On invertible hypercyclic operators

Abstract

Let A A be an invertible (bounded linear) operator acting on a complex Banach space X \mathcal {X} . A A is called hypercyclic if there is a vector y y in X \mathcal {X} such that the orbit Orb ⁡ ( A ; y ) := { y , A y , A 2 , y , … } \operatorname {Orb}(A;y): = \{ y,Ay,{A^2},y, \ldots \} is dense in X \mathcal {X} . ( X \mathcal {X} is necessarily separable and infinite dimensional.) Theorem 1. The following are equivalent for an invertible operator A acting on X : ( i ) A \mathcal {X}:({\text {i}})A or A − 1 {A^{ - 1}} is hypercyclic; (ii) A A and A − 1 {A^{ - 1}} are hypercyclic; (iii) there is a vector z z such that Orb ⁡ ( A ; z ) − = Orb ⁡ ( A − 1 ;z ) − = X \operatorname {Orb}(A;z)^ - = \operatorname {Orb}({A^{ - 1}}{\text {;z}})^ - = \mathcal {X} (the upper bar denotes norm-closure); (iv) there is a vector y y in X \mathcal {X} such that \[ [ Orb ⁡ ( A ; y ) ∪ Orb ⁡ ( A − 1 ; y ) ] − = X . [\operatorname {Orb}(A;y) \cup \operatorname {Orb}({A^{ - 1}};y)]^ - = \mathcal {X}. \] . Theorem 2. If A A is not hypercyclic, then A A and A − 1 {A^{ - 1}} have a common nontrivial invariant closed subset.