# Proper Splitting for the Generalized InverseAT,S(2) and Its Application on Banach Spaces

Authors: Xiaoji Liu; Shaowu Huang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A possible type of the operator splitting is studied. Using this operator splitting, we introduce some properties and representations of generalized inverses as well as iterative method for computing various solutions of the restricted linear operator systemAx=b,x∈T, where A∈ℒ(X,Y) and T is an arbitrary but fixed subspace of X.


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## Body

## 1. Introduction

The subject of splitting was investigated by numerous authors. And there are several papers concerning the iterative methods of the general formas follows:(1.1)Xi+1=MT,K(2)Xi+MT,K(2),i=0,1,2,…,
where A=M-N is a splitting of A, and K is an arbitrary closed subspaces of Y (see [1–4]). Particular results concerning the computation of the Drazin inverse and the Moore-Penrose inverse can be investigated in [5, 6].In this paper, we will consider the linear operator systemAx=b, where A∈ℒ(X,Y), T⊕L=X,  and AT⊕S=Y. The concept of a operator splitting can be used in characterizations of the generalized inverse AT,S(2) and in iterative method as follows:
(1.2)Xk+1=MT,K(2)NXk+MT,K(2)Gb,i=0,1,2,…
for solving linear operator system Ax=b. In particular, let K=L, the authors in [7] gave the iterative method
(1.3)Xi+1=MgNXi+MgGb,i=0,1,2,….
to solve linear system Ax=b.Now, we introduce some notations and terminologies.LetX,Y be Banach spaces and ℒ(X,Y) be the set of all bounded operators from X to Y. We use ℛ(A),𝒩(A) and ρ(A), respectively, to denote the range, the null space, and the spectral radius for an operator A∈ℒ(X,Y). Suppose that there exists an operator B∈ℒ(Y,X), such that
(1.4)BAB=B,R(B)=T,N(B)=S.
Then B is usually denoted by AT,S(2). Recall that the splitting A=U-V is called a proper splitting of A if ℛ(A)=ℛ(U) and 𝒩(A)=𝒩(U).LetT∈ℒ(X) if, for some nonnegative integer k≥0, there exists S∈ℒ(X) such that
(1.5)(1k)TSTk=Tk,(2)STS=S,(5)TS=ST,
then S is called the Drazin inverse of T, and the smallest n is called the index of T and will be denoted by ind(T). If an operator T∈ℒ(X) has a Drazin inverse, then it is unique and is denoted by Td (see [8] for details). Particularly, when ind(T)≤1, Td is called the group inverse of T and is denoted by Tg.The paper is organized as follows. In the remainder of this section, we will introduce some lemmas which are useful in the proofs. In Section2, We express some properties and representations of generalized inverses based on the operator splitting. Moreover, by using these representations, we introduce iterative method for computing various solutions of linear operator system. In Section 3, we give a numerical example to demonstrate one of the results in Section 2.Basic auxiliary results are summarized in the following lemmas.Lemma 1.1 (see [9, page 43]).
LetA∈ℒ(X,Y)  and S,T be closed subspaces of X and Y such that T⊕L=X and AT⊕S=Y. Then A has the following matrix form:
(1.6)A=[A1A20A3]:[TL]⟶[ATS],
where A1 is invertible. Moreover,
(1.7)AT,S(2)=[A1-1000]:[ATS]⟶[TL].Lemma 1.2 (see [9, lemma 3.5.2]).
LetB∈ℒ(X). L and M are closed subspaces of X such that X=L⊕M and PL,M is a projection from X onto Y parallel to M. Then, one has (i)
PL,MB=B if and only if ℛ(B)⊆L,(ii)
BPL,M=B if and only if 𝒩(B)⊇M.Lemma 1.3 (see [6]).
LetA∈ℂrm×n be factorized in the following form:
(1.8)A=U[IC]A11[IB]V,
where A11 is an r×r nonsingular matrix, and U and V are permutation matrices. Then
(1.9)N(M)=N(A),R(M)=R(A)
(i.e., A=M-N is a proper splitting) if and only if
(1.10)M=U[IC]M11[IB]V,
where M11 is a nonsingular matrix of order r.Lemma 1.4 (see [10]).
LetA∈ℂrm×n with ind(A)=1 and A be partitioned as
(1.11)A=U[IC]A11[IB]V,
where U,V and A11 are the same as Lemma 1.3. Then the group inverse Ag exists if and only if
(1.12)VU+BVUCisnonsingular,Ag=U[IC](VU+BVUC)-1A11-1(VU+BVUC)-1[IB]V.


