# Perturbation Bound of the Group Inverse and the Generalized Schur Complement in Banach Algebra

Authors: Xiaoji Liu; Yonghui Qin; Hui Wei
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We investigate the relative perturbation bound of the group inverse and also consider the perturbation bound of the generalized Schur complement in a Banach algebra.


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## Body

## 1. Introduction

Let𝒜 denote a Banach algebra with unit 1. The symbols 𝒜-1, 𝒜D, 𝒜d, 𝒜g, 𝒜nil, 𝒜qnil, and 𝒜• stand for the sets of all invertible, Drazin invertible, generalized Drazin invertible, group invertible, nilpotent, quasinilpotent, and idempotent elements of a Banach algebra 𝒜, respectively.Some definitions will be given in the following.Lettinga∈𝒜D, there is an unique element x∈𝒜 such that
(1.1)ak+1x=ak,xax=x,ax=xa.
Then x is called the Drazin inverse of a, denoted by aD. The smallest nonnegative integer k which satisfies (1.1) is called the index of a, denoted by Ind(a)=k. If Ind(a)≤1, then aD=a♯(orag).Leta∈𝒜, if the conditions (1.1) are replaced by
(1.2)axa=a,xax=x,ax=xa.
Then x is called the group inverse of a, denoted by x=a♯. If the conditions (1.1) are replaced by
(1.3)xax=x,ax=xa,a(1-ax)isquasinilpotent.
Then x is called the generalized Drazin inverse of a, denoted by x=ad.Some notations of the Schur complement are given in the following.For a2×2 block complex matrix M is defined as
(1.4)M=[ABCD],
where A∈Cm×m, D∈Cp×p, B∈Cm×p, and C∈Cp×m. If A is nonsingular, then the classical Schur complement of A in M is given as follows (see [1]):
(1.5)S=D-CA-1B.In [2], Benítez and Thome considered the expression
(1.6)N=[A-+A-BS-CA--A-BS--S-CA-S-],
and N is called the generalized Schur form of the matrix M given in (1.4) being S=D-CA-B for some fixed generalized inverses A-∈A{1}, S-∈S{1}, where S is called generalized Schur complement of A in M. In [2, Theorem 2], Benítez and Thome investigated the expression of the group inverse of M in (1.4) by the generalized Schur complement, where (1.5) is replaced by
(1.7)S=D-CA♯B.
Similar results also were given by Sheng and Chen in [3, Theorem 3.2]. The Drazin inverse of a 2 × 2 block complex square matrix in (1.4) with a singular generalized Schur complement was considered in [4–6], where
(1.8)S=A-CADD.
For the expression of a 2 × 2 block operator matrix was investigated by Deng and Wei in [7].Some notations for the block matrix form of a given elementa∈𝒜 are introduced in [8]. Let a∈𝒜 and s∈𝒜• (see [8, Chapter VII]) which denotes the set of all idempotent elements in 𝒜. Then we write
(1.9)a=sas+sa(1-s)+(1-s)as+(1-s)a(1-s)
and use the notations
(1.10)a11=sas,a12=sa(1-s),a21=(1-s)as,a22=(1-s)a(1-s).
For a representation of arbitrary element a∈𝒜 is given as the following matrix form:
(1.11)a=[sassa(1-s)(1-s)as(1-s)a(1-s)]=[a11a12a21a22]s.In this paper, we will consider some results on the relative perturbation bounds of group inverse and also give the perturbation bounds of the generalized Schur complement of an elementa∈𝒜 under some certain conditions in a Banach algebra.


## 2. Perturbation Bound of∥(a+b)♯-ad∥ in Banach Algebra

In recent years, perturbation theory for the Drazin inverse of a given matrixA∈Cn×n and its applications have been considered in [9–20]. In [12], Yimin and Guorong gave a perturbation result for the Drazin inverse under condition (𝒲) (see [12] for details). In [8, Ch 5], Djordjević and Rakočević extended the perturbation bound of Yimin and Guorong [12] to Banach algebra. In [13], Wei had discussed the upper perturbation bound of ∥B♯-A♯∥/∥A♯∥ with B=A+E and had answered the question of Campbell and Meyer [21] when Ind(A)=1. In [14], Wei and Wu presented the perturbation upper bounds of ∥BD-AD∥/∥AD∥ under the weaker condition core-rankB=core-rankA and completely answered the question of Campbell and Meyer in [21]. In [16], Wei derived a relative perturbation upper bound of ∥B♯-AD∥/∥AD∥ by Jordan canonical of A. In [5], Li gave sharper upper bounds for ∥B♯-AD∥ under weaker conditions: rank(B)=rank(Ak) and ∥AD∥∥E∥<1/(1+∥AD∥∥A∥). In [17], Wei et al. derived constructive perturbation bound of the Drazin inverse of a square matrix by using a technique proposed by Stewart and based on perturbation theory for invariant subspaces. In [18], Xu et al. gave some upper bounds for ∥BD-AD∥/∥AD∥ only under the condition that B is a stable perturbation of A. In [22], González and Koliha investigated the perturbation of the Drazin inverse of a closed linear operator and derived explicit bounds for the perturbations under certain restrictions on the perturbing operators. In [23], González and Vélez-Cerrada analyzed the perturbation of the Drazin inverse and also gave explicit upper bounds of ∥B♯-AD∥ and ∥BB♯-AAD∥ and obtained a result on the continuity of the group inverse for operators on Banach space.In this section, we will investigate the relative perturbation bound of the group inverse in Banach algebra.At first, we will give some concepts and lemmas as follows.Fora∈𝒜d, let p=aad and p∈𝒜• (see [24]):
(2.1)a=[a100a2]p,ad=[a1-1000]p,aπ=1-p=[0001-p]p,
where a1∈p𝒜p is invertible and a2∈(1-p)𝒜(1-p) is quasinilpotent.For anya∈𝒜d, we write σ(a), ρ(a), and r(a) for the spectrum, the resolvent set, and the spectral radius of a, respectively. For λ∈ρ(a) and let R(λ,a)=(λ-a)-1. If 0 is an isolated point of σ(a), then the spectral idempotent corresponding to the set {0} is defined by
(2.2)aπ=12πi∫γ‍R(λ,a)dλ,
where γ is a small circle surrounding 0 and separating 0 from σ(a)/{0}.Some lemmas will be useful for the following proof in this paper.Lemma 2.1 (see [24, Theorem  2.3]).
