Property (ω1) and Weyl type theorem

In this note we define the property (ω1), a variant of Weyl's theorem, and establish for a bounded linear operator defined on a Banach space the sufficient and necessary conditions for which property (ω1) holds by means of the variant of the essential approximate point spectrum σ1(⋅). In addition, the relation between property (ω1) and hypercyclicity (or supercyclicity) is discussed.