U-Operators

Bernal–González, Luis; Prado-Tendero, J. A.

doi:10.1017/s1446788700015561

Cited by 5 publications

(3 citation statements)

References 24 publications

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“…Remark 3.13. The conclusion of Theorem 3.12 can be also deduced by applying Lemma 5.1 in [7], which in turn is a modified version of Theorem 3.1 in [6], to appropriate sequences of linear continuous mappings.…”

Section: Simultaneous Approximations and Universal Seriesmentioning

confidence: 96%

Real Taylor shifts and simultaneous approximation

Klaja

Mouze

2018

Mathematische Nachrichten

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We are interested in the Taylor shift operator acting on the space of infinitely differentiable functions. In particular if we choose a countable family of centers of Taylor expansion, we prove that the associated family of real Taylor shifts fulfills the approximation of any given family of infinitely differentiable functions with a common subsequence of iterates applied on a common vector. We obtain similar conclusions in the context of universal series improving recent statements. Finally we introduce the notion of doubly universal Taylor shift. All these results give new and natural examples of disjoint universality.

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Section: Simultaneous Approximations and Universal Seriesmentioning

confidence: 96%

Real Taylor shifts and simultaneous approximation

Klaja

Mouze

2018

Mathematische Nachrichten

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“…Fortunately, lineability properties hold for the families of universal vectors of sequences of mappings under not too strong restrictions on the spaces and the mappings. In the following theorem a number of related results, starting from 1999, due to Bernal, Calderón, and Prado-Tendero [61,74,85] are collected. Theorem 3.2.…”

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

confidence: 99%

“…For instance, the sets of functions which are respectively universal for the sequences (C ϕ n ), (C ψ n ), (Φ n (D)) considered in Section 3.1-under the restrictions imposed there-are dense-lineable (see also [85] for combinations of composition operators with other kinds of operators). In particular, U((τ a n )) is dense-lineable provided that (a n ) is an unbounded sequence in C. With more sophisticated techniques it can be proved that for a fixed large set A ⊂ C (belonging to a wide class) the family of Birkhoff-universal entire functions presenting rapid decay along A is dense-lineable; see [76] for details.…”

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.

197

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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On Bounded Universal Functions

Vogt

2012

Comput. Methods Funct. Theory

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U-Operators

Cited by 5 publications

References 24 publications

Real Taylor shifts and simultaneous approximation

Real Taylor shifts and simultaneous approximation

Linear subsets of nonlinear sets in topological vector spaces

On Bounded Universal Functions

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