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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 356105, 5 pages
http://dx.doi.org/10.1155/2013/356105
Research Article

Higher-Order Convergent Iterative Method for Computing the Generalized Inverse over Banach Spaces

College of Science, Guangxi University for Nationalities, Nanning 530006, China

Received 16 May 2013; Revised 24 September 2013; Accepted 24 September 2013

Academic Editor: Qing-Wen Wang

Copyright © 2013 Xiaoji Liu and Fu Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A higher-order convergent iterative method is provided for calculating the generalized inverse over Banach spaces. We also use this iterative method for computing the generalized Drazin inverse in Banach algebra. Moreover, we estimate the error bounds of the iterative methods for approximating or .

1. Introduction

It is well known that the outer generalized inverse has been widely used in various fields, for instance, in statistics, control theory, power systems, nonlinear equations, optimization and numerical analysis, and so on (see [115]). Recently, in [16], the authors discussed the iteration (1) for computing of a given matrix.

Throughout this paper, let and be arbitrary Banach spaces. Then, the symbol denotes the set of all bounded linear operators from to , in particular, . For any , we denote its range, null space, and norm by , , and , respectively. Further, we say that is regular if there exists an such that and that has a  (or outer) inverse if there exists an such that . If , then we denote its spectrum and spectral radius by and , respectively. Let the symbol denote that is a subspace of . If and , then the restriction of on is defined by , . Let with . Then, the symbol stands for an operator that is called a projection from onto if it is a bounded linear map from onto and . It is well known that a closed subspace of a Banach space is complemented in if and only if there exists a projection from onto .

Let be close; there exists a unique operator such that Then, is called the Moore-Penrose inverse of , denoted by . It is well known that is regular is closed exists.

Throughout this paper, let be a complex Banach algebra with the unit . The symbols and , respectively, stand for the left and right annihilators of in . Let be idempotent. Then, is a subalgebra of with unit . Thus, for , if there exists an element such that , then we say that is invertible in , and is denoted by . Recall that an element is the generalized Drazin inverse of (or Koliha-Drazin inverse of ), if the following hold: If the generalized Drazin inverse of exists, then it is denoted by (see [15] for more details). In particular, if and , then is called the group inverse of and is denoted by .

In [17], W. G. Li and Z. Li defined the iterative formula In [18], Chen and Wang extended the iterative method (3) proposed by W. G. Li and Z. Li to compute the Moore-Penrose inverse of a matrix. In [19], Liu et al provided the higher-order convergent iterative method (3) in order to calculate the generalized inverse of a given matrix. In this paper, we will extend the iterative method proposed by W. G. Li and Z. Li in [17] to compute the -inverse, generalized inverse over Banach space and also consider the iterative scheme for computing the generalized Drazin inverse in Banach algebra.

The paper is organized as follows. Some lemmas will be presented in the remainder of this section. In Section 2, we consider iterative scheme of [19] to compute the generalized inverses in Banach space. In Section 3, we present iterative formulas for computing the generalized Drazin inverse of Banach algebra element .

The following lemmas are needed in what follows.

Lemma 1 (see [14, Chapter 1]). Let . Then(i) is a nonempty closed subset of .(ii)(Spectral mapping theorem for polynomials) if is a polynomial, then (iii) if and only if .

Lemma 2 (see [15, Section 4]). Let and be Banach spaces, and let , and , respectively, be closed subspaces of and . Then, the following statements are equivalent.(i) has a -inverse such that and .(ii) is a complemented subspace of , is closed, is invertible, and .
In the case when or holds, is unique and is denoted by .

Lemma 3. Suppose that the conditions of Lemma 2 are satisfied. Then, and where . Moreover, for any , ; .

2. Higher-Order Convergent Iterative Method for Computing the Generalized Inverse over Banach Spaces

In this section, we will consider higher-order convergent iterative method for computing the generalized inverse over Banach spaces. First, we deduce convergent conditions and error bounds of our iterative methods.

