Property and Weyl type theorem

In this note we define the property ( ω 1 ) , a variant of Weylʹs theorem, and establish for a bounded linear operator defined on a Banach space the sufficient and necessary conditions for which property ( ω 1 ) holds by means of the variant of the essential approximate point spectrum σ 1 ( ⋅ ) . In addition, the relation between property ( ω 1 ) and hypercyclicity (or supercyclicity) is discussed.