The Expression of the Generalized Drazin Inverse of 𝐴−𝐶𝐵

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 𝜎(𝐴) and acc(𝜎(𝐴)) stand for its spectrum and the set of all accumulation points of 𝜎(𝐴), respectively.

Recall the concept of the generalized Drazin inverse introduced by Koliha [1] that the element 𝑇_𝑑∈ℬ(𝒳) is called the generalized Drazin inverse of 𝑇∈ℬ(𝒳) provided it satisfies𝑇𝑇_𝑑=𝑇_𝑑𝑇,𝑇_𝑑𝑇𝑇_𝑑=𝑇_𝑑,𝑇−𝑇²𝑇_𝑑isquasinilpotent.(1.1) If it exists then it is unique. The Drazin index Ind(𝑇) of 𝑇 is the least positive integer 𝑘 if (𝑇−𝑇²𝑇_𝑑)^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 𝑇_𝑑 exists and 𝑇_𝑑=0. It is well known that the generalized Drazin inverse of 𝑇∈ℬ(𝒳) exists if and only if 0∉acc(𝜎(𝑇)) (see [1,Theorem  4.2]). If 𝑇 is generalized Drazin invertible, then the spectral idempotent 𝑇^𝜋 of 𝑇 corresponding to 0 is given by 𝑇^𝜋=𝐼−𝑇𝑇^𝑑.

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=𝐵^𝜋∞_𝑛=0𝐵^𝑛𝐴^𝑑𝑛+1+^∞_𝑛=0𝐵^𝑑𝑛+1𝐴^𝑛𝐴^𝜋.(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 𝑀=⎛⎜⎜⎝𝐴0𝐶𝐵⎞⎟⎟⎠(1.3) is also generalized Drazin invertible and 𝑀^𝑑=⎛⎜⎜⎝𝐴^𝑑𝑂𝐵𝑆_𝑑⎞⎟⎟⎠,(1.4) where 𝑆=𝐴^2𝑑^∞_𝑛=0𝐴^𝑛𝑑𝐶𝐵^𝑛𝐵^𝜋+𝐴^𝜋^∞_𝑛=0𝐴^𝑛𝐶𝐵^𝑛𝑑𝐵^2𝑑−𝐴_𝑑𝐶𝐵_𝑑.