Abstract and Applied Analysis

Abstract and Applied Analysis / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 412872 | https://doi.org/10.1155/2012/412872

Xiaoji Liu, Dengping Tu, Yaoming Yu, "The Expression of the Generalized Drazin Inverse of 𝐴−𝐶𝐵", Abstract and Applied Analysis, vol. 2012, Article ID 412872, 10 pages, 2012. https://doi.org/10.1155/2012/412872

The Expression of the Generalized Drazin Inverse of 𝐴−𝐶𝐵

Academic Editor: Ondřej Došlý
Received05 Aug 2011
Accepted05 Dec 2011
Published16 Feb 2012

Abstract

We investigate the generalized Drazin inverse of 𝐴−𝐶𝐵 over Banach spaces stemmed from the Drazin inverse of a modified matrix and present its expressions under some conditions.

1. Introduction

Let 𝒳 and 𝒴 be Banach spaces. We denote the set of all bounded linear operators from 𝒳 to 𝒴 by ℬ(𝒳,𝒴). In particular, we write ℬ(𝒳) instead of ℬ(𝒳,𝒳).

For any 𝐴∈ℬ(𝒳,𝒴), ℛ(𝐴) and 𝒩(𝐴) represent its range and null space, respectively. If 𝐴∈ℬ(𝒳), the symbols ğœŽ(𝐴) and acc(ğœŽ(𝐴)) stand for its spectrum and the set of all accumulation points of ğœŽ(𝐴), respectively.

Recall the concept of the generalized Drazin inverse introduced by Koliha [1] that the element 𝑇𝑑∈ℬ(𝒳) is called the generalized Drazin inverse of 𝑇∈ℬ(𝒳) provided it satisfies𝑇𝑇𝑑=𝑇𝑑𝑇,𝑇𝑑𝑇𝑇𝑑=𝑇𝑑,𝑇−𝑇2𝑇𝑑isquasinilpotent.(1.1) If it exists then it is unique. The Drazin index Ind(𝑇) of 𝑇 is the least positive integer 𝑘 if (𝑇−𝑇2𝑇𝑑)𝑘=0, and otherwise Ind(𝑇)=+∞.

From the definition of the generalized Drazin inverse, it is easy to see that if 𝑇 is a quasinilpotent operator, then 𝑇𝑑 exists and 𝑇𝑑=0. It is well known that the generalized Drazin inverse of 𝑇∈ℬ(𝒳) exists if and only if 0∉acc(ğœŽ(𝑇)) (see [1,Theorem  4.2]). If 𝑇 is generalized Drazin invertible, then the spectral idempotent 𝑇𝜋 of 𝑇 corresponding to 0 is given by 𝑇𝜋=𝐼−𝑇𝑇𝑑.

The generalized Drazin inverse is widely investigated because of its applications in singular differential difference equations, Markor chains, (semi-) iterative method numerical analysis (see, for example, [1–5, 7], and references therein).

In this paper, we aim to discuss the generalized Drazin inverse of 𝐴−𝐶𝐵 over Banach spaces. This question stems from the Drazin inverse of a modified matrix (see, e.g., [6]). In [3], Deng studied the generalized Drazin inverse of 𝐴−𝐶𝐵. Here we research the problem under more general conditions than those in [3]. Our results extend the relative results in [3, 4].

In this section, we will list some lemmas. In next section, we will present the expressions of the generalized Drazin inverse of 𝐴−𝐶𝐵. In final section, we illustrate a simple example.

Lemma 1.1 (see [4, Theorem  2.3]). Let 𝐴,𝐵∈ℬ(𝒳) be the generalized Drazin invertible. If 𝐴𝐵=0, then 𝐴+𝐵 is generalized Drazin invertible and (𝐴+𝐵)𝑑=ğµğœ‹âˆžî“ğ‘›=0𝐵𝑛𝐴𝑑𝑛+1+îƒ©âˆžî“ğ‘›=0𝐵𝑑𝑛+1𝐴𝑛𝐴𝜋.(1.2)

