Abstract: In this paper, a new spectrum is defined according to the property of the consistent Fredholm index. We establish the sufficient and necessary conditions for a bounded linear operator defined on a Hilbert space that satisfies the property (ω₁). In addition, the paper explores the relationship between the property (ω₁) and hypercyclic operators (or supercyclic operators). Meanwhile, new conditions for hypercyclic operators and supercyclic operators are given.

Key words: property (ω₁), hypercyclic operator, property of the consistent Fredholm index, spectrum