# Spectrum of k-quasi-class A n operators

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### New Zealand Journal of Mathematics

Vol. 50, (2020), Pages 61-70

M. H. M. Rashid

Department of Mathematics and Statistics

Faculty of Science

Mu'tah University

Alkarak

Jordan

Abstract In this paper, we introduce a new class of operators, called k-quasi-class An operators, which is a superclass of class A and a subclass of (n,k)-quasiparanormal operators. We will show basic structural properties and some spectral properties of this class of operators. We show that, if T is of k-quasi-class An then T − λ has finite ascent for all $\lambda\in{\rm C}$. Also, we will prove T is polaroid and Weyl's theorem holds for T and f(T), where f is an analytic function in a neighborhood of the spectrum of T. Moreover, we show that if λ is an isolated point of σ(T) and E is the Riesz idempotent of the spectrum of a k-quasi-class An operator T, then EH = ker(T − λ) if $\lambda\neq 0$ and EH = ker(Tn + 1) if λ = 0.

Keywords Class An, k-quasi-class An, Fuglede-Putnam Theorem, Riesz Idempotent, Polaroid operators.

Classification (MSC2000) 47B20, 47A10, 47A11.