Consistent Invertibility's Application

In this paper, we describe the property of consistency in invertibility and give the sufficient and necessary conditions for a bounded linear operator defined on a Hilbert space which has this property. And, we study a-Weyl's theorem and its variants in terms of the relation of the spectrumσCI(·) defined in view of the property of consistency in invertibility and other spectrums. In addition, the relationships among a-Weyl's theorem and its variants are also discussed.Three chapters are contained in this article:The chapter 1, In here, a bounded linear operator on a Hilbert space having consistent invertibility (briefly CI operator) is introduced. Then, sufficient and necessary conditions for property CI are given, and we define a new spectrumσCI(·) corresponding to the property CI and study the spectral mapping theorem for this new spectrum. Meanwhile, the property CI for upper triangular operator matrix is considered.The chapter 2, The new spectrumσCI(·) contributes the conditions equivalent to a-Weyl type theorem; And, using the relation of the new spectrumσCI(·) and the variant of the essential approximate point spectrumσ1(·), we study a bounded linear operator which has property (ω), a variant of a-Weyl's theorem; In addition, the relation between property (ω) and hypercyclicity is discussed. Moreover, the a-Browder theorem and property (ω1), a variant of a-Weyl's theorem, of operational calculus of operators are also discussed.The chapter 3, In this note, the sufficient and necessary conditions for which a-Weyl's theorem holds for a bounded linear operator defined on a Hilbert space are given in view of the relation between the new spectrumσCI(·) and topological uniform descent. Also, the equivalence of a-Weyl's theorem for an operator and its adjoint is provided. In additional, the hypercyclicity for operator matrices is considered.