Hypercyclic operators on non-normable Fréchet spaces
type: Item
instance of: scholarly article
title: Hypercyclic operators on non-normable Fréchet spaces
zbMATH Open document ID: 0926.47011
published in: Journal of Functional Analysis
publication date: 1999-11-16T00:00:00Z
review text: An operator \(T\) on a locally convex space (lcs) \(E\) is called hypercyclic if \[ \text{Orb} (T,x):=\{x,Tx,T^2x,\ldots\} \] is dense in \(E\) for some \(x\in E\). In this case \(x\) is a hypercyclic vector for \(T.\) Hypercyclic vectors are of importance in connection with invariant subspaces. It is known that an operator lacks closed non-ntrivial invariant subsets iff every non-zero vector is hypercyclic. \textit{S. Ansari} [J. Funct. Anal. 148, 384-390 (1997; Zbl 0898.47019)] proved that every infinite dimensional separable Banach space admits a hypercyclic operator. A corollary of the main result of this paper asserts that every infinite dimensional separable Frechet space admits a hypercyclic operator. The proof of this result contains a gap in the case of nonnormable Frechet space. The main purpose of this paper is to show that the result claimed by Ansari does hold for arbitrary Frechet spaces. An example of a complete separable lcs (more precisely, an inductive limit of Banach spaces) for which there is no hypercyclic operator defined on it is also given.
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47A16
zbMATH DE Number: 1268057
zbMATH Keywords: Frechet space
zbMATH Keywords: hypercyclic operator
zbMATH Keywords: hypercyclic vectors
zbMATH Keywords: invariant subspaces
author: José Bonet
author: A. Peris
reviewed by: D. Zarnadze
MaRDI profile type: Publication
full work available at URL: jfan.1998.3315
OpenAlex ID: W2026691296
cites work: Existence of hypercyclic operators on topological vector spaces
cites work: An operator without invariant subspaces on a nuclear Fréchet space
cites work: On hypercyclic operators on Banach spaces
cites work: A Projective Description of Weighted Inductive Limits
cites work: Spaces of operators between Fréchet spaces
cites work: On the invariant subspace problem for Banach spaces
cites work: Countable inductive limits of Fréchet algebras
cites work: Universal Vectors for Operators on Spaces of Holomorphic Functions
cites work: Operators with dense, invariant, cyclic vector manifolds
cites work: Q5183990
cites work: Q5182704
cites work: Q5569793
cites work: Q4020232
cites work: Q4722576
cites work: A Solution to the Invariant Subspace Problem on the Space l 1
cites work: Hypercyclic Weighted Shifts
DOI: 10.1006/JFAN.1998.3315
description: scientific article; zbMATH DE number 1268057