Questions in linear recurrence: From the T ⊕ T -problem to lineability

Grivaux, Sophie; López-Martínez, Antoni; Peris, Alfred

Document type :

Pré-publication ou Document de travail

Title :

Questions in linear recurrence: From the T ⊕ T -problem to lineability

Author(s) :

Grivaux, Sophie [Auteur]

Laboratoire Paul Painlevé - UMR 8524 [LPP]
López-Martínez, Antoni [Auteur]
Peris, Alfred [Auteur]

English keyword(s) :

Linear dynamical systems
hypercyclicity
recurrence
quasi-rigidity
lineability.

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

We study, for a continuous linear operator T on an F-space X, when the direct sum operator T ⊕ T is recurrent on X ⊕ X. In particular: we establish the analogous notion, for recurrence, to that of (topological) weak-mixing ...
Show more >We study, for a continuous linear operator T on an F-space X, when the direct sum operator T ⊕ T is recurrent on X ⊕ X. In particular: we establish the analogous notion, for recurrence, to that of (topological) weak-mixing for transitivity/hypercyclicity, namely quasi-rigidity; and we construct a recurrent but not quasi-rigid operator on each separable infinite-dimensional Banach space, solving the T ⊕ T-recurrence problem in the negative way. The quasi-rigidity notion is closely related to the dense lineability of the set of recurrent vectors, and using similar conditions we study the lineability and dense lineability properties for the set of F-recurrent vectors, under very weak assumptions on the Furstenberg family F.Show less >

Language :

Anglais

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

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