Letx,y∈𝒜, and let p∈𝒜•. Assume that
(2.3)x=[ac0b]p,y=[b0ca]1-p.(i)
Ifa∈(p𝒜p)d and b∈((1-p)𝒜(1-p))d, then x,y∈𝒜d and
(2.4)xd=[adu0bd]p,yd=[bd0uad]1-p,
where u=∑n=0∞‍(ad)n+2cbnbπ+∑n=0∞‍aπanc(bd)n+2-adcbd.(ii)
Ifx∈𝒜d and a∈(p𝒜p)d, then b∈[(1-p)𝒜(1-p)]d and xd is given by (2.4).Lemma 2.2 (see [24, Corollary  3.4]).
Ifa,b∈𝒜 are generalized Drazin invertible, b is quasinilpotent, and ab=0, then a+b is generalized Drazin invertible and
(2.5)(a+b)d=∑n=0∞‍bn(ad)n+1.The following lemma is a generalization of [25, Theorem 1].Lemma 2.3.
Leta,b∈𝒜d such that ab=ba. Then a+b∈𝒜d if and only if 1+adb∈𝒜d. In this case
(2.6)(a+b)d=ad(1+adb)dbbd+∑n=0∞‍bπ(-b)n(ad)n+1+∑n=0∞‍(bd)n+1(-a)naπ.Now we will state a lemma for the representation of the group inverse of an elementa∈𝒜d with 2×2 block form in Banach algebra (see [26, Theorem 7.7.7] and [23, Theorem 2.2.] which were established for a finite dimensional case and partitioned operators matrix, resp.).Lemma 2.4.
Letz∈𝒜, and it has the block matrix form as z=[z1z12z21z2]p, where p∈𝒜• is an idempotent element, z1 is invertible in p𝒜p, and z2=z21z1-1z12. Let δ=p+z1-1z12z21z1-1. Then z is group invertible if and only if δ is an invertible element in p𝒜p. In this case
(2.7)z♯=[(δz1δ)-1(δz1δ)-1z1-1z12z21z1-1(δz1δ)-1z21z1-1(δz1δ)-1z1-1z12]p,zπ=[p-δ-1-δ-1z1-1z12-z21z1-1δ-11-p-z21z1-1δ-1z1-1z12]p.Letb∈𝒜 be a perturbation element of a. According to (2.1), we obtain
(2.8)b=[b1b12b21b2]p,a+b=[a1+b1b12b21a2+b2]p,
where p=aad.Theorem 2.5.
Leta∈𝒜d and b∈𝒜 be a perturbation element of a, a and a+b which are defined as (2.1) and (2.8), respectively. If ∥adbaad∥<1, then a1+b1 is invertible in subalgebra p𝒜p. Furthermore let a2+b2=b21(a1+b1)-1b12 and δ=p+[p(a+b)p]db(1-p)b[p(a+b)p]d∈p𝒜p. Then a+b is group invertible if and only if δ∈p𝒜p is invertible and δ is invertible if and only if [p(a+b)p]2+pb(1-p)bp∈p𝒜p is invertible. In this case,
(2.9)‖(a+b)♯-ad‖‖ad‖≤T121-‖a1-1b1‖(‖b12‖+‖b21‖)+T12‖a1-1‖3(1-‖a1-1b1‖)4‖b12‖‖b21‖+(T1T2+T2+‖a1-1b1‖1-‖a1-1b1‖T1)T1,
where
(2.10)a1=pap,b1=pbp,b12=pb(1-p),b21=(1-p)bp,T1=‖a1+b1‖2‖((a1+b1)2+b12b21)-1‖,T2=‖a1-1‖‖b12b21‖1-‖a1-1b1‖.Proof.
Letp=aad. Then a, ad, b and a+b have the matrix form as (2.1) and (2.8), respectively, where a1 is invertible in p𝒜p and a2 is quasinilpotent in (1-p)𝒜(1-p).
