Inner Generalized Inverses with Prescribed Idempotents

We define and characterize inner generalized inverses with prescribed idempotents. These classes of generalized inverses are natural algebraic extension of generalized inverses of linear operators with prescribed range and kernel. We consider the reverse order rule for inner generalized inverses of elements of a ring, some perturbation bounds, and we construct an iterative method for computing a (p, q)-inner inverse in Banach algebras.     

Additive results for the generalized Drazin inverse in a Banach algebra

In this paper we investigate additive properties of the generalized Drazin inverse in a Banach algebra. We find some new conditions under which the generalized Drazin inverse of the sum a + b could be explicitly expressed in terms of a, a^d, b, b^d. Also, some recent results of Castro and Koliha [New additive results for the g-Drazin inverse, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 1085-1097] are extended.     

Additive property of Drazin invertibility of elements in a ring

In this article, we investigate additive properties on the Drazin inverse of elements in rings. Under the commutative condition of ab?=?ba, we show that a?+?b is Drazin invertible if and only if 1?+?a ^D b is Drazin invertible. Not only the explicit representations of the Drazin inverse (a?+?b) ^D in terms of a, a ^D , b and b ^D , but also (1?+?a ^D b) ^D is given. Further, the same property is inherited by the generalized Drazin invertibility in a Banach algebra and is extended to bounded linear operators.     

New additive results for the generalized Drazin inverse

In this paper, we investigate additive properties of generalized Drazin inverse of two Drazin invertible linear operators in Banach spaces. Under the commutative condition of PQ=QP, we give explicit representations of the generalized Drazin inverse d(P+Q) in term of P, Pd, Q and Qd. We consider some applications of our results to the perturbation of the Drazin inverse and analyze a number of special cases.     

Analytic perturbation of generalized inverses

We investigate the analytic perturbation of generalized inverses. Firstly we analyze the analytic perturbation of the Drazin generalized inverse (also known as reduced resolvent in operator theory). Our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent. It allows us to treat regular and singular perturbations in a unified framework. We provide an algorithm for computing the coefficients of the Laurent series of the perturbed Drazin generalized inverse. In particular, the regular part coefficients can be efficiently calculated by recursive formulae. Finally we apply the obtained results to the perturbation analysis of the Moore–Penrose generalized inverse in the real domain.     

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 =[abcd]in a Banach algebra A in terms of a~d and(bc)~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.     

Formulae for the generalized Drazin inverse of a block matrix in terms of Banachiewicz–Schur forms

We introduce new expressions for the generalized Drazin inverse of a block matrix with the generalized Schur complement being generalized Drazin invertible in a Banach algebra under some conditions. We generalized some recent results for Drazin inverse and group inverse of complex matrices.     

Generalized stable ranks for commutative Banach algebras

We introduce the notion of generalized E-stable ranks for commutative unital Banach algebras and determine these ranks for the disk-algebra A(\mathbbD){A(\mathbb{D})}, many of its subalgebras, and the algebra H ^∞ of bounded holomorphic functions in the unit disk. Relations to L-sets and separating algebras, notions due to Csordas and Reiter, are given, too. Finally we show that the absolute stable rank of A(\mathbbD){A(\mathbb{D})} and H ^∞ is bigger than 2.     

ON THE GENERALIZED RESOLVENT OF LINEAR PENCILS IN BANACH SPACES

Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvent of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil λ→ T λ S are provided and an explicit expression of the generalized resolvent is also given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators and semi-Fredholm operators are also considered. The results obtained in this paper extend and improve many results in this area.     

On the Drazin inverse of the sum of two operators and its application to operator matrices

Given two bounded linear operators F,G on a Banach space X such that G²F=GF²=0, we derive an explicit expression for the Drazin inverse of F+G. For this purpose, firstly, we obtain a formula for the resolvent of an auxiliary operator matrix in the form . From the provided representation of D(F+G) several special cases are considered. In particular, we recover the case GF=0 studied by Hartwig et al. [R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001) 207-217] for matrices and by Djordjevi? and Wei [D.S. Djordjevi?, Y. Wei, Additive results for the generalized Drazin inverse, J. Aust. Math. Soc. 73 (1) (2002) 115-126] for operators. Finally, we apply our results to obtain representations for the Drazin inverse of operator matrices in the form which are extensions of some cases given in the literature.     

A note on the Drazin inverse of an anti-triangular matrix

In this paper we give formulae for the generalized Drazin inverse M^d of an anti-triangular matrix M under some conditions. Moreover, some particular cases of these results are also considered.     

Left and right generalized Drazin invertible operators

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.     

Generalized notions of character amenability

In this paper, the concepts of approximate character amenability （contractibility）, uniform approximate character amenability （contractibility） and w^＊-approximate character amenability are introduced. We are concerned with the relations among the generalized concepts of character amenability for Banach algebras. We show that approximate character amenability, w^＊-approximate character amenability and approximate character contractibility are the same properties, as uniform approximate character amenability and character amenability as uniform approximate character contractibility and character contractibility. The general theory for these concepts is also developed. Moreover, approximate character amenability of several concrete classes of Banach algebras related to locally compact groups and also some discrete semigroups is considered.     

Hilbert-Carleman and regularized determinants for linear operators

A general theory of regularized and Hilbert-Carleman determinants in normed algebras of operators acting in Banach spaces is proposed. In this approach regularized determinants are defined as continuous extensions of the corresponding determinants of finite dimensional operators. We characterize the algebras for which such extensions exist, describe the main properties of the extended determinants, obtain Cramer's rule and the formulas for the resolvent which are expressed via the extended tracestr(A ^k ) of iterations and regularized determinants.This paper is a continuation of the paper [GGKr].     

有界线性算子和的Drazin逆表示

本文讨论了两个有界线性算子和的Drazin可逆性及其表达式.在PQ~3=0,P~2Q=0,QPQ~2=0的条件下,采用预解式的Laurent展开方法,证明了P+Q是Drazin可逆的,并得到了P+Q的Drazin逆的表达式.同时,还确定出P+Q的指标的范围ind(P+Q)≤2t+r+s—1,给出数值算例说明结论的有效性.     

Representation for the W-weighted Drazin inverse of linear operators

In this paper we study the W-weighted Drazin inverse of the bounded linear operators between Banach spaces and its representation theorem. Based on this representation, utilizing the spectral theory of Banach space operators, we derive an approximating expression of the W-weighted Drazin inverse and an error bound. Also, a perturbation theorem for the W-weighted Drazin inverse is uniformly obtained from the representation theorem.     

Note on the iterative methods for computing the generalized inverse over Banach spaces

In this paper, we construct iterative methods for computing the generalized inverse A over Banach spaces, and also for computing the generalized Drazin inverses a^d of Banach algebra element a. Moreover, we estimate the error bounds of the iterative methods for approximating A or a^d. Copyright © 2011 John Wiley & Sons, Ltd.     

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 (A) 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.     

Solving the Inverse Generalized Problem of Vertical Seismic Profiling

The dynamic inverse seismics problem is considered in a generalized setting. We investigate whether the wave propagation problem in a vertically nonhomogeneous medium is well-posed. We show that the regular part of the solution is an L ₂ function and the inverse problem, i.e., the determination of the reflection coefficient, is thus reducible to minimizing the error functional. The gradient of the functional is obtained in explicit form from the conjugate problem, and approximate formulas for its evaluation are derived. A regularization algorithm for the solution of the inverse problem is considered; simulation results using various excitation sources are reported.