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.

1. Introduction

Let π’œ denote a Banach algebra with unit 1. The symbols π’œβˆ’1, π’œπ·, π’œπ‘‘, π’œπ‘”, π’œnil, π’œqnil, and π’œβ€’ stand for the sets of all invertible, Drazin invertible, generalized Drazin invertible, group invertible, nilpotent, quasinilpotent, and idempotent elements of a Banach algebra π’œ, respectively.

Some definitions will be given in the following.

Letting π‘Žβˆˆπ’œπ·, there is an unique element π‘₯βˆˆπ’œ such that π‘Žπ‘˜+1π‘₯=π‘Žπ‘˜,π‘₯π‘Žπ‘₯=π‘₯,π‘Žπ‘₯=π‘₯π‘Ž.(1.1) Then π‘₯ is called the Drazin inverse of π‘Ž, denoted by π‘Žπ·. The smallest nonnegative integer π‘˜ which satisfies (1.1) is called the index of π‘Ž, denoted by Ind(π‘Ž)=π‘˜. If Ind(π‘Ž)≀1, then π‘Žπ·=π‘Žβ™―(orπ‘Žπ‘”).

Let π‘Žβˆˆπ’œ, if the conditions (1.1) are replaced by π‘Žπ‘₯π‘Ž=π‘Ž,π‘₯π‘Žπ‘₯=π‘₯,π‘Žπ‘₯=π‘₯π‘Ž.(1.2) Then π‘₯ is called the group inverse of π‘Ž, denoted by π‘₯=π‘Žβ™―. If the conditions (1.1) are replaced by π‘₯π‘Žπ‘₯=π‘₯,π‘Žπ‘₯=π‘₯π‘Ž,π‘Ž(1βˆ’π‘Žπ‘₯)isquasinilpotent.(1.3) Then π‘₯ is called the generalized Drazin inverse of π‘Ž, denoted by π‘₯=π‘Žπ‘‘.

Some notations of the Schur complement are given in the following.

For a 2Γ—2 block complex matrix 𝑀 is defined as ⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦π‘€=𝐴𝐡𝐢𝐷,(1.4) where π΄βˆˆπΆπ‘šΓ—π‘š, π·βˆˆπΆπ‘Γ—π‘, π΅βˆˆπΆπ‘šΓ—π‘, and πΆβˆˆπΆπ‘Γ—π‘š. If 𝐴 is nonsingular, then the classical Schur complement of 𝐴 in 𝑀 is given as follows (see [1]): 𝑆=π·βˆ’πΆπ΄βˆ’1𝐡.(1.5)

In [2], BenΓ­tez and Thome considered the expression βŽ‘βŽ’βŽ’βŽ£π΄π‘=βˆ’+π΄βˆ’π΅π‘†βˆ’πΆπ΄βˆ’βˆ’π΄βˆ’π΅π‘†βˆ’βˆ’π‘†βˆ’πΆπ΄βˆ’π‘†βˆ’βŽ€βŽ₯βŽ₯⎦,(1.6) and 𝑁 is called the generalized Schur form of the matrix 𝑀 given in (1.4) being 𝑆=π·βˆ’πΆπ΄βˆ’π΅ for some fixed generalized inverses π΄βˆ’βˆˆπ΄{1}, π‘†βˆ’βˆˆπ‘†{1}, where 𝑆 is called generalized Schur complement of 𝐴 in 𝑀. In [2, Theorem 2], BenΓ­tez 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 𝑆=π·βˆ’πΆπ΄β™―π΅.(1.7) Similar results also were given by Sheng and Chen in [3, Theorem 3.2]. The Drazin inverse of a 2 Γ— 2 block complex square matrix in (1.4) with a singular generalized Schur complement was considered in [4–6], where 𝑆=π΄βˆ’πΆπ΄π·π·.(1.8) For the expression of a 2 Γ— 2 block operator matrix was investigated by Deng and Wei in [7].

Some notations for the block matrix form of a given element π‘Žβˆˆπ’œ are introduced in [8]. Let π‘Žβˆˆπ’œ and π‘ βˆˆπ’œβ€’ (see [8, Chapter VII]) which denotes the set of all idempotent elements in π’œ. Then we write π‘Ž=π‘ π‘Žπ‘ +π‘ π‘Ž(1βˆ’π‘ )+(1βˆ’π‘ )π‘Žπ‘ +(1βˆ’π‘ )π‘Ž(1βˆ’π‘ )(1.9) and use the notations π‘Ž11=π‘ π‘Žπ‘ ,π‘Ž12=π‘ π‘Ž(1βˆ’π‘ ),π‘Ž21=(1βˆ’π‘ )π‘Žπ‘ ,π‘Ž22=(1βˆ’π‘ )π‘Ž(1βˆ’π‘ ).(1.10) For a representation of arbitrary element π‘Žβˆˆπ’œ is given as the following matrix form: ⎑⎒⎒⎣(⎀βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ£π‘Žπ‘Ž=π‘ π‘Žπ‘ π‘ π‘Ž(1βˆ’π‘ )1βˆ’π‘ )π‘Žπ‘ (1βˆ’π‘ )π‘Ž(1βˆ’π‘ )11π‘Ž12π‘Ž21π‘Ž22⎀βŽ₯βŽ₯βŽ¦π‘ .(1.11)

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 element π‘Žβˆˆπ’œ under some certain conditions in a Banach algebra.

2. Perturbation Bound of β€–(π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘β€– in Banach Algebra

In recent years, perturbation theory for the Drazin inverse of a given matrix π΄βˆˆπΆπ‘›Γ—π‘› 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], DjordjeviΔ‡ and RakočeviΔ‡ extended the perturbation bound of Yimin and Guorong [12] to Banach algebra. In [13], Wei had discussed the upper perturbation bound of β€–π΅β™―βˆ’π΄β™―β€–/‖𝐴♯‖ with 𝐡=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 β€–π΅π·βˆ’π΄π·β€–/‖𝐴𝐷‖ under the weaker condition coreβˆ’rank𝐡=coreβˆ’rank𝐴 and completely answered the question of Campbell and Meyer in [21]. In [16], Wei derived a relative perturbation upper bound of β€–π΅β™―βˆ’π΄π·β€–/‖𝐴𝐷‖ by Jordan canonical of 𝐴. In [5], Li gave sharper upper bounds for β€–π΅β™―βˆ’π΄π·β€– under weaker conditions: rank(𝐡)=rank(π΄π‘˜) 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 β€–π΅π·βˆ’π΄π·β€–/‖𝐴𝐷‖ only under the condition that 𝐡 is a stable perturbation of 𝐴. In [22], GonzΓ‘lez 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], GonzΓ‘lez and VΓ©lez-Cerrada analyzed the perturbation of the Drazin inverse and also gave explicit upper bounds of β€–π΅β™―βˆ’π΄π·β€– and β€–π΅π΅β™―βˆ’π΄π΄π·β€– 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.

For π‘Žβˆˆπ’œπ‘‘, let 𝑝=π‘Žπ‘Žπ‘‘ and π‘βˆˆπ’œβ€’ (see [24]): βŽ‘βŽ’βŽ’βŽ£π‘Žπ‘Ž=100π‘Ž2⎀βŽ₯βŽ₯βŽ¦π‘,π‘Žπ‘‘=βŽ‘βŽ’βŽ’βŽ£π‘Ž1βˆ’10⎀βŽ₯βŽ₯⎦00𝑝,π‘Žπœ‹βŽ‘βŽ’βŽ’βŽ£βŽ€βŽ₯βŽ₯⎦=1βˆ’π‘=0001βˆ’π‘π‘,(2.1) where π‘Ž1βˆˆπ‘π’œπ‘ is invertible and π‘Ž2∈(1βˆ’π‘)π’œ(1βˆ’π‘) is quasinilpotent.

For any π‘Žβˆˆπ’œπ‘‘, we write 𝜎(π‘Ž), 𝜌(π‘Ž), and π‘Ÿ(π‘Ž) for the spectrum, the resolvent set, and the spectral radius of π‘Ž, respectively. For πœ†βˆˆπœŒ(π‘Ž) and let 𝑅(πœ†,π‘Ž)=(πœ†βˆ’π‘Ž)βˆ’1. If 0 is an isolated point of 𝜎(π‘Ž), then the spectral idempotent corresponding to the set {0} is defined by π‘Žπœ‹=1ξ€œ2πœ‹π‘–π›Ύπ‘…(πœ†,π‘Ž)π‘‘πœ†,(2.2) where 𝛾 is a small circle surrounding 0 and separating 0 from 𝜎(π‘Ž)/{0}.

Some lemmas will be useful for the following proof in this paper.

Lemma 2.1 (see [24, Theorem  2.3]). Let π‘₯,π‘¦βˆˆπ’œ, and let π‘βˆˆπ’œβ€’. Assume that ⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦π‘₯=π‘Žπ‘0π‘π‘βŽ‘βŽ’βŽ’βŽ£βŽ€βŽ₯βŽ₯⎦,𝑦=𝑏0π‘π‘Ž1βˆ’π‘.(2.3)(i)If π‘Žβˆˆ(π‘π’œπ‘)𝑑 and π‘βˆˆ((1βˆ’π‘)π’œ(1βˆ’π‘))𝑑, then π‘₯,π‘¦βˆˆπ’œπ‘‘ and π‘₯𝑑=βŽ‘βŽ’βŽ’βŽ£π‘Žπ‘‘π‘’0π‘π‘‘βŽ€βŽ₯βŽ₯βŽ¦π‘,𝑦𝑑=βŽ‘βŽ’βŽ’βŽ£π‘π‘‘0π‘’π‘Žπ‘‘βŽ€βŽ₯βŽ₯⎦1βˆ’π‘,(2.4) where βˆ‘π‘’=βˆžπ‘›=0(π‘Žπ‘‘)𝑛+2π‘π‘π‘›π‘πœ‹+βˆ‘βˆžπ‘›=0π‘Žπœ‹π‘Žπ‘›π‘(𝑏𝑑)𝑛+2βˆ’π‘Žπ‘‘π‘π‘π‘‘. (ii)If π‘₯βˆˆπ’œπ‘‘ and π‘Žβˆˆ(π‘π’œπ‘)𝑑, then π‘βˆˆ[(1βˆ’π‘)π’œ(1βˆ’π‘)]𝑑 and π‘₯𝑑 is given by (2.4).

