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 Td∈ℬ(𝒳) is called the generalized Drazin inverse of T∈ℬ(𝒳) provided it satisfiesTTd=TdT,TdTTd=Td,T-T2Tdis quasinilpotent.
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(σ(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 [<xref ref-type="bibr" rid="B4">4</xref>, 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=Bπ∑n=0∞BnAdn+1+(∑n=0∞Bdn+1An)Aπ.

Lemma 1.2 (see [<xref ref-type="bibr" rid="B3">7</xref>, Theorem 5.1]).

If A∈ℬ(𝒳) and B∈ℬ(𝒴) are generalized Drazin invertible and C∈ℬ(𝒴,𝒳), then
M=(AC0B)
is also generalized Drazin invertible and
Md=(AdSOBd),
where
S=Ad2(∑n=0∞AdnCBn)Bπ+Aπ(∑n=0∞AnCBdn)Bd2-AdCBd.

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(Rn+VRn-1+V2Rn-2)]Rπ-(AP)d[VRd+V2Rd2+(AP)dV2Rd]+(AP)π∑n=0∞(AP)n(Rdn+1+VRdn+2+V2Rdn+3),
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,RP=(I-P)(A-CB)P=0,A-CB=AP+A(I-P)-CB(I-P)=S+T
since AP=PAP and BP=0. So, by Lemma 1.1,
(T+S)d=Sπ∑n=0∞SnTdn+1+∑n=0∞Sdn+1TnTπ.

Next, we will give the representations of Td, Tn, and Tdn. In order to obtain the expression of Td, rewrite T as
T=R+PA-PCB-PAP=R+V.
Since VP=PAP-AP2=PAP(I-P),
V2P=(PA-PCB-AP)PAP(I-P)=(PAPAP-APPAP)(I-P)=0,
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+VRd2+V2Rd3,
and then
TTd=RRd+VRd+V2Rd2.
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.
From RdV=0, it is easy to verify that
Tdn=(Rd+VRd2+V2Rd3)n=Rdn+VRdn+1+V2Rdn+2.
Hence,
(∑n=0∞Sdn+1Tn)Tπ=(AP)d[I+(AP)d(R+V)+(AP)d2(R2+VR+V2)]×(Rπ-VRd-V2Rd2)+∑n=3∞(AP)dn+1(Rn+VRn-1+V2Rn-2)Rπ=(AP)d[I+(AP)d(R+V)+(AP)d2(R2+VR+V2)]Rπ-(AP)d(VRd+V2Rd2+(AP)dV2Rd)+∑n=3∞(AP)dn+1(Rn+VRn-1+V2Rn-2)Rπ,Sπ∑n=0∞SnTdn+1=(AP)π∑n=0∞(AP)n(Rdn+1+VRdn+2+V2Rdn+3).
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=[∑n=0k-1(AP)dn+1(Rn+VRn-1+V2Rn-2)]Rπ-(AP)d[VRd+V2Rd2+(AP)dV2Rd]+(AP)π∑n=0h-1(AP)n(Rdn+1+VRdn+2+V2Rdn+3),
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=∑n=0∞(AP)n(Rdn+1+VRdn+2+V2Rdn+3),
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)+∑n=0∞Adn+2(I-P)(A-CB)P(A-CB)nRπ+Aπ∑n=0∞An(I-P)(A-CB)PRdn+2-Ad(I-P)(A-CB)Rd.

Proof.

Since P2=P, we have 𝒳=ℛ(P)⨁𝒩(P) and can write P in the following matrix form:
P=(I000).
The condition PA=PAP, therefore, yields the matrix form of A as follows:
A=(A10A3A2).
From σ(A)=σ(A1)∪σ(A2) and the hypothesis that Ad 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,
Ad=(A1d0WA2d),
where W is some operator. Since
A(I-P)=(000A2),(A(I-P))d exists and
(A(I-P))d=(000A2d)=Ad(I-P).

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)d2(R+V)Rπ+[∑n=2∞(AQ)dn+1(Rn+VRn-1+V2Rn-2)]Rπ-(AQ)d[VRd+V2Rd2+(AQ)dV2Rd]+(AQ)π(Rd+VRd2+V2Rd3)+(AQ)π∑n=1∞(AP)n(Rdn+1+VRdn+2+V2Rdn+3),
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)d2VRπ+[∑n=2∞(AQ)dn+1VRn-1]Rπ-(AQ)dVRd+Rd+(AQ)πVRd2+(AQ)π∑n=1∞(AQ)nVRdn+2=(AQ)d+[∑n=0∞(AQ)dn+2VRn]Rπ-(AQ)dVRd+Rd+(AQ)π∑n=0∞(AQ)nV(Rd)n+2.
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)+∑n=0∞Adn+2(I-P)AP(A-CB)nRπ+Aπ∑n=0∞An(I-P)APRdn+2-Ad(I-P)ARd.

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+∑n=0∞Adn+2(I-P)(A-CB)PAnAπ+Aπ∑n=0∞An(I-P)(A-CB)PAdn+2-Ad(I-P)(A-CB)PAd.

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+∑n=0∞Rdn+2P(A-CB)(I-P)AnAπ+Rπ∑n=0∞(A-CB)nP(A-CB)(I-P)Adn+2-Rd(A-CB)(I-P)Ad.

Remark 2.8 (see [<xref ref-type="bibr" rid="B4">4</xref>, 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+∑n=0∞Adn+2P(A-CB)(I-P)AnAπ+Aπ∑n=0∞AnP(A-CB)(I-P)Adn+2-AdP(A-CB)(I-P)Ad.

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=Rπ∑n=0∞(Rn+Rn-1V+Rn-2V2)(PA)dn+1-[RdV+Rd2V2+RdV2(PA)d](PA)d+[∑n=0∞(Rdn+1+Rdn+2V+Rdn+3V2)(PA)n](PA)π,
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=Rπ∑n=0k-1(Rn+Rn-1V+Rn-2V2)(PA)dn+1-[RdV+Rd2V2+RdV2(PA)d](PA)d+[∑n=0h-1(Rdn+1+Rdn+2V+Rdn+3V2)(PA)n](PA)π,
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 [<xref ref-type="bibr" rid="B2">3</xref>, 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+Rπ∑n=0∞(A-CB)nVAdn+2+∑n=0∞Rdn+2VAnAπ,
where V=-CBP.

3. Example

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

Example 3.1.

Let
A=(12410-1100-1100000),B=(0001),C=(1-100).
Then
CB=(0001000-100000000),A-CB=(12400-1110-1100000).
We will compute the Drazin inverse of A-CB. To do this, we choose the matrix
P=(100001000-1200000).
Apparently, P is not idempotent and PA≠AP. But BP=0 and
AP=PAP=(1-2800-2200-2200000).
Obviously, Ind(AP)=2. Computing
R=(I-P)(A-CB)=(0000000000010000),Rd=(0000000000000000),V=PA-PCB-AP=(04-4001-1101-1-10000),
we have Ind(R)=2. So, by Corollary 2.2,
(A-CB)d=(1-410-4000000000000).

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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