ON SUPERCYCLIC OPERATORS WITH ADJOINTS HAVING NONEMPTY POINT SPECTRUM

Abstract

Let T be a bounded linear operator on an infinite dimensional, separable Banach space X. We consider a class of supercyclic operators, whose point spectrum of their adjoints are nonempty, and prove that under certain conditions, the orbit of every supercyclic vector for such an operator is unbounded. This result has some nice applications: (1) We obtain some conditions equivalent to the supercyclicity of extreme points of the closed unit ball of [Formula: see text] on a separable infinite dimensional Hilbert space; this helps us to characterize all supercyclic operators in this class. (2) The adjoint of composition operators on certain weighted Hardy spaces are never supercyclic. Next, we turn our attention to the commutant and show that if T is an operator in the mentioned class, then every operator in the commutant of T is not hypercyclic.