Lemma 2.2 (see [24, Corollary  3.4]). If π‘Ž,π‘βˆˆπ’œ are generalized Drazin invertible, 𝑏 is quasinilpotent, and π‘Žπ‘=0, then π‘Ž+𝑏 is generalized Drazin invertible and (π‘Ž+𝑏)𝑑=βˆžξ“π‘›=0π‘π‘›ξ€·π‘Žπ‘‘ξ€Έπ‘›+1.(2.5)

The following lemma is a generalization of [25, Theorem 1].

Lemma 2.3. Let π‘Ž,π‘βˆˆπ’œπ‘‘ such that π‘Žπ‘=π‘π‘Ž. Then π‘Ž+π‘βˆˆπ’œπ‘‘ if and only if 1+π‘Žπ‘‘π‘βˆˆπ’œπ‘‘. In this case (π‘Ž+𝑏)𝑑=π‘Žπ‘‘ξ€·1+π‘Žπ‘‘π‘ξ€Έπ‘‘π‘π‘π‘‘+βˆžξ“π‘›=0π‘πœ‹(βˆ’π‘)π‘›ξ€·π‘Žπ‘‘ξ€Έπ‘›+1+βˆžξ“π‘›=0𝑏𝑑𝑛+1(βˆ’π‘Ž)π‘›π‘Žπœ‹.(2.6)

Now we will state a lemma for the representation of the group inverse of an element π‘Žβˆˆπ’œπ‘‘ with 2Γ—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. Let π‘§βˆˆπ’œ, and it has the block matrix form as 𝑧=𝑧1𝑧12𝑧21𝑧2𝑝, where π‘βˆˆπ’œβ€’ is an idempotent element, 𝑧1 is invertible in π‘π’œπ‘, and 𝑧2=𝑧21𝑧1βˆ’1𝑧12. Let 𝛿=𝑝+𝑧1βˆ’1𝑧12𝑧21𝑧1βˆ’1. Then 𝑧 is group invertible if and only if 𝛿 is an invertible element in π‘π’œπ‘. In this case 𝑧♯=⎑⎒⎒⎣(𝛿𝑧1𝛿)βˆ’1(𝛿𝑧1𝛿)βˆ’1𝑧1βˆ’1𝑧12𝑧21𝑧1βˆ’1(𝛿𝑧1𝛿)βˆ’1𝑧21𝑧1βˆ’1(𝛿𝑧1𝛿)βˆ’1𝑧1βˆ’1𝑧12⎀βŽ₯βŽ₯βŽ¦π‘,π‘§πœ‹=βŽ‘βŽ’βŽ’βŽ£π‘βˆ’π›Ώβˆ’1βˆ’π›Ώβˆ’1𝑧1βˆ’1𝑧12βˆ’π‘§21𝑧1βˆ’1π›Ώβˆ’11βˆ’π‘βˆ’π‘§21𝑧1βˆ’1π›Ώβˆ’1𝑧1βˆ’1𝑧12⎀βŽ₯βŽ₯βŽ¦π‘.(2.7)

Let π‘βˆˆπ’œ be a perturbation element of π‘Ž. According to (2.1), we obtain βŽ‘βŽ’βŽ’βŽ£π‘π‘=1𝑏12𝑏21𝑏2⎀βŽ₯βŽ₯βŽ¦π‘βŽ‘βŽ’βŽ’βŽ£π‘Ž,π‘Ž+𝑏=1+𝑏1𝑏12𝑏21π‘Ž2+𝑏2⎀βŽ₯βŽ₯βŽ¦π‘,(2.8) where 𝑝=π‘Žπ‘Žπ‘‘.

Theorem 2.5. Let π‘Žβˆˆπ’œπ‘‘ and π‘βˆˆπ’œ be a perturbation element of π‘Ž, π‘Ž and π‘Ž+𝑏 which are defined as (2.1) and (2.8), respectively. If β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–<1, then π‘Ž1+𝑏1 is invertible in subalgebra π‘π’œπ‘. Furthermore let π‘Ž2+𝑏2=𝑏21(π‘Ž1+𝑏1)βˆ’1𝑏12 and 𝛿=𝑝+[𝑝(π‘Ž+𝑏)𝑝]𝑑𝑏(1βˆ’π‘)𝑏[𝑝(π‘Ž+𝑏)𝑝]π‘‘βˆˆπ‘π’œπ‘. Then π‘Ž+𝑏 is group invertible if and only if π›Ώβˆˆπ‘π’œπ‘ is invertible and 𝛿 is invertible if and only if [𝑝(π‘Ž+𝑏)𝑝]2+𝑝𝑏(1βˆ’π‘)π‘π‘βˆˆπ‘π’œπ‘ is invertible. In this case, β€–β€–(π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘β€–β€–β€–β€–π‘Žπ‘‘β€–β€–β‰€π‘‡21β€–β€–π‘Ž1βˆ’1βˆ’1𝑏1‖‖‖‖𝑏12β€–β€–+‖‖𝑏21β€–β€–ξ€Έ+𝑇21β€–β€–π‘Ž1βˆ’1β€–β€–3ξ€·β€–β€–π‘Ž1βˆ’1βˆ’1𝑏1β€–β€–ξ€Έ4‖‖𝑏12‖‖‖‖𝑏21β€–β€–+𝑇1𝑇2+𝑇2+β€–β€–π‘Ž1βˆ’1𝑏1β€–β€–β€–β€–π‘Ž1βˆ’1βˆ’1𝑏1‖‖𝑇1ξƒͺ𝑇1,(2.9) where π‘Ž1=π‘π‘Žπ‘,𝑏1=𝑝𝑏𝑝,𝑏12=𝑝𝑏(1βˆ’π‘),𝑏21𝑇=(1βˆ’π‘)𝑏𝑝,1=β€–β€–π‘Ž1+𝑏1β€–β€–2β€–β€–β€–ξ‚€ξ€·π‘Ž1+𝑏1ξ€Έ2+𝑏12𝑏21ξ‚βˆ’1β€–β€–β€–,𝑇2=β€–β€–π‘Ž1βˆ’1‖‖‖‖𝑏12𝑏21β€–β€–β€–β€–π‘Ž1βˆ’1βˆ’1𝑏1β€–β€–.(2.10)

