Triple reverse order law for the Drazin inverse

In this paper,we investigate the reverse order law for Drazin inverse of three bound-ed linear operators under some commutation relations.Moreover,the Drazin invertibility of sum is also obtained for two bounded linear operators and its expression is presented.

Representations of the Drazin inverse involving idempotents in a ring

An element a of a ring R is called Drazin invertible if there exists b∈R such that ab =ba,bab =b,and a -a2 b is nilpotent.The element b above is unique if it exists and is denoted as aD .The equivalent conditions of the Drazin inverse involving idempotents in R are established.As applications, some formulae for the Drazin inverse of the difference and the product of idempotents in a ring are given.Hence,a number of results of bounded linear operators in Banach spaces are extended to the ring case.

On Characterizations of Drazin Inverse

Let φ be a pre-additive category. Assume that φ： X→X is a morphism of φ. In this paper, we give the necessary and sufficient conditions for φ to have the Drazin inverse by using the von Neumann regular inverse for the φ^k, and extend a result by Puystjens and Hartwig from the group inverse to Drazin inverse.

A NOTE ON THE DRAZIN INVERSE OF A MODIFIED MATRIX

In this article, the expression for the Drazin inverse of a modified matrix is considered and some interesting results are established. This contributes to certain recent results obtained by Y.Wei [9].

A NOTE ON THE REPRESENTATIONS FOR THE GENERALIZED DRAZIN INVERSE OF BLOCK MATRICES

We present some representations for the generalized Drazin inverse of a block matrix x =[cd ab]in a Banach algebra ~4 in terms of ad and （be）d under certain conditions,extending some recent result related to the generalized Drazin inverse of an anti-triangular operator matrix. Also, several particular cases of this result are considered.

Some additive results on Drazin inverse

In this paper, we investigate additive results of the Drazin inverse of elements in a ring R. Under the condition ab = ha, we show that a ＋ b is Drazin invertible if and only if aaD （a ＋ b） is Drazin invertible, where the superscript D means the Drazin inverse. Furthermore we find an expression of （a ＋ b）D. As an application we give some new representations for the Drazin inverse of a 2 × 2 block matrix.

Centrally clean elements and central Drazin inverses in a ring

Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u.Moreover,a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element of R is centrally clean.First,some characterizations of centrally clean elements are given.Furthermore,some properties of centrally clean rings,as well as the necessary and sufficient conditions for R to be a centrally clean ring are investigated.Centrally clean rings are closely related to the central Drazin inverses.Then,in terms of centrally clean decomposition,the necessary and sufficient conditions for the existence of central Drazin inverses are presented.Moreover,the central cleanness of special rings,such as corner rings,the ring of formal power series over ring R,and a direct product ∏ R_(α) of ring R_(α),is analyzed.Furthermore,the central group invertibility of combinations of two central idempotents in the algebra over a field is investigated.Finally,as an application,an example that lists all invertible,central group invertible,group invertible,central Drazin invertible elements,and centrally clean elements of the group ring Z_(2)S_(3) is given.

THE DRAZIN INVERSE OF MATRICES OF QUATERNIONS

Let H be the real quaternion field,C and R be the complex and real field respectively.Clearly R（?）C（?）H. Let Hm×n denote the set of all m×n matrices over H.If A=(ars)∈Hm×n,then there exist A1 and A2∈Cm×n such that A=A1+A2j.Let AC denote the complexrepresentation of A,that is the 2m×2n complex matrix Ac=（（A1/A2）（-A2/A1））（see[1,2]）.We denote by AD the Drazin inverse of A∈Hm×n which is the unique solution of the e-

The Pseudo Drazin inverses in Banach Algebras

Let A be a complex Banach algebra and J be the Jacobson radical of A.(1)We firstly show that a is generalized Drazin invertible in A if and only if a+J is generalized Drazin invertible in A/J.Then we prove that a is pseudo Drazin invertible in si if and only if a+J is Drazin invertible in A/J.As its application,the pseudo Drazin invertibility of elements in a Banach algebra is explored.(2)The pseudo Drazin order is introduced in A.We give the necessary and sufficient conditions under which elements in A have pseudo Drazin order,then we prove that the pseudo Drazin order is a pre-order.

A NOTE ON CLINE'S FORMULA FOR TWO SUBCLASSES OF GENERALIZED DRAZIN INVERSES

For a Banach algebra A with identity and a, b, c, d ∈ A, the relations between the extended g-Drazin inverse(resp. generalized strong Drazin inverse)of ac and that of bd are given, when bac = bdb and cac = cdb.

Perturbation Analysis for the Drazin Inverses of Bounded Linear Operators on a Banach Space

Let X be a Banach space over the complex field C and let T : X→X be a bounded linear operator with Ind(T) = k and R(Tk) closed. Denote the Drazin inverse of T by TD. Let T = T + δT, then TD has the simple expression TD = TD(I + δTTD)-1 = (I + TDδT)-1TD under certain hypotheses. The upper bound for the relative error ‖TD-TD‖/‖TD‖and for the solution to the operator equation: Tx = u (u∈R(TD)) is also considered.

