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Journal of Mathematical Analysis and Applications, 2002
A bounded linear operator $T$ on a separable Banach space $\mathcal B$ is said to be hypercyclic if there is $x \in \mathcal B$, also called hypercyclic, such that the elements in the orbit $\{T^ n x\}_{n\geq 0}$ are dense in $\mathcal B$. Hypercyclicity is one of the strongest forms of cyclicity. \textit{S. Rolewicz} [Stud. Math. 32, 17--22 (
Nathan S Feldman
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Remarks on common hypercyclic vectors [PDF]

Journal of Functional Analysis, 2010
We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator $T$ on a complex Fréchet space $X$ and a set $Λ\subseteq \R_+\times\C$ which is not of zero three-dimensional Lebesgue measure, the family $\{aT+bI:(a,b)\inΛ\}$ has no common hypercyclic ...
Stanislav Shkarin
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Algebras of frequently hypercyclic vectors [PDF]

Mathematische Nachrichten, 2020
AbstractWe show that the multiples of the backward shift operator on the spaces , , or c0, when endowed with coordinatewise multiplication, do not possess frequently hypercyclic algebras. More generally, we characterize the existence of algebras of ‐hypercyclic vectors for these operators.
Javier Falco, Karl-G Grosse-Erdmann
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Common hypercyclic vectors for the conjugate class of a hypercyclic operator

Journal of Mathematical Analysis and Applications, 2011
Let $X$ be an infinite-dimensional, separable Banach space and $B(X)$ the algebra of all bounded linear operators on $X$. The authors show that, if $T$ is a continuous hypercyclic operator on $X$, then the conjugate set $S(T):=\{L^{-1}TL: L\in B(X)$ invertible\} contains a path $\{F_t\in B(X)$, $t\in [1,\infty[\}$ of operators which is ...
Kit Chan, Rebecca Sanders
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Common hypercyclic vectors for the unitary orbit of a hypercyclic operator

Journal of Mathematical Analysis and Applications, 2012
The authors continue their recent work on common hypercyclic vectors for paths and similarity orbits of operators $T$ on a separable Hilbert space $H$ [J. Oper. Theory 61, 191--233 (2009); ibid. 66, No.~1, 107--124 (2011; Zbl 1230.47021); J. Math. Anal. Appl. 375, No.~1, 139--148 (2011; Zbl 1208.47013); Integral Equations Oper.
Kit Chan, Rebecca Sanders
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Common Hypercyclic Vectors and the Hypercyclicity Criterion

Integral Equations and Operator Theory, 2009
An operator on a separable, infinite dimensional Banach space satisfies the Hypercyclicity Criterion if and only if the associated left multiplication operator is hypercyclic; see [14], [16], [29]. By examining paths of operators where each operator along the path satisfies the criterion, we provide necessary and sufficient conditions for a path of ...
Rebecca Sanders
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Common hypercyclic vectors for multiples of backward shift

Journal of Functional Analysis, 2003
We prove that the space $l^2$ contains a dense set of vectors which are hypercyclic simultaneously for all multiples of the backward shift operator by constants of absolute value greater than 1.
Evgeny Abakumov, Julia Gordon
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Hypercyclic and Cyclic Vectors

Journal of Functional Analysis, 1995
Let $\mathcal X$ denote a separable complex Banach space. A vector $x\in {\mathcal X}$ is said to be hypercyclic for an operator $T$ on $\mathcal X$ if the set $\{T^n x: n\in \mathbb{N}\}$ is norm dense in $\mathcal X$. We say that $x$ is supercyclic if the set $\{aT^n x: n\in \mathbb{N}, a\in \mathbb{C}\}$ is norm dense. An operator is
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