The Expression of the Generalized Drazin Inverse of A −

and Applied Analysis 3 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 thatAP 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 n 1 d ( R VRn−1 V 2Rn−2 )] R − AP d [ VRd V Rd AP dV Rd ] AP π ∞ ∑ n 0 AP n ( R 1 d VR n 2 d V 2Rn 3 d ) , 2.1 where V PA − PCB −AP and the symbols V R 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 0 SnT 1 d ∞ ∑ n 0 S 1 d T T. 2.5 Next, we will give the representations of Td, T, and T d . In order to obtain the expression of Td, rewrite T as T R PA − PCB − PAP R V. 2.6 Since VP PAP −AP 2 PAP I − P , V 2P 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 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 VR 2 d V Rd, 2.8


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 satisfies If it exists then it is unique.The Drazin index Ind T of T is the least positive integer k if T − T 2 T 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.
is also generalized Drazin invertible and where

Main Results
We start with our main result.
where V PA − PCB − AP and the symbols V i R j 0, i 1, 2, if j < 0.
Next, we will give the representations of T d , T n , and T n d .In order to obtain the expression of T d , rewrite T as and then

2.10
From R d V 0, it is easy to verify that Hence,
When Ind AP , Ind R < ∞, we have the following corollary.

2.13
where V PA − PCB − AP and the symbols If an operator T is quasinilpotent, T d 0 and T π I. So, the following corollary follows from Theorem 2.1.

2.15
Proof.Since P 2 P , we have X R P N P and can write P in the following matrix form: The condition PA PAP, therefore, yields the matrix form of A as follows:

2.17
From σ A σ A 1 ∪ σ A 2 and the hypothesis that A d exists, A 1 ∈ B R P and A 2 ∈ B N P are generalized Drazin invertible since 0 / ∈ acc σ A if and only if 0 / ∈ acc σ A 1 and 0 / ∈ acc σ A 2 .And, by Lemma 1.2, where W is some operator.Since where V QA − QCB − AQ.
Since P 2 P and Q 2 Q and then V Q 0 and V QV .So V 2 0. Note that QR 0 and then QR d 0 and AQ d R 0. Thus it follows from 2.21 that

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

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

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

2.26
Similar to Theorem 2.1 and Corollary 2.2, we can show the following two results.

2.27
where V AP − CBP − PA and the symbols R i V j 0, j 1, 2, if i < 0.

2.28
where V AP − CBP − PA and the symbols R i V j 0, j 1, 2, if i < 0. When where V −CBP .

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

3 A 2 . 4 since
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.− CB AP A I − P − CB I − P S T AP PAP and BP 0. So, by Lemma 1.1,

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

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

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 the Drazin inverse of A − CB.To do this, we choose the matrix P is not idempotent and PA / AP .But BP 0 and 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

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

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 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 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 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 PA AP and P 2 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