Infinite-dimensional Frechet Space Research Articles

On a semigroup problem

If \begin{document}$ S $\end{document} is a semigroup in \begin{document}$ \mathbb{R}^n $\end{document} that is not separated by a linear functional, then it is known that the closure of \begin{document}$ S $\end{document} is a group. We investigate a similar statement in an infinite dimensional topological vector space \begin{document}$ X $\end{document} . We show that if \begin{document}$ X $\end{document} is an infinite dimensional Banach space, then there exists a semigroup \begin{document}$ S\subset X $\end{document} , not separated by the continuous functionals supported by the closed linear span of \begin{document}$ S $\end{document} , for which the closure of the semigroup is not a group. If \begin{document}$ X $\end{document} is an infinite dimensional Frechet space, then the closure of a semigroup that is not separated is always a group if and only if \begin{document}$ X $\end{document} is \begin{document}$ \mathbb{R}^{\omega} $\end{document} , the countably infinite direct product of lines. Other infinite dimensional topological vector spaces, such as \begin{document}$ \mathbb{R}^{\infty} $\end{document} , the countably infinite direct sum of lines, are discussed. The Semigroup Problem has applications to the study of certain dynamical systems, in particular for the construction of topologically transitive extensions of hyperbolic systems. Some examples are shown in the paper.

Sequential Analogues of the Lyapunov and Krein–Milman Theorems in Fréchet Spaces

In this paper we develop the theory of anti-compact sets we introduced earlier. We describe the class of Frechet spaces where anti-compact sets exist. They are exactly the spaces that have a countable set of continuous linear functionals. In such spaces we prove an analogue of the Hahn–Banach theorem on extension of a continuous linear functional from the original space to a space generated by some anti-compact set. We obtain an analogue of the Lyapunov theorem on convexity and compactness of the range of vector measures, which establishes convexity and a special kind of relative weak compactness of the range of an atomless vector measure with values in a Frechet space possessing an anti-compact set. Using this analogue of the Lyapunov theorem, we prove the solvability of an infinite-dimensional analogue of the problem of fair division of resources. We also obtain an analogue of the Lyapunov theorem for nonadditive analogues of measures that are vector quasi-measures valued in an infinite-dimensional Frechet space possessing an anti-compact set. In the class of Frechet spaces possessing an anti-compact set, we obtain analogues of the Krein–Milman theorem on extreme points for convex bounded sets that are not necessarily compact. A special place is occupied by analogues of the Krein–Milman theorem in terms of extreme sequences introduced in the paper (the so-called sequential analogues of the Krein–Milman theorem).

Existence Theorems in Linear Chaos

Chaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. In this survey paper, we treat questions of characterizing which of the spaces from a given class support a semigroup of prescribed shape satisfying a given topological ergodic property. In particular, we characterize countable inductive limits of separable Banach spaces that admit a hypercyclic operator, show that there is a non-mixing hypercyclic operator on a separable infinite dimensional complex Frechet space X if and only if X is non-isomorphic to the space ω of all sequences with coordinatewise convergence topology. It is also shown for any k ∈ N, any separable infinite dimensional Frechet space X non-isomorphic to ω admits a mixing uniformly continuous group {Tt}t∈Cn T of continuous linear operators and that there is no supercyclic strongly continuous operator semigroup {Tt}t≥0 on ω. We specify a wide class of Frechet spaces X, including all infinite dimensional Banach spaces with separable dual, such that there is a hypercyclic operator T on X for which the dual operator T′ is also hypercyclic. An extension of the Salas theorem on hypercyclicity of a perturbation of the identity by adding a backward weighted shift is presented and its various applications are outlined.

Existence and nonexistence of hypercyclic semigroups

In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinite-dimensional Banach space that is very different from-and considerably shorter than-the one recently given by Bermudez, Bonilla and Martinon. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite-dimensional Frechet space. This complements recent results due to Bes and Chan. Moreover, we discuss the Hypercyclicity Criterion for semigroups and we give an example of a separable infinite-dimensional locally convex space which supports no supercyclic strongly continuous semigroup of operators.

TRANSITIVE AND HYPERCYCLIC OPERATORS ON LOCALLY CONVEX SPACES

Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Frechet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space ). (2) The space of all test functions for distributions, which is also a complete direct sum of Frechet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Frechet space contains a dense hyperplane that admits no transitive operator.

On Symmetric Schauder Bases in Non-Archimedean Fréchet Spaces

It is proved that any infinite-dimensional non-archimedean Frechet space with a symmetric basis is isomorphic to c 0 or ?ℕ. A similar result is shown for homogeneous bases. It is also proved that any infinite-dimensional nuclear non-archimedean Frechet space with a subsymmetric basis is isomorphic to ?ℕ. In fact, much stronger results are obtained.

SOME RESULTS ON SOLVABILITY OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS IN LOCALLY CONVEX SPACES

Let be the class of sequentially complete locally convex spaces such that an existence theorem holds for the linear Cauchy problem , , with respect to functions . It is proved that if , then for an arbitrary set . It is also proved that a topological product of infinitely many infinite-dimensional Frechet spaces, each not isomorphic to , does not belong to .

Any Infinite-Dimensional Frechet Space Homeomorphic with its Countable Product is Topologically a Hilbert Space

Any Infinite-Dimensional Frechet Space Homeomorphic with its Countable Product is Topologically a Hilbert Space