It follows from the hypothesis∥adbaad∥<1 that ∥a1-1b1∥<1. Thus, it implies that p+a1-1b1∈p𝒜p is invertible. It is easy to see that (a1+b1)-1=(p+a1-1b1)-1a1-1. Let δ=p+[p(a+b)p]db(1-p)b[p(a+b)p]d∈p𝒜p; that is, we have δ=p+(a1+b1)-1b12b21(a1+b1)-1. Therefore, we have
(2.11)δ=(a1+b1)-1[(a1+b1)p(a1+b1)+b12b21](a1+b1)-1=(a1+b1)-1[(a1+b1)2+b12b21](a1+b1)-1=[p(a+b)p]d[[p(a+b)p]2+pb(1-p)bp][p(a+b)p]d.
From the previous equations, we get that δ is invertible if and only if [p(a+b)p]2+pb(1-p)bp is invertible. Since a2+b2=b21(a1+b1)-1b12 and by Lemma 2.4, we obtain that a+b is group invertible if and only if δ∈p𝒜p is invertible.
In the following, we consider the upper bound of∥(a+b)♯-ad∥/∥ad∥.
Applying Lemma2.4, we obtain
(2.12)(a+b)♯=[ηη(a1+b1)-1b12b21(a1+b1)-1ηb21(a1+b1)-1η(a1+b1)-1b12]p,
where η=(δ(a1+b1)δ)-1.
Note that(2.13)η-a1-1=(δ(a1+b1)1δ)-1-a1-1=δ-1(a1+b1)-1δ-1-a1-1=δ-1a1-1δ-1+δ-1∑n=1∞‍(a1-1b1)na1-1δ-1-a1-1=δ-1(a1-1-δa1-1δ)δ-1+δ-1∑n=1∞‍(a1-1b1)na1-1δ-1=δ-1(a1-1-(p+θ)a1-1(p+θ))δ-1+δ-1∑n=1∞‍(a1-1b1)na1-1δ-1=δ-1[a1-1-(pa1-1p+θa1-1p+pa1-1θ+θa1-1θ)]δ-1+δ-1∑n=1∞‍(a1-1b1)na1-1δ-1=-δ-1(θa1-1p+pa1-1θ+θa1-1θ)δ-1+δ-1∑n=1∞‍(a1-1b1)na1-1δ-1=-δ-1(θa1-1p+δa1-1θ)δ-1+δ-1∑n=1∞‍(a1-1b1)na1-1δ-1=-δ-1θa1-1δ-1-a1-1θδ-1+δ-1∑n=1∞‍(a1-1b1)na1-1δ-1,(2.14)‖δ-1‖=‖[p+(a1+b1)-1b12b21(a1+b1)-1]-1‖≤‖a1+b1‖2‖[(a1+b1)2+b12b21]-1‖=T1,
where θ=(a1+b1)-1b12b21(a1+b1)-1.
It shows from∥adbaad∥<1 (i.e., ∥a1-1b1∥<1) that
(2.15)‖θ‖=‖(a1+b1)-1b12b21(a1+b1)-1‖≤‖a1-1‖‖b12b21‖1-‖a1-1b1‖=T2.
From (2.13), (2.14), (2.15), and by ∥a1-1b1∥<1, we obtain that
(2.16)‖η-a1-1‖=‖-δ-1θa1-1δ-1-a1-1θδ-1+δ-1∑n=1∞‍(a1-1b1)na1-1δ-1‖≤‖δ-1θa1-1δ-1‖+‖a1-1θδ-1‖+‖δ-1∑n=1∞‍(a1-1b1)na1-1δ-1‖≤(T1T2+T2+‖a1-1b1‖1-‖a1-1b1‖T1)T1‖a1-1‖.
It follows from (2.12) that
(2.17)(a+b)♯-ad=[η-a1-1η(a1+b1)-1b12b21(a1+b1)-1ηb21(a1+b1)-1η(a1+b1)-1b12]p.
Therefore, according to (2.14), (2.15), and (2.16), we obtain
(2.18)‖(a+b)♯-ad‖=‖[η-a1-1η(a1+b1)-1b12b21(a1+b1)-1ηb21(a1+b1)-1η(a1+b1)-1b12]p‖≤‖η-a1-1‖+‖η(a1+b1)-1b12‖+‖b21(a1+b1)-1η‖+‖b21(a1+b1)-1η(a1+b1)-1b12‖≤‖a1-1‖T121-‖a1-1b1‖(‖b12‖+‖b21‖)+T12(‖a1-1‖1-‖a1-1b1‖)4‖b12‖‖b21‖+(T1T2+T2+‖a1-1b1‖1-‖a1-1b1‖T1)T1‖a1-1‖.
Since ∥a1-1∥=∥ad∥ and by (2.18), it is easy to see that the conclusion holds.
Thus, we complete the proof.LetA,E∈B(X) be both bounded linear operators with B=A+E on Banach space, where X denotes Banach space. If ∥ADE∥+∥Bπ-Aπ∥<1 is satisfied, (it implies that ∥ADE∥<1 and ∥ADEAAD∥<1), then we have the remark.Remark 2.6 (see [23, Theorem  4.2]).
