Characterisation of the isometric composition operators on the Bloch space

Colonna, Flavia

doi:10.1017/s0004972700035073

Cited by 37 publications

(41 citation statements)

References 5 publications

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“…Then ψ = ϕ • L α is an analytic function of ∆ into itself such that ψ(0) = 0 and β ψ = 1. As mentioned above, by Theorem 5 of[8], C ψ preserves the Bloch norm, i.e. f • ψ = f for all f ∈ B.…”

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confidence: 82%

“…In Theorem 5 of [8], the second author proved that every analytic function ϕ of ∆ into itself mapping 0 to 0 and having Bloch norm equal to one induces an isometric composition operator on the Bloch space.…”

Section: One-point Sets Of Bloch and Normal Functions 521mentioning

confidence: 99%

“…Analytic functions mapping ∆ into itself with semi-norm one and having 0 as a fixed point play a special role in composition operator theory [8]. A useful reference on composition operators on spaces of analytic functions is [9].…”

Section: Introductionmentioning

confidence: 99%

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Preimages of One-Point Sets of Bloch and Normal Functions

Cohen

Colonna

2006

MedJM

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We call a sequence {zn} in the open unit disk ∆ almost thin if lim sup n→∞ Π(zn) = 1, where Π(zn) is the product of the pseudo-hyperbolic dis-tances between zn and the other elements in the sequence. We call the sequence thick if lim n→∞ Π(zn) = 0. In this paper, we use elementary complex variables methods to prove a sharper form of a characterization of the symbols of the isometric composition operators on the Bloch space proved by Martín and Vukotic in terms of the preimages of one-point sets. Specifically, we show that a bounded analytic function ϕ from ∆ into itself has Bloch semi-norm equal to one (which is the largest possible) if and only if it is a conformal automorphism of ∆ or if the preimage of each point of ∆ contains a sequence along which the hyperbolic derivative of ϕ approaches one (condition (M )). In particular, the preimage of each point of ∆ is an almostthin sequence. Using similar techniques we also show that a bounded analytic function f is in the little Bloch space if and only if for each a ∈ C such that f −1 (a) = {zn : n ∈ N} is infinite lim n→∞ (1 − |zn| 2 )|f (zn)| = 0.Consequently, an infinite Blaschke product whose left composition by a conformal automorphism of ∆ is always a Blaschke product belongs to the little Bloch space if and only if the preimage of every point in ∆ is a thick sequence. We give an example to show that this characterization does not extend to unbounded Bloch functions. We also prove an analogous characterization of the little normal functions. Finally, we give a sufficient condition on a holomorphic self-map of the polydisk ∆ n to be the symbol of an isometric composition operator on the Bloch space of ∆ n and discuss a higher-dimensional analogue of condition (M ). Mathematics Subject Classification (2000). Primary: 30D45, 30D50; Secondary: 32A18.

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confidence: 82%

Section: One-point Sets Of Bloch and Normal Functions 521mentioning

confidence: 99%

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Preimages of One-Point Sets of Bloch and Normal Functions

Cohen

Colonna

2006

MedJM

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“…In [10] and [2], it was shown that there is a large class of isometric composition operators on the Bloch space of the disk and more generally on bounded homogeneous domains that have the disk as a factor. We shall now prove that there are no isometries among the composition operators between the Hardy space H ∞ and the Bloch space of the disk.…”

Section: Ieotmentioning

confidence: 99%

Weighted Composition Operators from H ∞ to the Bloch Space of a Bounded Homogeneous Domain

Allen

Colonna

2010

Integr. Equ. Oper. Theory

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Let D be a bounded homogeneous domain in C n . In this paper, we study the bounded and the compact weighted composition operators mapping the Hardy space H ∞ (D) into the Bloch space of D. We characterize the bounded weighted composition operators, provide operator norm estimates, and give sufficient conditions for compactness. We prove that these conditions are necessary in the case of the unit ball and the polydisk. We then show that if D is a bounded symmetric domain, the bounded multiplication operators from H ∞ (D) to the Bloch space of D are the operators whose symbol is bounded.Mathematics Subject Classification (2010). Primary 47B38; Secondary 32A18, 30D45.

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“…In [3] The isometric composition operators on the Bloch spaces in the unit disk were discussed by Martín and Vukotić [7], Colonna [8], Allen and Colonna [9,10], Li and Zhou [11]. The same problems were studied on the Bloch spaces in the unit polydisk by Cohen and Colonna [12], in unit ball by Li [13], and Li and Ruan [14].…”

Section: Introductionmentioning

confidence: 99%