Further Results on the Drazin Inverse of Tensors via Einstein Product

In this paper,we give further results on the Drazin inverse of tensors via the Einstein product.We give a limit formula for the Drazin inverse of tensors.By using this formula,the representations for the Drazin inverse of several block tensor are obtained.Further,we give the Drazin inverse of the sum of two tensors based on the representation for the Drazin inverse of,a block tensor.

Drazin invertibility for matrices over an arbitrary ring

In order to study the Drazin invertibility of a matrix with the generalized factorization over an arbitrary ring, the necessary and sufficient conditions for the existence of the Drazin inverse of a matrix are given by the properties of the generalized factorization. Let T = PAQ be a square matrix with the generalized factorization, then T has Drazin index k if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible. The formulae to compute the Drazin inverse are also obtained. These results generalize recent results obtained for the Drazin inverse of a matrix with a universal factorization.

Generalized Cayley-Hamilton theorem for core-EP inverse matrix and DMP inverse matrix

By using the classical Cayley-Hamilton theorem,the polynomial equations of the core-EP inverse matrix and Drazin-Moore-Penrose(DMP)inverse matrix are given,respectively.If the characteristic polynomial of the singular matrix A,p A(s)=det(s E n-A)=s n+a n-1 s n-1+…+a 1 s,is given,then f A(A)=0 and f A(A d,+)=0 in which f A(A)=a 1 x n+a 2 x n-1+…+a n-1 x 2+x,and A and A d,+are the core-EP inverse and the DMP inverse of A,respectively.Furthermore,some properties of the characteristic polynomials of A D∈C n,n and A∈C n,n are derived.

Jacobson's Lemma for Generalized Drazin–Riesz Inverses

For bounded linear operators A,B,C and D on a Banach space X,we show that if BAC=BDB and CDB=CAC then I-AC is generalized Drazin-Riesz invertible if and only if I-BD is generalized Drazin-Riesz invertible,which gives a positive answer to Question 4.9 in Yan,Zeng and Zhu[Complex Anal.Oper.Theory 14,Paper No.12(2020)].In particular,we show that Jacobson’s lemma holds for generalized Drazin-Riesz inverses.

Generalized Inverses and Units in a Unitary Ring

Let R be a unitary ring and a,b∈R with ab=0.We find the 2/3 property of Drazin invertibility:if any two of a,b and a+b are Drazin invertible,then so is the third one.Then,we combine the 2/3 property of Drazin invertibility to characterize the existence of generalized inverses by means of units.As applications,the need for two invertible morphisms used by You and Chen to characterize the group invertibility of a sum of morphisms is reduced to that for one invertible morphism,and the existence and expression of the inverse along a product of two regular elements are obtained,which generalizes the main result of Mary and Patricio(2016)about the group inverse of a product.

Characterizations of m-weak group inverses

To characterize m-weak group inverses,several algebraic methods are used,such as the use of idempotents,one-side principal ideals,and units.Consider an element a within a unitary ring that possesses Drazin invertibility and an involution.This paper begins by outlining the conditions necessary for the existence of the m-weak group inverse of a.Moreover,it explores the criteria under which a can be considered pseudo core invertible and weak group invertible.In the context of a weak proper*-ring,it is proved that a is weak group invertible if,and only if,a D can serve as the weak group inverse of au,where u represents a specially invertible element closely associated with a D.The paper also introduces a counterexample to illustrate that a D cannot universally serve as the pseudo core inverse of another element.This distinction underscores the nuanced differences between pseudo core inverses and weak group inverses.Ultimately,the discussion expands to include the commuting properties of weak group inverses,extending these considerations to m-weak group inverses.Several new conditions on commuting properties of generalized inverses are given.These results show that pseudo core inverses,weak group inverses,and m-weak group inverses are not only closely linked but also have significant differences that set them apart.

A Finite Method for Computing the Drazin and Core-EP Inverses of Matrices Based on Partial Full-Rank Factorization

This paper presents a formula for the Drazin inverses of matrices based on a sequence of partial full-rank factorizations which theoretically extends the classic full-rank factorization method for computing the Drazin inverses established by R.E.Cline.The result is then extended to the core-EP inverses.

On a Comparison Property of M-matrices

In this paper, we present a spectral property of a comparison matrix, which improves and generalize the comparison theorem of nonsingular M-matrices.

Generalized Drazin spectrum of upper triangular matrices in Banach algebras

Let A be a Banach algebra with unit e and a,b,c∈A,Mc=(a c 0 b)∈M_(2)(A).The concepts of left and right generalized Drazin invertible of elements in a Banach algebra are proposed.A generalized Drazin spectrum of is defined byσ_(gD)(α)={λ∈C:α-λe is not generalized Drazin invertible}.It is shown thatσ_(gD)(a)∪σ_(gD)(b)=σ_(gD)(M_(C))∪W_(2),where W_(g) is a union of certain holes σ_(gD) and W_(g)■σ_(gD)(a)∩σ_(gD)(b),or more finely.In addition,some properties of generalized Drazin spectrum of elements in a Banach algebra are studied.