Strong hypercyclicity of Banach space operators
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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.
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.
참고문헌 (16)
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