Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 412872, 10 pages
doi:10.1155/2012/412872
Research Article
The Expression of the Generalized Drazin Inverse
of A − CB
Xiaoji Liu,1, 2 Dengping Tu,1 and Yaoming Yu3
1
School of Science, Guangxi University for Nationalities, Nanning 530006, China
Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis, Nanning 530006, China
3
School of Mathematical Sciences, Monash University, Caulfield East, VIC 3800, Australia
2
Correspondence should be addressed to Xiaoji Liu, liuxiaoji.2003@yahoo.com.cn
Received 5 August 2011; Accepted 5 December 2011
Academic Editor: Ondřej Došlý
Copyright q 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.
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 BX, Y. In particular, we write BX instead of BX, X.
For any A ∈ BX, Y, RA and NA represent its range and null space, respectively.
If A ∈ BX, the symbols σA and accσA stand for its spectrum and the set of all
accumulation points of σA, respectively.
Recall the concept of the generalized Drazin inverse introduced by Koliha 1 that the
element Td ∈ BX is called the generalized Drazin inverse of T ∈ BX provided it satisfies
T Td Td T,
T d T Td Td ,
T − T 2 Td is quasinilpotent.
1.1
If it exists then it is unique. The Drazin index IndT of T is the least positive integer k if
T − T 2 Td k 0, and otherwise IndT ∞.
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 ∈ BX exists if and only if 0 ∈
/ accσT see 1, Theorem 4.2.
2
Abstract and Applied Analysis
If T is generalized Drazin invertible, then the spectral idempotent T π of T corresponding
to 0 is given by T π I − T T 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 ∈ BX be the generalized Drazin invertible. If AB 0, then A B is generalized Drazin invertible and
A Bd B
π
∞
B
n
n0
An1
d
∞
n1 n
Bd A Aπ .
1.2
n0
Lemma 1.2 see 7, Theorem 5.1. If A ∈ BX and B ∈ BY are generalized Drazin invertible
and C ∈ BY, X, then
M
A C
0 B
1.3
is also generalized Drazin invertible and
Md Ad S
,
O Bd
1.4
where
S
A2d
∞
n0
2. Main Results
We start with our main result.
And CBn
∞
n
n
B A
A CBd Bd2 − Ad CBd .
π
π
n0
1.5
Abstract and Applied Analysis
3
Theorem 2.1. Let A ∈ BX be the generalized Drazin invertible, C ∈ BX, Y, and B ∈ BY, X.
Suppose that there exists a P ∈ BX such that AP P AP and BP 0. If R I − P A − CB and
AP are generalized Drazin invertible, then A − CB is generalized Drazin invertible and
A − CBd ∞
AP n1
d
n
n−1
2 n−2
R VR V R
Rπ
n0
− AP d V Rd V 2 R2d AP d V 2 Rd
∞
n2
2 n3
,
AP π AP n Rn1
d V Rd V Rd
2.1
n0
where V P A − P CB − AP and the symbols V i Rj 0, i 1, 2, if j < 0.
Proof. Let S : AP and T : A − CBI − P . Then
T S A − CBI − P AP 0,
2.2
RP I − P A − CBP 0,
2.3
A − CB AP AI − P − CBI − P S T
2.4
since AP P AP and BP 0. So, by Lemma 1.1,
T Sd Sπ
∞
Sn Tdn1 n0
∞
Sn1
T nT π .
d
n0
2.5
Next, we will give the representations of Td , T n , and Tdn . In order to obtain the expression of Td , rewrite T as
T R P A − P CB − P AP R V.
2.6
Since V P P AP − AP 2 P AP I − P ,
V 2 P P A − P CB − AP P AP I − P P AP AP − AP P AP I − P 0,
2.7
and then V n 0 for n > 2 since V P A − CB − AP . So Vd exists and Vd 0. By 2.3,
RV RP A − CB − AP 0 and then Rd V Rd Rd RV 0. So, by Lemma 1.1,
Td R V d Rd V R2d V 2 R3d ,
2.8
T Td RRd V Rd V 2 R2d .
