Hypercyclic subspaces in omega
type: Item
instance of: scholarly article
title: Hypercyclic subspaces in omega
zbMATH Open document ID: 1094.47011
author: J. Bès
author: José Alberto Conejero
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-03-29T00:00:00Z
review text: A continuous linear operator \(T\) acting on a Fréchet space \(X\) is called hypercyclic if there exists a vector \(x\in X\) whose orbit Orb\((T,x)=\{x, T(x), T^2(x), \ldots\}\) is dense in \(X\). Such a vector \(x\) is called a hypercyclic vector for \(T\). A hypercyclic manifold for \(T\) is a dense, invariant subspace of \(X\) consisting entirely -- except for the origin -- of hypercyclic vectors for \(T\). A hypercyclic subspace for \(T\) is a closed, infinite-dimensional subspace of \(T\) consisting entirely -- except for the origin -- of hypercyclic vectors for \(T\). It is well-known that once an operator on a Fréchet space has a hypercyclic vector, the smallest manifold invariant for \(T\) containing that vector is a hypercyclic manifold [see \textit{P.~Bourdon}, Proc.\ Am. Math. Soc. 118, 845--847 (1993; Zbl 0809.47005); \textit{D.~A. Herrero}, J. Funct. Anal.\ 99, 179--190 (1991; Zbl 0758.47016); \textit{J.~Wengenroth}, Proc. Am. Math.\ Soc. 131, 1759--1761 (2003; Zbl 1039.47009)]. The situation for hypercyclic subspaces is different. Indeed, \textit{A.~Montes--Rodríguez} [Mich.\ Math. J. 43, 419--436 (1996; Zbl 0907.47023)] showed that no scalar multiple \(\lambda B\), \(| \lambda | >1\), of the backward shift \(B(x_1,x_2,x_3, \ldots)=(x_2,x_3,x_4, \ldots)\) on \(\ell^2\) has a hypercyclic subspace. In particular, \textit{F.~Léon-Saavedra} and \textit{A.~Montes-Rodríguez} [J.\ Funct. Anal. 148, 524--545 (1997; Zbl 0999.47009)] showed that every separable, infinite-dimensional Banach space supports an operator with a hypercyclic subspace. This result holds also in the setting of Fréchet spaces with a continuous norm, as \textit{L.~Bernal-Gonzáles} [Proc. Am. Math. Soc. 134, No.~7, 1955--1961 (2006; Zbl 1094.47010)] and, independently, \textit{H.~Petersson} [J. Math. Anal. Appl. 319, No.~2, 764--782 (2006; Zbl 1101.47006)] proved. In the paper under review, the authors show that the space \(\omega\), i.e., the countably infinite product of the real or compl…
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 46A04
zbMATH DE Number: 5016164
zbMATH Keywords: hypercyclic operators
zbMATH Keywords: hypercyclic subspaces
zbMATH Keywords: Fréchet spaces
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.04.083
OpenAlex ID: W1982521776
cites work: Common hypercyclic vectors for multiples of backward shift.
cites work: Existence of hypercyclic operators on topological vector spaces
cites work: Q5492480
cites work: Common hypercyclic subspaces
cites work: On hypercyclic operators on Banach spaces
cites work: Non-finite dimensional closed vector spaces of universal functions for composition operators
cites work: Approximation by chaotic operators and by conjugate classes.
cites work: Hereditarily hypercylic operators
cites work: Linear Chaos on Fréchet Spaces
cites work: Universal and chaotic multipliers on spaces of operators
cites work: Hypercyclic operators on non-normable Fréchet spaces
cites work: Invariant Manifolds of Hypercyclic Vectors
cites work: Genericity of wild holomorphic functions and common hypercyclic vectors.
cites work: Universal families and hypercyclic operators
cites work: Q4652147
cites work: Semi-Fredholm theory: hypercyclic and supercyclic subspaces
cites work: Most Similarity Orbits are Strongly Dense
cites work: Limits of hypercyclic and supercyclic operators
cites work: On endomorphisms with dense orbits
cites work: Linear structure of hypercyclic vectors
cites work: Spectral theory and hypercyclic subspaces
cites work: CHAOS FOR BACKWARD SHIFT OPERATORS
cites work: Q4020232
cites work: Banach spaces of hypercyclic vectors
cites work: The invariant subspace problem for a class of Banach spaces. II: Hypercyclic operators
cites work: On orbits of elements
cites work: Supercyclicity and weighted shifts
cites work: Hypercyclic operators on non-locally convex spaces
DOI: 10.1016/J.JMAA.2005.04.083
description: scientific article; zbMATH DE number 5016164