We characterize the weighted generalized core-EP inverse via the canonical decomposition, utilizing a weighted core invertible element and a quasinilpotent. We then offer a polar-like characterization for the weighted generalized core-EP inverse. The representations of weighted generalized core-EP inverse by leveraging the weighted generalized Drazin inverse are thereby presented. These lead to new representations and properties for the weighted core-EP inverse.

The generalized Drazin inverse of the sum of two elements in a Banach algebra

The purpose of this paper is to explore more properties and representations of the W-weighted m-weak group (in short, W-m-WG) inverse. We first explore an interesting relation between two projectors with respect to the W-m-WG inverse. Then, the W-m-WG inverse is represented by various generalized inverses, including W-weighted Drazin inverse, W-weighted weak group inverse, W-weighted core inverse, etc. We also give three concise explicit expressions for the W-m-WG inverse. Moreover, a canonical form of the W-m-WG inverse is presented in terms of the singular value decomposition. Finally, several numerical examples are designed to illustrate some results given in the paper.

By employing the minimal rank weak Drazin inverse and the minimal rank right weak Drazin inverse, we study extended systems of matrix equations that generalize those whose solutions are A † A D and A D A † . We verify that new extended systems have solutions based on the Moore–Penrose inverse, minimal rank weak Drazin inverse and minimal rank right weak Drazin inverse of A, and these solutions are called Moore–Penrose weak Drazin and weak Drazin Moore–Penrose matrices, whose particular types are A † A D and A D A † . We investigate properties and characterizations of Moore–Penrose weak Drazin and weak Drazin Moore–Penrose matrices. As applications, we prove the solvability of some linear equations in terms of Moore–Penrose weak Drazin and weak Drazin Moore–Penrose matrices. Consequently, we get new characterizations of A † A D and A D A † as well as of new special cases of Moore–Penrose weak Drazin and weak Drazin Moore–Penrose matrices.

On the Normality of the Sum of an Operator with its Drazin Inverse

Properties of one-sided generalized Drazin inverses in Banach algebras

In this paper, we investigate the connections between certain spectra arising from Fredholm theory of a generalized Drazin invertible bounded linear operator and those of its generalized Drazin inverse. Furthermore, we analyze the transfer of Browder’s theorem and its generalized form from such an operator to its corresponding generalized Drazin inverse. Applications to left, right, and multiplication operators are also presented.

Abstract We generalize the systems of equations, which introduced the MPCEP and *CEPMP inverses, using a minimal rank weak Drazin inverse and a minimal rank right weak Drazin inverse. In order to solve new generalized systems of matrix equations, we define new types of generalized inverses, the so-called weak MPCEP and *CEPMP inverses. The DMP, MPD, MPCEP and *CEPMP inverses are particular kinds of weak MPCEP and *CEPMP inverses. We show characterizations and formulae for weak MPCEP and *CEPMP inverses as well as their perturbation results. As application of weak MPCEP and *CEPMP inverses, we prove solvability of certain linear equations and recover the main application of the Moore–Penrose inverse.

We present a general nonperturbative method to determine the exact steady state of open quantum systems under perturbation. The method works for systems with a unique steady state and the perturbation may be time-independent or periodic, and of arbitrarily large amplitude. Using the Drazin inverse and a single diagonalization, we construct an operator that generates the entire dependence of the steady state on the perturbation parameter. The approach also enables exact analytic operations—such as differentiation, integration, and ensemble averaging—with respect to the parameter, even when the steady state is computed numerically. We apply the method to three nontrivial open quantum systems, showing that it achieves exact results, with a computational speedup of one to several orders of magnitude for calculations requiring large sampling, compared to previous approaches.

In this paper we investigate the g-Drazin invertibility of an anti-triangular block-operator matrix $\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)$ with $F^{\pi}EF^d=0$ and $F^{\pi}EF^iE=0$ for all $i\in {\mathbb N}$. This generalizes the main results of [Guo, Zou and Chen, Hacet. J. Math. Stat., 49(3)(2020), 1134-1149] and [Chen and Sheibani, Appl. Math. Comput., 463(2024), 128368 (12 pp)] to a wider case.

