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 \(|T^{1+n}|^{2/(1+n)} \geq |T|^{2}\), which is a generalization of class A and a subclass of n-paranormal operators, i.e., \({\parallel}T^{1+n}x{\parallel}^{1/(1+n)} \geq {\parallel}Tx{\parallel}\) 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.