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International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 66 | Issue 4 | Year 2020 | Article Id. IJMTT-V66I4P521 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I4P521

On Hypercyclic Operators


Varughese Mathew
Abstract

The Set { Tnx:n≥0} is called the orbit of a vector x in a linear topological space X under a linear map T and is denoted by Orb(T,x). If dense orbits exist, T is called a hypercyclic operator. In this paper, a hypercyclic vector is constructed based on the existing sufficient condition for hypercyclicity of an operator T. From a separable Banach space Y and a sequence of real numbers {β(n)}, a sequence space (Y)β is defined and proved that the backward shift operator acting on (Y)β is hypercyclic.

Keywords
Orbit of a vector, Hypercyclic operator, Invariant subspace problem.
References

[1] G. D. Birkhoff,”Demonstration dun theorem element sur les fonctions entieres”, C. R. Acad. Sci. Paris, 189, pp. 473-475, 1929.
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[3] R. M. Gethner and J. H. Shapiro,“Universal vectors for operators on spaces of holomorphic functions”,Proc. Am. Math. Soc., pp.281- 288, 1987.
[4] G. Godefroy and J. H. Shapiro, “Operators with dense, invariant cyclic vector manifolds”, J. Funct. Anal., 98, pp.229-269, 1991.
[5] C. W. Groetsch, Generalized Inverses of Linear Operators, CRC Press, 1st Edition, 1977.
[6] Kuikui Liu, “Existence of Linear Hypercyclic Operators on Infinite-Dimensional Banach Spaces”, https://sites.math.washington.edu, 2015.
[7] C. Kitai, “Invariant closed sets for linear operators”, Thesis, Univ. of Toronto, 1982.
[8] G. R. MacLane, “Sequences of derivatives and normal families”, J. Analyze Math., pp.72-87, 1952.
[9] S. Rolewicz, “On orbits of elements”, Studia Math., pp. 17-22, 1969.

Citation :

Varughese Mathew, "On Hypercyclic Operators," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 4, pp. 172-176, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I4P521

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