Weyl Spectrum of Class A(n) and n-Paranormal Operators

Abstract

Let n be a positive integer, an operator T belongs to class A(n) if |T 1+n | 2/(1+n) ≥ |T | 2 , which is a generalization of class A and a subclass of n-paranormal operators, i.e., T 1+n x 1/(1+n) ≥ T x for unit vector x. It is showed that if T is a class A(n) or n-paranormal operator, then the spectral mapping theorem on Weyl spectrum of T holds. If T belongs to class A(n), then the nonzero points of its point spectrum and joint point spectrum are identical, the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical.

ON n-*-PARANORMAL OPERATORS

“…Since T ∈ C( n), we have T ∈ P( n + 1) by Theorem 3.8, and so the result follows by Lemma 2.6 of [31].…”

Reducibility of Invariant Subspaces of Operators Related to k-Quasiclass-A(n) Operators

“…The class k-Q A(1) is equal to k-Q A [6,11,15] and 0-quasiclass A(n) means class A(n) [17]. It is well-known that, for each n, class A(n) includes every p-hyponormal operators [4,5].…”

“…A (n, 0)-quasiparanormal operator means a n-paranormal operator [17], the class of n-paranormal operator includes all class A(n) operators, and every n-paranormal operator is normaloid [9, Theorem 1].…”

Finite Operators and Weyl Type Theorems for Quasi-*-n-Paranormal Operators

“…(see [6]) (3) T is n-paranormal if ||T 1+n x|| 1 1+n ≥ ||T x|| for unit vector x. (see [17]) (4) T is normaloid if ||T n || = ||T || n for n ∈ N.(see [3]) In this paper, we generalize * -n-paranormal operators to quasi- * -n-paranormal operators as follows. Using the same method as that in Lemma 1.1 [19] we have …”