Left and right generalized inverses

Let R denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators. For a bounded linear operator T on a Banach space X we prove that $$T=T_M\oplus T_N$$
 with $$T_M \in $$
 R and $$T_N$$
 quasinilpotent (nilpotent) if and only if T admits a generalized Kato decomposition (T is of Kato type) and 0 is not an interior point of the corresponding spectrum $$\sigma _\mathbf{R}(T) =\{\lambda \in \mathbb {C}: T-\lambda \notin \mathbf{R}\}$$
 . Moreover, we prove that if $$T-\lambda _0$$
 admits a generalized Kato decomposition, then $$\sigma _\mathbf{R}(T)$$
 does not cluster at $$\lambda _0$$
 if and only if $$\lambda _0$$
 is not an interior point of $$\sigma _\mathbf{R}(T)$$
 . As a consequence we get several results on cluster points of essential spectra. In that way we extend some results regarding the approximate point spectrum and the surjective spectrum given by Aiena and Rosas (J. Math. Anal. Appl. 279:180–188, 2003), as well as results given by Jiang and Zhong (J. Math. Anal. Appl. 356:322–327, 2009) to the cases of essential spectra.

Generalized Kato Decomposition and Essential Spectra

In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting finite rank perturbations. Furthermore, we investigate the transmission of some local spectral properties from a bounded linear operator, as the SVEP, Dunford property $(C)$, and property $(\beta)$, to its generalized Drazin inverse.

Left and right generalized Drazin invertible operators and Local spectral theory

‎‎In this paper‎, ‎we relate the existence of certain projections‎, ‎commuting with a bounded linear operator $T\in L(X)$ acting on Banach space $X$‎, ‎with the generalized Kato decomposition of $T$‎. ‎We also relate the existence of these projections with some properties of the quasi-nilpotent part $H_0(T)$ and the analytic core $K(T)$‎. ‎Further results are given for the isolated points of some parts of the spectrum‎.

Projections and isolated points of parts of the spectrum

In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting finite rank perturbations. Furthermore, we investigate the transmission of some local spectral properties from a bounded linear operator, as the SVEP, Dunford property (C), and property (β), to its generalized Drazin inverse.

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Left and right generalized Drazin invertible operators and local spectral theory

In this paper, we define and study the left and the right generalized Drazin inverse of bounded operators in a Banach space We show that the left (resp the right) generalized Drazin inverse is a sum of a left invertible (resp a right invertible) operator and a quasi-nilpotent one In particular, we define the left and the right generalized Drazin spectra of a bounded operator and also show that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations Furthermore, we prove that a bounded operator is left generalized Drazin invertible if and only if its adjoint is right generalized Drazin invertible An equivalent definition of the pseudo-Fredholm operators in terms of the left generalized Drazin invertible operators is also given Our obtained results are used to investigate some relationships between the left and right generalized Drazin spectra and other spectra founded in Fredholm theory

Left and right generalized Drazin invertible operators

In this paper, we are interested in the perturbation of generalized Drazin operators by compact, polynomially compact and polynomially Riesz operators. We show that the generalized Drazin spectra are invariant under commuting perturbation of these classes. We describe some relevant cases especially the case of finite spectrum.

The perturbation classes problem for generalized Drazin invertible operators I

Let $A$ and $B$ be given linear operators on Banach spaces $X$ and $Y$, we denote by $M_C$ the operator defined on $X \oplus Y$ by $M_{C}=\begin{pmatrix}A & C \\ 0 & B%\end{pmatrix}.$In this paper, we study an abstract boundaryvalue matrix problems with a spectral parameter described by Drazin invertibile operators of the form $$\begin{cases}U_L=\lambda M_{C}w+F, & \\\Gamma w=\Phi, & \end{cases}%$$where $U_L , M_C$ are upper triangular operators matrices $(2\times 2)$ acting in Banach spaces, $\Gamma$ is boundary operator, $F$ and $\Phi $ are given vectors and $\lambda $ is a complex spectral parameter.We introduce theconcept of initial boundary operators adapted to the Drazin invertibility andwe present a spectral approach for solving the problem. It can be shown thatthe considered boundary value problems are uniquely solvable and that theirsolutions are explicitly calculated. As an application we give an example to illustrate our results.

Kouider Miloud Hocine

Papers

Left and right generalized Drazin invertible operators

Left and right generalized Drazin invertible operators and Local spectral theory

Left and right generalized Drazin invertible operators and local spectral theory

The perturbation classes problem for generalized Drazin invertible operators I

Boundary value matrix problems and Drazin invertible operators