LetA,B∈B(X) be Drazin invertible and group invertible, respectively. If ∥ADE∥+∥Bπ-Aπ∥<1, then
(2.19)‖B#-AD‖‖AD‖≤‖ADE‖+2‖Bπ-Aπ‖1-‖ADE‖-‖Bπ-Aπ‖.Leta=a1⊕a2 and ad=[a1]p-1=a1#; if we put δa=b+a2, then 1+δaa is invertible in 𝒜 when ∥adbaad∥<1. From the Proposition 2.2 (5) of [20], we have a2+b2=b21(a1+b1)-1b12 when (a1+δa)𝒜∩(1-aad)𝒜={0} for [a1]p[a1]p-1=p=aad. Therefore, for ∥a1#∥∥b+a2∥<(1+∥1-aad∥)-1, we arrive at [20, Theorem 4.2]. In fact, the following remark implies that Theorem 2.5 improves the upper bound of ∥(a+b)#-a#∥ of [20, Theorem 4.2].Remark 2.7 (see [20, Theorem  4.2]).
Leta∈G(𝒜) and let a¯=a+δa∈𝒜 with 𝒦#ϵa<(1+∥aπ∥). Assume that a¯𝒜∩(1-aa#)𝒜={0}. Then a¯∈G(𝒜) and
(2.20)‖a¯#-a#‖≤(1+2‖aπ‖)‖a#‖K#(a)ϵa[1-(1+‖aπ‖)‖a#‖K#(a)ϵa]2,
where 𝒦#=∥a∥∥a#∥ and ϵa=∥δa∥∥a∥-1.Theorem 2.8.
Leta,b∈𝒜 be generalized Drazin invertible and satisfy the conditions
(2.21)‖adbaad‖<1,aπba=0.
Then (a+b)♯ exists if and only if aπ(a+b) is group invertible. In this case,
(2.22)‖(a+b)♯-ad‖‖ad‖≤‖ad‖(1-‖adb‖)2[‖b‖+‖baπ‖∑n=0∞‍‖an+1‖‖(bd)n+1‖]+[‖ad‖(1-‖adb‖)2‖baπb‖+‖baπ‖1-‖adb‖]∑n=0∞‍‖an‖‖(bd)n+1‖+‖aπ‖‖ad‖-1∑n=0∞‍‖an‖‖(bd)n+1‖+‖adb‖1-‖adb‖.Proof.
Sincead exists, ad is defined as (2.1). Let b have the block matrix form as
(2.23)b=[b1b3b4b2]p.
Applying the conditionaπba=0, we have b2a2=0 and
(2.24)aπbaada=[00b4a10]p=0.
It follows from (2.24) that a1∈p𝒜p is invertible, b4=0, and
(2.25)b=[b1b30b2]p.
Combining (2.1) and (2.25), we obtain
(2.26)a+b=[a1+b1b30a2+b2]p.
The condition∥adbaad∥<1 implies ∥a1-1b1∥<1 in the subalgebra p𝒜p. Therefore, we conclude that a1+b1∈p𝒜p is invertible and Ind(a1+b1)=0. According to (2.26) and by Lemma 2.1, one observes that (a+b)d exists if and only if (a2+b2)d also. Thus, (a+b)♯ exists if and only if aπ(a+b) is group invertible.
Ifaπ(a+b) is group invertible and by Lemma 2.1, we obtain
(2.27)(a+b)♯=[(a1+b1)-1x0(a2+b2)♯]p,
where x=(a1+b1)-2b3(a2+b2)π-(a1+b1)-1b3(a2+b2)♯.
Sinceb2a2=0 and a2 is quasinilpotent, by Lemma 2.2, we obtain
(2.28)(a2+b2)♯=∑n=0∞‍a2n(b2d)n+1=aπ∑n=0∞‍an(bd)n+1.
From∥adbaad∥<1, one easily has
(2.29)(a1+b1)-1⊕0=∑n=0∞‍(a1-1b1)na1-1⊕0=∑n=0∞‍(adb)nad=a1-1(1+b1a1-1)-1⊕0=ad∑n=0∞‍(bad)n.
It follows from (2.27) and (2.29) that
(2.30)x=(a1+b1)-2b3(a2+b2)π-(a1+b1)-1b3(a2+b2)♯=(∑n=0∞‍(adb)nad)2[b-baπ∑n=0∞‍an+1(bd)n+1]+[(∑n=0∞‍(adb)nad)2baπb-∑n=0∞‍(adb)nadbaπ]∑n=0∞‍an(bd)n+1.
Combining (2.27), (2.28), and (2.29), we obtain
(2.31)(a+b)♯=[∑n=0∞‍a1-1(b1a1-1)nx0∑n=0∞‍a2n(b2d)n+1]p=(∑n=0∞‍(adbad)n)2[b-baπ∑n=0∞‍an+1(bd)n+1]+[(∑n=0∞‍(adb)nad)2baπb-∑n=0∞‍(adb)nadbaπ]∑n=0∞‍an(bd)n+1+∑n=0∞‍(adb)nad+aπ∑n=0∞‍an(bd)n+1.
From (2.31), we derive
(2.32)(a+b)♯-ad=(∑n=0∞‍(adb)nad)2[b-baπ∑n=0∞‍an+1(bd)n+1]+[(∑n=0∞‍(adb)nad)2baπb-∑n=0∞‍(adb)nadbaπ]∑n=0∞‍an(bd)n+1+aπ∑n=0∞‍an(bd)n+1+∑n=1∞‍(adb)nad.