Proof. Let 𝑝=π‘Žπ‘Žπ‘‘. Then π‘Ž, π‘Žπ‘‘, 𝑏 and π‘Ž+𝑏 have the matrix form as (2.1) and (2.8), respectively, where π‘Ž1 is invertible in π‘π’œπ‘ and π‘Ž2 is quasinilpotent in (1βˆ’π‘)π’œ(1βˆ’π‘).
It follows from the hypothesis β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–<1 that β€–π‘Ž1βˆ’1𝑏1β€–<1. Thus, it implies that 𝑝+π‘Ž1βˆ’1𝑏1βˆˆπ‘π’œπ‘ is invertible. It is easy to see that (π‘Ž1+𝑏1)βˆ’1=(𝑝+π‘Ž1βˆ’1𝑏1)βˆ’1π‘Ž1βˆ’1. Let 𝛿=𝑝+[𝑝(π‘Ž+𝑏)𝑝]𝑑𝑏(1βˆ’π‘)𝑏[𝑝(π‘Ž+𝑏)𝑝]π‘‘βˆˆπ‘π’œπ‘; that is, we have 𝛿=𝑝+(π‘Ž1+𝑏1)βˆ’1𝑏12𝑏21(π‘Ž1+𝑏1)βˆ’1. Therefore, we have ξ€·π‘Žπ›Ώ=1+𝑏1ξ€Έβˆ’1π‘Žξ€Ίξ€·1+𝑏1ξ€Έπ‘ξ€·π‘Ž1+𝑏1ξ€Έ+𝑏12𝑏21π‘Žξ€»ξ€·1+𝑏1ξ€Έβˆ’1=ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1ξ‚ƒξ€·π‘Ž1+𝑏1ξ€Έ2+𝑏12𝑏21ξ‚„ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1=[]𝑝(π‘Ž+𝑏)𝑝𝑑[]𝑝(π‘Ž+𝑏)𝑝2ξ€»[]+𝑝𝑏(1βˆ’π‘)𝑏𝑝𝑝(π‘Ž+𝑏)𝑝𝑑.(2.11) From the previous equations, we get that 𝛿 is invertible if and only if [𝑝(π‘Ž+𝑏)𝑝]2+𝑝𝑏(1βˆ’π‘)𝑏𝑝 is invertible. Since π‘Ž2+𝑏2=𝑏21(π‘Ž1+𝑏1)βˆ’1𝑏12 and by Lemma 2.4, we obtain that π‘Ž+𝑏 is group invertible if and only if π›Ώβˆˆπ‘π’œπ‘ is invertible.
In the following, we consider the upper bound of β€–(π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘β€–/β€–π‘Žπ‘‘β€–.
Applying Lemma 2.4, we obtain (π‘Ž+𝑏)β™―=βŽ‘βŽ’βŽ’βŽ£ξ€·π‘Žπœ‚πœ‚1+𝑏1ξ€Έβˆ’1𝑏12𝑏21ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1πœ‚π‘21ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1πœ‚ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1𝑏12⎀βŽ₯βŽ₯βŽ¦π‘,(2.12) where πœ‚=(𝛿(π‘Ž1+𝑏1)𝛿)βˆ’1.
Note that πœ‚βˆ’π‘Ž1βˆ’1=ξ€·π›Ώξ€·π‘Ž1+𝑏1ξ€Έ1π›Ώξ€Έβˆ’1βˆ’π‘Ž1βˆ’1=π›Ώβˆ’1ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1π›Ώβˆ’1βˆ’π‘Ž1βˆ’1=π›Ώβˆ’1π‘Ž1βˆ’1π›Ώβˆ’1+π›Ώβˆžβˆ’1𝑛=1ξ€·π‘Ž1βˆ’1𝑏1ξ€Έπ‘›π‘Ž1βˆ’1π›Ώβˆ’1βˆ’π‘Ž1βˆ’1=π›Ώβˆ’1ξ€·π‘Ž1βˆ’1βˆ’π›Ώπ‘Ž1βˆ’1π›Ώξ€Έπ›Ώβˆ’1+π›Ώβˆžβˆ’1𝑛=1ξ€·π‘Ž1βˆ’1𝑏1ξ€Έπ‘›π‘Ž1βˆ’1π›Ώβˆ’1=π›Ώβˆ’1ξ€·π‘Ž1βˆ’1βˆ’(𝑝+πœƒ)π‘Ž1βˆ’1𝛿(𝑝+πœƒ)βˆ’1+π›Ώβˆžβˆ’1𝑛=1ξ€·π‘Ž1βˆ’1𝑏1ξ€Έπ‘›π‘Ž1βˆ’1π›Ώβˆ’1=π›Ώβˆ’1ξ€Ίπ‘Ž1βˆ’1βˆ’ξ€·π‘π‘Ž1βˆ’1𝑝+πœƒπ‘Ž1βˆ’1𝑝+π‘π‘Ž1βˆ’1πœƒ+πœƒπ‘Ž1βˆ’1πœƒπ›Ώξ€Έξ€»βˆ’1+π›Ώβˆžβˆ’1𝑛=1ξ€·π‘Ž1βˆ’1𝑏1ξ€Έπ‘›π‘Ž1βˆ’1π›Ώβˆ’1=βˆ’π›Ώβˆ’1ξ€·πœƒπ‘Ž1βˆ’1𝑝+π‘π‘Ž1βˆ’1πœƒ+πœƒπ‘Ž1βˆ’1πœƒξ€Έπ›Ώβˆ’1+π›Ώβˆžβˆ’1𝑛=1ξ€·π‘Ž1βˆ’1𝑏1ξ€Έπ‘›π‘Ž1βˆ’1π›Ώβˆ’1=βˆ’π›Ώβˆ’1ξ€·πœƒπ‘Ž1βˆ’1𝑝+π›Ώπ‘Ž1βˆ’1πœƒξ€Έπ›Ώβˆ’1+π›Ώβˆžβˆ’1𝑛=1ξ€·π‘Ž1βˆ’1𝑏1ξ€Έπ‘›π‘Ž1βˆ’1π›Ώβˆ’1=βˆ’π›Ώβˆ’1πœƒπ‘Ž1βˆ’1π›Ώβˆ’1βˆ’π‘Ž1βˆ’1πœƒπ›Ώβˆ’1+π›Ώβˆžβˆ’1𝑛=1ξ€·π‘Ž1βˆ’1𝑏1ξ€Έπ‘›π‘Ž1βˆ’1π›Ώβˆ’1,‖‖𝛿(2.13)βˆ’1β€–β€–=β€–β€–β€–ξ‚ƒξ€·π‘Žπ‘+1+𝑏1ξ€Έβˆ’1𝑏12𝑏21ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1ξ‚„βˆ’1β€–β€–β€–β‰€β€–β€–π‘Ž1+𝑏1β€–β€–2β€–β€–β€–ξ‚ƒξ€·π‘Ž1+𝑏1ξ€Έ2+𝑏12𝑏21ξ‚„βˆ’1β€–β€–β€–=𝑇1,(2.14) where πœƒ=(π‘Ž1+𝑏1)βˆ’1𝑏12𝑏21(π‘Ž1+𝑏1)βˆ’1.
It shows from β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–<1 (i.e., β€–π‘Ž1βˆ’1𝑏1β€–<1) that β€–β€–β€–ξ€·π‘Žπœƒβ€–=1+𝑏1ξ€Έβˆ’1𝑏12𝑏21ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1β€–β€–β‰€β€–β€–π‘Ž1βˆ’1‖‖‖‖𝑏12𝑏21β€–β€–β€–β€–π‘Ž1βˆ’1βˆ’1𝑏1β€–β€–=𝑇2.(2.15)
From (2.13), (2.14), (2.15), and by β€–π‘Ž1βˆ’1𝑏1β€–<1, we obtain that β€–β€–πœ‚βˆ’π‘Ž1βˆ’1β€–β€–=β€–β€–β€–β€–βˆ’π›Ώβˆ’1πœƒπ‘Ž1βˆ’1π›Ώβˆ’1βˆ’π‘Ž1βˆ’1πœƒπ›Ώβˆ’1+π›Ώβˆžβˆ’1𝑛=1ξ€·π‘Ž1βˆ’1𝑏1ξ€Έπ‘›π‘Ž1βˆ’1π›Ώβˆ’1β€–β€–β€–β€–β‰€β€–β€–π›Ώβˆ’1πœƒπ‘Ž1βˆ’1π›Ώβˆ’1β€–β€–+β€–β€–π‘Ž1βˆ’1πœƒπ›Ώβˆ’1β€–β€–+β€–β€–β€–β€–π›Ώβˆžβˆ’1𝑛=1ξ€·π‘Ž1βˆ’1𝑏1ξ€Έπ‘›π‘Ž1βˆ’1π›Ώβˆ’1‖‖‖‖≀𝑇1𝑇2+𝑇2+β€–β€–π‘Ž1βˆ’1𝑏1β€–β€–β€–β€–π‘Ž1βˆ’1βˆ’1𝑏1‖‖𝑇1ξƒͺ𝑇1β€–β€–π‘Ž1βˆ’1β€–β€–.(2.16)
It follows from (2.12) that (π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘=βŽ‘βŽ’βŽ’βŽ£πœ‚βˆ’π‘Ž1βˆ’1πœ‚ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1𝑏12𝑏21ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1πœ‚π‘21ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1πœ‚ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1𝑏12⎀βŽ₯βŽ₯βŽ¦π‘.(2.17)
Therefore, according to (2.14), (2.15), and (2.16), we obtain β€–β€–(π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘β€–β€–=β€–β€–β€–β€–β€–βŽ‘βŽ’βŽ’βŽ£πœ‚βˆ’π‘Ž1βˆ’1πœ‚(π‘Ž1+𝑏1)βˆ’1𝑏12𝑏21ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1πœ‚π‘21ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1πœ‚ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1𝑏12⎀βŽ₯βŽ₯βŽ¦π‘β€–β€–β€–β€–β€–β‰€β€–β€–πœ‚βˆ’π‘Ž1βˆ’1β€–β€–+β€–β€–πœ‚ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1𝑏12β€–β€–+‖‖𝑏21ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1πœ‚β€–β€–+‖‖𝑏21ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1πœ‚ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1𝑏12β€–β€–β‰€β€–β€–π‘Ž1βˆ’1‖‖𝑇21β€–β€–π‘Ž1βˆ’1βˆ’1𝑏1‖‖‖‖𝑏12β€–β€–+‖‖𝑏21β€–β€–ξ€Έ+𝑇21ξƒ©β€–β€–π‘Ž1βˆ’1β€–β€–β€–β€–π‘Ž1βˆ’1βˆ’1𝑏1β€–β€–ξƒͺ4‖‖𝑏12‖‖‖‖𝑏21β€–β€–+𝑇1𝑇2+𝑇2+β€–β€–π‘Ž1βˆ’1𝑏1β€–β€–β€–β€–π‘Ž1βˆ’1βˆ’1𝑏1‖‖𝑇1ξƒͺ𝑇1β€–β€–π‘Ž1βˆ’1β€–β€–.(2.18) Since β€–π‘Ž1βˆ’1β€–=β€–π‘Žπ‘‘β€– and by (2.18), it is easy to see that the conclusion holds.
Thus, we complete the proof.

Let 𝐴,𝐸∈𝐡(𝑋) be both bounded linear operators with 𝐡=𝐴+𝐸 on Banach space, where 𝑋 denotes Banach space. If ‖𝐴𝐷𝐸‖+β€–π΅πœ‹βˆ’π΄πœ‹β€–<1 is satisfied, (it implies that ‖𝐴𝐷𝐸‖<1 and ‖𝐴𝐷𝐸𝐴𝐴𝐷‖<1), then we have the remark.

Remark 2.6 (see [23, Theorem  4.2]). Let 𝐴,𝐡∈𝐡(𝑋) be Drazin invertible and group invertible, respectively. If ‖𝐴𝐷𝐸‖+β€–π΅πœ‹βˆ’π΄πœ‹β€–<1, then ‖‖𝐡#βˆ’π΄π·β€–β€–β€–β€–π΄π·β€–β€–β‰€β€–β€–π΄π·πΈβ€–β€–+2β€–π΅πœ‹βˆ’π΄πœ‹β€–β€–β€–π΄1βˆ’π·πΈβ€–β€–βˆ’β€–π΅πœ‹βˆ’π΄πœ‹β€–.(2.19)

Let π‘Ž=π‘Ž1βŠ•π‘Ž2 and π‘Žπ‘‘=[π‘Ž1]π‘βˆ’1=π‘Ž#1; if we put π›Ώπ‘Ž=𝑏+π‘Ž2, then 1+π›Ώπ‘Žπ‘Ž is invertible in π’œ when β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–<1. From the Proposition 2.2 (5) of [20], we have π‘Ž2+𝑏2=𝑏21(π‘Ž1+𝑏1)βˆ’1𝑏12 when (π‘Ž1+π›Ώπ‘Ž)π’œβˆ©(1βˆ’π‘Žπ‘Žπ‘‘)π’œ={0} for [π‘Ž1]𝑝[π‘Ž1]π‘βˆ’1=𝑝=π‘Žπ‘Žπ‘‘. Therefore, for β€–π‘Ž#1‖‖𝑏+π‘Ž2β€–<(1+β€–1βˆ’π‘Žπ‘Žπ‘‘β€–)βˆ’1, we arrive at [20, Theorem 4.2]. In fact, the following remark implies that Theorem 2.5 improves the upper bound of β€–(π‘Ž+𝑏)#βˆ’π‘Ž#β€– of [20, Theorem 4.2].