Lemma 1.2 (see [7, Theorem  5.1]). If 𝐴∈ℬ(𝒳) and 𝐵∈ℬ(𝒴) are generalized Drazin invertible and 𝐶∈ℬ(𝒴,𝒳), then âŽ›âŽœâŽœâŽâŽžâŽŸâŽŸâŽ ğ‘€=𝐴𝐶0𝐵(1.3) is also generalized Drazin invertible and 𝑀𝑑=âŽ›âŽœâŽœâŽğ´ğ‘‘ğ‘†ğ‘‚ğµğ‘‘âŽžâŽŸâŽŸâŽ ,(1.4) where 𝑆=𝐴2ğ‘‘îƒ©âˆžî“ğ‘›=0𝐴𝑛𝑑𝐶𝐵𝑛𝐵𝜋+ğ´ğœ‹îƒ©âˆžî“ğ‘›=0𝐴𝑛𝐶𝐵𝑛𝑑𝐵2𝑑−𝐴𝑑𝐶𝐵𝑑.(1.5)

2. Main Results

We start with our main result.

Theorem 2.1. Let 𝐴∈ℬ(𝒳) be the generalized Drazin invertible, 𝐶∈ℬ(𝒳,𝒴), and 𝐵∈ℬ(𝒴,𝒳). Suppose that there exists a 𝑃∈ℬ(𝒳) such that 𝐴𝑃=𝑃𝐴𝑃 and 𝐵𝑃=0. If 𝑅=(𝐼−𝑃)(𝐴−𝐶𝐵) and 𝐴𝑃 are generalized Drazin invertible, then 𝐴−𝐶𝐵 is generalized Drazin invertible and (𝐴−𝐶𝐵)𝑑=îƒ¬âˆžî“ğ‘›=0(𝐴𝑃)𝑑𝑛+1𝑅𝑛+𝑉𝑅𝑛−1+𝑉2𝑅𝑛−2𝑅𝜋−(𝐴𝑃)𝑑𝑉𝑅𝑑+𝑉2𝑅2𝑑+(𝐴𝑃)𝑑𝑉2𝑅𝑑+(𝐴𝑃)ğœ‹âˆžî“ğ‘›=0(𝐴𝑃)𝑛𝑅𝑑𝑛+1+𝑉𝑅𝑑𝑛+2+𝑉2𝑅𝑑𝑛+3,(2.1) where 𝑉=𝑃𝐴−𝑃𝐶𝐵−𝐴𝑃 and the symbols 𝑉𝑖𝑅𝑗=0,𝑖=1,2, if 𝑗<0.

