36 research outputs found
Strongly -supercyclic operators
In this paper, we are interested in the properties of a new class of
operators, recently introduced by Shkarin, called strongly -supercyclic
operators. This notion is stronger than -supercyclicity. We prove that such
operators have interesting spectral properties and give examples and
counter-examples answering some natural questions asked by Shkarin
Operators commuting with the Volterra operator are not weakly supercyclic
We prove that any bounded linear operator on for ,
commuting with the Volterra operator , is not weakly supercyclic, which
answers affirmatively a question raised by L\'eon-Saavedra and Piqueras-Lerena.
It is achieved by providing an algebraic flavored condition on an operator
which prevents it from being weakly supercyclic and is satisfied for any
operator commuting with .Comment: Submitted to LM
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Mini-Workshop: Hypercyclicity and Linear Chaos
The mini-workshop was devoted to the study of hypercyclic and chaotic operators within the wider framework of linear dynamical systems. Topics discussed included common hypercyclic vectors; hypercyclic and supercyclic subspaces; extensions of hypercyclicity like Cesàro-, Faber-, and disjoint hypercyclicity; hypercyclic N -tuples and hypercyclic direct sums; hypercyclic C0 -semigroups and hypercyclic polynomials; hypercyclicity in nonmetrizable spaces; weak supercyclicity; hypercyclic composition operators; and the influence of the norms kT n k on the dynamical behaviour of T . A list of open problems is included in the report
Simultaneous universality
In this paper, the notion of simultaneous universality is introduced, concerning operators having orbits that simultaneously approximate any given vector. This notion is related to the well known concepts of universality and disjoint universality. Several criteria are provided, and several applications to specific operators or sequences of operators are performed, mainly in the setting of sequence spaces or spaces of holomorphic functions.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y CompetitividadDeutsche Forschungsgemeinschaf
Common hypercyclic vectors for families of operators
We provide a criterion for the existence of a residual set of common hypercyclic vectors for an uncountable family of hypercyclic operators which is based on a previous one given by Costakis and Sambarino. As an application, we get common hypercyclic vectors for a particular family of hypercyclic scalar multiples of the adjoint of a multiplier in the Hardy space, generalizing recent results by Abakumov and Gordon and also Bayart. The criterion is applied to other specific families of operators
Hypercyclic subspaces in Fréchet spaces
In this note, we show that every infinite-dimensional separable Fr´echet space
admitting a continuous norm supports an operator for which there is an infinite-dimensional closed subspace consisting, except for zero, of hypercyclic vectors.
The family of such operators is even dense in the space of bounded operators when endowed with the strong operator topology. This completes earlier work of several authors.Plan Andaluz de Investigación (Junta de Andalucía)Dirección General de Enseñanza Superior (DGES). Españ