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1
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Topological tensor products of a Fréchet-Schwartz space and a Banach space

100%
Peris A.
Studia Mathematica
|
1993
|
tom 106
|
nr 2
189-196
EN
We exhibit examples of countable injective inductive limits E of Banach spaces with compact linking maps (i.e. (DFS)-spaces) such that $E ⊗_{ε} X$ is not an inductive limit of normed spaces for some Banach space X. This solves in the negative open questions of Bierstedt, Meise and Hollstein. As a consequence we obtain Fréchet-Schwartz spaces F and Banach spaces X such that the problem of topologies of Grothendieck has a negative answer for $F ⨶_π X$. This solves in the negative a question of Taskinen. We also give examples of Fréchet-Schwartz spaces and (DFS)-spaces without the compact approximation property and with the compact approximation property but without the approximation property.
2
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Frequently hypercyclic semigroups

63%
Mangino E., Peris A.
Studia Mathematica
|
2011
|
tom 202
|
nr 3
227-242
EN
We study frequent hypercyclicity in the context of strongly continuous semigroups of operators. More precisely, we give a criterion (sufficient condition) for a semigroup to be frequently hypercyclic, whose formulation depends on the Pettis integral. This criterion can be verified in certain cases in terms of the infinitesimal generator of the semigroup. Applications are given for semigroups generated by Ornstein-Uhlenbeck operators, and especially for translation semigroups on weighted spaces of p-integrable functions, or continuous functions that, multiplied by the weight, vanish at infinity.
3
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Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces

51%
Bermúdez T., Bonilla A., Conejero J., Peris A.
Studia Mathematica
|
2005
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tom 170
|
nr 1
57-75
EN
Our aim in this paper is to prove that every separable infinite-dimensional complex Banach space admits a topologically mixing holomorphic uniformly continuous semigroup and to characterize the mixing property for semigroups of operators. A concrete characterization of being topologically mixing for the translation semigroup on weighted spaces of functions is also given. Moreover, we prove that there exists a commutative algebra of operators containing both a chaotic operator and an operator which is not a multiple of the identity and no multiple of which is chaotic. This gives a negative answer to a question of deLaubenfels and Emamirad.
4
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Linear dynamics of semigroups generated by differential operators

51%
Alberto Conejero J., Lizama C., Murillo-Arcila M., Peris A.
Open Mathematics
|
2017
|
tom 15
|
nr 1
745-767
EN
During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C0-semigroups of linear and continuous operators. We will review some of these notions and we will discuss basic properties of the dynamics of C0-semigroups. We will also study in detail the dynamics of the translation C0-semigroup on weighted spaces of integrable functions and of continuous functions vanishing at infinity. Using the comparison lemma, these results can be transferred to the solution C0-semigroups of some partial differential equations. Additionally, we will also visit the chaos for infinite systems of ordinary differential equations, that can be of interest for representing birth-and-death process or car-following traffic models.
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