Proof. Let 𝑆∶=𝐴𝑃 and 𝑇∶=(𝐴−𝐶𝐵)(𝐼−𝑃). Then ,𝑇𝑆=(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴𝑃=0(2.2)𝑅𝑃=(𝐼−𝑃)(𝐴−𝐶𝐵)𝑃=0,(2.3)𝐴−𝐶𝐵=𝐴𝑃+𝐴(𝐼−𝑃)−𝐶𝐵(𝐼−𝑃)=𝑆+𝑇(2.4) since 𝐴𝑃=𝑃𝐴𝑃 and 𝐵𝑃=0. So, by Lemma 1.1, (𝑇+𝑆)𝑑=ğ‘†ğœ‹âˆžâˆ‘ğ‘›=0𝑆𝑛𝑇𝑑𝑛+1+âˆžâˆ‘ğ‘›=0𝑆𝑑𝑛+1𝑇𝑛𝑇𝜋.(2.5)
Next, we will give the representations of 𝑇𝑑, 𝑇𝑛, and 𝑇𝑛𝑑. In order to obtain the expression of 𝑇𝑑, rewrite 𝑇 as 𝑇=𝑅+𝑃𝐴−𝑃𝐶𝐵−𝑃𝐴𝑃=𝑅+𝑉.(2.6) Since 𝑉𝑃=𝑃𝐴𝑃−𝐴𝑃2=𝑃𝐴𝑃(𝐼−𝑃), 𝑉2𝑃=(𝑃𝐴−𝑃𝐶𝐵−𝐴𝑃)𝑃𝐴𝑃(𝐼−𝑃)=(𝑃𝐴𝑃𝐴𝑃−𝐴𝑃𝑃𝐴𝑃)(𝐼−𝑃)=0,(2.7) and then 𝑉𝑛=0 for 𝑛>2 since 𝑉=𝑃𝐴−𝐶𝐵−𝐴𝑃. So 𝑉𝑑 exists and 𝑉𝑑=0. By (2.3), 𝑅𝑉=𝑅𝑃(𝐴−𝐶𝐵−𝐴𝑃)=0 and then 𝑅𝑑𝑉=𝑅𝑑𝑅𝑑𝑅𝑉=0. So, by Lemma 1.1, 𝑇𝑑=(𝑅+𝑉)𝑑=𝑅𝑑+𝑉𝑅2𝑑+𝑉2𝑅3𝑑,(2.8) and then 𝑇𝑇𝑑=𝑅𝑅𝑑+𝑉𝑅𝑑+𝑉2𝑅2𝑑.(2.9) Since 𝑅(𝑅+𝑉)𝑘=𝑅𝑘+1 and 𝑉2(𝑅+𝑉)𝑘=𝑉2𝑅𝑘 for 𝑘≥1, 𝑇𝑛=(𝑅+𝑉)𝑛=𝑅2+𝑉𝑅+𝑉2(𝑅+𝑉)𝑛−2=𝑅𝑛+𝑉𝑅𝑛−1+𝑉2𝑅𝑛−2,𝑛≥2.(2.10) From 𝑅𝑑𝑉=0, it is easy to verify that 𝑇𝑛𝑑=𝑅𝑑+𝑉𝑅2𝑑+𝑉2𝑅3𝑑𝑛=𝑅𝑛𝑑+𝑉𝑅𝑑𝑛+1+𝑉2𝑅𝑑𝑛+2.(2.11) Hence, îƒ©âˆžî“ğ‘›=0𝑆𝑑𝑛+1𝑇𝑛𝑇𝜋=(𝐴𝑃)𝑑𝐼+(𝐴𝑃)𝑑(𝑅+𝑉)+(𝐴𝑃)2𝑑𝑅2+𝑉𝑅+𝑉2×𝑅𝜋−𝑉𝑅𝑑−𝑉2𝑅2𝑑+âˆžî“ğ‘›=3(𝐴𝑃)𝑑𝑛+1𝑅𝑛+𝑉𝑅𝑛−1+𝑉2𝑅𝑛−2𝑅𝜋=(𝐴𝑃)𝑑𝐼+(𝐴𝑃)𝑑(𝑅+𝑉)+(𝐴𝑃)2𝑑𝑅2+𝑉𝑅+𝑉2𝑅𝜋−(𝐴𝑃)𝑑𝑉𝑅𝑑+𝑉2𝑅2𝑑+(𝐴𝑃)𝑑𝑉2𝑅𝑑+âˆžî“ğ‘›=3(𝐴𝑃)𝑑𝑛+1𝑅𝑛+𝑉𝑅𝑛−1+𝑉2𝑅𝑛−2𝑅𝜋,ğ‘†ğœ‹âˆžî“ğ‘›=0𝑆𝑛𝑇𝑑𝑛+1=(𝐴𝑃)ğœ‹âˆžî“ğ‘›=0(𝐴𝑃)𝑛𝑅𝑑𝑛+1+𝑉𝑅𝑑𝑛+2+𝑉2𝑅𝑑𝑛+3.(2.12) Therefore, we reach (2.1).

When Ind(𝐴𝑃),Ind(𝑅)<+∞, we have the following corollary.

