The specification property in linear dynamics

Abstract

[EN] The dynamics of linear operators, namely linear dynamics, is mainly concerned with the behaviour of iterates of linear transformations. Hypercyclicity is the study of linear operators that possess a dense orbit. Although the first examples of hypercyclic operators are due to G. D. Birkhoff (in 1929), G. R. MacLane (in 1952) and S. Rolewicz (in 1969), we can date the birth of the linear dynamics in 1982 with the unpublished PhD thesis of C. Kitai. Since then, many mathematicians have contributed to the development of this flourishing new area of the analysis. Linear dynamics connects functional analysis and dynamics. As for the classical dynamical systems, one can study the dynamics of linear operators from a topological point of view. In this context, we state that an operator has the specification property (SP). Precisely, the aim of this PhD thesis is to study the specification property on linear dynamical systems. A continuous map on a compact metric space satisfies the specification property if one can approximate pieces of orbits by a single periodic orbits with a certain uniformity. This Doctoral dissertation is a compendium of articles on the specification property. It is structured in four parts preceded by a chapter which introduces the notation, definitions and the basic results that will be needed throughout the thesis. The shift operators on sequence spaces constitute one of the most important test ground for discrete linear dynamical systems. Due to its simple structure, every time you introduce a new property in linear dynamics it is common to check it on weighted shifts operators. It is for this reason that the first part of this research work is devoted to study the specification property for unilateral and bilateral backward shift operators on weighted l^p-spaces and the relationship with other dynamical properties. In Chapter 3 we extend the results on the SP to shift operators on separable sequence F-spaces. An F-space is a vector space that is endowed with an F-norm and that is complete under the induced metric. The notion of an F-norm has the advantage that one can largely argue as if one was working in a Banach space. One need to be aware of the fact that the positive homogeneity of a norm is no longer available. The spaces l^p with 0