## 2. Main Results

In this section, the representation of the generalized inverseAT,S(2) of a linear operator is studied. Moreover, we consider the fundamental problem of solving a general linear operator equation of the type Ax=b, where A∈ℒ(X,Y).Now we are ready to present the representation theorem.Theorem 2.1 (representation theorem).
LetA∈ℒ(X,Y) be given and T and S, respectively, be closed subspaces of X and Y such that there exists the generalized inverse AT,S(2). And let G∈ℒ(Y,X) with ℛ(G)=T, 𝒩(G)=S, GA=M-N be a proper splitting of GA, that is, 𝒩(M)=𝒩(GA),  ℛ(M)=ℛ(GA). Then, one has
(2.1)AT,S(2)=(I-MT,K(2)N)-1MT,K(2)G,
where K is a subspace of X.Proof.
By Lemma1.2, it is easy to verify that
(2.2)(I-MT,K(2)N)AT,S(2)=AT,S(2)-MT,K(2)NAT,S(2)=AT,S(2)-MT,K(2)(M-GA)AT,S(2)=AT,S(2)-MT,K(2)MAT,S(2)+MT,K(2)GAAT,S(2)=MT,K(2)G.
Now, we will show that I-MT,K(2)N is invertible. Notice that
(2.3)I-MT,K(2)N=I-MT,K(2)M+MT,K(2)GA.
Since I-MT,K(2)M is a projection from Y onto L parallel to T and MT,K(2)GA is an invertible operator from T to T, we get that I-MT,K(2)M+MT,K(2)GA is invertible. Thus, the proof is completed.Now, we are in position to state the main result of this section.Theorem 2.2.
Under the hypotheses of Theorem2.1, we have that
(2.4)Xk+1=MT,K(2)NXk+MT,K(2)G
converges to AT,S(2) for every X0∈X if and only if ρ(MT,K(2)N)<1. Then we get that Xk has the error estimation
(2.5)‖Xk+1-Xk‖≤‖MT,K(2)N‖k‖X1-X0‖.
Moreover, one has
(2.6)‖Xk-AT,S(2)‖≤‖MT,K(2)N‖k+1‖X0-AT,S(2)‖,
where K is a subspace of X.Proof.
“⇐” Suppose that ρ(MT,K(2)N)<1. According to Representation theorem, we obtain that
(2.7)Xk+1-AT,S(2)=MT,K(2)NXk+MT,K(2)G-(I-MT,K(2)N)-1MT,K(2)G=MT,K(2)NXk+MT,K(2)G-∑k=0∞(MT,K(2)N)kMT,K(2)G‍=MT,K(2)NXk-∑k=1∞(MT,K(2)N)kMT,K(2)G‍=MT,K(2)N(Xk-∑k=0∞(MT,K(2)N)kMT,K(2)G‍)=MT,K(2)N(Xk-AT,S(2))=(MT,K(2)N)k+1(X0-AT,S(2)).
By the hypothesis and (2.7), it is to see that Xk converges to AT,S(2) for every X0∈X.
“⇒” A simple computation shows that
(2.8)Xk+1-Xk=MT,K(2)NXk-MT,K(2)NXk-1=MT,K(2)N(Xk-Xk-1)=⋯=(MT,K(2)N)k(X1-X0),
it follows that (MT,K(2)N)k→0 as k→∞, which implies that ρ(MT,K(2)N)<1. Now the results follow immediately.From Theorem2.2, we can immediately obtain the following theorem.Theorem 2.3.
Under the hypotheses of Theorem2.1, we have that
(2.9)Xk+1=MT,K(2)NXk+MT,K(2)Gb
converges to AT,S(2)b   (the unique solution of linear operator system Ax=b.) for every X0∈X if and only if ρ(MT,K(2)N)<1. Then we get that Xk has the error bounded
(2.10)‖Xk+1-Xk‖≤‖MT,K(2)N‖k‖X1-X0‖.
Moreover, one has
(2.11)‖Xk-AT,S(2)b‖≤‖MT,K(2)N‖k+1‖X0-AT,S(2)b‖,
where K is a subspace of X.Proof.
The proof is similar to that of Theorem2.2.Similar to [7, Theorems 3.1 and 3.3]. In the case when K=L, the previous theorems reduce to the following corollary.Corollary 2.4.
LetA, T and S be the same as Theorem 2.1 and let G∈ℒ(X,Y), ℛ(G)=T and 𝒩(G)=S. Let GA=M-N be a proper splitting of GA, that is,
(2.12)N(M)=N(GA),R(M)=R(GA).
Then, one has(i)
ind(M)=1,(ii)
AT,S(2)=(I-MgN)-1MgG, and(iii)
the iterationXi+1=MgNXi+MgG converges to AT,S(2) for each X0∈X if and only if ρ(MgN)<1.Now, we will consider the proper splitting ofAG.Theorem 2.5.
LetA∈ℒ(X,Y) be given and T and S, respectively, be closed subspaces of X and Y, such that there exists the generalized inverse AT,S(2). And let G∈ℒ(Y,X) with ℛ(G)=T, 𝒩(G)=S, AG=M-N be a proper splitting of AG, that is, 𝒩(M)=𝒩(AG),  ℛ(M)=ℛ(AG). Then, one has
(2.13)AT,S(2)=GMK,S(2)(I-NMK,S(2))-1,
where K is a subspace of Y.Theorem 2.6.
Under the hypotheses of Theorem2.5, we have that
(2.14)Xk+1=NMK,S(2)Xk+GMK,S(2)
converges to AT,S(2) for every X0∈X if and only if ρ(NMK,S(2))<1. Then we get that Xk has the error estimation as follows:
(2.15)‖Xk-AT,S(2)‖≤‖NMK,S(2)‖k+1‖X0-AT,S(2)‖.
Moreover, consider that
(2.16)‖Xk+1-Xk‖≤‖NMK,S(2)‖k‖X1-X0‖,
where K is a subspace of Y.It is well known thatAd=AR(Ak),N(Ak)(2). Let G=Ak; thus, the following result follows from Corollary 2.4.Corollary 2.7.
LetA∈ℒ(X,Y),G∈ℒ(Y,X), T, S be closed subspaces of X and Y such that AT,S(2) exists with Ind(A)=k. Suppose that R(Ak)=T,N(Ak)=S and Ak+1=M-N be a proper splitting of GA, that is,
(2.17)N(M)=N(Ak+1),R(M)=R(Ak+1).
Then the iteration
(2.18)xi+1=MgNxi+MgAk
or
(2.19)xi+1=MgNxi+MgAkb
converges to Ad or Adb for every X0∈X if and only if ρ(MgN)<1.Remark 2.8.
In [11], the author present {T,S} splitting for computing the generalized inverse AT,S(2), particular result concerning the computation Drazin inverse. Let T=ℛ(Ak),S=𝒩(Ak),k=ind(A), and A=U-V be a {T,S} splitting of A. Then the iteration
(2.20)Xi+1=UdVXi+Udb
converges to Adb for every X0∈X if and only if ρ(UdV)<1. Moreover, the iteration (2.20) can be reduced to
(2.21)R(Uk)=R(Ak),N(Uk)=N(Ak).
The splitting (2.17) is more practical than (2.21). Since M in (2.17) can easily be calculated by Lemmas 1.3 and 1.4, while U in (2.21) is more difficult to be getten.


