Toledo, Toledo, Ohio, USA, 1969.[19] F. Forelli, “The isometries ofHp,” Canadian Journal of Mathematics, vol. 16, pp. 721–728, 1964.[20] J. A. Cima and W. R. Wogen, “On isometries of the Bloch space,” Illinois Journal of Mathematics, vol.

24, no. 2, pp. 313–316, 1980.[21] R. J. Fleming and J. E. Jamison, Isometries on Banach Spaces: Function Spaces, vol. 129 of Chapman &

Hall/CRCMonographs and Surveys in Pure and AppliedMathematics, Chapman&Hall/CRC, Boca Raton,Fla, USA, 2003.

[22] W. Hornor and J. E. Jamison, “Isometries of some Banach spaces of analytic functions,” IntegralEquations and Operator Theory, vol. 41, no. 4, pp. 410–425, 2001.

[23] M. J. Martın and D. Vukotic, “Isometries of the Bloch space among the composition operators,”Bulletin of the London Mathematical Society, vol. 39, no. 1, pp. 151–155, 2007.

[24] L. J. Patton and M. E. Robbins, “Composition operators that are m-isometries,” Houston Journal ofMathematics, vol. 31, no. 1, pp. 255–266, 2005.

[25] S. Stevic and S. I. Ueki, “Isometries of a Bergman-Privalov-type space on the unit ball,” DiscreteDynamics in Nature and Society, vol. 2009, Article ID 725860, 16 pages, 2009.

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