Moreover, by (2.32) we get
(2.33)‖(a+b)♯-ad‖≤(‖ad‖1-‖adb‖)2[‖b‖+‖baπ‖∑n=0∞‍‖an+1‖‖(bd)n+1‖]+[(‖ad‖1-‖adb‖)2‖baπb‖+‖ad‖‖baπ‖1-‖adb‖]∑n=0∞‍‖an‖‖(bd)n+1‖+‖aπ‖∑n=0∞‍‖an‖‖(bd)n+1‖+‖ad‖‖adb‖1-‖adb‖.
Finally, from (2.33) we easily finish the proof.Corollary 2.9.
Leta∈𝒜g and let b∈𝒜d. If a,b satisfy the conditions
(2.34)‖a♯baa♯‖<1,aπba=0,
then (a+b)♯ exists if and only if aπb is group invertible. In this case,
(2.35)‖(a+b)♯-a♯‖‖a♯‖≤‖b‖‖(aπb)π‖‖a♯‖(1-‖a♯b‖)2+‖baπb♯‖1-‖a♯b‖+‖aπb♯‖‖a♯‖+‖a♯b‖1-‖a♯b‖.The conditions of Theorem2.8  ∥adbaad∥<1,aπba=0 are weaker than the conditions (𝒲) (see [12, Theorem 3.2] for finite dimensional cases and [8, Theorem 5.3.2 and Corollary 5.3.3] for Banach algebra). According to aπba=0, we obtain that (2.26) holds. However, in view of (𝒲), we have
(2.36)a+b=[a1+b100a2]p.
Thus, by the conditions (𝒲), we know that a and a+b have the same Drazin invertible property (see [12, Theorem 3.1]). Thus, if a is group invertible, then (a+b) is group invertible. It is easy to see that ∥adbaad∥<1, aπba=0 are weaker than the conditions (𝒲). From [8, Theorem 5.3.2 and Corollary 5.3.3], we easily state the following remark.Remark 2.10.
Leta∈𝒜g and let b∈𝒜d. If a,b satisfy the condition (𝒲)
(2.37)‖a♯baa♯‖<1,b=aa♯baa♯,
then a+b is group invertible and
(2.38)‖(a+b)♯-a♯‖‖a♯‖≤‖a♯b‖1-‖a♯b‖.Theorem 2.11.
Leta,b∈𝒜 be generalized Drazin invertible and satisfy the conditions
(2.39)max⁡{‖adbaad‖,‖aπa‖‖aπbd‖}<1,aπba=0.
Then (a+b)♯ exists if and only if aπ(a+b) is group invertible. In this case,
(2.40)‖(a+b)♯-ad‖‖ad‖≤‖ad‖(1-‖adb‖)2[‖b‖+‖baπ‖‖a‖‖bd‖1-‖a‖‖bd‖]+[‖ad‖(1-‖adb‖)2‖baπb‖+‖baπ‖1-‖adb‖]‖a‖‖bd‖1-‖a‖‖bd‖+‖ad‖-1‖aπ‖‖bd‖1-‖a‖‖bd‖+‖adb‖1-‖adb‖.Proof.
The notations are taken as Theorem2.8, and the rest of proof of theorem is similar to Theorem 2.8. Now, we only consider the perturbation of a2+b2. From (2.28) and the first condition of (2.39), we have ∥a2∥∥b2d∥<1 and
(2.41)‖(a2+b2)♯‖=‖∑n=0∞‍a2n(b2d)n+1‖≤‖aπ‖‖bd‖1-‖a‖‖bd‖.
Thus, from (2.41) we completed the proof.Theorem 2.12.
Leta,b∈𝒜 be generalized Drazin invertible and satisfy the conditions
(2.42)‖adbaad‖<1,aπba=abaπ.
Then (a+b)♯ exists if and only if aπ(a+b) is group invertible. In this case,
(2.43)‖(a+b)♯-ad‖‖ad‖≤‖adb‖1-‖adb‖+‖aπ‖‖ad‖-1∑n=0∞‖(bd)n+1‖‖an‖.Proof.
Lettingp=aad, and it is similar to Theorem 2.8, we obtain that a, ad, and p have the matrix forms as (2.1). Here b is taken as (2.23) in the proof of Theorem 2.8. The condition aπba=abaπ implies that
(2.44)aπba=[0001-p]p[b1b3b4b2]p[a100a2]p=[00b4a1b2a2]p,abaπ=[a100a2]p[b1b3b4b2]p[0001-p]p=[0a1b30a2b2]p.
Thus, according to (2.44), we obtain b4a1=0, a1b3=0 and a2b2=b2a2. Because a1 is invertible in subalgebra p𝒜p, we have b3=b4=0. Thus, b, a+b have the matrix forms as follows:
(2.45)b=[b100b2]p,a+b=[a1+b100a2+b2]p.
It follows from the condition∥adbaad∥<1 that ∥a1-1b1∥<1. Thus, it shows from ∥a1-1b1∥<1 that a1+b1 is invertible in subalgebra p𝒜p. Therefore, easily we observe that a+b is Drazin invertible if and only if a2+b2∈(1-p)𝒜(1-p) is Drazin invertible. That is, (a+b)♯ exists if and only if aπ(a+b) is group invertible.
In the following, we will consider the perturbation ofa2.
Leta2+b2 be group invertible. The condition aπba=abaπ implies that a2b2=b2a2 holds. Since a2 is quasinilpotent in subalgebra (1-p)𝒜(1-p) and by Lemma 2.3, we get
(2.46)(a2+b2)♯=∑n=0∞‍(b2d)n+1(-a2)n=aπ∑n=0∞‍(bd)n+1(-a)n.
