Eigenvalues and hypercyclicity in omega
type: Item
instance of: scholarly article
title: Eigenvalues and hypercyclicity in omega
zbMATH Open document ID: 1266.47017
author: Héctor N. Salas
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2012-12-05T00:00:00Z
review text: It is well known that an infinite matrix \(A=(a_{i,j})\) defines a continuous linear operator on the Fréchet space \(\omega\) of all sequences if and only if each row \(i\) has a finite number of coordinates different from zero. Letting \(n_i\) denote the maximum \(j\) such that \(a_{i,j} \neq 0\), the author defines upper staircase matrices as those matrices \(A\) defining a continuous operator on \(\omega\) such that the sequence \((n_i)\) is strictly increasing and \(i < n_i\) for each row \(i\). The set \(S\) of upper staircase matrices is a noncommutative semigroup. It is proved that upper staircase matrices define continuous linear operators on \(\omega\) that have many eigenvalues and eigenvectors, that are hypercyclic, and even chaotic in the complex case. Extending a theorem by Bès and Conejero, the author proves that countably many strictly upper triangular matrices on \(\omega\) which are also upper staircase have a common hypercyclic subspace. The article concludes with a list of open questions.
reviewed by: José Bonet
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 6112715
zbMATH Keywords: eigenvalues
zbMATH Keywords: eigenvectors
zbMATH Keywords: hypercyclic
zbMATH Keywords: upper staircase matrices
MaRDI profile type: Publication
full work available at URL: s13398-011-0008-8
OpenAlex ID: W1965429154
cites work: Common hypercyclic vectors for multiples of backward shift.
cites work: Frequently hypercyclic operators
cites work: Q3638111
cites work: Hypercyclic subspaces in omega
cites work: Hypercyclic operators on non-normable Fréchet spaces
cites work: Frequently hypercyclic operators and vectors
cites work: Operators with dense, invariant, cyclic vector manifolds
cites work: Semi-Fredholm theory: hypercyclic and supercyclic subspaces
cites work: Eigenvalues of infinite matrices
cites work: Hypercyclicity in omega
cites work: Q3626812
cites work: Eigenvalues for infinite matrices
DOI: 10.1007/S13398-011-0008-8
description: scientific article; zbMATH DE number 6112715