Corollary 2.2. Let 𝐴∈ℬ(𝒳) be generalized Drazin invertible. 𝐶∈ℬ(𝒳,𝒴), and 𝐵∈ℬ(𝒴,𝒳). Suppose that there exists a 𝑃∈ℬ(𝒳) such that 𝐴𝑃=𝑃𝐴𝑃 and 𝐵𝑃=0. If 𝑅=(𝐼−𝑃)(𝐴−𝐶𝐵) and 𝐴𝑃 are generalized Drazin invertible and Ind(𝑅)=𝑘<+∞ and Ind(𝐴𝑃)=ℎ<+∞, then 𝐴−𝐶𝐵 is generalized Drazin invertible and (𝐴−𝐶𝐵)𝑑=𝑘−1𝑛=0(𝐴𝑃)𝑑𝑛+1𝑅𝑛+𝑉𝑅𝑛−1+𝑉2𝑅𝑛−2𝑅𝜋−(𝐴𝑃)𝑑𝑉𝑅𝑑+𝑉2𝑅2𝑑+(𝐴𝑃)𝑑𝑉2𝑅𝑑+(𝐴𝑃)ğœ‹â„Žâˆ’1𝑛=0(𝐴𝑃)𝑛𝑅𝑑𝑛+1+𝑉𝑅𝑑𝑛+2+𝑉2𝑅𝑑𝑛+3,(2.13) where 𝑉=𝑃𝐴−𝑃𝐶𝐵−𝐴𝑃 and the symbols 𝑉𝑖𝑅𝑗=0,𝑖=1,2, if 𝑗<0.

If an operator 𝑇 is quasinilpotent, 𝑇𝑑=0 and 𝑇𝜋=𝐼. So, the following corollary follows from Theorem 2.1.

Corollary 2.3. Let 𝐴∈ℬ(𝒳) be generalized Drazin invertible, 𝐶∈ℬ(𝒳,𝒴), and 𝐵∈ℬ(𝒴,𝒳). Suppose that there exists a 𝑃∈ℬ(𝒳) such that 𝐴𝑃=𝑃𝐴𝑃 and 𝐵𝑃=0. If 𝑅=(𝐼−𝑃)(𝐴−𝐶𝐵) is generalized Drazin invertible and 𝐴𝑃 is a quasinilpotent operator, then 𝐴−𝐶𝐵 is generalized Drazin invertible and (𝐴−𝐶𝐵)𝑑=âˆžî“ğ‘›=0(𝐴𝑃)𝑛𝑅𝑑𝑛+1+𝑉𝑅𝑑𝑛+2+𝑉2𝑅𝑑𝑛+3,(2.14) where 𝑉=𝑃𝐴−𝑃𝐶𝐵−𝐴𝑃.

Theorem 2.4. Let 𝐴∈ℬ(𝒳) be generalized Drazin invertible, 𝐶∈ℬ(𝒳,𝒴), and 𝐵∈ℬ(𝒴,𝒳). Suppose that there exists an idempotent 𝑃∈ℬ(𝒳) such that 𝑃𝐴=𝑃𝐴𝑃 and 𝐵𝑃=𝐵. If 𝑅=𝑃(𝐴−𝐶𝐵) is generalized Drazin invertible, then 𝐴−𝐶𝐵 is generalized Drazin invertible and (𝐴−𝐶𝐵)𝑑=𝑅𝑑+𝐴𝑑(𝐼−𝑃)+âˆžî“ğ‘›=0𝐴𝑑𝑛+2(𝐼−𝑃)(𝐴−𝐶𝐵)𝑃(𝐴−𝐶𝐵)𝑛𝑅𝜋+ğ´ğœ‹âˆžî“ğ‘›=0𝐴𝑛(𝐼−𝑃)(𝐴−𝐶𝐵)𝑃𝑅𝑑𝑛+2−𝐴𝑑(𝐼−𝑃)(𝐴−𝐶𝐵)𝑅𝑑.(2.15)

