Strong hypercyclicity of Banach space operators
J. Korean Math. Soc.
Published online September 25, 2020
Mohammad Ansari, Karim Hedayatian, and Bahram Khani-Robati
Azad University of Gachsaran, 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 Leon-Muller 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 : 47A16, 47A15
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