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.

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.

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.

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.

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.

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.

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.

Additive Preservers of Drazin Invertible Operators with Bounded Index

Let B（X） be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. Given an integer n 〉 1, we show that an additive surjective map Ф on B（X） preserves Drazin invertible operators of index non-greater than n in both directions if and only if Ф is either of the form Ф（T） = aATA-1 or of the form Ф（T） = aBT＊B-1 where a is a non-zero scalar, A ： X → X and B ： X＊ → X are two bounded invertible linear or conjugate linear operators.

Perturbations of Drazin invertible operators

The necessary and sufficient conditions for the small norm perturba- tion of a Drazin invertible operator to be still Drazin invertible and the sufficient conditions for the finite rank perturbation of a Drazin invertible operator to be still Drazin invertible are established.