Proof. Since 𝑃2=𝑃, we have ⨁𝒳=ℛ(𝑃)𝒩(𝑃) and can write 𝑃 in the following matrix form: ⎛⎜⎜⎝⎞⎟⎟⎠.𝑃=𝐼000(2.16) The condition 𝑃𝐴=𝑃𝐴𝑃, therefore, yields the matrix form of 𝐴 as follows: âŽ›âŽœâŽœâŽğ´ğ´=10𝐴3𝐴2⎞⎟⎟⎠.(2.17) From ğœŽ(𝐴)=ğœŽ(𝐴1)âˆªğœŽ(𝐴2) and the hypothesis that 𝐴𝑑 exists, 𝐴1∈ℬ(ℛ(𝑃)) and 𝐴2∈ℬ(𝒩(𝑃)) are generalized Drazin invertible since 0∉acc(ğœŽ(𝐴)) if and only if 0∉acc(ğœŽ(𝐴1)) and 0∉acc(ğœŽ(𝐴2)). And, by Lemma 1.2, 𝐴𝑑=âŽ›âŽœâŽœâŽğ´ğ‘‘10𝑊𝐴𝑑2⎞⎟⎟⎠,(2.18) where 𝑊 is some operator. Since âŽ›âŽœâŽœâŽğ´(𝐼−𝑃)=000𝐴2⎞⎟⎟⎠,(2.19)(𝐴(𝐼−𝑃))𝑑 exists and (𝐴(𝐼−𝑃))𝑑=⎛⎜⎜⎝000𝐴𝑑2⎞⎟⎟⎠=𝐴𝑑(𝐼−𝑃).(2.20)
To use Theorem 2.1 to complete the proof, let 𝑄=(𝐼−𝑃). So 𝑅=(𝐼−𝑄)(𝐴−𝐶𝐵) and 𝐴𝑄 are generalized Drazin invertible. And from the conditions 𝑃𝐴=𝑃𝐴𝑃 and 𝐵𝑃=𝐵, we can obtain 𝐴𝑄=𝑄𝐴𝑄 and 𝐵𝑄=0. Thus, by Theorem 2.1, we have (𝐴−𝐶𝐵)𝑑=(𝐴𝑄)𝑑𝑅𝜋+(𝐴𝑄)2𝑑(𝑅+𝑉)𝑅𝜋+îƒ¬âˆžî“ğ‘›=2(𝐴𝑄)𝑑𝑛+1𝑅𝑛+𝑉𝑅𝑛−1+𝑉2𝑅𝑛−2𝑅𝜋−(𝐴𝑄)𝑑𝑉𝑅𝑑+𝑉2𝑅2𝑑+(𝐴𝑄)𝑑𝑉2𝑅𝑑+(𝐴𝑄)𝜋𝑅𝑑+𝑉𝑅2𝑑+𝑉2𝑅3𝑑+(𝐴𝑄)ğœ‹âˆžî“ğ‘›=1(𝐴𝑃)𝑛𝑅𝑑𝑛+1+𝑉𝑅𝑑𝑛+2+𝑉2𝑅𝑑𝑛+3,(2.21) where 𝑉=𝑄𝐴−𝑄𝐶𝐵−𝐴𝑄.
Since 𝑃2=𝑃 and 𝑄2=𝑄 and then 𝑉𝑄=0 and 𝑉=𝑄𝑉. So 𝑉2=0. Note that 𝑄𝑅=0 and then 𝑄𝑅𝑑=0 and (𝐴𝑄)𝑑𝑅=0. Thus it follows from (2.21) that (𝐴−𝐶𝐵)𝑑=(𝐴𝑄)𝑑+(𝐴𝑄)2𝑑𝑉𝑅𝜋+îƒ¬âˆžî“ğ‘›=2(𝐴𝑄)𝑑𝑛+1𝑉𝑅𝑛−1𝑅𝜋−(𝐴𝑄)𝑑𝑉𝑅𝑑+𝑅𝑑+(𝐴𝑄)𝜋𝑉𝑅2𝑑+(𝐴𝑄)ğœ‹âˆžî“ğ‘›=1(𝐴𝑄)𝑛𝑉𝑅𝑑𝑛+2=(𝐴𝑄)𝑑+îƒ¬âˆžî“ğ‘›=0(𝐴𝑄)𝑑𝑛+2𝑉𝑅𝑛𝑅𝜋−(𝐴𝑄)𝑑𝑉𝑅𝑑+𝑅𝑑+(𝐴𝑄)ğœ‹âˆžî“ğ‘›=0(𝐴𝑄)𝑛𝑉𝑅𝑑𝑛+2.(2.22) Since 𝑉=𝑄(𝐴−𝐶𝐵)−(𝐴−𝐶𝐵)𝑄=(𝐴−𝐶𝐵)(𝐼−𝑄)−(𝐼−𝑄)(𝐴−𝐶𝐵), 𝑉𝑅=𝑄(𝐴−𝐶𝐵)𝑅 and 𝑄𝑉=𝑄(𝐴−𝐶𝐵)(𝐼−𝑄). Note that 𝑅𝑛=𝑃(𝐴−𝐶𝐵)𝑛 and (𝐴𝑄)𝑛=𝐴𝑛𝑄. Substituting 𝑉 and 𝑄=𝐼−𝑃 in (2.22) yields (2.15).

