Results 101 to 110 of about 142 (132)

Multi-hypercyclic operators are hypercyclic

Mathematische Zeitschrift, 2001
An operator \(T\) on a separable complex Hilbert space \(\mathcal H\) space is said to be hypercyclic if there is a vector \(x\) such that the orbit \(\{T^nx: n=0,1,\ldots\}\) is dense in \(\mathcal H\). An operator is said to be supercyclic if there is a vector \(x\) such that the scalar multiples of the elements in the orbit are dense in \(\mathcal H\
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Rotations of Hypercyclic and Supercyclic Operators

Integral Equations and Operator Theory, 2004
A (bounded linear) operator \(T\) on a Banach space \(X\) is called hypercyclic if there is a vector \(x \in X\) such that its orbit \(\{T^n(x) \;| \;n=0,1,2,... \}\) is dense in \(X\); the vector \(x\) is called hypercyclic for \(T\). The operator \(T\) is called supercyclic if \(\{ \alpha T^n(x) \;| \alpha \in \mathbb C, n \in \mathbb N \}\) is dense
León-Saavedra, Fernando, Müller, Vladimír   +1 more
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Faber-hypercyclic operators

Israel Journal of Mathematics, 2008
Let \(X\) be a complex infinite-dimensional separable Banach space and \(T\) be a bounded linear operator on \(X\). Let \(\Omega\) be a bounded domain of the complex plane whose boundary is a closed Jordan curve and \((F_n^{\Omega})_{n\geq 0}\) be the sequence of Faber polynomials of \(\Omega\).
Badea, Catalin, Grivaux, Sophie
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Residuality of Sets of Hypercyclic Operators

Integral Equations and Operator Theory, 2011
Let \(X\) be a separable metrizable topological vector space and let \(L(X)\) denote the set of continuous and linear operators from \(X\) to \(X\). An operator \(T\in L(X)\) is called \textit{hypercyclic} if there exists a vector \(x\in X\) such that \(\{T^n x:n\in\mathbb{N}\}\) is dense in \(X\), and it is said to be \textit{supercyclic} if there ...
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Hypercyclic and Chaotic Convolution Operators

Journal of the London Mathematical Society, 2000
Every convolution operator on a space of ultradifferentiable functions of Beurling or Roumieu type and on the corresponding space of ultradistributions is hypercyclic and chaotic (i.e., it is transitive and has a dense set of periodic points) when it is not a multiple of the identity.
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A strictly weakly hypercyclic operator with a hypercyclic subspace

Journal of Operator Theory
An interesting topic of study for a hypercyclic operator T:X→X on a topological vector space X has been whether X has an infinite\hyp{}dimensional, closed subspace consisting entirely, except for the zero vector, of hypercyclic vectors of T. These subspaces are called hypercyclic subspaces. It has been known that there is an operator T:H→H on a Hilbert
Chan, Kit C., Madarasz, Zeno
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Existence of hypercyclic operators

2011
In this chapter we obtain, among other things, the Ansari–Bernal theorem that every infinite-dimensional separable Banach space supports a hypercyclic operator. In contrast, some infinite-dimensional separable Banach spaces do not support any chaotic operator. We also discuss here the richness of the set of hypercyclic operators in two ways: it forms a
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot   +1 more
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Powers of Hypercyclic Functions for Some Classical Hypercyclic Operators

Integral Equations and Operator Theory, 2007
We show that no power of any entire function is hypercyclic for Birkhoff’s translation operator on $$\mathcal{H}(\mathbb{C})$$ . On the other hand, we see that the set of functions whose powers are all hypercyclic for MacLane’s differentiation operator is a Gδ ...
R. M. Aron, J. A. Conejero, A. Peris, J. B. Seoane–Sepúlveda   +3 more
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Pathological hypercyclic operators

Archiv der Mathematik, 2006
We exhibit a hypercyclic operator whose square is not hypercyclic. Our operator is necessarily unbounded since a result of S. Ansari asserts that powers of a hypercyclic bounded operator are also hypercyclic. We also exhibit an unbounded Hilbert space operator whose non-zero vectors are hypercyclic.
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Hypercyclic and chaotic operators

2011
In this chapter, the notions and results from the first chapter are revisited in the context of linearity. We introduce the notion of a hypercyclic operator and that of a chaotic operator. Among other things it is proved that the classical operators of Birkhoff, MacLane and Rolewicz are chaotic; it is shown that every hypercyclic operator possesses a ...
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot   +1 more
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