J. Korean Math. Soc. 58 (2021), No. 1, pp. 91–107 https://doi.org/10.4134/JKMS.j190850 pISSN: 0304-9914 / eISSN: 2234-3008 STRONG HYPERCYCLICITY OF BANACH SPACE OPERATORS Mohammad Ansari, Karim Hedayatian, and Bahram Khani-Robati Dedicated to Kit Chan and Vladimir Troitsky Abstract. A bounded linear operator T on a separable infinite dimen- sional Banach space X is called strongly hypercyclic if X\{0}⊆ ∞  n=0 T n (U ) for all nonempty open sets U ⊆ X. We show that if T is strongly hy- percyclic, then so are T n and cT for every n ≥ 2 and each unimodular complex number c. These results are similar to the well known Ansari andLe´on-M¨ uller 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. 1. Introduction Let X be a separable infinite dimensional Banach space and B(X) be the space of all bounded linear operators on X. An operator T ∈ B(X) is said to be hypercyclic if there is some x ∈ X for which orb(T,x)= {T n x : n ∈ N 0 } is dense in X. In that case, x is called a hypercyclic vector for T . Here N 0 is the set of all nonnegative integers and T 0 = I , the identity operator on X. The set of all hypercyclic vectors for T is denoted by HC (T ) and it is known that if T is hypercyclic, then HC (T ) is dense in X. An operator T ∈ B(X) is called hypertransitive if HC (T )= X\{0}. A set M ⊆ X is called an invariant subset for T if T (M ) ⊆ M . It is clear that T is hypertransitive if and only if T lacks nontrivial closed invariant subsets. The Read operator on ℓ 1 is an example of such operators [14]. An operator T is called topologically transitive if, for any nonempty open sets U,V ⊆ X, there is some n ∈ N 0 such that T n (U ) ∩ V = ∅. It is well known Received December 18, 2019; Revised June 29, 2020; Accepted July 17, 2020. 2010 Mathematics Subject Classification. Primary 47A16; Secondary 47A15. Key words and phrases. Strongly hypercyclic, strongly supercyclic, hypertransitive, in- variant subset. c 2021 Korean Mathematical Society 91