Remark 2.7 (see [20, Theorem  4.2]). Let π‘ŽβˆˆπΊ(π’œ) and let π‘Ž=π‘Ž+π›Ώπ‘Žβˆˆπ’œ with 𝒦#πœ–π‘Ž<(1+β€–π‘Žπœ‹β€–). Assume that π‘Žπ’œβˆ©(1βˆ’π‘Žπ‘Ž#)π’œ={0}. Then π‘ŽβˆˆπΊ(π’œ) and β€–β€–π‘Ž#βˆ’π‘Ž#‖‖≀(1+2β€–π‘Žπœ‹β€–)β€–β€–π‘Ž#‖‖𝒦#(π‘Ž)πœ–π‘Žξ€Ί1βˆ’(1+β€–π‘Žπœ‹β€–)β€–β€–π‘Ž#‖‖𝒦#(π‘Ž)πœ–π‘Žξ€»2,(2.20) where 𝒦#=β€–π‘Žβ€–β€–π‘Ž#β€– and πœ–π‘Ž=β€–π›Ώπ‘Žβ€–β€–π‘Žβ€–βˆ’1.

Theorem 2.8. Let π‘Ž,π‘βˆˆπ’œ be generalized Drazin invertible and satisfy the conditions β€–β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–β€–<1,π‘Žπœ‹π‘π‘Ž=0.(2.21) Then (π‘Ž+𝑏)β™― exists if and only if π‘Žπœ‹(π‘Ž+𝑏) is group invertible. In this case, β€–β€–(π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘β€–β€–β€–β€–π‘Žπ‘‘β€–β€–β‰€β€–β€–π‘Žπ‘‘β€–β€–ξ€·β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–ξ€Έ2‖𝑏‖+β€–π‘π‘Žπœ‹β€–βˆžξ“π‘›=0β€–β€–π‘Žπ‘›+1‖‖‖‖𝑏𝑑𝑛+1β€–β€–ξƒ­+ξƒ¬β€–β€–π‘Žπ‘‘β€–β€–ξ€·β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–ξ€Έ2β€–π‘π‘Žπœ‹π‘β€–+β€–π‘π‘Žπœ‹β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–ξƒ­βˆžξ“π‘›=0β€–π‘Žπ‘›β€–β€–β€–ξ€·π‘π‘‘ξ€Έπ‘›+1β€–β€–+β€–π‘Žπœ‹β€–β€–β€–π‘Žπ‘‘β€–β€–βˆžβˆ’1𝑛=0β€–π‘Žπ‘›β€–β€–β€–ξ€·π‘π‘‘ξ€Έπ‘›+1β€–β€–+β€–β€–π‘Žπ‘‘π‘β€–β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–.(2.22)

Proof. Since π‘Žπ‘‘ exists, π‘Žπ‘‘ is defined as (2.1). Let 𝑏 have the block matrix form as βŽ‘βŽ’βŽ’βŽ£π‘π‘=1𝑏3𝑏4𝑏2⎀βŽ₯βŽ₯βŽ¦π‘.(2.23)
Applying the condition π‘Žπœ‹π‘π‘Ž=0, we have 𝑏2π‘Ž2=0 and π‘Žπœ‹π‘π‘Žπ‘Žπ‘‘βŽ‘βŽ’βŽ’βŽ£π‘π‘Ž=004π‘Ž10⎀βŽ₯βŽ₯βŽ¦π‘=0.(2.24) It follows from (2.24) that π‘Ž1βˆˆπ‘π’œπ‘ is invertible, 𝑏4=0, and βŽ‘βŽ’βŽ’βŽ£π‘π‘=1𝑏30𝑏2⎀βŽ₯βŽ₯βŽ¦π‘.(2.25) Combining (2.1) and (2.25), we obtain βŽ‘βŽ’βŽ’βŽ£π‘Žπ‘Ž+𝑏=1+𝑏1𝑏30π‘Ž2+𝑏2⎀βŽ₯βŽ₯βŽ¦π‘.(2.26)
The condition β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–<1 implies β€–π‘Ž1βˆ’1𝑏1β€–<1 in the subalgebra π‘π’œπ‘. Therefore, we conclude that π‘Ž1+𝑏1βˆˆπ‘π’œπ‘ is invertible and Ind(π‘Ž1+𝑏1)=0. According to (2.26) and by Lemma 2.1, one observes that (π‘Ž+𝑏)𝑑 exists if and only if (π‘Ž2+𝑏2)𝑑 also. Thus, (π‘Ž+𝑏)β™― exists if and only if π‘Žπœ‹(π‘Ž+𝑏) is group invertible.
If π‘Žπœ‹(π‘Ž+𝑏) is group invertible and by Lemma 2.1, we obtain (π‘Ž+𝑏)β™―=βŽ‘βŽ’βŽ’βŽ£ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1π‘₯0ξ€·π‘Ž2+𝑏2ξ€Έβ™―βŽ€βŽ₯βŽ₯βŽ¦π‘,(2.27) where π‘₯=(π‘Ž1+𝑏1)βˆ’2𝑏3(π‘Ž2+𝑏2)πœ‹βˆ’(π‘Ž1+𝑏1)βˆ’1𝑏3(π‘Ž2+𝑏2)β™―.
Since 𝑏2π‘Ž2=0 and π‘Ž2 is quasinilpotent, by Lemma 2.2, we obtain ξ€·π‘Ž2+𝑏2ξ€Έβ™―=βˆžξ“π‘›=0π‘Žπ‘›2𝑏𝑑2𝑛+1=π‘Žπœ‹βˆžξ“π‘›=0π‘Žπ‘›ξ€·π‘π‘‘ξ€Έπ‘›+1.(2.28)
From β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–<1, one easily has ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1βŠ•0=βˆžξ“π‘›=0ξ€·π‘Ž1βˆ’1𝑏1ξ€Έπ‘›π‘Ž1βˆ’1βŠ•0=βˆžξ“π‘›=0ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘=π‘Ž1βˆ’1ξ€·1+𝑏1π‘Ž1βˆ’1ξ€Έβˆ’1βŠ•0=π‘Žπ‘‘βˆžξ“π‘›=0ξ€·π‘π‘Žπ‘‘ξ€Έπ‘›.(2.29) It follows from (2.27) and (2.29) that ξ€·π‘Žπ‘₯=1+𝑏1ξ€Έβˆ’2𝑏3ξ€·π‘Ž2+𝑏2ξ€Έπœ‹βˆ’ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1𝑏3ξ€·π‘Ž2+𝑏2ξ€Έβ™―=ξƒ©βˆžξ“π‘›=0ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘ξƒͺ2ξƒ¬π‘βˆ’π‘π‘Žπœ‹βˆžξ“π‘›=0π‘Žπ‘›+1𝑏𝑑𝑛+1ξƒ­+βŽ‘βŽ’βŽ’βŽ£ξƒ©βˆžξ“π‘›=0ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘ξƒͺ2π‘π‘Žπœ‹π‘βˆ’βˆžξ“π‘›=0ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘π‘π‘Žπœ‹βŽ€βŽ₯βŽ₯βŽ¦βˆžξ“π‘›=0π‘Žπ‘›ξ€·π‘π‘‘ξ€Έπ‘›+1.(2.30)
Combining (2.27), (2.28), and (2.29), we obtain (π‘Ž+𝑏)β™―=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£βˆžξ“π‘›=0π‘Ž1βˆ’1𝑏1π‘Ž1βˆ’1𝑛π‘₯0βˆžξ“π‘›=0π‘Žπ‘›2𝑏𝑑2𝑛+1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘=ξƒ©βˆžξ“π‘›=0ξ€·π‘Žπ‘‘π‘π‘Žπ‘‘ξ€Έπ‘›ξƒͺ2ξƒ¬π‘βˆ’π‘π‘Žπœ‹βˆžξ“π‘›=0π‘Žπ‘›+1𝑏𝑑𝑛+1ξƒ­+βŽ‘βŽ’βŽ’βŽ£ξƒ©βˆžξ“π‘›=0ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘ξƒͺ2π‘π‘Žπœ‹π‘βˆ’βˆžξ“π‘›=0ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘π‘π‘Žπœ‹βŽ€βŽ₯βŽ₯βŽ¦βˆžξ“π‘›=0π‘Žπ‘›ξ€·π‘π‘‘ξ€Έπ‘›+1+βˆžξ“π‘›=0ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘+π‘Žπœ‹βˆžξ“π‘›=0π‘Žπ‘›ξ€·π‘π‘‘ξ€Έπ‘›+1.(2.31)
From (2.31), we derive (π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘=ξƒ©βˆžξ“π‘›=0ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘ξƒͺ2ξƒ¬π‘βˆ’π‘π‘Žπœ‹βˆžξ“π‘›=0π‘Žπ‘›+1𝑏𝑑𝑛+1ξƒ­+βŽ‘βŽ’βŽ’βŽ£ξƒ©βˆžξ“π‘›=0ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘ξƒͺ2π‘π‘Žπœ‹π‘βˆ’βˆžξ“π‘›=0ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘π‘π‘Žπœ‹βŽ€βŽ₯βŽ₯βŽ¦βˆžξ“π‘›=0π‘Žπ‘›ξ€·π‘π‘‘ξ€Έπ‘›+1+π‘Žπœ‹βˆžξ“π‘›=0π‘Žπ‘›ξ€·π‘π‘‘ξ€Έπ‘›+1+βˆžξ“π‘›=1ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘.(2.32) Moreover, by (2.32) we get β€–β€–(π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘β€–β€–β‰€ξƒ©β€–β€–π‘Žπ‘‘β€–β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–ξƒͺ2‖𝑏‖+β€–π‘π‘Žπœ‹β€–βˆžξ“π‘›=0β€–β€–π‘Žπ‘›+1‖‖‖‖𝑏𝑑𝑛+1β€–β€–ξƒ­+βŽ‘βŽ’βŽ’βŽ£ξƒ©β€–β€–π‘Žπ‘‘β€–β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–ξƒͺ2β€–π‘π‘Žπœ‹β€–β€–π‘Žπ‘β€–+π‘‘β€–β€–β€–π‘π‘Žπœ‹β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–βŽ€βŽ₯βŽ₯βŽ¦βˆžξ“π‘›=0β€–π‘Žπ‘›β€–β€–β€–ξ€·π‘π‘‘ξ€Έπ‘›+1β€–β€–+β€–π‘Žπœ‹β€–βˆžξ“π‘›=0β€–π‘Žπ‘›β€–β€–β€–ξ€·π‘π‘‘ξ€Έπ‘›+1β€–β€–+β€–β€–π‘Žπ‘‘β€–β€–β€–β€–π‘Žπ‘‘π‘β€–β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–.(2.33)
Finally, from (2.33) we easily finish the proof.