Adding the condition 𝑃𝐶=𝐶 in Theorem 2.4 yields a result below.

Corollary 2.5. Let 𝐴∈ℬ(𝒳) be generalized Drazin invertible, 𝐶∈ℬ(𝒳,𝒴), and 𝐵∈ℬ(𝒵,𝒳). Suppose that there exists an idempotent 𝑃∈ℬ(𝒳) such that 𝑃𝐴=𝑃𝐴𝑃, 𝐵𝑃=𝐵, and 𝑃𝐶=𝐶. If 𝑅=𝑃(𝐴−𝐶𝐵) is generalized Drazin invertible, then 𝐴−𝐶𝐵 is generalized Drazin invertible and (𝐴−𝐶𝐵)𝑑=𝑅𝑑+𝐴𝑑(𝐼−𝑃)+âˆžî“ğ‘›=0𝐴𝑑𝑛+2(𝐼−𝑃)𝐴𝑃(𝐴−𝐶𝐵)𝑛𝑅𝜋+ğ´ğœ‹âˆžî“ğ‘›=0𝐴𝑛(𝐼−𝑃)𝐴𝑃𝑅𝑑𝑛+2−𝐴𝑑(𝐼−𝑃)𝐴𝑅𝑑.(2.23)

Adding the condition 𝑃𝐶=0 in Theorem 2.4 yields 𝑅=𝑃𝐴. So similar to the proof of (𝐴(𝐼−𝑃))𝑑=𝐴𝑑(𝐼−𝑃) in Theorem 2.4, we can gain (𝑃𝐴)𝑑=𝑃𝐴𝑑.

Corollary 2.6. Let 𝐴∈ℬ(𝒳) be generalized Drazin invertible, 𝐶∈ℬ(𝒳,𝒴), and 𝐵∈ℬ(𝒵,𝒳). Suppose that there exists an idempotent 𝑃∈ℬ(𝒳) such that 𝑃𝐴=𝑃𝐴𝑃, 𝐵𝑃=𝐵, and 𝑃𝐶=0; then 𝐴−𝐶𝐵 is generalized Drazin invertible and (𝐴−𝐶𝐵)𝑑=𝐴𝑑+âˆžî“ğ‘›=0𝐴𝑑𝑛+2(𝐼−𝑃)(𝐴−𝐶𝐵)𝑃𝐴𝑛𝐴𝜋+ğ´ğœ‹âˆžî“ğ‘›=0𝐴𝑛(𝐼−𝑃)(𝐴−𝐶𝐵)𝑃𝐴𝑑𝑛+2−𝐴𝑑(𝐼−𝑃)(𝐴−𝐶𝐵)𝑃𝐴𝑑.(2.24)

Analogously, we can deduce Theorem 2.7 and Corollary 2.9 below.

Theorem 2.7. Let 𝐴∈ℬ(𝒳) be generalized Drazin invertible, 𝐶∈ℬ(𝒳,𝒴), and 𝐵∈ℬ(𝒴,𝒳). Suppose that there exists an idempotent 𝑃∈ℬ(𝒳) such that 𝐴𝑃=𝑃𝐴𝑃 and 𝑃𝐶=𝐶. If 𝑅=(𝐴−𝐶𝐵)𝑃 is generalized Drazin invertible, then 𝐴−𝐶𝐵 is generalized Drazin invertible and (𝐴−𝐶𝐵)𝑑=𝑅𝑑+(𝐼−𝑃)𝐴𝑑+âˆžî“ğ‘›=0𝑅𝑑𝑛+2𝑃(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴𝑛𝐴𝜋+ğ‘…ğœ‹âˆžî“ğ‘›=0(𝐴−𝐶𝐵)𝑛𝑃(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴𝑑𝑛+2−𝑅𝑑(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴𝑑.(2.25)

Remark 2.8 (see [4, Theorem  2.4]). It is a special case of Theorem 2.7.