By virtue of∥a1-1b1∥<1, we get that
(2.47)[aad(a+b)]-1=(a1+b1)-1=∑n=0∞‍(a1-1b1)na1-1=∑n=0∞‍(adb)nad.
It follows from (2.46) and (2.47) that
(2.48)‖[aad(a+b)]-1-[aada]p-1‖=‖∑n=1∞‍(adb)nad‖≤‖ad‖‖adb‖1-‖adb‖,‖[aπ(a+b)]d‖=‖(a2+b2)♯‖≤‖aπ‖∑n=0∞‍‖(bd)n+1‖‖an‖.
Next, according to (2.48), we obtain
(2.49)‖(a+b)♯-ad‖≤‖ad‖‖adb‖1-‖adb‖+‖aπ‖∑n=1∞‍‖(bd)n+1‖‖an‖.
Finally, using (2.49) the proof is finished.Corollary 2.13.
Leta∈𝒜g and let b∈𝒜d. If a,b satisfy the conditions
(2.50)‖a♯baa♯‖<1,aπba=abaπ.
Then (a+b)♯ exists if and only if aπb is group invertible. In this case,
(2.51)‖(a+b)♯-a♯‖‖a♯‖≤‖a♯b‖1-‖a♯b‖+‖aπ‖‖bd‖‖a♯‖.LetA,E∈Cn×n with B=A+E, and let
(2.52)A=P-1[A100A2]P,E=P-1[E1E12E21E2]P.
If Bπ=Aπ (see [10, Theorem 2.1] ), then
(2.53)B=P-1[B100B2]P,A+E=P-1[A1+E100A2+E2]P,
where B1 is invertible and B2=A2 is quasinilpotent (it follows that E2=0). It follows from (2.53) that Aπ=Bπ implies that AπBA=ABAπ, (i.e., AπEA=AEAπ). If A is group invertible, then B is group invertible and
(2.54)B♯=P-1[B1-1000]P,
where B1=A1+E1.By virtue ofAπ=Bπ and ∥AD(B-A)∥<1 (see [10]), we give the following remark.Remark 2.14 (see [10, Theorem  3.1]).
LetA,B∈Cd×d with Aπ=Bπ. Then
(2.55)‖AD‖1+‖AD(B-A)‖≤‖BD‖.
If ∥AD(B-A)∥<1, then
(2.56)‖BD‖≤‖AD‖1-‖AD(B-A)‖,‖BD-AD‖‖AD‖≤‖AD(B-A)‖1-‖AD(B-A)‖.Theorem 2.15.
Leta,b∈𝒜 be generalized Drazin invertible and satisfy the conditions
(2.57)max⁡{‖aπabd‖,‖adbaad‖}<1,aπba=abaπ.
Then (a+b)♯ exists if and only if aπ(a+b) is group invertible. In this case,
(2.58)‖(a+b)♯-ad‖‖ad‖≤‖adb‖1-‖adb‖+‖aπ‖‖bd‖‖ad‖(1-‖bda‖).Proof.
Similarly to Theorem2.12, we have that the formulas (2.45) hold. The details will be omitted. In the following we only give the simple proof.
By the condition(2.59)max⁡{‖aπabd‖,‖adbaad‖}<1,
it shows that ∥a1-1b1∥<1 and ∥a2b2d∥<1. Thus, the first result shows that a1+b1∈p𝒜p is invertible. In view of Lemma 2.1, one concludes that a+b is Drazin invertible if and only if a2+b2 is Drazin invertible. That is, (a+b)♯ exists if and only if aπ(a+b) is group invertible.
In the following, we consider the perturbation ofa2.
After application of the hypothesisaπba=abaπ, we find that a2b2=b2a2. It follows from Theorem 2.12 and Lemma 2.3 that
(2.60)(a2+b2)♯=∑n=0∞‍(b2d)n+1(-a2)n=aπ∑n=0∞‍(-1)n(bd)n+1an.
It follows from the condition∥a2b2d∥<1 and
(2.61)σ(a2b2d)∪{0}=σ(b2da2)∪{0}.
It implies that ∥b2da2∥<1 and
(2.62)‖(a2+b2)♯‖≤‖aπ‖‖bd‖1-‖bda‖.
Therefore, combining (2.49) with (2.62), we have
(2.63)‖(a+b)♯-ad‖≤‖ad‖‖adb‖1-‖adb‖+‖aπ‖‖bd‖1-‖bda‖.
Thus, by (2.63), we complete the proof.


## 3. Perturbation Bound of the Generalized Schur Complement

The perturbation bounds of the Schur complement are investigated in [29–31]. In [29] Stewart gave perturbation bounds for the Schur complement of a positive definite matrix in a positive semidefinite matrix. In [30] Wei and Wang generalized the results in [29] and enrich the perturbation theory for the Schur complement. In [31] the authors derived some new norm upper bounds for Schur complements of a positive semidefinite operator matrix. In this section, we consider the perturbation bounds of the generalized Schur complement in Banach algebra.Some notations of the generalized Schur complement over Banach algebra will be stated in the following.Leta∈𝒜, and let it be written in the form as follows:
(3.1)a=a11+a12+a21+a22.