(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=^∞_𝑛=0(𝐴𝑃)^𝑑𝑛+1𝑅^𝑛+𝑉𝑅^𝑛−1+𝑉²𝑅^𝑛−2𝑅^𝜋−+(𝐴(𝐴𝑃𝑃)_𝑑)𝑉_𝜋∞𝑅_𝑑+𝑉²𝑅^2𝑑+(𝐴𝑃)_𝑑𝑉²𝑅_𝑑_𝑛=0(𝐴𝑃)^𝑛𝑅^𝑑𝑛+1+𝑉𝑅^𝑑𝑛+2+𝑉²𝑅^𝑑𝑛+3,(2.1) where 𝑉=𝑃𝐴−𝑃𝐶𝐵−𝐴𝑃 and the symbols 𝑉^𝑖𝑅^𝑗=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=𝑆∑^𝜋∞_𝑛=0𝑆^𝑛𝑇^𝑑𝑛+1+∑^∞_𝑛=0𝑆^𝑑𝑛+1𝑇^𝑛𝑇^𝜋.(2.5)
Next, we will give the representations of 𝑇_𝑑, 𝑇^𝑛, and 𝑇^𝑛𝑑. In order to obtain the expression of 𝑇_𝑑, 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 𝑉^𝑛=0 for 𝑛>2 since 𝑉=𝑃𝐴−𝐶𝐵−𝐴𝑃. So 𝑉_𝑑 exists and 𝑉_𝑑=0. By (2.3), 𝑅𝑉=𝑅𝑃(𝐴−𝐶𝐵−𝐴𝑃)=0 and then 𝑅_𝑑𝑉=𝑅_𝑑𝑅_𝑑𝑅𝑉=0. So, by Lemma 1.1, 𝑇_𝑑=(𝑅+𝑉)_𝑑=𝑅_𝑑+𝑉𝑅^2𝑑+𝑉²𝑅^3𝑑,(2.8) and then 𝑇𝑇_𝑑=𝑅𝑅_𝑑+𝑉𝑅_𝑑+𝑉²𝑅^2𝑑.(2.9) Since 𝑅(𝑅+𝑉)^𝑘=𝑅^𝑘+1 and 𝑉²(𝑅+𝑉)^𝑘=𝑉²𝑅^𝑘 for 𝑘≥1, 𝑇^𝑛=(𝑅+𝑉)^𝑛=𝑅²+𝑉𝑅+𝑉²(𝑅+𝑉)^𝑛−2=𝑅^𝑛+𝑉𝑅^𝑛−1+𝑉²𝑅^𝑛−2,𝑛≥2.(2.10) From 𝑅_𝑑𝑉=0, it is easy to verify that 𝑇^𝑛𝑑=𝑅_𝑑+𝑉𝑅^2𝑑+𝑉²𝑅^3𝑑^𝑛=𝑅^𝑛𝑑+𝑉𝑅^𝑑𝑛+1+𝑉²𝑅^𝑑𝑛+2.(2.11) Hence, ^∞_𝑛=0𝑆^𝑑𝑛+1𝑇^𝑛𝑇^𝜋=×(𝐴𝑃𝑅)_𝑑𝐼+(𝐴𝑃)_𝑑(𝑅+𝑉)+(𝐴𝑃)^2𝑑𝑅²+𝑉𝑅+𝑉²_𝜋−𝑉𝑅_𝑑−𝑉²𝑅^2𝑑+^∞_𝑛=3(𝐴𝑃)^𝑑𝑛+1𝑅^𝑛+𝑉𝑅^𝑛−1+𝑉²𝑅^𝑛−2𝑅^𝜋=(𝐴𝑃)_𝑑𝐼+(𝐴𝑃)_𝑑(𝑅+𝑉)+(𝐴𝑃)^2𝑑𝑅²+𝑉𝑅+𝑉²𝑅^𝜋+−(_∞𝐴𝑃)_𝑑𝑉𝑅_𝑑+𝑉²𝑅^2𝑑+(𝐴𝑃)_𝑑𝑉²𝑅_𝑑_𝑛=3𝑆_𝜋∞(𝐴𝑃)^𝑑𝑛+1𝑅^𝑛+𝑉𝑅^𝑛−1+𝑉²𝑅^𝑛−2𝑅^𝜋,_𝑛=0𝑆^𝑛𝑇^𝑑𝑛+1=(𝐴𝑃)^𝜋∞_𝑛=0(𝐴𝑃)^𝑛𝑅^𝑑𝑛+1+𝑉𝑅^𝑑𝑛+2+𝑉²𝑅^𝑑𝑛+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𝑅^𝑛+𝑉𝑅^𝑛−1+𝑉²𝑅^𝑛−2𝑅^𝜋−+((𝐴𝐴𝑃𝑃))_𝑑𝑉_𝜋ℎ−1𝑅_𝑑+𝑉²𝑅^2𝑑+(𝐴𝑃)_𝑑𝑉²𝑅_𝑑_𝑛=0(𝐴𝑃)^𝑛𝑅^𝑑𝑛+1+𝑉𝑅^𝑑𝑛+2+𝑉²𝑅^𝑑𝑛+3,(2.13) where 𝑉=𝑃𝐴−𝑃𝐶𝐵−𝐴𝑃 and the symbols 𝑉^𝑖𝑅^𝑗=0,𝑖=1,2, if 𝑗<0.