Abstract The solution to the system of equations of the descriptor linear discrete-time with different fractional orders is derived by the use of the Drazin inverse of matrices. This solution is applied to analysis of the pointwise completeness and the pointwise degeneracy of the descriptor discrete-time linear systems with different fractional orders. Necessary and sufficient conditions for the pointwise completeness and the pointwise degeneracy of the descriptor discrete–time linear systems with different fractional orders are established. The proposed methods are illustrated by numerical examples.

In this paper, we obtain some equivalent characterizations of the generalized [Formula: see text]-strong Drazin invertibility in Banach algebras. By utilizing Harte’s techniques, we demonstrate some additive results for two generalized [Formula: see text]-strong Drazin invertible elements under certain conditions. In particular, we prove that [Formula: see text] is generalized [Formula: see text]-strong Drazin invertible if and only if [Formula: see text] possesses the same property, provided that [Formula: see text] and [Formula: see text] commute in Banach algebras. Finally, we derive the common generalized [Formula: see text]-strong Drazin invertibility of two elements under certain intertwined conditions.

The defect set, a fundamental concept in local spectral theory, serves as a well-established criterion in the study of generalized Drazin invertibility. This paper emphasizes its significance by examining the left and right generalized Drazin invertibility of upper triangular operator matrices within Banach spaces. Additionally, it aims to build on the recent advancements made by Bahloul and Walha [Generalized Drazin invertibility of operator matrices, Numer. Funct. Anal. Opt. 43(16) (2022) 1836–1847]. Our approach focuses on establishing sufficient conditions, using the defect set within local spectral theory, to explore the relationship between the generalized Drazin-type spectra of [Formula: see text] upper triangular block operator matrices and those associated with their diagonal entries. Specifically, this contribution addresses the questions raised by Zguitti [A note on Drazin invertibility for upper triangular block operators, Mediterr. J. Math. 10 (2013) 1497–1507] and tackles the challenging problem outlined by Campbell [The Drazin inverse and systems of second order linear differential equations, Linear Multilinear Algebra 14 (1983) 195–198] regarding the representation of left and right generalized Drazin spectra for [Formula: see text] block operator matrices.

Researches on Drazin inverse of operators in Banach space

Left and right $W$-weighted $G$-Drazin inverses and new matrix partial orders

Let [Formula: see text] be a Banach algebra, and let [Formula: see text] satisfying [Formula: see text] for some [Formula: see text]. Under this condition, we show that Cline’s formula holds for some new generalized inverses, so [Formula: see text] has strong Drazin inverse (respectively, generalized strong Drazin inverse, Hirano inverse and generalized Hirano inverse) if and only if [Formula: see text] has strong Drazin inverse (respectively, generalized strong Drazin inverse, Hirano inverse and generalized Hirano inverse). As a particular case, some results for Jacobson’s lemma are given.

The aim of this work is to offer a method for studying the consistence and computing the set of solutions of some infinite systems of differential equations from the exponential map of a finite potent endomorphism. This method is related with the Drazin inverse of a finite potent operator on an infinite-dimensional vector space that was introduced by the author in 2019. Moreover, explicit examples of the exponential of a finite potent endomorphism and the set of solutions of an infinite system of differential equations are provided.

Some Additive Properties of the Drazin Inverse and Generalized Drazin Inverse

Abstract In this article, we construct a new explicit formula for the Drazin inverse of a sum of two matrices P, Q ∈ ℂ n×n under conditions weaker than those used in some recent papers, and modify an invalid formula in [Dopazo et al.: Block representations for the Drazin inverse of anti-triangular matrices, Filomat 30 (2016), 3897–3906]. Furthermore, we apply our results to obtain some new representations for the Drazin inverse of a 2 × 2 block matrix. Some numerical examples are given to illustrate our results.

Abstract The principal goal of this paper is to present one-sided versions of the extended g-Drazin inverse as weaker kinds of the extended g-Drazin inverse. Precisely, we define left and right extended g-Drazin inverses for Banach algebra elements. A number of characterizations for left and right extended g-Drazin inverses are developed. Since we obtain equivalence between concepts of left (or right) extended g-Drazin invertibility and left (or right) generalized Drazin invertibility, in fact, we get new characterizations for left (or right) generalized Drazin invertibility as consequences. As particular cases of one-sided extended g-Drazin inverses, we investigate one-sided extended Drazin inverses.