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 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 Full-Text :