If an operator 𝑇 is quasinilpotent, 𝑇_𝑑=0 and 𝑇^𝜋=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=^∞_𝑛=0(𝐴𝑃)^𝑛𝑅^𝑑𝑛+1+𝑉𝑅^𝑑𝑛+2+𝑉²𝑅^𝑑𝑛+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=𝑅_𝑑+𝐴_𝑑(𝐼−𝑃)+^∞_𝑛=0𝐴^𝑑𝑛+2(𝐼−𝑃)(𝐴−𝐶𝐵)𝑃(𝐴−𝐶𝐵)^𝑛𝑅^𝜋+𝐴^𝜋∞_𝑛=0𝐴^𝑛(𝐼−𝑃)(𝐴−𝐶𝐵)𝑃𝑅^𝑑𝑛+2−𝐴_𝑑(𝐼−𝑃)(𝐴−𝐶𝐵)𝑅_𝑑.(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: 𝐴=⎛⎜⎜⎝𝐴¹𝐴₃0𝐴₂⎞⎟⎟⎠.(2.17) From 𝜎(𝐴)=𝜎(𝐴₁)∪𝜎(𝐴₂) and the hypothesis that 𝐴_𝑑 exists, 𝐴₁∈ℬ(ℛ(𝑃)) and 𝐴₂∈ℬ(𝒩(𝑃)) are generalized Drazin invertible since 0∉acc(𝜎(𝐴)) if and only if 0∉acc(𝜎(𝐴₁)) and 0∉acc(𝜎(𝐴₂)). And, by Lemma 1.2, 𝐴_𝑑=⎛⎜⎜⎝𝐴^𝑑1𝑊𝐴0_𝑑2⎞⎟⎟⎠,(2.18) where 𝑊 is some operator. Since 𝐴(𝐼−𝑃)=⎛⎜⎜⎝00𝐴0₂⎞⎟⎟⎠,(2.19)(𝐴(𝐼−𝑃))_𝑑 exists and (𝐴(𝐼−𝑃))_𝑑=⎛⎜⎜⎝00𝐴0_𝑑2⎞⎟⎟⎠=𝐴_𝑑(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)_𝑑𝑅^𝜋+(𝐴𝑄)^2𝑑(𝑅+𝑉)𝑅^𝜋+^∞_𝑛=2(𝐴𝑄)^𝑑𝑛+1𝑅^𝑛+𝑉𝑅^𝑛−1+𝑉²𝑅^𝑛−2𝑅^𝜋−+((𝐴𝐴𝑄𝑄))^𝑑_𝜋∞𝑉𝑅_𝑑+𝑉²𝑅^2𝑑+(𝐴𝑄)_𝑑𝑉²𝑅_𝑑+(𝐴𝑄)^𝜋𝑅_𝑑+𝑉𝑅^2𝑑+𝑉²𝑅^3𝑑_𝑛=1(𝐴𝑃)^𝑛𝑅^𝑑𝑛+1+𝑉𝑅^𝑑𝑛+2+𝑉²𝑅^𝑑𝑛+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 𝑄𝑅_𝑑=0 and (𝐴𝑄)_𝑑𝑅=0. Thus it follows from (2.21) that (𝐴−𝐶𝐵)_𝑑=(𝐴𝑄)_𝑑+(𝐴𝑄)^2𝑑𝑉𝑅^𝜋+^∞_𝑛=2(𝐴𝑄)^𝑑𝑛+1𝑉𝑅^𝑛−1𝑅^𝜋−(𝐴𝑄)_𝑑𝑉𝑅_𝑑+𝑅_𝑑+(𝐴𝑄)^𝜋𝑉𝑅^2𝑑+(𝐴𝑄)^𝜋∞_𝑛=1(𝐴𝑄)^𝑛𝑉𝑅^𝑑𝑛+2=(𝐴𝑄)_𝑑+^∞_𝑛=0(𝐴𝑄)^𝑑𝑛+2𝑉𝑅^𝑛𝑅^𝜋−(𝐴𝑄)_𝑑𝑉𝑅_𝑑+𝑅_𝑑+(𝐴𝑄)^𝜋∞_𝑛=0(𝐴𝑄)^𝑛𝑉𝑅_𝑑^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 𝑅^𝑛=𝑃(𝐴−𝐶𝐵)^𝑛 and (𝐴𝑄)^𝑛=𝐴^𝑛𝑄. 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=𝑅_𝑑+𝐴_𝑑(𝐼−𝑃)+^∞_𝑛=0𝐴^𝑑𝑛+2(𝐼−𝑃)𝐴𝑃(𝐴−𝐶𝐵)^𝑛𝑅^𝜋+𝐴^𝜋∞_𝑛=0𝐴^𝑛(𝐼−𝑃)𝐴𝑃𝑅^𝑑𝑛+2−𝐴_𝑑(𝐼−𝑃)𝐴𝑅_𝑑.(2.23)