2.9
and then
4
Abstract and Applied Analysis
Since RR V k Rk1 and V 2 R V k V 2 Rk for k ≥ 1,
T n R V n R2 V R V 2 R V n−2 Rn V Rn−1 V 2 Rn−2 ,
n ≥ 2.
2.10
From Rd V 0, it is easy to verify that
n
2 n2
Tdn Rd V R2d V 2 R3d Rnd V Rn1
d V Rd .
2.11
Hence,
∞
n0
n
Sn1
d T
T π AP d I AP d R V AP 2d R2 V R V 2
∞
Rn V Rn−1 V 2 Rn−2 Rπ
× Rπ − V Rd − V 2 R2d AP n1
d
n3
AP d I AP d R V AP 2d R2 V R V 2 Rπ
− AP d V Rd V 2 R2d AP d V 2 Rd
∞
Rn V Rn−1 V 2 Rn−2 Rπ ,
AP n1
d
2.12
n3
∞
∞
n2
2 n3
.
Sπ Sn Tdn1 AP π AP n Rn1
d V Rd V Rd
n0
n0
Therefore, we reach 2.1.
When IndAP , IndR < ∞, we have the following corollary.
Corollary 2.2. Let A ∈ BX be generalized Drazin invertible. C ∈ BX, Y, and B ∈ BY, X.
Suppose that there exists a P ∈ BX such that AP P AP and BP 0. If R I − P A − CB and
AP are generalized Drazin invertible and IndR k < ∞ and IndAP h < ∞, then A − CB
is generalized Drazin invertible and
A − CBd k−1
AP n1
d
R VR
n
n−1
V R
2
n−2
Rπ
n0
− AP d V Rd V 2 R2d AP d V 2 Rd
AP π
h−1
n0
2.13
n2
2 n3
,
AP n Rn1
d V Rd V Rd
where V P A − P CB − AP and the symbols V i Rj 0, i 1, 2, if j < 0.
If an operator T is quasinilpotent, Td 0 and T π I. So, the following corollary
follows from Theorem 2.1.
Abstract and Applied Analysis
5
Corollary 2.3. Let A ∈ BX be generalized Drazin invertible, C ∈ BX, Y, and B ∈ BY, X.
Suppose that there exists a P ∈ BX such that AP P AP 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 − CBd ∞
n2
2 n3
,
V
R
V
R
AP n Rn1
d
d
d
2.14
n0
where V P A − P CB − AP .
Theorem 2.4. Let A ∈ BX be generalized Drazin invertible, C ∈ BX, Y, and B ∈ BY, X.
Suppose that there exists an idempotent P ∈ BX such that P A P AP and BP B. If R P A − CB is generalized Drazin invertible, then A − CB is generalized Drazin invertible and
A − CBd Rd Ad I − P ∞
n0
n π
An2
d I − P A − CBP A − CB R
∞
Aπ An I − P A − CBP Rn2
d − Ad I − P A − CBRd .
2.15
n0
Proof. Since P 2 P , we have X RP NP and can write P in the following matrix form:
P
I 0
.
0 0
2.16
The condition P A P AP , therefore, yields the matrix form of A as follows:
A
A1 0
.
A 3 A2
2.17
From σA σA1 ∪ σA2 and the hypothesis that Ad exists, A1 ∈ BRP and A2 ∈
BNP are generalized Drazin invertible since 0 ∈
/ accσA if and only if 0 ∈
/ accσA1 and 0 ∈
/ accσA2 . And, by Lemma 1.2,
d
A1 0
Ad ,
W Ad2
2.18
where W is some operator. Since
AI − P 0 0
,
0 A2
2.19
AI − P d exists and
AI − P d 0 0
0 Ad2
Ad I − P .