Corollary 2.9. Let π‘Žβˆˆπ’œπ‘” and let π‘βˆˆπ’œπ‘‘. If π‘Ž,𝑏 satisfy the conditions β€–β€–π‘Žβ™―π‘π‘Žπ‘Žβ™―β€–β€–<1,π‘Žπœ‹π‘π‘Ž=0,(2.34) then (π‘Ž+𝑏)β™― exists if and only if π‘Žπœ‹π‘ is group invertible. In this case, β€–β€–(π‘Ž+𝑏)β™―βˆ’π‘Žβ™―β€–β€–β€–β€–π‘Žβ™―β€–β€–β‰€β€–π‘β€–β€–(π‘Žπœ‹π‘)πœ‹β€–β€–β€–π‘Žβ™―β€–β€–ξ€·β€–β€–π‘Ž1βˆ’β™―π‘β€–β€–ξ€Έ2+β€–β€–π‘π‘Žπœ‹π‘β™―β€–β€–β€–β€–π‘Ž1βˆ’β™―π‘β€–β€–+β€–β€–π‘Žπœ‹π‘β™―β€–β€–β€–β€–π‘Žβ™―β€–β€–+β€–β€–π‘Žβ™―π‘β€–β€–β€–β€–π‘Ž1βˆ’β™―π‘β€–β€–.(2.35)

The conditions of Theorem 2.8β€‰β€‰β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–<1,π‘Žπœ‹π‘π‘Ž=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 π‘Žπœ‹π‘π‘Ž=0, we obtain that (2.26) holds. However, in view of (𝒲), we have βŽ‘βŽ’βŽ’βŽ£π‘Žπ‘Ž+𝑏=1+𝑏100π‘Ž2⎀βŽ₯βŽ₯βŽ¦π‘.(2.36) Thus, by the conditions (𝒲), we know that π‘Ž and π‘Ž+𝑏 have the same Drazin invertible property (see [12, Theorem 3.1]). Thus, if π‘Ž is group invertible, then (π‘Ž+𝑏) is group invertible. It is easy to see that β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–<1, π‘Žπœ‹π‘π‘Ž=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. Let π‘Žβˆˆπ’œπ‘” and let π‘βˆˆπ’œπ‘‘. If π‘Ž,𝑏 satisfy the condition (𝒲) β€–β€–π‘Žβ™―π‘π‘Žπ‘Žβ™―β€–β€–<1,𝑏=π‘Žπ‘Žβ™―π‘π‘Žπ‘Žβ™―,(2.37) then π‘Ž+𝑏 is group invertible and β€–β€–(π‘Ž+𝑏)β™―βˆ’π‘Žβ™―β€–β€–β€–β€–π‘Žβ™―β€–β€–β‰€β€–β€–π‘Žβ™―π‘β€–β€–β€–β€–π‘Ž1βˆ’β™―π‘β€–β€–.(2.38)

Theorem 2.11. Let π‘Ž,π‘βˆˆπ’œ be generalized Drazin invertible and satisfy the conditions ξ€½β€–β€–π‘Žmaxπ‘‘π‘π‘Žπ‘Žπ‘‘β€–β€–,β€–π‘Žπœ‹β€–β€–π‘Žπ‘Žβ€–πœ‹π‘π‘‘β€–β€–ξ€Ύ<1,π‘Žπœ‹π‘π‘Ž=0.(2.39) Then (π‘Ž+𝑏)β™― exists if and only if π‘Žπœ‹(π‘Ž+𝑏) is group invertible. In this case, β€–β€–(π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘β€–β€–β€–β€–π‘Žπ‘‘β€–β€–β‰€β€–β€–π‘Žπ‘‘β€–β€–ξ€·β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–ξ€Έ2‖𝑏‖+β€–π‘π‘Žπœ‹β€–β€–β€–π‘β€–π‘Žβ€–π‘‘β€–β€–β€–β€–π‘1βˆ’β€–π‘Žβ€–π‘‘β€–β€–ξƒ­+ξƒ¬β€–β€–π‘Žπ‘‘β€–β€–ξ€·β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–ξ€Έ2β€–π‘π‘Žπœ‹π‘β€–+β€–π‘π‘Žπœ‹β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–ξƒ­β€–β€–β€–π‘π‘Žβ€–π‘‘β€–β€–β€–β€–π‘1βˆ’β€–π‘Žβ€–π‘‘β€–β€–+β€–β€–π‘Žπ‘‘β€–β€–βˆ’1β€–π‘Žπœ‹β€–β€–β€–π‘π‘‘β€–β€–β€–β€–π‘1βˆ’β€–π‘Žβ€–π‘‘β€–β€–+β€–β€–π‘Žπ‘‘π‘β€–β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–.(2.40)

Proof. The notations are taken as Theorem 2.8, and the rest of proof of theorem is similar to Theorem 2.8. Now, we only consider the perturbation of π‘Ž2+𝑏2. From (2.28) and the first condition of (2.39), we have β€–π‘Ž2‖‖𝑏𝑑2β€–<1 and β€–β€–ξ€·π‘Ž2+𝑏2ξ€Έβ™―β€–β€–=β€–β€–β€–β€–βˆžξ“π‘›=0π‘Žπ‘›2𝑏𝑑2𝑛+1β€–β€–β€–β€–β‰€β€–π‘Žπœ‹β€–β€–β€–π‘π‘‘β€–β€–β€–β€–π‘1βˆ’β€–π‘Žβ€–π‘‘β€–β€–.(2.41) Thus, from (2.41) we completed the proof.

Theorem 2.12. Let π‘Ž,π‘βˆˆπ’œ be generalized Drazin invertible and satisfy the conditions β€–β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–β€–<1,π‘Žπœ‹π‘π‘Ž=π‘Žπ‘π‘Žπœ‹.(2.42) Then (π‘Ž+𝑏)β™― exists if and only if π‘Žπœ‹(π‘Ž+𝑏) is group invertible. In this case, β€–β€–(π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘β€–β€–β€–β€–π‘Žπ‘‘β€–β€–β‰€β€–β€–π‘Žπ‘‘π‘β€–β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–+β€–π‘Žπœ‹β€–β€–β€–π‘Žπ‘‘β€–β€–βˆžβˆ’1𝑛=0‖‖𝑏𝑑𝑛+1β€–β€–β€–π‘Žπ‘›β€–.(2.43)

Proof. Letting 𝑝=π‘Žπ‘Žπ‘‘, and it is similar to Theorem 2.8, we obtain that π‘Ž, π‘Žπ‘‘, 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 π‘Žπœ‹π‘π‘Ž=π‘Žπ‘π‘Žπœ‹ implies that π‘Žπœ‹βŽ‘βŽ’βŽ’βŽ£βŽ€βŽ₯βŽ₯βŽ¦π‘π‘Ž=0001βˆ’π‘π‘βŽ‘βŽ’βŽ’βŽ£π‘1𝑏3𝑏4𝑏2⎀βŽ₯βŽ₯βŽ¦π‘βŽ‘βŽ’βŽ’βŽ£π‘Ž100π‘Ž2⎀βŽ₯βŽ₯βŽ¦π‘=βŽ‘βŽ’βŽ’βŽ£π‘004π‘Ž1𝑏2π‘Ž2⎀βŽ₯βŽ₯βŽ¦π‘,π‘Žπ‘π‘Žπœ‹=βŽ‘βŽ’βŽ’βŽ£π‘Ž100π‘Ž2⎀βŽ₯βŽ₯βŽ¦π‘βŽ‘βŽ’βŽ’βŽ£π‘1𝑏3𝑏4𝑏2⎀βŽ₯βŽ₯βŽ¦π‘βŽ‘βŽ’βŽ’βŽ£βŽ€βŽ₯βŽ₯⎦0001βˆ’π‘π‘=⎑⎒⎒⎣0π‘Ž1𝑏30π‘Ž2𝑏2⎀βŽ₯βŽ₯βŽ¦π‘.(2.44) Thus, according to (2.44), we obtain 𝑏4π‘Ž1=0, π‘Ž1𝑏3=0 and π‘Ž2𝑏2=𝑏2π‘Ž2. Because π‘Ž1 is invertible in subalgebra π‘π’œπ‘, we have 𝑏3=𝑏4=0. Thus, 𝑏, π‘Ž+𝑏 have the matrix forms as follows: βŽ‘βŽ’βŽ’βŽ£π‘π‘=100𝑏2⎀βŽ₯βŽ₯βŽ¦π‘βŽ‘βŽ’βŽ’βŽ£π‘Ž,π‘Ž+𝑏=1+𝑏100π‘Ž2+𝑏2⎀βŽ₯βŽ₯βŽ¦π‘.(2.45)
It follows from the condition β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–<1 that β€–π‘Ž1βˆ’1𝑏1β€–<1. Thus, it shows from β€–π‘Ž1βˆ’1𝑏1β€–<1 that π‘Ž1+𝑏1 is invertible in subalgebra π‘π’œπ‘. Therefore, easily we observe that π‘Ž+𝑏 is Drazin invertible if and only if π‘Ž2+𝑏2∈(1βˆ’π‘)π’œ(1βˆ’π‘) is Drazin invertible. That is, (π‘Ž+𝑏)β™― exists if and only if π‘Žπœ‹(π‘Ž+𝑏) is group invertible.
In the following, we will consider the perturbation of π‘Ž2.
Let π‘Ž2+𝑏2 be group invertible. The condition π‘Žπœ‹π‘π‘Ž=π‘Žπ‘π‘Žπœ‹ implies that π‘Ž2𝑏2=𝑏2π‘Ž2 holds. Since π‘Ž2 is quasinilpotent in subalgebra (1βˆ’π‘)π’œ(1βˆ’π‘) and by Lemma 2.3, we get ξ€·π‘Ž2+𝑏2ξ€Έβ™―=βˆžξ“π‘›=0𝑏𝑑2𝑛+1ξ€·βˆ’π‘Ž2𝑛=π‘Žπœ‹βˆžξ“π‘›=0𝑏𝑑𝑛+1(βˆ’π‘Ž)𝑛.(2.46)
By virtue of β€–π‘Ž1βˆ’1𝑏1β€–<1, we get that ξ€Ίπ‘Žπ‘Žπ‘‘(ξ€»π‘Ž+𝑏)βˆ’1=ξ€·π‘Ž1+𝑏1ξ€Έβˆ’1=βˆžξ“π‘›=0ξ€·π‘Ž1βˆ’1𝑏1ξ€Έπ‘›π‘Ž1βˆ’1=βˆžξ“π‘›=0ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘.(2.47)
It follows from (2.46) and (2.47) that β€–β€–ξ€Ίπ‘Žπ‘Žπ‘‘(ξ€»π‘Ž+𝑏)βˆ’1βˆ’ξ€Ίπ‘Žπ‘Žπ‘‘π‘Žξ€»π‘βˆ’1β€–β€–=β€–β€–β€–β€–βˆžξ“π‘›=1ξ€·π‘Žπ‘‘π‘ξ€Έπ‘›π‘Žπ‘‘β€–β€–β€–β€–β‰€β€–β€–π‘Žπ‘‘β€–β€–β€–β€–π‘Žπ‘‘π‘β€–β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–,β€–β€–[π‘Žπœ‹](π‘Ž+𝑏)𝑑‖‖=β€–β€–ξ€·π‘Ž2+𝑏2ξ€Έβ™―β€–β€–β‰€β€–π‘Žπœ‹β€–βˆžξ“π‘›=0‖‖𝑏𝑑𝑛+1β€–β€–β€–π‘Žπ‘›β€–.(2.48)
Next, according to (2.48), we obtain β€–β€–(π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘β€–β€–β‰€β€–β€–π‘Žπ‘‘β€–β€–β€–β€–π‘Žπ‘‘π‘β€–β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–+β€–π‘Žπœ‹β€–βˆžξ“π‘›=1‖‖𝑏𝑑𝑛+1β€–β€–β€–π‘Žπ‘›β€–.(2.49)
Finally, using (2.49) the proof is finished.