Corollary 2.9. Let 𝐴∈ℬ(𝒳) be generalized Drazin invertible, 𝐶∈ℬ(𝒳,𝒴), and 𝐵∈ℬ(𝒵,𝒳). Suppose that there exists an idempotent 𝑃∈ℬ(𝒳) such that 𝐴𝑃=𝑃𝐴𝑃, 𝑃𝐶=𝐶, and 𝐵𝑃=0; then 𝐴−𝐶𝐵 is generalized Drazin invertible and (𝐴−𝐶𝐵)𝑑=𝐴𝑑+âˆžî“ğ‘›=0𝐴𝑑𝑛+2𝑃(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴𝑛𝐴𝜋+ğ´ğœ‹âˆžî“ğ‘›=0𝐴𝑛𝑃(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴𝑑𝑛+2−𝐴𝑑𝑃(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴𝑑.(2.26)

Similar to Theorem 2.1 and Corollary 2.2, we can show the following two results.

Theorem 2.10. Let 𝐴∈ℬ(𝒳) be generalized Drazin invertible, 𝐶∈ℬ(𝒳,𝒴), and 𝐵∈ℬ(𝒴,𝒳). Suppose that there exists a 𝑃∈ℬ(𝒳) such that 𝑃𝐴=𝑃𝐴𝑃 and 𝑃𝐶=0. If 𝑅=(𝐴−𝐶𝐵)(𝐼−𝑃) and 𝑃𝐴 are generalized Drazin invertible, then 𝐴−𝐶𝐵 is generalized Drazin invertible and (𝐴−𝐶𝐵)𝑑=ğ‘…ğœ‹âˆžî“ğ‘›=0𝑅𝑛+𝑅𝑛−1𝑉+𝑅𝑛−2𝑉2(𝑃𝐴)𝑑𝑛+1−𝑅𝑑𝑉+𝑅2𝑑𝑉2+𝑅𝑑𝑉2(𝑃𝐴)𝑑(𝑃𝐴)𝑑+îƒ¬âˆžî“ğ‘›=0𝑅𝑑𝑛+1+𝑅𝑑𝑛+2𝑉+𝑅𝑑𝑛+3𝑉2(𝑃𝐴)𝑛(𝑃𝐴)𝜋,(2.27) where 𝑉=𝐴𝑃−𝐶𝐵𝑃−𝑃𝐴 and the symbols 𝑅𝑖𝑉𝑗=0,𝑗=1,2, if 𝑖<0.

Corollary 2.11. Let 𝐴∈ℬ(𝒳) be generalized Drazin invertible. 𝐶∈ℬ(𝒳,𝒴), and 𝐵∈ℬ(𝒴,𝒳). Suppose that there exists a 𝑃∈ℬ(𝒳) such that 𝑃𝐴=𝑃𝐴𝑃 and 𝑃𝐶=0. If 𝑅=(𝐴−𝐶𝐵)(𝐼−𝑃) and 𝑃𝐴 are generalized Drazin invertible and Ind(𝑅)=𝑘<+∞ and Ind(𝑃𝐴)=ℎ<+∞, then 𝐴−𝐶𝐵 is generalized Drazin invertible and (𝐴−𝐶𝐵)𝑑=𝑅𝜋𝑘−1𝑛=0𝑅𝑛+𝑅𝑛−1𝑉+𝑅𝑛−2𝑉2(𝑃𝐴)𝑑𝑛+1−𝑅𝑑𝑉+𝑅2𝑑𝑉2+𝑅𝑑𝑉2(𝑃𝐴)𝑑(𝑃𝐴)𝑑+ℎ−1𝑛=0𝑅𝑑𝑛+1+𝑅𝑑𝑛+2𝑉+𝑅𝑑𝑛+3𝑉2(𝑃𝐴)𝑛(𝑃𝐴)𝜋,(2.28) where 𝑉=𝐴𝑃−𝐶𝐵𝑃−𝑃𝐴 and the symbols 𝑅𝑖𝑉𝑗=0,𝑗=1,2, if 𝑖<0.

