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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.
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참고문헌 (16)
참고문헌 신청
M. Ansari / 2018 / Strong topological transitivity of some classes of operators / Bull. Belg. Math. Soc. Simon Stevin 25(5):677~685
S.I. Ansari / 1995 / Hypercyclic and Cyclic Vectors / Journal of Functional Analysis 128(2):374~383
Mohammad ANSARI / 2018 / On the density and transitivity of sets of operators / TURKISH JOURNAL OF MATHEMATICS 42:181~189
F. Bayart / 2009 / Dynamics of linear operators / Cambridge University Press
Juan Bès / 1999 / Hereditarily Hypercyclic Operators / Journal of Functional Analysis 167(1):94~112
S.I. Ansari / 1995 / Hypercyclic and Cyclic Vectors / Journal of Functional Analysis 128(2):374~383
Mohammad ANSARI / 2018 / On the density and transitivity of sets of operators / TURKISH JOURNAL OF MATHEMATICS 42:181~189
F. Bayart / 2009 / Dynamics of linear operators / Cambridge University Press
Juan Bès / 1999 / Hereditarily Hypercyclic Operators / Journal of Functional Analysis 167(1):94~112
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