Corollary 2.13. Let π‘Žβˆˆπ’œπ‘” and let π‘βˆˆπ’œπ‘‘. If π‘Ž,𝑏 satisfy the conditions β€–β€–π‘Žβ™―π‘π‘Žπ‘Žβ™―β€–β€–<1,π‘Žπœ‹π‘π‘Ž=π‘Žπ‘π‘Žπœ‹.(2.50) Then (π‘Ž+𝑏)β™― exists if and only if π‘Žπœ‹π‘ is group invertible. In this case, β€–β€–(π‘Ž+𝑏)β™―βˆ’π‘Žβ™―β€–β€–β€–β€–π‘Žβ™―β€–β€–β‰€β€–β€–π‘Žβ™―π‘β€–β€–β€–β€–π‘Ž1βˆ’β™―π‘β€–β€–+β€–π‘Žπœ‹β€–β€–β€–π‘π‘‘β€–β€–β€–β€–π‘Žβ™―β€–β€–.(2.51)

Let 𝐴,πΈβˆˆπΆπ‘›Γ—π‘› with 𝐡=𝐴+𝐸, and let 𝐴=π‘ƒβˆ’1⎑⎒⎒⎣𝐴100𝐴2⎀βŽ₯βŽ₯βŽ¦π‘ƒ,𝐸=π‘ƒβˆ’1⎑⎒⎒⎣𝐸1𝐸12𝐸21𝐸2⎀βŽ₯βŽ₯βŽ¦π‘ƒ.(2.52) If π΅πœ‹=π΄πœ‹ (see [10, Theorem 2.1] ), then 𝐡=π‘ƒβˆ’1⎑⎒⎒⎣𝐡100𝐡2⎀βŽ₯βŽ₯βŽ¦π‘ƒ,𝐴+𝐸=π‘ƒβˆ’1⎑⎒⎒⎣𝐴1+𝐸100𝐴2+𝐸2⎀βŽ₯βŽ₯βŽ¦π‘ƒ,(2.53) where 𝐡1 is invertible and 𝐡2=𝐴2 is quasinilpotent (it follows that 𝐸2=0). It follows from (2.53) that π΄πœ‹=π΅πœ‹ implies that π΄πœ‹π΅π΄=π΄π΅π΄πœ‹, (i.e., π΄πœ‹πΈπ΄=π΄πΈπ΄πœ‹). If 𝐴 is group invertible, then 𝐡 is group invertible and 𝐡♯=π‘ƒβˆ’1⎑⎒⎒⎣𝐡1βˆ’10⎀βŽ₯βŽ₯⎦00𝑃,(2.54) where 𝐡1=𝐴1+𝐸1.

By virtue of π΄πœ‹=π΅πœ‹ and ‖𝐴𝐷(π΅βˆ’π΄)β€–<1 (see [10]), we give the following remark.

Remark 2.14 (see [10, Theorem  3.1]). Let 𝐴,π΅βˆˆπΆπ‘‘Γ—π‘‘ with π΄πœ‹=π΅πœ‹. Then ‖‖𝐴𝐷‖‖‖‖𝐴1+𝐷‖‖≀‖‖𝐡(π΅βˆ’π΄)𝐷‖‖.(2.55) If ‖𝐴𝐷(π΅βˆ’π΄)β€–<1, then ‖‖𝐡𝐷‖‖≀‖‖𝐴𝐷‖‖‖‖𝐴1βˆ’π·β€–β€–,‖‖𝐡(π΅βˆ’π΄)π·βˆ’π΄π·β€–β€–β€–β€–π΄π·β€–β€–β‰€β€–β€–π΄π·β€–β€–(π΅βˆ’π΄)‖‖𝐴1βˆ’π·(β€–β€–.π΅βˆ’π΄)(2.56)

Theorem 2.15. Let π‘Ž,π‘βˆˆπ’œ be generalized Drazin invertible and satisfy the conditions ξ€½β€–β€–π‘Žmaxπœ‹π‘Žπ‘π‘‘β€–β€–,β€–β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–β€–ξ€Ύ<1,π‘Žπœ‹π‘π‘Ž=π‘Žπ‘π‘Žπœ‹.(2.57) Then (π‘Ž+𝑏)β™― exists if and only if π‘Žπœ‹(π‘Ž+𝑏) is group invertible. In this case, β€–β€–(π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘β€–β€–β€–β€–π‘Žπ‘‘β€–β€–β‰€β€–β€–π‘Žπ‘‘π‘β€–β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–+β€–π‘Žπœ‹β€–β€–β€–π‘π‘‘β€–β€–β€–β€–π‘Žπ‘‘β€–β€–ξ€·β€–β€–π‘1βˆ’π‘‘π‘Žβ€–β€–ξ€Έ.(2.58)

Proof. Similarly to Theorem 2.12, 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 ξ€½β€–β€–π‘Žmaxπœ‹π‘Žπ‘π‘‘β€–β€–,β€–β€–π‘Žπ‘‘π‘π‘Žπ‘Žπ‘‘β€–β€–ξ€Ύ<1,(2.59) it shows that β€–π‘Ž1βˆ’1𝑏1β€–<1 and β€–π‘Ž2𝑏𝑑2β€–<1. Thus, the first result shows that π‘Ž1+𝑏1βˆˆπ‘π’œπ‘ is invertible. In view of Lemma 2.1, one concludes that π‘Ž+𝑏 is Drazin invertible if and only if π‘Ž2+𝑏2 is Drazin invertible. That is, (π‘Ž+𝑏)β™― exists if and only if π‘Žπœ‹(π‘Ž+𝑏) is group invertible.
In the following, we consider the perturbation of π‘Ž2.
After application of the hypothesis π‘Žπœ‹π‘π‘Ž=π‘Žπ‘π‘Žπœ‹, we find that π‘Ž2𝑏2=𝑏2π‘Ž2. It follows from Theorem 2.12 and Lemma 2.3 that ξ€·π‘Ž2+𝑏2ξ€Έβ™―=βˆžξ“π‘›=0𝑏𝑑2𝑛+1ξ€·βˆ’π‘Ž2𝑛=π‘Žπœ‹βˆžξ“π‘›=0(βˆ’1)𝑛𝑏𝑑𝑛+1π‘Žπ‘›.(2.60)
It follows from the condition β€–π‘Ž2𝑏𝑑2β€–<1 and πœŽξ€·π‘Ž2𝑏𝑑2ξ€Έβˆͺ𝑏{0}=πœŽπ‘‘2π‘Ž2ξ€Έβˆͺ{0}.(2.61) It implies that ‖𝑏𝑑2π‘Ž2β€–<1 and β€–β€–ξ€·π‘Ž2+𝑏2ξ€Έβ™―β€–β€–β‰€β€–π‘Žπœ‹β€–β€–β€–π‘π‘‘β€–β€–β€–β€–π‘1βˆ’π‘‘π‘Žβ€–β€–.(2.62)
Therefore, combining (2.49) with (2.62), we have β€–β€–(π‘Ž+𝑏)β™―βˆ’π‘Žπ‘‘β€–β€–β‰€β€–β€–π‘Žπ‘‘β€–β€–β€–β€–π‘Žπ‘‘π‘β€–β€–β€–β€–π‘Ž1βˆ’π‘‘π‘β€–β€–+β€–π‘Žπœ‹β€–β€–β€–π‘π‘‘β€–β€–β€–β€–π‘1βˆ’π‘‘π‘Žβ€–β€–.(2.63) 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.

Let π‘Žβˆˆπ’œ, and let it be written in the form as follows: π‘Ž=π‘Ž11+π‘Ž12+π‘Ž21+π‘Ž22.(3.1) It has the following matrix form: βŽ‘βŽ’βŽ’βŽ£π‘Žβ„³=11π‘Ž12π‘Ž21π‘Ž22⎀βŽ₯βŽ₯βŽ¦π‘ ,(3.2) where π‘ βˆˆπ’œβ€’ is idempotent element in π’œ and π‘Žπ‘–π‘— is taken as (1.10).

