Let A be an invertible (bounded linear) operator acting on a complex Banach space 3f . A is called hypercyclic if there is a vector y in 3f such that the orbit Orb(^ ; y) := {y, Ay, A2y, ...} is dense in 3f . (ßC is necessarily separable and infinite dimensional.) Theorem 1. The following are equivalent for an invertible operator A acting on 3f : (i) A or A~l is hypercyclic; (ii) A and A~l are hypercyclic; (iii) there is a vector z such that Orb(A ; z)~ = Orb(A~^ ; z)~~ = 3? (the upper bar

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... es norm-closure); (iv) there is a vector y in 3? such that \Ox\s(A;y)\JOx\)(A-x ; y)]~ = 3f. Theorem 2. If A is not hypercyclic, then A and A~] have a common nontrivial invariant closed subset.