Supercyclicity and Hypercyclicity of an Isometry Plus a Nilpotent

Abstract

Suppose thatXis a separable normed space and the operatorsAandQare bounded onX. In this paper, it is shown that ifAQ=QA,Ais an isometry, andQis a nilpotent then the operatorA+Qis neither supercyclic nor weakly hypercyclic. Moreover, if the underlying space is a Hilbert space andAis a co-isometric operator, then we give sufficient conditions under which the operatorA+Qsatisfies the supercyclicity criterion.

Power Regularity of Isometric N-Jordan Operators

“…Also, we see that if T is normaloid or 2-isometry then it is an isometry. It is proved in [7] that the eigenvectors of an isometric N -Jordan operator corresponding to distinct eigenvalues are orthogonal. In the next proposition, we generalize this result to distinct approximate eigenvalues.…”

“…By Proposition 1 of [7], σ ap (T ) = σ ap (A) and so |λ| = 1. Therefore, since A is isometric, for every integer k we have…”

“…Note that the notions of isometric 1-Jordan and isometry coincide. The dynamic and spectral properties of N -Jordan operators have been studied in [7]. It follows from Proposition 1.1 of [7] that these operators are injective.…”

“…The dynamic and spectral properties of N -Jordan operators have been studied in [7]. It follows from Proposition 1.1 of [7] that these operators are injective. We note that for an N -Jordan operator T , T * T is invertible.…”

“…We note that for an N -Jordan operator T , T * T is invertible. Indeed, Corollary 1.2 of [7] states that the operator T is bounded below, and so for every h ∈ H,…”