It has the following matrix form:
(3.2)M=[a11a12a21a22]s,
where s∈𝒜• is idempotent element in 𝒜 and aij is taken as (1.10).The formulas (1.5) and (1.6) are written in Banach algebra, respectively:
(3.3)s1=a22-a21a11-1a12,M¯=[a11-+a11-a12s1-a21a11--a11-a12s1--s1-a21a11-s1-]s.
Similarly, the generalized Schur complement in (1.7) and (1.8) is defined in the following over Banach algebra, respectively:
(3.4)s1=a22-a21a11♯a12,s1=a22-a21a11da12,
where s1 denotes the generalized Schur complement of a11 in ℳ.Theorem 3.1.
Letℳ be given as (3.2) let and
(3.5)M¯=[a11+Δa11a12+Δa12a21+Δa21a22+Δa22]s=[a¯11a¯12a¯21a¯22]s
be perturbed version of ℳ, and the following conditions are satisfied:
(3.6)‖Δa11‖≤ϵ‖a11‖,‖Δa12‖≤ϵ‖a12‖,‖Δa21‖≤ϵ‖a21‖,‖Δa22‖≤ϵ‖a22‖,
where ϵ>0. If a11, Δa11 and a¯11 satisfy the conditions of Theorem 2.8, then
(3.7)‖s¯1-s1‖≤ϵ‖a22‖+‖a21‖‖a12‖(θ(2ϵ+ϵ2)+‖a11π‖η2+ϵ‖a11d‖‖a11‖η1)+ϵ‖a21‖‖a11‖‖a12‖(η12+‖a11π‖η1η2+(ϵ+ϵ2)‖a11π‖‖a11‖η12η2),
where
(3.8)η1=‖a11d‖1-‖a11dΔa11‖,η2=∑n=0∞‍‖a11n+1‖‖[(Δa11)d]n+1‖,(3.9)θ=ϵ‖a11‖η12(1+‖a11π‖η2)+ϵ‖a11‖‖a11π‖η1η2(1+ϵη1‖a11‖)+‖a11π‖η2+ϵ‖a11d‖‖a11‖η1,
and  s¯1 and s1 are Schur complement of a11 in ℳ and Schur complement of  a¯11 in ℳ¯, respectively.Proof.
Sincea11, Δa11 and a¯11 satisfy Theorem 2.8, according to (2.31), we obtain
(3.10)a¯11♯=(∑n=0∞‍(a11dΔa11)na11d)2[Δa11-Δa11a11π∑n=0∞‍a11n+1[(Δa11)d]n+1]+[(∑n=0∞‍(a11dΔa11)na11d)2Δa11a11πΔa11-∑n=0∞‍(a11dΔa11)na11dΔa11a11π]∑n=0∞‍a11n[(Δa11)d]n+1+∑n=0∞‍(a11dΔa11)na11d+a11π∑n=0∞‍a11n[(Δa11)d]n+1.
Therefore, it is easy to see that
(3.11)s¯1=a22+Δa22-(a21+Δa21)a¯11♯(a12+Δa12)=a22+Δa22-a21a¯11♯a12-Δa21a¯11♯a12-a21a¯11♯Δa12-Δa21a¯11♯Δa12=s1+Δa22-Δa21a¯11♯a12-a21a¯11♯Δa12-Δa21a¯11♯Δa12-a21a11π∑n=0∞‍a11n[(Δa11)d]n+1a12+a21∑n=1∞‍(a11dΔa11)na11da12+a21(∑n=0∞‍(a11dΔa11)na11d)2Δa11[1-a11π(a11+Δa11)∑n=0∞‍a11n[(Δa11)d]n+1]a12-a21∑n=0∞‍(a11dΔa11)na11dΔa11a11π∑n=0∞‍a11n[(Δa11)d]n+1a12,‖a¯11♯‖≤ϵ‖a11‖(‖a11d‖1-‖a11dΔa11‖)2[1+‖a11π‖∑n=0∞‍‖a11n+1‖‖[(Δa11)d]n+1‖]+[ϵ2(‖a11d‖1-‖a11dΔa11‖)2‖a11π‖‖a11‖2+ϵ‖a11d‖‖a11‖‖a11π‖1-‖a11dΔa11‖]∑n=0∞‖a11n‖‖[(Δa11)d]n+1‖+‖a11π‖∑n=0∞‖a11n‖‖[(Δa11)d]n+1‖+ϵ‖a11d‖2‖a11‖1-‖a11dΔa11‖.
From (3.11) and by the conditions (3.6), we obtain(3.12)‖s¯1-s1‖≤‖Δa22‖+θ(2ϵ+ϵ2)‖a21‖‖a12‖+‖a21‖‖a12‖(‖a11π‖∑n=0∞‍‖a11n‖‖[(Δa11)d]n+1‖+ϵ‖a11d‖2‖a11‖1-‖a11dΔa11‖)+ϵ‖a21‖‖a11‖‖a12‖(‖a11d‖1-‖a11dΔa11‖)2[1+(1+ϵ)‖a11‖∑n=0∞‖a11n‖‖[(Δa11)d]n+1‖]+ϵ‖a21‖‖a12‖‖a11d‖‖a11‖‖a11π‖1-‖a11dΔa11‖∑n=0∞‖a11n‖‖[(Δa11)d]n+1‖,
where
(3.13)θ=ϵ‖a11‖(‖a11d‖1-‖a11dΔa11‖)2[1+‖a11π‖∑n=0∞‍‖a11n+1‖‖[(Δa11)d]n+1‖]+ϵ‖a11d‖‖a11‖‖a11π‖1-‖a11dΔa11‖[1+ϵ‖a11d‖‖a11‖‖a11π‖1-‖a11dΔa11‖]∑n=0∞‖a11n‖‖[(Δa11)d]n+1‖+‖a11π‖∑n=0∞‖a11n‖‖(Δa11d)n+1‖+ϵ‖a11d‖2‖a11‖1-‖a11dΔa11‖.
