# Spectrum of k-quasi-class A n operators

### From NZJM

**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 *A*_{n} 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 *A*_{n} then *T* − λ has finite ascent for all . 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 *A*_{n} operator *T*, then *E**H* = ker(*T* − λ) if and *E**H* = ker(*T*^{n + 1}) if λ = 0.

**Keywords** Class *A*_{n}, *k*-quasi-class *A*_{n}, Fuglede-Putnam Theorem, Riesz Idempotent, Polaroid operators.

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