Theorem 4. Let , , and let and both be complemented subspaces, respectively, with , . Define the sequence in in the following way: it converges to and with if and only if for some scalar , where is an arbitrary positive integer, , and is projection from onto . Moreover,(i)if , then exists if and only if for ;(ii)if , then exists.
In particular, if exists, and , then

Proof. From (5), we obtain Note that , from (7). Similarly, it is easy to prove that , .
Since and , then for .
From (5), we have By (8), we get . Premultiplying (9) by , then (9) yields
Next, we will investigate the necessary and sufficient condition for the convergent property of the iterative scheme (5). Assume that exists; denote by and . Then, . Thus, and ; we obtain , a projection from onto , and by (8).
Since , and by (10). Thus, and then .
Conversely, suppose that for some scalar , where denotes a projection from to and is complement. Then, by (10), and therefore and .
By (8), because is close, and then . Hence, we obtain . Thus, . It is easy to know that if , then . Thus, exists.
Assume that exists. By (8), because is closed complement. If , then for some . Thus, . Thus, . Therefore, and then by Lemma 2. Consequently, .
Since , . Thus, postmultiplying (10) by yields Since , we have Hence, we get (6).

Similarly, we have the dual result as below.

Theorem 5. Let , , and let and both be closed, respectively, with , . Define the sequence such that it converges to and with if and only if for some scalar , where is an arbitrary positive integer, , and is a projection from onto . Moreover,(i)if , then exists if and only if for ;(ii)if , then exists.
In particular, if exists, and , then , .

Remark 6. Now, we consider how to choose a suitable scalar for the iterative scheme (5) such that it converges more faster to .
Since and for any , . Therefore, if and only if and . Thus, there exists with .
Let and , where , . Then, . Thus, if and only if and .
Hence, by and satisfing , we have . In practice, once such a is determined, is taken to satisfy and . If is a subset of , then we take satisfying and , where , so as to ensure that .
Assume that hold. In the following, we will obtain the best value such that minimizes for achieving good convergence. Unfortunately, it may be rather difficult. If is a subset of and analogous to [8, Example 4.1], we can have In practice, because is not easily obtained, we often utilize instead of it in the above inequations and (15) to choose , which is followed from .

3. Higher-Order Convergent Iterative Method for Computing the Generalized Inverse over Banach Algebra

In the section, we will investigate a higher-order convergent iterative method for computing the generalized Drazin inverse over Banach algebra.

Theorem 7. Let , be idempotents with , and with . Define the sequence in such that where and . Then the iteration (16) converges to and if and only if . In this case, assume that . Then(i) exists and the iteration (16) converges to if and only if is quasinilpotent in ;(ii)if , then .

Proof. (i) By and , it implies that . By induction on , we have By (16), we obtain From (17) and (18), we get
The right-hand side of the last equality of (19) implies that By (20), we easily have .
Conversely, assume that . Since and , , and then is invertible in . We will show that is invertible in . Clearly, , if for some , then . Hence, and . Hence, , and then is invertible in .(i) In the following, we will consider the result (i). It is similar to the deduction of (10), we can write (16) as Thus, postmultiplying (21) by yields to By Lemma 1 and (22), we prove that converges to and is denoted by . Therefore, in ; then Thus, we obtain . Since , we have that if and only if is quasinilpotent in .(ii) Since is idempotent and , and then Therefore, we obtain in . By (10), we have Hence, by the argument in (i) and (26), we have Taking limit in (27), then it reduces to (ii).

Similarly, we have the following.

Theorem 8. Let be idempotents with , and with . Define the sequence in such that where and . Then, the iteration (28) converges to and if and only if . In this case, assume that . Then(i) exists and the iteration (28) converges to if and only if is quasinilpotent in ;(ii)if , then .

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11061005) and the Ministry of Education Science and Grants (HCIC201103) of Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis Open Fund.

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