Weighted core–EP inverse of an operator between Hilbert spaces

We introduce and study the weighted core–EP inverse of an operator between two Hilbert spaces as a generalization of the weighted core–EP inverse for a rectangular matrix. Several new properties of weighted core–EP inverses are given and some known results are extended. Using a weighted operator, the core–EP pre-order and the minus partial order of corresponding operators, we define new pre-orders on the set of all Wg–Drazin invertible operators between two Hilbert spaces. As consequences of our results, we present a new characterization and new representations of the core–EP inverse, new characterizations of the core–EP pre-order and extend the core–EP pre-order to a partial order.     

Characterizations of m-EP elements in rings

Let R be a ring with involution. In this paper, we extend the notions of m-EP matrices and m-EP operators to an arbitrary ring case. A number of new characterizations of m-EP elements in rings are presented. In particular, the existence criteria for 1-EP (i.e. EP) elements are obtained by means of the group inverse, Moore–Penrose inverse, and core inverse. Some properties of 2-EP are also given.     

New results for EP matrices in indefinite inner product spaces

In this paper we study J-EP matrices, as a generalization of EP-matrices in indefinite inner product spaces, with respect to indefinite matrix product. We give some properties concerning EP and J-EP matrices and find connection between them. Also, we present some results for reverse order law for Moore-Penrose inverse in indefinite setting. Finally, we deal with the star partial ordering and improve some results given in the “EP matrices in indefinite inner product spaces” (2012), by relaxing some conditions.     

The CMP inverse for rectangular matrices

We extend the notation of the CMP inverse for a square matrix to a rectangular matrix. Precisely, we define and characterize a new generalized inverse called the weighted CMP inverse. Also, we investigate properties of the weighted CMP inverse using a representation by block matrices. Some new characterizations and properties of the CMP inverse are obtained.     

Pseudo core inverses in rings with involution

Let R be a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse in R. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then it was generalized to an arbitrary ?-ring case by Raki?, Din?i? and Djordjevi? in 2014. Our definition of pseudo core inverse extends the notion of core inverse to elements of an arbitrary index in R. Meanwhile, it generalizes the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for matrices in 2014, to the case of ?-ring. Some equivalent characterizations for elements in R to be pseudo core invertible are given and expressions are presented especially in terms of Drazin inverse and {1,3}-inverse. Then, we investigate the relationship between pseudo core inverse and other generalized inverses. Further, we establish several properties of the pseudo core inverse. Finally, the computations for pseudo core inverses of matrices are exhibited.     

Core–EP inverse

In this work, we introduce a notion of ‘core–EP inverse’ for a square matrix which is not essentially of index one. This extends the notion of ‘core inverse’, which was initially defined for the matrices of index one. The properties of matrices having ‘core–EP inverse’ and ‘core–EP generalized inverse’ are studied, and obtained a formula to compute the core–EP generalized inverse from a particular linear combination of minors of given matrix.     

Note on the weighted generalized inverse of the product of matrices

The weighted generalized inverse has several important applications in the theoretical research and numerical computations. In this paper, we present a representation for the weighted generalized inverse of the product of two matrices A and B.     

On Semiprime Goldie Modules

In this article, we investigate some properties of right core inverses. Particularly, new characterizations and expressions for right core inverses are given, using projections and {1, 3}-inverses. Also, we introduced and investigated a new generalized right core inverse which is called right pseudo core inverse. Then, we provide the relation schema of (one-sided) core inverses, (one-sided) pseudo core inverses, and EP elements.     

On Complex Split Quaternion Matrices

In this paper, we present some important properties of complex split quaternions and their matrices. We also prove that any complex split quaternion has a 4 × 4 complex matrix representation. On the other hand, we give answers to the following two basic questions “If AB =  I, is it true that BA =  I for complex split quaternion matrices?” and “How can the inverse of a complex split quaternion matrix be found?”. Finally, we give an explicit formula for the inverse of a complex split quaternion matrix by using complex matrices.     

Group inverse and core inverse in Banach and C*-algebras

Abstract

In this paper, we have focused our study on the acute perturbation of the group inverse for the elements of Banach algebra with respect to the spectral radius. We also give perturbation analysis for the core inverse in C*- algebra. The perturbation bounds for the core inverse under some conditions are presented. Additionally, this paper extends the results obtained in [11, 14].     

The group inverse and core inverse of sums of two elements in a ring

Abstract

In Dedekind-finite ring, we present the group inverse of sum of two group invertible elements under different conditions. Then, the core inverse of a sum of two core invertible elements is investigated. Furthermore, the core inverse of the difference of two core invertible elements is presented. These results generalized the corresponding results of complex matrices.     

Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares (?) solutions to a multilinear system and establish the relationship between the minimum-norm (N) least-squares (?) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.     

Rank equalities related to outer inverses of matrices and applications

A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices.     

The spinL-function on the symplectic groupGSp(6)

In this paper, we will study some essential analytic properties of the “spin”L-function on the symplectic groupGSp (6) (which is associated with the eight-dimensional spin representation of theL-group Gspin (7, ℂ), namely, uniqueness of a bilinear form on an irreducible admissible representation ofGSp (6)×GL(2), local functional equation, and meromorphic continuation, non-vanishing properties at non-archimedean places as well as at archimedean places. Consequently, we will determine the location of the possible poles of the global spinL-function of a generic automorphic cuspidal representation ofGSp(6).     

环上矩阵的加权广义逆

该文从另一方面给出了Hartwig R E提出的公开问题的解答.设R是带有对合*的任意环,作者定义了环R上的一种新的加权广义逆,记为A_(P,Q)~+,并给出了A_(P,Q)~+存在的充要条件.同时,得到了一些重要的性质.     

广义Bott-Duffin逆及加权Drazin逆的若干性质

本文给出了L-零矩阵的广义Bott-Duffin逆及矩阵的加权Drazin逆的若干性质及表达形式．     

Some results on the Drazin inverse of anti-triangular matrices

Abstract

The representations for the Drazin inverse of anti-triangular matrices are obtained under some conditions. Applying these representations, we give a necessary condition for a class of block matrices to have signed Drazin inverse.     

Jacobson’s lemma and Cline’s formula for generalized inverses in a ring with involution

Abstract

In a ring with involution, we prove that a Drazin invertible element is pseudo core invertible if and only if its spectral idempotent is {1, 4}-invertible. As its applications, we obtain necessary and sufficient conditions for 1???ba (resp., ba) being pseudo core invertible while 1???ab (resp., ab) has pseudo core inverse, and the pseudo core inverse of 1???ba (resp., ba) is given in terms of 1???ab (resp., ab). Inspired by the above idea, Jacobson’s lemma for Moore-Penrose inverse is considered.