On k-quasi-paranormal operators

Abstract

For a positive integer k, an operator T ∈ B(H) is called k -quasi-paranormal if ||Tk+1x||2 ≤ ||Tk+2x||||Tkx|| for all x ∈ H, which is a common generalization of paranormal and quasi-paranormal. In this paper, firstly we prove some inequalities of this class of operators; secondly we give a necessary and sufficient condition for T to be k -quasi-paranormal. Using these results, we prove that: (1) if ||Tn+1|| = ||T||n+1 for some positive integer n ≥ k, then a k -quasi-paranormal operator T is normaloid; (2) if E is the Riesz idempotent for an isolated point λ0 of the spectrum of a k -quasi-paranormal operator T, then (i) if λ0 ≠ 0, then EH = ker(T -λ0); (ii) if λ0 = 0, then EH = ker(Tk+1). © Zagreb Paper JMI-08-11.