Irregular vectors of Hilbert space operators

A vector x in a Hilbert space H is called irregular for an operator T : H → H provided that sup n ‖ T n x ‖ = ∞ and inf n ‖ T n x ‖ = 0 . We establish some basic properties of operators having irregular vectors and present examples that highlight the relationship, or lack thereof, between irregularity and hypercyclicity.