Thus, we finish the proof.Similar to Theorem3.1. It follows from the proof of Theorem 2.11 that the results are given as follow.Theorem 3.2.
Letℳ and ℳ¯ be taken as Theorem 3.1, and let the relations in (3.6) be satisfied, where ϵ>0. If a11, Δa11 and  a¯11 satisfy the conditions of Theorem 2.11, then
(3.14)‖s¯1-s‖≤ϵ‖a22‖+‖a21‖‖a12‖(θ(2ϵ+ϵ2)+‖a11π‖η2+ϵ‖a11d‖‖a11‖η1)+ϵ‖a21‖‖a11‖‖a12‖(η12+‖a11π‖η1η2+(ϵ+ϵ2)‖a11π‖‖a11‖η12η2),
where η1,θ,s¯1 and s1 are taken as Theorem 3.1.Theorem 3.3.
Letℳ and ℳ¯ be taken as Theorem 3.1, and let the relations in (3.6) be satisfied, where ϵ>0. If a11, Δa11 and a¯11 satisfy the conditions of Theorem 2.12, then
(3.15)‖s¯1-s‖≤ϵ‖a22‖+[(1+2ϵ+ϵ2)δ1+ϵ‖a11d‖2‖a11‖]‖a12‖‖a21‖,
where
(3.16)δ1={‖a11d‖1-‖a11dΔa11‖+‖a11π‖∑n=0∞‍‖[(Δa11)d]n+1a11n‖},
and s¯1 and s1 are taken as Theorem 3.1.Proof.
Similar to the proof of Theorem3.1 the details are omitted. A simple proof is given as follows.
By (2.45), (2.46), and (2.47), we obtain
(3.17)a¯11♯=∑n=0∞‍(a11dΔa11)na11d+a11π∑n=0∞‍[(Δa11)d]n+1(-a11)n.
In view of (3.17), we easily have(3.18)s¯1=a22+Δa22-(a21+Δa21)a¯11♯(a12+Δa12)=s1+Δa22-a21∑n=0∞‍(a11dΔa11)na11da12+a21a11π∑n=0∞‍[(Δa11)d]n+1(-a11)na12-a21a11dΔa11a11da12-Δa21a¯11♯a12-a21a¯11♯Δa12-Δa21a¯11♯Δa12,‖a¯11♯‖≤‖∑n=0∞‍(a11dΔa11)na11d‖+‖a11π∑n=0∞‍[(Δa11)d]n+1(-a11)n‖≤‖a11d‖1-‖a11dΔa11‖+‖a11π‖∑n=0∞‍‖[(Δa11)d]n+1a11n‖.
It shows from (3.18) that
(3.19)‖s¯1-s1‖≤ϵ‖a22‖+‖a11d‖‖a21‖‖a12‖1-‖a11dΔa11‖+‖a21‖‖a11π‖∑n=0∞‍‖[(Δa11)d]n+1a11n‖‖a12‖+(2ϵ+ϵ2)‖a21‖‖a¯11#‖‖a12‖+ϵ‖a11d‖‖a11‖‖a12‖‖a21‖≤ϵ‖a22‖+{ϵ‖a11d‖‖a11‖+‖a11d‖1-‖a11dΔa11‖+‖a11π‖∑n=0∞‍‖[(Δa11)d]n+1a11n‖}‖a21‖‖a12‖+(2ϵ+ϵ2)‖a21‖‖a12‖{‖a11π‖∑n=0∞‍‖[(Δa11)d]n+1a11n‖+‖a11d‖1-‖a11dΔa11‖}=ϵ‖a22‖+‖a21‖‖a12‖[(1+2ϵ+ϵ2)δ1+ϵ‖a11d‖‖a11‖],
where
(3.20)δ1={‖a11d‖1-‖a11dΔa11‖+‖a11π‖∑n=0∞‍‖[(Δa11)d]n+1a11n‖}.
Therefore, we complete the proof.Similar to Theorems3.1 and 3.3. The proof of the following theorem follows from Theorem 2.15.Theorem 3.4.
Letℳ and ℳ¯ be taken as Theorem 3.1, and let the relations in (3.6) be satisfied, where ϵ>0. If a11, Δa11 and  a¯11 satisfy the conditions of Theorem 2.15, then
(3.21)‖s¯1-s‖≤ϵ‖a22‖+(1+2ϵ+ϵ2)‖a21‖‖a12‖δ1,
where
(3.22)δ1={‖a11d‖1-‖a11dΔa11‖+‖a11π‖‖(Δa11)d‖1-‖(Δa11)da11‖},
and s¯1 and s1 are taken as Theorem 3.1.



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*Source: 629178-2012-08-30.xml*