When 𝑃𝐴=𝐴𝑃 and 𝑃2=𝑃 in Theorem 2.10, we can obtain the following result since 𝑅𝑛=(𝐴−𝐶𝐵)𝑛(𝐼−𝑃).

Corollary 2.12 (see [3, Theorem  4.3]). Let 𝐴∈ℬ(𝒳) be the generalized Drazin invertible, 𝐶∈ℬ(𝒳,𝒴), and 𝐵∈ℬ(𝒴,𝒳). Suppose that there exists an idempotent 𝑃∈ℬ(𝒳) commuting with 𝐴 such that 𝑃𝐶=0. If 𝑅=(𝐴−𝐶𝐵)(𝐼−𝑃) is generalized Drazin invertible, then 𝐴−𝐶𝐵 is the generalized Drazin invertible and (𝐴−𝐶𝐵)𝑑=𝑅𝑑+𝑃𝐴𝑑−𝑅𝑑𝑉𝐴𝑑+ğ‘…ğœ‹âˆžî“ğ‘›=0(𝐴−𝐶𝐵)𝑛𝑉𝐴𝑑𝑛+2+âˆžî“ğ‘›=0𝑅𝑑𝑛+2𝑉𝐴𝑛𝐴𝜋,(2.29) where 𝑉=−𝐶𝐵𝑃.

3. Example

Before ending this paper, we give an example as follows.

Example 3.1. Let ⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝100⎞⎟⎟⎟⎟⎟⎟⎠.𝐴=12410−1100−1100000,𝐵=0001,𝐶=−1(3.1) Then ⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠.𝐶𝐵=0001000−100000000,𝐴−𝐶𝐵=12400−1110−1100000(3.2) We will compute the Drazin inverse of 𝐴−𝐶𝐵. To do this, we choose the matrix ⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠.𝑃=100001000−1200000(3.3) Apparently, 𝑃 is not idempotent and 𝑃𝐴≠𝐴𝑃. But 𝐵𝑃=0 and ⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠.𝐴𝑃=𝑃𝐴𝑃=1−2800−2200−2200000(3.4) Obviously, Ind(𝐴𝑃)=2. Computing âŽ›âŽœâŽœâŽœâŽœâŽœâŽœâŽâŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽŸâŽ ğ‘…=(𝐼−𝑃)(𝐴−𝐶𝐵)=0000000000010000,𝑅𝑑=⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠,0000000000000000(3.5)⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠,𝑉=𝑃𝐴−𝑃𝐶𝐵−𝐴𝑃=04−4001−1101−1−10000(3.6) we have Ind(𝑅)=2. So, by Corollary 2.2, (𝐴−𝐶𝐵)𝑑=⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠.1−410−4000000000000(3.7)

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (11061005), the Ministry of Education Science and Technology Key Project under Grant no. 210164, and Grants HCIC201103 of Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis Open Fund.

References

  1. J. J. Koliha, “A generalized drazin inverse,” Glasgow Mathematical Journal, vol. 38, no. 3, pp. 367–381, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. N. Castro-González, E. Dopazo, and M. F. Martínez-Serrano, “On the Drazin inverse of the sum of two operators and its application to operator matrices,” Journal of Mathematical Analysis and Applications, vol. 350, no. 1, pp. 207–215, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. C. Y. Deng, “On the invertibility of the operator A−XB,” Numerical Linear Algebra with Applications, vol. 16, no. 10, pp. 817–831, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. D. S. Djordjević and Y. Wei, “Additive results for the generalized Drazin inverse,” Journal of the Australian Mathematical Society, vol. 73, no. 1, pp. 115–125, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. X. Liu, Y. Yu, and C. Hu, “The iterative methods for computing the generalized inverse AT,S(2) of the bounded linear operator between Banach spaces,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 391–410, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. Y. Wei, “The Drazin inverse of a modified matrix,” Applied Mathematics and Computation, vol. 125, no. 2-3, pp. 295–301, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. D. S. Djordjević and P. S. Stanimirović, “On the generalized Drazin inverse and generalized resolvent,” Czechoslovak Mathematical Journal, vol. 51, no. 3, pp. 617–634, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 2012 Xiaoji Liu et al. 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.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views687
Downloads475
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.