2.20
6
Abstract and Applied Analysis
To use Theorem 2.1 to complete the proof, let Q I − P . So R I − QA − CB and
AQ are generalized Drazin invertible. And from the conditions P A P AP and BP B, we
can obtain AQ QAQ and BQ 0. Thus, by Theorem 2.1, we have
A − CBd AQd R π
AQ2d R
V R π
∞
AQn1
d
R VR
n
n−1
V R
2
n−2
n2
− AQd V Rd V 2 R2d AQd V 2 Rd AQπ Rd V R2d V 2 R3d
AQπ
Rπ
2.21
∞
n2
2 n3
,
AP n Rn1
d V Rd V Rd
n1
where V QA − QCB − AQ.
Since P 2 P and Q2 Q and then V Q 0 and V QV . So V 2 0. Note that QR 0
and then QRd 0 and AQd R 0. Thus it follows from 2.21 that
A − CBd AQd AQ2d V Rπ
∞
n1
n−1
Rπ − AQd V Rd
AQd V R
n2
Rd AQπ V R2d AQπ
∞
AQn V Rn2
d
n1
∞
n2
n
AQd AQd V R Rπ − AQd V Rd Rd
2.22
n0
AQ
∞
π
AQn V Rd n2 .
n0
Since V QA − CB − A − CBQ A − CBI − Q − I − QA − CB, V R QA − CBR
and QV QA − CBI − Q. Note that Rn P A − CBn and AQn An Q. Substituting V
and Q I − P in 2.22 yields 2.15.
Adding the condition P C C in Theorem 2.4 yields a result below.
Corollary 2.5. Let A ∈ BX be generalized Drazin invertible, C ∈ BX, Y, and B ∈ BZ, X.
Suppose that there exists an idempotent P ∈ BX such that P A P AP , BP B, and P C C. If
R P A − CB is generalized Drazin invertible, then A − CB is generalized Drazin invertible and
A − CBd Rd Ad I − P ∞
n π
An2
d I − P AP A − CB R
n0
∞
Aπ An I − P AP Rn2
d − Ad I − P ARd .
2.23
n0
Adding the condition P C 0 in Theorem 2.4 yields R P A. So similar to the proof of
AI − P d Ad I − P in Theorem 2.4, we can gain P Ad P Ad .
Abstract and Applied Analysis
7
Corollary 2.6. Let A ∈ BX be generalized Drazin invertible, C ∈ BX, Y, and B ∈ BZ, X.
Suppose that there exists an idempotent P ∈ BX such that P A P AP , BP B, and P C 0; then
A − CB is generalized Drazin invertible and
A − CBd Ad ∞
n0
n π
π
An2
d I − P A − CBP A A A
∞
An I − P A − CBP An2
d
n0
2.24
− Ad I − P A − CBP Ad .
Analogously, we can deduce Theorem 2.7 and Corollary 2.9 below.
Theorem 2.7. Let A ∈ BX be generalized Drazin invertible, C ∈ BX, Y, and B ∈ BY, X.
Suppose that there exists an idempotent P ∈ BX such that AP P AP and P C C. If R A − CBP is generalized Drazin invertible, then A − CB is generalized Drazin invertible and
A − CBd Rd I − P Ad ∞
n π
Rn2
d P A − CBI − P A A
n0
∞
Rπ A − CBn P A − CBI − P An2
d − Rd A − CBI − P Ad .
2.25
n0
Remark 2.8 see 4, Theorem 2.4. It is a special case of Theorem 2.7.
Corollary 2.9. Let A ∈ BX be generalized Drazin invertible, C ∈ BX, Y, and B ∈ BZ, X.
Suppose that there exists an idempotent P ∈ BX such that AP P AP , P C C, and BP 0; then
A − CB is generalized Drazin invertible and
A − CBd Ad Aπ
∞
n0
∞
n0
n π
An2
d P A − CBI − P A A
2.26
An P A − CBI − P An2
d − Ad P A − CBI − P Ad .
Similar to Theorem 2.1 and Corollary 2.2, we can show the following two results.
Theorem 2.10. Let A ∈ BX be generalized Drazin invertible, C ∈ BX, Y, and B ∈ BY, X.