## 3. Illustration Example

The following matrixA is from [11]. Let
(3.1)A=[246514540-1-10-1-2-3-3]∈R4×4,
where ind(A)=2, rank(A2)=2.We will use Corollary2.7 to compute the unique solution Adb of the restricted linear equation Ax=b,x∈ℛ(A2)=T and  b=(8,7,-3,-3)T∈ℛ(A2). Take
(3.2)N=[0-8-8-8-20-2-20.41.62.02.00.41.62.02.0],M=[30330777-0.6-1.4-2.0-2.0-0.6-1.4-2.0-2.0]
which satisfies the conditions of Corollary 2.7. Therefore, by Lemmas 1.3 and 1.4,
(3.3)Md=[25720745745720755211152111521-97-2321-5021-5021-97-2321-5021-5021],Ad=[3-1222133-10-1-1-10-1-1].Denote thatrk=∥xk+1-xk∥ and Rk=∥xk-ADb∥. We have Table 1 for the norm ∥·∥2, and from which we conclude that Xk is an approximation of the exact solution x of Ax=b.Table 1
Convergence of (2.19) with any choice of initial x0.
x
0
T
k
r
k
R
k
X
k
T
[
10
;
2
;
-
5
;
9
]
98
0.0016
0.0014
[
5.0001
,
5.0001
,
-
2.0000
,
-
2.0000
]
[
1
;
20
;
84
;
9
]
98
0.0039
0.0027
[
5.0008,5.0009
,
-
2.0003
,
-
2.0003
]
[
55
;
6
;
5
;
2
;
4
]
98
0.0065
0.0013
[
4.9990,4.9993
,
-
1.9997
,
-
1.9997
]
[
0
;
45
;
600
;
87
]
98
0.0222
0.0088
[
5.0058,5.0058
,
-
2.0023
,
-
2.0023
]
[
66
;
22
;
1
;
-
9
]
98
0
3.1765
e
-
004
[
4.9998,4.9998
,
-
1.9999
,
-
1.9999
]
[
-
1
;
-
99
;
8
;
-
7
]
98
0.0023
6.3226
e
-
004
[
4.9997,4.9995
,
-
1.9998
,
-
1.9998
]
[
-
555
;
0
;
99
;
34
]
98
0.0004
0.0019
[
4.9984,4.9993
,
-
1.9995
,
-
1.9995
]



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*Source: 736929-2012-07-03.xml*