Hypercyclic tuples of operators and somewhere dense orbits
type: Item
instance of: scholarly article
title: Hypercyclic tuples of operators and somewhere dense orbits
zbMATH Open document ID: 1148.47008
author: Nathan S. Feldman
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-07-21T00:00:00Z
review text: An \(n\)-tuple of operators is a finite sequence of length \(n\) of commuting continuous linear operators on a locally convex space \(X\). Let \(T=(T_1,T_2,\dots,T_n)\) be an \(n\)-tuple of operators on a locally convex space \(X\) and let \({\mathcal F}={\mathcal F}_T=\{T_1^{k_1}T_2^{k_2}\dots T_n^{k_n}:k_i\geq 0\}\) be the semigroup generated by \(T\). Then the tuple \(T\) is called hypercyclic if there is a vector \(x\in X\) whose orbit \(\text{Orb}(T,x)=\text{Orb}({\mathcal F},x)=\{Sx:S\in {\mathcal F}\}\) under \(T\) is dense in \(X\). In the paper under review, the author gives some examples of hypercyclic tuples of operators. In particular, he shows that there are hypercyclic tuples of diagonal matrices on \({\mathbb R}^n\) and \({\mathbb C}^n\). He uses this result to show that there are no hypercyclic tuples of normal operators on an infinite-dimensional Hilbert space, and even no hypercyclic tuples of subnormal operators having commuting subnormal extensions. The author also proves that on Hilbert spaces over the field of real numbers, there are tuples of operators that have a somewhere dense orbit that is not dense. Furthermore, examples are given of tuples \(T\) that are hypercyclic, while \(T^n\) is not hypercyclic.
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 47A13
Mathematics Subject Classification ID: 46C99
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 5303195
zbMATH Keywords: tuple
zbMATH Keywords: hypercyclic
zbMATH Keywords: semigroup
zbMATH Keywords: orbit
zbMATH Keywords: somewhere dense orbit
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.04.027
OpenAlex ID: W2021531877
cites work: Hypercyclic and cyclic vectors
cites work: \(\mathbb C\)-supercyclic versus \({\mathbb R}^+\)-supercyclic operators
cites work: Orbits of hyponormal operators
cites work: Somewhere dense orbits are everywhere dense
cites work: Cyclic phenomena for composition operators
cites work: Towards a Functional Calculus for Subnormal Tuples: The Minimal Normal Extension
cites work: Q4000087
cites work: On a conjecture of D. Herrero concerning hypercyclic operators
cites work: Hypercyclic pairs of coanalytic Toeplitz operators
cites work: Q4780307
cites work: Operators with dense, invariant, cyclic vector manifolds
cites work: Q3856819
cites work: Q4299845
cites work: On the multiplicity of T\(\oplus T\oplus \cdot \cdot \cdot \oplus T\)
cites work: Q5696184
cites work: Q4165788
cites work: Multi-hypercyclic operators are hypercyclic
cites work: On orbits of elements
cites work: Hypercyclic Weighted Shifts
DOI: 10.1016/J.JMAA.2008.04.027
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description: scientific article; zbMATH DE number 5303195