Suppose that there exists a P ∈ BX such that P A P AP and P C 0. If R A − CBI − P and P A are generalized Drazin invertible, then A − CB is generalized Drazin invertible and
A − CBd Rπ
∞ Rn Rn−1 V Rn−2 V 2 P An1
d
n0
− Rd V R2d V 2 Rd V 2 P Ad P Ad
∞ n0
2.27
n2
n3 2
Rn1
P An P Aπ ,
d Rd V Rd V
where V AP − CBP − P A and the symbols Ri V j 0, j 1, 2, if i < 0.
8
Abstract and Applied Analysis
Corollary 2.11. Let A ∈ BX be generalized Drazin invertible. C ∈ BX, Y, and B ∈ BY, X.
Suppose that there exists a P ∈ BX such that P A P AP and P C 0. If R A − CBI − P and
P A are generalized Drazin invertible and IndR k < ∞ and IndP A h < ∞, then A − CB
is generalized Drazin invertible and
A − CBd Rπ
k−1 Rn Rn−1 V Rn−2 V 2 P An1
d
n0
− Rd V R2d V 2 Rd V 2 P Ad P Ad
h−1 n
n1
n2
n3 2
Rd Rd V Rd V P A P Aπ ,
2.28
n0
where V AP − CBP − P A and the symbols Ri V j 0, j 1, 2, if i < 0.
When P A AP and P 2 P in Theorem 2.10, we can obtain the following result since
R A − CBn I − P .
n
Corollary 2.12 see 3, Theorem 4.3. Let A ∈ BX be the generalized Drazin invertible, C ∈
BX, Y, and B ∈ BY, X. Suppose that there exists an idempotent P ∈ BX commuting with A
such that P C 0. If R A − CBI − P is generalized Drazin invertible, then A − CB is the
generalized Drazin invertible and
A − CBd Rd P Ad − Rd V Ad Rπ
∞
∞
n π
Rn2
A − CBn V An2
d
d VA A ,
n0
2.29
n0
where V −CBP .
3. Example
Before ending this paper, we give an example as follows.
Example 3.1. Let
⎛
1
⎜0
A⎜
⎝0
0
2
−1
−1
0
4
1
1
0
⎞
1
0⎟
⎟,
0⎠
0
0
0
0
0
0
0
0
0
⎛
B 0 0 0 1 ,
⎞
1
⎜−1⎟
⎟
C⎜
⎝ 0 ⎠.
0
3.1
Then
⎛
0
⎜0
CB ⎜
⎝0
0
⎞
1
−1⎟
⎟,
0⎠
0
⎛
1
⎜0
A − CB ⎜
⎝0
0
2
−1
−1
0
4
1
1
0
⎞
0
1⎟
⎟.
0⎠
0
3.2
Abstract and Applied Analysis
9
We will compute the Drazin inverse of A − CB. To do this, we choose the matrix
⎛
1
⎜0
P ⎜
⎝0
0
0
1
−1
0
0
0
2
0
⎞
0
0⎟
⎟.
0⎠
0
3.3
Apparently, P is not idempotent and P A /
AP . But BP 0 and
⎛
1
⎜0
AP P AP ⎜
⎝0
0
−2
−2
−2
0
8
2
2
0
⎞
0
0⎟
⎟.
0⎠
3.4
0
Obviously, IndAP 2. Computing
⎛
0
⎜0
R I − P A − CB ⎜
⎝0
0
0
0
0
0
0
0
0
0
⎛
⎞
0
0⎟
⎟,
1⎠
0
⎛
0
⎜0
V P A − P CB − AP ⎜
⎝0
0
0
⎜0
Rd ⎜
⎝0
0
4
1
1
0
−4
−1
−1
0
⎞
0
1⎟
⎟,
−1⎠
0
0
0
0
0
0
0
0
⎞
0
0⎟
⎟,
0⎠
0
3.5
3.6
0
we have IndR 2. So, by Corollary 2.2,
⎛
1
⎜0
A − CBd ⎜
⎝0
0
−4
0
0
0
10
0
0
0
⎞
−4
0⎟
⎟.
0⎠
0
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.
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10
Abstract and Applied Analysis
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