A new characterization of generalized Browder’s theorem and a cline’s formula for generalized Drazin-eromorphic inverses

In this paper we give a new characterization of generalized Browder?s theorem by considering equality between the generalized Drazin-meromorphic Weyl spectrum and the generalized Drazinmeromorphic spectrum. Also, wegeneralize Cline?s formula to the case of generalized Drazin-meromorphic invertibility under the assumption that AkBkAk = Ak+1 for some positive integer k.

Upper Semi-Weyl and Upper Semi-Browder Spectra of Unbounded Upper Triangular Operator Matrices

In this paper, we study the unbounded upper triangular operator matrix with diagonal domain. Some sufficient and necessary conditions are given under which upper semi-Weyl spectrum (resp. upper semi-Browder spectrum) of such operator matrix is equal to the union of the upper semi-Weyl spectra (resp. the upper semi-Browder spectra) of its diagonal entries. As an application, the corresponding spectral properties of Hamiltonian operator matrix are obtained.

On new strong versions of Browder type theorems

An operator T acting on a Banach space X satisfies the property (UWΠ) if σa(T)∖ $\begin{array}{} \sigma_{SF_{+}^{-}} \end{array} $(T) = Π(T), where σa(T) is the approximate point spectrum of T, $\begin{array}{} \sigma_{SF_{+}^{-}} \end{array} $(T) is the upper semi-Weyl spectrum of T and Π(T) the set of all poles of T. In this paper we introduce and study two new spectral properties, namely (VΠ) and (VΠa), in connection with Browder type theorems introduced in [1], [2], [3] and [4]. Among other results, we have that T satisfies property (VΠ) if and only if T satisfies property (UWΠ) and σ(T) = σa(T).

Symmetric difference between pseudo B-Fredholm spectrum and spectra originated from fredholm theory

In this paper, we continue the study of the pseudo B-Fredholm operators of Boasso, and the pseudo B-Weyl spectrum of Zariouh and Zguitti; in particular we find that the pseudo B-Weyl spectrum is empty whenever the pseudo B-Fredholm spectrum is, and look at the symmetric differences between the pseudo B-Weyl and other spectra.

Weyl Spectral Identity and Biquasitriangularity

Let A and B be operators acting on infinite-dimensional complex Banach spaces. We say that the Weyl spectral identity holds for the tensor product A⊗B if σw(A⊗B) = σw(A)·σ(B)∪σ(A)·σw(B), where σ(·) and σw(·) stand for the spectrum and the Weyl spectrum, respectively. Conditions on A and B for which the Weyl spectral identity holds are investigated. Especially, it is shown that if A and B are biquasitriangular (in particular, if the spectra of A and B have empty interior), then the Weyl spectral identity holds. It is also proved that if A and B are biquasitriangular, then the tensor product A ⊗ B is biquasitriangular.

Weyl's theorem for the square of operator and perturbations

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ∈ B(H) satisfies Weyl's theorem if σ(T)\σw(T) = π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum of T, respectively, π00(T) = {λ ∈ iso σ(T) : 0 < dim N(T - λI) < ∞}. T ∈ B(H) is said to have the stability of Weyl's theorem if T + K satisfies Weyl's theorem for all compact operator K ∈ B(H). In this paper, we characterize the operator T on H satisfying the stability of Weyl's theorem holds for T2.

Localization and computation in an approximation of eigenvalues

In this note we consider the problem of localization and approximation of eigenvalues of operators on infinite dimensional Banach and Hilbert spaces. This problem has been studied for operators of finite rank but it is seldom investigated in the infinite dimensional case. The eigenvalues of an operator (between infinite dimensional vector spaces) can be positioned in different parts of the spectrum of the operator, even it is not necessary to be isolated points in the spectrum. Also, an isolated point in the spectrum is not necessary an eigenvalue. One method that we can apply is using Weyl?s theorem for an operator, which asserts that every point outside the Weyl spectrum is an isolated eigenvalue.