The formulas (1.5) and (1.6) are written in Banach algebra, respectively: 𝑠1=π‘Ž22βˆ’π‘Ž21π‘Žβˆ’111π‘Ž12,βŽ‘βŽ’βŽ’βŽ£π‘Žβ„³=βˆ’11+π‘Žβˆ’11π‘Ž12π‘ βˆ’1π‘Ž21π‘Žβˆ’11βˆ’π‘Žβˆ’11π‘Ž12π‘ βˆ’1βˆ’π‘ βˆ’1π‘Ž21π‘Žβˆ’11π‘ βˆ’1⎀βŽ₯βŽ₯βŽ¦π‘ .(3.3) Similarly, the generalized Schur complement in (1.7) and (1.8) is defined in the following over Banach algebra, respectively: 𝑠1=π‘Ž22βˆ’π‘Ž21π‘Žβ™―11π‘Ž12,𝑠1=π‘Ž22βˆ’π‘Ž21π‘Žπ‘‘11π‘Ž12,(3.4) where 𝑠1 denotes the generalized Schur complement of π‘Ž11 in β„³.

Theorem 3.1. Let β„³ be given as (3.2) let and βŽ‘βŽ’βŽ’βŽ£π‘Žβ„³=11+Ξ”π‘Ž11π‘Ž12+Ξ”π‘Ž12π‘Ž21+Ξ”π‘Ž21π‘Ž22+Ξ”π‘Ž22⎀βŽ₯βŽ₯βŽ¦π‘ =βŽ‘βŽ’βŽ’βŽ£π‘Ž11π‘Ž12π‘Ž21π‘Ž22⎀βŽ₯βŽ₯βŽ¦π‘ (3.5) be perturbed version of β„³, and the following conditions are satisfied: β€–β€–Ξ”π‘Ž11β€–β€–β€–β€–π‘Žβ‰€πœ–11β€–β€–,β€–β€–Ξ”π‘Ž12β€–β€–β€–β€–π‘Žβ‰€πœ–12β€–β€–,β€–β€–Ξ”π‘Ž21β€–β€–β€–β€–π‘Žβ‰€πœ–21β€–β€–,β€–β€–Ξ”π‘Ž22β€–β€–β€–β€–π‘Žβ‰€πœ–22β€–β€–,(3.6) where πœ–>0. If π‘Ž11, Ξ”π‘Ž11 and π‘Ž11 satisfy the conditions of Theorem 2.8, then ‖‖𝑠1βˆ’π‘ 1β€–β€–β€–β€–π‘Žβ‰€πœ–22β€–β€–+β€–β€–π‘Ž21β€–β€–β€–β€–π‘Ž12β€–β€–ξ€·πœƒξ€·2πœ–+πœ–2ξ€Έ+β€–β€–π‘Žπœ‹11β€–β€–πœ‚2β€–β€–π‘Ž+πœ–π‘‘11β€–β€–β€–β€–π‘Ž11β€–β€–πœ‚1ξ€Έβ€–β€–π‘Ž+πœ–21β€–β€–β€–β€–π‘Ž11β€–β€–β€–β€–π‘Ž12β€–β€–ξ€·πœ‚21+β€–β€–π‘Žπœ‹11β€–β€–πœ‚1πœ‚2+ξ€·πœ–+πœ–2ξ€Έβ€–β€–π‘Žπœ‹11β€–β€–β€–β€–π‘Ž11β€–β€–πœ‚21πœ‚2ξ€Έ,(3.7) where πœ‚1=β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–,πœ‚2=βˆžξ“π‘›=0β€–β€–π‘Žπ‘›+111β€–β€–β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1β€–β€–β€–,β€–β€–π‘Ž(3.8)πœƒ=πœ–11β€–β€–πœ‚21ξ€·β€–β€–π‘Ž1+πœ‹11β€–β€–πœ‚2ξ€Έβ€–β€–π‘Ž+πœ–11β€–β€–β€–β€–π‘Žπœ‹11β€–β€–πœ‚1πœ‚2ξ€·1+πœ–πœ‚1β€–β€–π‘Ž11β€–β€–ξ€Έ+β€–β€–π‘Žπœ‹11β€–β€–πœ‚2β€–β€–π‘Ž+πœ–π‘‘11β€–β€–β€–β€–π‘Ž11β€–β€–πœ‚1,(3.9) and  𝑠1 and 𝑠1 are Schur complement of π‘Ž11 in β„³ and Schur complement ofβ€‰β€‰π‘Ž11 in β„³, respectively.

Proof. Since π‘Ž11, Ξ”π‘Ž11 and π‘Ž11 satisfy Theorem 2.8, according to (2.31), we obtain π‘Žβ™―11=ξƒ©βˆžξ“π‘›=0ξ€·π‘Žπ‘‘11Ξ”π‘Ž11ξ€Έπ‘›π‘Žπ‘‘11ξƒͺ2ξƒ¬Ξ”π‘Ž11βˆ’Ξ”π‘Ž11π‘Žπœ‹βˆž11𝑛=0π‘Žπ‘›+111ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1ξƒ­+βŽ‘βŽ’βŽ’βŽ£ξƒ©βˆžξ“π‘›=0ξ€·π‘Žπ‘‘11Ξ”π‘Ž11ξ€Έπ‘›π‘Žπ‘‘11ξƒͺ2Ξ”π‘Ž11π‘Žπœ‹11Ξ”π‘Ž11βˆ’βˆžξ“π‘›=0ξ€·π‘Žπ‘‘11Ξ”π‘Ž11ξ€Έπ‘›π‘Žπ‘‘11Ξ”π‘Ž11π‘Žπœ‹11⎀βŽ₯βŽ₯βŽ¦βˆžξ“π‘›=0π‘Žπ‘›11ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1+βˆžξ“π‘›=0ξ€·π‘Žπ‘‘11Ξ”π‘Ž11ξ€Έπ‘›π‘Žπ‘‘11+π‘Žπœ‹βˆž11𝑛=0π‘Žπ‘›11ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1.(3.10) Therefore, it is easy to see that 𝑠1=π‘Ž22+Ξ”π‘Ž22βˆ’ξ€·π‘Ž21+Ξ”π‘Ž21ξ€Έπ‘Žβ™―11ξ€·π‘Ž12+Ξ”π‘Ž12ξ€Έ=π‘Ž22+Ξ”π‘Ž22βˆ’π‘Ž21π‘Žβ™―11π‘Ž12βˆ’Ξ”π‘Ž21π‘Žβ™―11π‘Ž12βˆ’π‘Ž21π‘Žβ™―11Ξ”π‘Ž12βˆ’Ξ”π‘Ž21π‘Žβ™―11Ξ”π‘Ž12=𝑠1+Ξ”π‘Ž22βˆ’Ξ”π‘Ž21π‘Žβ™―11π‘Ž12βˆ’π‘Ž21π‘Žβ™―11Ξ”π‘Ž12βˆ’Ξ”π‘Ž21π‘Žβ™―11Ξ”π‘Ž12βˆ’π‘Ž21π‘Žπœ‹βˆž11𝑛=0π‘Žπ‘›11ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1π‘Ž12+π‘Žβˆž21𝑛=1ξ€·π‘Žπ‘‘11Ξ”π‘Ž11ξ€Έπ‘›π‘Žπ‘‘11π‘Ž12+π‘Ž21ξƒ©βˆžξ“π‘›=0ξ€·π‘Žπ‘‘11Ξ”π‘Ž11ξ€Έπ‘›π‘Žπ‘‘11ξƒͺ2Ξ”π‘Ž111βˆ’π‘Žπœ‹11ξ€·π‘Ž11+Ξ”π‘Ž11ξ€Έβˆžξ“π‘›=0π‘Žπ‘›11ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1ξƒ­π‘Ž12βˆ’π‘Žβˆž21𝑛=0ξ€·π‘Žπ‘‘11Ξ”π‘Ž11ξ€Έπ‘›π‘Žπ‘‘11Ξ”π‘Ž11π‘Žπœ‹βˆž11𝑛=0π‘Žπ‘›11ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1π‘Ž12,β€–β€–π‘Žβ™―11β€–β€–β€–β€–π‘Žβ‰€πœ–11β€–β€–ξƒ©β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–ξƒͺ2ξƒ¬β€–β€–π‘Ž1+πœ‹11β€–β€–βˆžξ“π‘›=0β€–β€–π‘Žπ‘›+111β€–β€–β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1β€–β€–β€–ξƒ­+βŽ‘βŽ’βŽ’βŽ£πœ–2ξƒ©β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–ξƒͺ2β€–β€–π‘Žπœ‹11β€–β€–β€–β€–π‘Ž11β€–β€–2+πœ–β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž11β€–β€–β€–β€–π‘Žπœ‹11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–βŽ€βŽ₯βŽ₯βŽ¦βˆžξ“π‘›=0β€–β€–π‘Žπ‘›11β€–β€–β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1β€–β€–β€–+β€–β€–π‘Žπœ‹11β€–β€–βˆžξ“π‘›=0β€–β€–π‘Žπ‘›11β€–β€–β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1β€–β€–β€–+πœ–β€–β€–π‘Žπ‘‘11β€–β€–2β€–β€–π‘Ž11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–.(3.11)
From (3.11) and by the conditions (3.6), we obtain‖‖𝑠1βˆ’π‘ 1β€–β€–β‰€β€–β€–Ξ”π‘Ž22β€–β€–ξ€·+πœƒ2πœ–+πœ–2ξ€Έβ€–β€–π‘Ž21β€–β€–β€–β€–π‘Ž12β€–β€–+β€–β€–π‘Ž21β€–β€–β€–β€–π‘Ž12β€–β€–ξƒ©β€–β€–π‘Žπœ‹11β€–β€–βˆžξ“π‘›=0β€–β€–π‘Žπ‘›11β€–β€–β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1β€–β€–β€–+πœ–β€–β€–π‘Žπ‘‘11β€–β€–2β€–β€–π‘Ž11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–ξƒͺβ€–β€–π‘Ž+πœ–21β€–β€–β€–β€–π‘Ž11β€–β€–β€–β€–π‘Ž12β€–β€–ξƒ©β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–ξƒͺ2ξƒ¬β€–β€–π‘Ž1+(1+πœ–)11β€–β€–βˆžξ“π‘›=0β€–β€–π‘Žπ‘›11β€–β€–β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1β€–β€–β€–ξƒ­β€–β€–π‘Ž+πœ–21β€–β€–β€–β€–π‘Ž12β€–β€–β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž11β€–β€–β€–β€–π‘Žπœ‹11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–βˆžξ“π‘›=0β€–β€–π‘Žπ‘›11β€–β€–β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1β€–β€–β€–,(3.12) where β€–β€–π‘Žπœƒ=πœ–11β€–β€–ξƒ©β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–ξƒͺ2ξƒ¬β€–β€–π‘Ž1+πœ‹11β€–β€–βˆžξ“π‘›=0β€–β€–π‘Žπ‘›+111β€–β€–β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1β€–β€–β€–ξƒ­+πœ–β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž11β€–β€–β€–β€–π‘Žπœ‹11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–ξƒ¬πœ–β€–β€–π‘Ž1+𝑑11β€–β€–β€–β€–π‘Ž11β€–β€–β€–β€–π‘Žπœ‹11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–ξƒ­βˆžξ“π‘›=0β€–β€–π‘Žπ‘›11β€–β€–β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1β€–β€–β€–+β€–β€–π‘Žπœ‹11β€–β€–βˆžξ“π‘›=0β€–β€–π‘Žπ‘›11β€–β€–β€–β€–ξ€·Ξ”π‘Žπ‘‘11𝑛+1β€–β€–+πœ–β€–β€–π‘Žπ‘‘11β€–β€–2β€–β€–π‘Ž11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–.(3.13) Thus, we finish the proof.

