Large subspaces of compositionally universal functions with maximal cluster sets

Bernal–González, Luis; Calderón-Moreno, M. C.; Prado-Bassas, J. A.

doi:10.1016/j.jat.2011.10.005

Cited by 9 publications

(3 citation statements)

References 25 publications

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“…In 2011, Menet [220] has proved the assertion if X is a Fréchet space admitting a continuous norm, and the same author [221] has recently shown the same result for Fréchet spaces admitting a continuous seminorm p with codim(ker p) = ∞. In [73] and [80], respectively, the combined properties of maximality of cluster sets in each boundary point with either universality of Taylor series or compositional universality in H(G) considered there (where G is a Jordan domain in C) are proved to give spaceability. As a related result, in [79] the authors provided a large family of classical operators (for instance, differential or composition operators) T : H(G) → H(G) (with G a domain in C) which satisfied that, for any subset A ⊂ G that is not relatively compact in G, the set {f ∈ H(G) : (T f)(A) = C} is spaceable.…”

Section: Hypercyclicity and Spaceabilitymentioning

confidence: 97%

“…For instance, Bernal, Bonilla, Calderón, and Prado-Bassas [73] showed in 2009 that the family of universal Taylor series f having maximal (i.e., equal to C ∞ ) cluster set C(f, γ, ξ) at each ξ ∈ ∂D along any curve γ ⊂ D tending to ∂D whose closure does not contain ∂D is dense-lineable in H(D). Incidentally, in [80] it has been shown the maximal-dense-lineability in H(G) of the class of functions f ∈ U ((C ϕ n )) satisfying that boundary property, where G is a Jordan domain and (C ϕ n ), is the sequence of composition operators generated by adequate holomorphic self-mappings ϕ n : G → G.…”

Section: Hypercyclity and Dense-lineability An Extreme Case Of Lineamentioning

confidence: 99%

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Linear subsets of nonlinear sets in topological vector spaces

Bernal–González

Pellegrino

Seoane‐Sepúlveda

2013

Bull. Amer. Math. Soc.

196

149

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For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability. 109 5. Some remarks and conclusions. General techniques 113 About the authors 117 Acknowledgments 118 References 118Theorem 1.2 (Gurariy, 1966 [178]). The set of continuous nowhere differentiable functions on [0, 1] is lineable.Let us also recall that, although the set of everywhere differentiable functions in R is, in itself, an infinite dimensional vector space, in 1966 Gurariy obtained the following analogue of Theorem 1.1.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LINEAR SUBSETS OF NONLINEAR SETS IN TOPOLOGICAL VECTOR SPACES 73 Theorem 1.3 (Gurariy, 1966 [178]). The set of everywhere differentiable functions on [0, 1] is not spaceable in C[0, 1]. Also, there exist closed infinite dimensional subspaces of C[0, 1] all of whose members are differentiable on (0, 1).Somehow, what we are seeing is that what one could expect to be an isolated phenomenon can actually even have a nice algebraic structure (in the form of infinite dimensional subspaces). Unfortunately, and as we mentioned above, the Baire category theorem cannot be employed in the search for large subspaces such as the ones mentioned in the previous results. Let us now provide a more formal and complete definition for the above concepts and some other ones. Definition 1.4 (Lineability and spaceability [22,181,253]). Let X be a topological vector space and M a subset of X. Let μ be a cardinal number.(1) M is said to be μ-lineable (μ-spaceable) if M ∪ {0} contains a vector space (resp. a closed vector space) of dimension μ. At times, we shall refer to the set M as simply lineable or spaceable if the existing subspace is infinite dimensional.(2) We also let λ(M ) be the maximum cardinality (if it exists) of such a vector space. 1 (3) When the above linear space can be chosen to be dense in X, we shall say that M is μ-dense-lineable.Moreover, Bernal introduced in [69] the notion of maximal-lineable (and those of maximal-dense-lineable and maximal-spaceable), meaning that, when keeping the above notation, the dimension of the existing linear space equals dim(X). Besides asking for linear spaces, one could also study other structures, such as algebrability and some related ones, which were presented in [25,27,253]. Definition 1.5. Given a Banach algebra A, a subset B ⊂ A, and two cardinal numbers α and β, we say that:

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Section: Hypercyclicity and Spaceabilitymentioning

confidence: 97%

Section: Hypercyclity and Dense-lineability An Extreme Case Of Lineamentioning

confidence: 99%

Linear subsets of nonlinear sets in topological vector spaces

Bernal–González

Pellegrino

Seoane‐Sepúlveda

2013

Bull. Amer. Math. Soc.

196

149

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“…The topic has been extensively continued in various directions later on, see for e.g. [1,3,4,5,6,13,14,15,17,22].…”

Section: Introductionmentioning

confidence: 99%

Universal sequences of composition operators

Charpentier¹,

Mouze²

2021

Preprint

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Let G and Ω be two planar domains. We give necessary and sufficient conditions on a sequence (φ n ) of eventually injective holomorphic mappings from G to Ω for the existence of a function f ∈ H(Ω) whose orbit under the composition by (φ n ) is dense in H(G). This extends a result of the same nature obtained by Grosse-Erdmann and Mortini when G = Ω. An interconnexion between the topological properties of G and Ω appears. Further, in order to exhibit in a natural way holomorphic functions with wild boundary behaviour on planar domains, we study a certain type of universality for sequences of continuous mappings from a union of Jordan curves to a domain.

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Universal Sequences of Composition Operators

Charpentier

Mouze

2023

J Geom Anal

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Large subspaces of compositionally universal functions with maximal cluster sets

Cited by 9 publications

References 25 publications

Linear subsets of nonlinear sets in topological vector spaces

Linear subsets of nonlinear sets in topological vector spaces

Universal sequences of composition operators

Universal Sequences of Composition Operators

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