The Hardy Class of Geometric Models and the Essential Spectral Radius of Composition Operators

Poggi‐Corradini, Pietro

doi:10.1006/jfan.1996.2978

Cited by 15 publications

(23 citation statements)

References 7 publications

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“…Theorem 4.7 below establishes necessity when ϕ is analytic on the closed disk. Furthermore, Pietro Poggi-Corradini has recently shown the sufficient condition of the Main Theorem is necessary when ϕ is univalent [19].…”

Section: Koenigs Eigenfunctions 301mentioning

confidence: 99%

“…The existence of Riesz composition operators that are not power compact is established in [2]. A characterization of Riesz composition operators with univalent symbol is contained in [19].…”

Section: Proof Suppose Thatmentioning

confidence: 99%

“…Over a century ago, Koenigs ([15]) showed that if f : U → C is a nonconstant holomorphic solution to Schroeder's functional equation, While the local study of Koenigs eigenfunctions plays a fundamental role in complex dynamics, the study of how global properties of ϕ influence those of σ is just beginning (see [2,18,19,28]). In this paper we obtain results showing how properties of ϕ influence the growth and mean growth of σ(z) as |z| → 1 − , mean growth being measured by Hardy-space membership.…”

Section: Introductionmentioning

confidence: 99%

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Mean growth of Koenigs eigenfunctions

Bourdon

Shapiro

1997

J. Amer. Math. Soc.

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Section: Koenigs Eigenfunctions 301mentioning

confidence: 99%

Section: Proof Suppose Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Mean growth of Koenigs eigenfunctions

Bourdon

Shapiro

1997

J. Amer. Math. Soc.

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“…Added in Proof. This question has been recently answered in [7]: σ satisfies the (NT S) condition if and only if C φ is a Riesz operator. 6.…”

Section: The No Twisted Sector Conditionmentioning

confidence: 99%

Hardy spaces and twisted sectors for geometric models

Poggi‐Corradini

1996

Trans. Amer. Math. Soc.

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Abstract. We study the one-to-one analytic maps σ that send the unit disc into a region G with the property that λG ⊂ G for some complex number λ, 0 < |λ| < 1. These functions arise in iteration theory, giving a model for the self-maps of the unit disk into itself, and in the study of composition operators as their eigenfunctions. We show that for such functions there are geometrical conditions on the image region G that characterize their rate of growth, i.e. we prove that σ ∈ p<∞ H p if and only if G does not contain a twisted sector. Then, we examine the connection with composition operators, and further investigate the no twisted sector condition. Finally, in the Appendix, we give a different proof of a result of J. Shapiro about the essential norm of a composition operator.

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“…Bourdon and Shapiro [4] and Poggi-Corradini [15] studied the range domains Ω of univalent Koenigs functions and found that the number h(Ω) can be described in terms of the essential norm of the associated composition operators.…”

Section: Introductionmentioning

confidence: 99%