Similar to Theorem 3.1. It follows from the proof of Theorem 2.11 that the results are given as follow.

Theorem 3.2. Let β„³ and β„³ be taken as Theorem 3.1, and let the relations in (3.6) be satisfied, where πœ–>0. If π‘Ž11, Ξ”π‘Ž11 andβ€‰β€‰π‘Ž11 satisfy the conditions of Theorem 2.11, then ‖‖𝑠1β€–β€–β€–β€–π‘Žβˆ’π‘ β‰€πœ–22β€–β€–+β€–β€–π‘Ž21β€–β€–β€–β€–π‘Ž12β€–β€–ξ€·πœƒξ€·2πœ–+πœ–2ξ€Έ+β€–β€–π‘Žπœ‹11β€–β€–πœ‚2β€–β€–π‘Ž+πœ–π‘‘11β€–β€–β€–β€–π‘Ž11β€–β€–πœ‚1ξ€Έβ€–β€–π‘Ž+πœ–21β€–β€–β€–β€–π‘Ž11β€–β€–β€–β€–π‘Ž12β€–β€–ξ€·πœ‚21+β€–β€–π‘Žπœ‹11β€–β€–πœ‚1πœ‚2+ξ€·πœ–+πœ–2ξ€Έβ€–β€–π‘Žπœ‹11β€–β€–β€–β€–π‘Ž11β€–β€–πœ‚21πœ‚2ξ€Έ,(3.14) where πœ‚1,πœƒ,𝑠1 and 𝑠1 are taken as Theorem 3.1.

Theorem 3.3. Let β„³ and β„³ be taken as Theorem 3.1, and let the relations in (3.6) be satisfied, where πœ–>0. If π‘Ž11, Ξ”π‘Ž11 and π‘Ž11 satisfy the conditions of Theorem 2.12, then ‖‖𝑠1β€–β€–β€–β€–π‘Žβˆ’π‘ β‰€πœ–22β€–β€–+1+2πœ–+πœ–2𝛿1β€–β€–π‘Ž+πœ–π‘‘11β€–β€–2β€–β€–π‘Ž11β€–β€–ξ‚„β€–β€–π‘Ž12β€–β€–β€–β€–π‘Ž21β€–β€–,(3.15) where 𝛿1=ξƒ―β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–+β€–β€–π‘Žπœ‹11β€–β€–βˆžξ“π‘›=0β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1π‘Žπ‘›11β€–β€–β€–ξƒ°,(3.16) and 𝑠1 and 𝑠1 are taken as Theorem 3.1.

Proof. Similar to the proof of Theorem 3.1 the details are omitted. A simple proof is given as follows.
By (2.45), (2.46), and (2.47), we obtain π‘Žβ™―11=βˆžξ“π‘›=0ξ€·π‘Žπ‘‘11Ξ”π‘Ž11ξ€Έπ‘›π‘Žπ‘‘11+π‘Žπœ‹βˆž11𝑛=0ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1ξ€·βˆ’π‘Ž11𝑛.(3.17)
In view of (3.17), we easily have𝑠1=π‘Ž22+Ξ”π‘Ž22βˆ’ξ€·π‘Ž21+Ξ”π‘Ž21ξ€Έπ‘Žβ™―11ξ€·π‘Ž12+Ξ”π‘Ž12ξ€Έ=𝑠1+Ξ”π‘Ž22βˆ’π‘Žβˆž21𝑛=0ξ€·π‘Žπ‘‘11Ξ”π‘Ž11ξ€Έπ‘›π‘Žπ‘‘11π‘Ž12+π‘Ž21π‘Žπœ‹βˆž11𝑛=0ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1ξ€·βˆ’π‘Ž11ξ€Έπ‘›π‘Ž12βˆ’π‘Ž21π‘Žπ‘‘11Ξ”π‘Ž11π‘Žπ‘‘11π‘Ž12βˆ’Ξ”π‘Ž21π‘Žβ™―11π‘Ž12βˆ’π‘Ž21π‘Žβ™―11Ξ”π‘Ž12βˆ’Ξ”π‘Ž21π‘Žβ™―11Ξ”π‘Ž12,β€–β€–π‘Žβ™―11β€–β€–β‰€β€–β€–β€–β€–βˆžξ“π‘›=0ξ€·π‘Žπ‘‘11Ξ”π‘Ž11ξ€Έπ‘›π‘Žπ‘‘11β€–β€–β€–β€–+β€–β€–β€–β€–π‘Žπœ‹βˆž11𝑛=0ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1ξ€·βˆ’π‘Ž11ξ€Έπ‘›β€–β€–β€–β€–β‰€β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–+β€–β€–π‘Žπœ‹11β€–β€–βˆžξ“π‘›=0β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1π‘Žπ‘›11β€–β€–β€–.(3.18)
It shows from (3.18) that ‖‖𝑠1βˆ’π‘ 1β€–β€–β€–β€–π‘Žβ‰€πœ–22β€–β€–+β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž21β€–β€–β€–β€–π‘Ž12β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–+β€–β€–π‘Ž21β€–β€–β€–β€–π‘Žπœ‹11β€–β€–βˆžξ“π‘›=0β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1π‘Žπ‘›11β€–β€–β€–β€–β€–π‘Ž12β€–β€–+ξ€·2πœ–+πœ–2ξ€Έβ€–β€–π‘Ž21β€–β€–β€–β€–π‘Ž#11β€–β€–β€–β€–π‘Ž12β€–β€–β€–β€–π‘Ž+πœ–π‘‘11β€–β€–β€–β€–π‘Ž11β€–β€–β€–β€–π‘Ž12β€–β€–β€–β€–π‘Ž21β€–β€–β€–β€–π‘Žβ‰€πœ–22β€–β€–+ξƒ―πœ–β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž11β€–β€–+β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–+β€–β€–π‘Žπœ‹11β€–β€–βˆžξ“π‘›=0β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1π‘Žπ‘›11β€–β€–β€–ξƒ°β€–β€–π‘Ž21β€–β€–β€–β€–π‘Ž12β€–β€–+ξ€·2πœ–+πœ–2ξ€Έβ€–β€–π‘Ž21β€–β€–β€–β€–π‘Ž12β€–β€–ξƒ―β€–β€–π‘Žπœ‹11β€–β€–βˆžξ“π‘›=0β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1π‘Žπ‘›11β€–β€–β€–+β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–ξƒ°β€–β€–π‘Ž=πœ–22β€–β€–+β€–β€–π‘Ž21β€–β€–β€–β€–π‘Ž12β€–β€–ξ€Ίξ€·1+2πœ–+πœ–2𝛿1β€–β€–π‘Ž+πœ–π‘‘11β€–β€–β€–β€–π‘Ž11β€–β€–ξ€»,(3.19) where 𝛿1=ξƒ―β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–+β€–β€–π‘Žπœ‹11β€–β€–βˆžξ“π‘›=0β€–β€–β€–ξ‚ƒξ€·Ξ”π‘Ž11𝑑𝑛+1π‘Žπ‘›11β€–β€–β€–ξƒ°.(3.20)
Therefore, we complete the proof.

Similar to Theorems 3.1 and 3.3. The proof of the following theorem follows from Theorem 2.15.

Theorem 3.4. Let β„³ and β„³ be taken as Theorem 3.1, and let the relations in (3.6) be satisfied, where πœ–>0. If π‘Ž11, Ξ”π‘Ž11 andβ€‰β€‰π‘Ž11 satisfy the conditions of Theorem 2.15, then ‖‖𝑠1β€–β€–β€–β€–π‘Žβˆ’π‘ β‰€πœ–22β€–β€–+ξ€·1+2πœ–+πœ–2ξ€Έβ€–β€–π‘Ž21β€–β€–β€–β€–π‘Ž12‖‖𝛿1,(3.21) where 𝛿1=⎧βŽͺ⎨βŽͺβŽ©β€–β€–π‘Žπ‘‘11β€–β€–β€–β€–π‘Ž1βˆ’π‘‘11Ξ”π‘Ž11β€–β€–+β€–β€–π‘Žπœ‹11β€–β€–β€–β€–ξ€·Ξ”π‘Ž11𝑑‖‖‖‖1βˆ’Ξ”π‘Ž11ξ€Έπ‘‘π‘Ž11β€–β€–βŽ«βŽͺ⎬βŽͺ⎭,(3.22) and 𝑠1 and 𝑠1 are taken as Theorem 3.1.

Acknowledgments

X. Liu is supported by the NSFC grant (11061005, 61165015), the Ministry of Education Science and Technology Key Project under Grant 210164 and Grants (HCIC201103) of Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis Open Fund and Key issues for Department of Education of Guangxi (201202ZD031). H. Wei is supported by 973 Program (Project no. 2010CB327900).