Property (ω) For Bounded Linear Operator

In this paper, we study the property (ω), a variant of Weyl's theorem intro-duced by Rakocevic. Meanwhile, we define the property (ω1), another variant of Weyl's theorem. We establish for a bounded linear operator defined on a Banach space the sufficient and necessary conditions for which both property (ω) and prop-erty (ω1) hold. The relationships among property (ω), property (ω1), single valued extension property and topological uniform descent are also discussed. In addition, the property (ω1) and property (ω) for 2×2 operator matrices are considered.This paper contains four chapters:In Chapter 1, we study the property (ω) by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a sufficient and necessary condition for which both property (ω) and a-Weyl's theorem hold. As a consequence of the main result, we study the property (ω) and a-Weyl's theorem for H(p) operators.In chapter 2, we define the property (ω1), a variant of Weyl's theorem, and establish for a bounded linear operator defined on Banach space the sufficient and necessary conditions for which property (ω1) holds by means of the variant of the essential approximate point spectrum. In addition, the relation between property (ω1) and hypercyclicity (or supercyclicity) is discussed.In chapter 3, we study the property (ω1) by means of the single valued extension property, and establish for a bounded linear operator defined on a Banach space the sufficient and necessary condition for which property (ω1) holds. As a consequence of the main result, the stability of property (ω1) is discussed.In chapter 4, we study the property (ω) and property (ω1) by means of the topological uniform descent, and establish for a bounded linear operator defined on a Hilbert space the sufficient and necessary conditions for which property (ω1) and property (ω) hold. Meanwhile, the property (ω1) and property (ω) for 2×2 operator matrices are discussed.