THE HYPERCYCLICITY CRITERION AND HYPERCYCLIC SEQUENCES OF MULTIPLES OF OPERATORS

Let T be a linear continuous operator acting on a Banach space X and {lambda(n)} a sequence of non-zero complex numbers satisfying lambda(n+1)/lambda(n) -> 1. In this article we look at sequences of operators of the form {lambda(n)T(n)}. In earlier work we showed that under the assumption that T is hypercyclic, if for some x is an element of X the set {lambda(n)T(n)x : n is an element of N} is somewhere dense then it is everywhere dense, a Bourdon-Feldman type theorem. In this article we show that this result fails to hold if the assumption of hypercyclicity for T is removed. A condition for the sequence {lambda(n)} under which an Ansari type theorem holds, namely, if {lambda(n)T(n)} is hypercyclic then {lambda(n)T(kn)} is hypercyclic for k = 2, 3, ..., is given. We show that if this condition is not satisfied, the result may fail to hold. Furthermore, we establish equivalences to the hypercyclicity criterion for this class of operator sequences.