Universal sequences of composition operators

Charpentier, Stéphane; Mouze, Augustin

doi:10.48550/arxiv.2111.05165

Cited by 2 publications

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References 14 publications

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“…S K , K ⊂ C \ D with connected complement). Let us observe that the sequences (T K ̺,n ) n are universal sequences of composition operators that fit within the framework of the recent paper [13]. Further, when I is not countable, standard examples of universal families (T i ) i∈I are given by semigroups.…”

Section: Introductionmentioning

confidence: 86%

Abel universal series

Charpentier¹,

Mouze²

2022

Preprint

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Given a sequence ̺ = (r n ) n ∈ [0, 1) tending to 1, we consider the set U A (D, ̺) of Abel universal series consisting of holomorphic functions f in the open unit disc D such that for any compact set K included in the unit circle T, different from T, the set {z →. We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at 0 are dense in C(K) for any compact set K ⊂ T different from T. Moreover we prove that the class of Abel universal series is not invariant under the action of the differentiation operator. Finally an Abel universal series can be viewed as a universal vector of the sequence of dilation operators T n : f → f (r n •) acting on H(D). Thus we study the dynamical properties of (T n ) n such as the multi-universality and the (common) frequent universality. All the proofs are constructive.

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Section: Introductionmentioning

confidence: 86%

Abel universal series

Charpentier¹,

Mouze²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…S , ⊂ C \ D with connected complement). Let us observe that the sequences ( , ) are universal sequences of composition operators that fit within the framework of the recent paper [16]. Further, when is not countable, standard examples of universal families ( ) ∈ are given by semigroups.…”

mentioning

confidence: 97%

Abel universal functions

Charpentier¹,

Mouze²

2022

Can. J. Math.-J. Can. Math.

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Given a sequence = ( ) ∈ [0, 1) tending to 1, we consider the set U (D, ) of Abel universal functions consisting of holomorphic functions in the open unit disc D such that for any compact set included in the unit circle T, different from T, the set { ↦ → ( •) | : ∈ N} is dense in the space C ( ) of continuous functions on . It is known that the set U (D, ) is residual in (D). We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it is not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at 0 are dense in C ( ) for any compact set ⊂ T different from T. Moreover we prove that the class of Abel universal functions is not invariant under the action of the differentiation operator. Finally an Abel universal function can be viewed as a universal vector of the sequence of dilation operators : ↦ → ( •) acting on (D). Thus we study the dynamical properties of ( ) such as the multi-universality and the (common) frequent universality. All the proofs are constructive.In complex function theory, it is of great interest to distinguish and study classes of holomorphic functions with a regular boundary behaviour. Regular boundary behaviour at a point of T can mean, for instance, convergence, Cesàro summability or Abel summability of the Taylor expansion at this point. We recall that a function in (D) is Abel summable at ∈ T if the quantity ( ), ∈ [0, 1), has a finite limit as → 1. Note that in the latter case, the cluster set of along the radius { : ∈ [0, 1)} is reduced to a single value. Yet, it is now well understood that non-regularity is a generic behaviour. We say that a property is generic in a Baire space if the set of those ∈ which satisfy

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Universal sequences of composition operators

Cited by 2 publications

References 14 publications

Abel universal series

Abel universal series

Abel universal functions

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