Weak supercyclicity: dynamics of paranormal operators

Abstract

If A is a Banach space operator with the single-valued extension property such that the isolated points of the spectrum of A are poles of (the resolvent of) A, then A is neither weakly hypercyclic nor n-supercyclic for any positive integer \(n\ge 1\). Furthermore, if also A and \(A^{-1}\) (whenever it exists) are normaloid, then A weakly supercyclic implies A is an invertible isometry. Translated to classes of Hilbert space operators, this implies that if an operator A is either paranormal or \(*\)-paranormal (or even M-hyponormal), then it is not weakly hypercyclic or n-supercyclic; furthermore, A weakly supercyclic implies A is a scalar multiple of a unitary operator [thereby answering a problem posed in Duggal et al. (J Math Anal Appl 427:107–113, 2015)].