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

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=𝐴_𝑑+^∞_𝑛=0𝐴^𝑑𝑛+2(𝐼−𝑃)(𝐴−𝐶𝐵)𝑃𝐴^𝑛𝐴^𝜋+𝐴^𝜋∞_𝑛=0𝐴^𝑛(𝐼−𝑃)(𝐴−𝐶𝐵)𝑃𝐴^𝑑𝑛+2−𝐴_𝑑(𝐼−𝑃)(𝐴−𝐶𝐵)𝑃𝐴_𝑑.(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=𝑅_𝑑+(𝐼−𝑃)𝐴_𝑑+^∞_𝑛=0𝑅^𝑑𝑛+2𝑃(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴^𝑛𝐴^𝜋+𝑅^𝜋∞_𝑛=0(𝐴−𝐶𝐵)^𝑛𝑃(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴^𝑑𝑛+2−𝑅_𝑑(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴_𝑑.(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=𝐴_𝑑+^∞_𝑛=0𝐴^𝑑𝑛+2𝑃(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴^𝑛𝐴^𝜋+𝐴^𝜋∞_𝑛=0𝐴^𝑛𝑃(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴^𝑑𝑛+2−𝐴_𝑑𝑃(𝐴−𝐶𝐵)(𝐼−𝑃)𝐴_𝑑.(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=𝑅^𝜋∞_𝑛=0𝑅^𝑛+𝑅^𝑛−1𝑉+𝑅^𝑛−2𝑉²(𝑃𝐴)^𝑑𝑛+1−𝑅_𝑑𝑉+𝑅^2𝑑𝑉²+𝑅_𝑑𝑉²(𝑃𝐴)_𝑑(𝑃𝐴)_𝑑+^∞_𝑛=0𝑅^𝑑𝑛+1+𝑅^𝑑𝑛+2𝑉+𝑅^𝑑𝑛+3𝑉²(𝑃𝐴)^𝑛(𝑃𝐴)^𝜋,(2.27) where 𝑉=𝐴𝑃−𝐶𝐵𝑃−𝑃𝐴 and the symbols 𝑅^𝑖𝑉^𝑗=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=𝑅^𝜋𝑘−1_𝑛=0𝑅^𝑛+𝑅^𝑛−1𝑉+𝑅^𝑛−2𝑉²(𝑃𝐴)^𝑑𝑛+1−𝑅_𝑑𝑉+𝑅^2𝑑𝑉²+𝑅_𝑑𝑉²(𝑃𝐴)_𝑑(𝑃𝐴)^𝑑+^ℎ−1_𝑛=0𝑅^𝑑𝑛+1+𝑅^𝑑𝑛+2𝑉+𝑅^𝑑𝑛+3𝑉²(𝑃𝐴)^𝑛(𝑃𝐴)^𝜋,(2.28) where 𝑉=𝐴𝑃−𝐶𝐵𝑃−𝑃𝐴 and the symbols 𝑅^𝑖𝑉^𝑗=0,𝑗=1,2, if 𝑖<0.

When 𝑃𝐴=𝐴𝑃 and 𝑃²=𝑃 in Theorem 2.10, we can obtain the following result since 𝑅^𝑛=(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=𝑅_𝑑+𝑃𝐴_𝑑−𝑅_𝑑𝑉𝐴_𝑑+𝑅^𝜋∞_𝑛=0(𝐴−𝐶𝐵)^𝑛𝑉𝐴^𝑑𝑛+2+^∞_𝑛=0𝑅^𝑑𝑛+2𝑉𝐴^𝑛𝐴^𝜋,(2.29) where 𝑉=−𝐶𝐵𝑃.

3. Example

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

Example 3.1. Let 𝐴=⎛⎜⎜⎜⎜⎜⎜⎝1000−−201141101000⎞⎟⎟⎟⎟⎟⎟⎠,𝐵=0001,𝐶=⎛⎜⎜⎜⎜⎜⎜⎝−1001⎞⎟⎟⎟⎟⎟⎟⎠.(3.1) Then 𝐶𝐵=⎛⎜⎜⎜⎜⎜⎜⎝000000000000−1001⎞⎟⎟⎟⎟⎟⎟⎠,𝐴−𝐶𝐵=⎛⎜⎜⎜⎜⎜⎜⎝1000−−201141100100⎞⎟⎟⎟⎟⎟⎟⎠.(3.2) We will compute the Drazin inverse of 𝐴−𝐶𝐵. To do this, we choose the matrix 𝑃=⎛⎜⎜⎜⎜⎜⎜⎝1000−010100200000⎞⎟⎟⎟⎟⎟⎟⎠.(3.3) Apparently, 𝑃 is not idempotent and 𝑃𝐴≠𝐴𝑃. But 𝐵𝑃=0 and 𝐴𝑃=𝑃𝐴𝑃=⎛⎜⎜⎜⎜⎜⎜⎝1000−−−022282200000⎞⎟⎟⎟⎟⎟⎟⎠.(3.4) Obviously, Ind(𝐴𝑃)=2. Computing 𝑅=(𝐼−𝑃)(𝐴−𝐶𝐵)=⎛⎜⎜⎜⎜⎜⎜⎝0000000000000010⎞⎟⎟⎟⎟⎟⎟⎠,𝑅_𝑑=⎛⎜⎜⎜⎜⎜⎜⎝0000000000000000⎞⎟⎟⎟⎟⎟⎟⎠,(3.5)𝑉=𝑃𝐴−𝑃𝐶𝐵−𝐴𝑃=⎛⎜⎜⎜⎜⎜⎜⎝00004110−−−0411−0101⎞⎟⎟⎟⎟⎟⎟⎠,(3.6) we have Ind(𝑅)=2. So, by Corollary 2.2, (𝐴−𝐶𝐵)_𝑑=⎛⎜⎜⎜⎜⎜⎜⎝1000−000410000−0004⎞⎟⎟⎟⎟⎟⎟⎠.(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.