Strong hypercyclicity of Banach space operators
J. Korean Math. Soc. 2021 Vol. 58, No. 1, 91-107
https://doi.org/10.4134/JKMS.j190850
Published online September 25, 2020
Printed January 1, 2021
Mohammad Ansari, Karim Hedayatian, Bahram Khani-Robati
Azad University of Gachsaran; Shiraz University; Shiraz University
Abstract : A bounded linear operator $T$ on a separable infinite dimensional Banach space $X$ is called strongly hypercyclic if $$X\backslash\{0\}\subseteq \bigcup_{n=0}^{\infty}T^n(U)$$ for all nonempty open sets $U\subseteq X$. We show that if $T$ is strongly hypercyclic, then so are $T^n$ and $cT$ for every $n\ge 2$ and each unimodular complex number $c$. These results are similar to the well known Ansari and Le\'{o}n-M\"{u}ller theorems for hypercyclic operators. We give some results concerning multiplication operators and weighted composition operators. We also present a result about the invariant subset problem.
Keywords : Strongly hypercyclic, strongly supercyclic, hypertransitive, invariant subset
MSC numbers : Primary 47A16; Secondary 47A15
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