Higher-Order Convergent Iterative Method for Computing the Generalized Inverse over Banach Spaces

Abstract

A higher-order convergent iterative method is provided for calculating the generalized inverse over Banach spaces. We also use this iterative method for computing the generalized Drazin inverse 𝑎^𝑑 in Banach algebra. Moreover, we estimate the error bounds of the iterative methods for approximating 𝐴⁽²⁾_𝑇,𝑆 or 𝑎^𝑑.

1. Introduction

It is well known that the outer generalized inverse has been widely used in various fields, for instance, in statistics, control theory, power systems, nonlinear equations, optimization and numerical analysis, and so on (see [1–15]). Recently, in [16], the authors discussed the iteration (1) for computing 𝐴⁽²⁾_𝑇,𝑆 of a given matrix.

Throughout this paper, let 𝑋 and 𝑌 be arbitrary Banach spaces. Then, the symbol ℬ(𝒳,𝒴) denotes the set of all bounded linear operators from 𝒳 to 𝒴, in particular, ℬ(𝒳)∶=ℬ(𝒳,𝒳). For any 𝐴∈ℬ(𝒳,𝒴), we denote its range, null space, and norm by ℛ(𝐴), 𝒩(𝐴), and ‖𝐴‖, respectively. Further, we say that 𝐴 is regular if there exists an 𝑋∈ℬ(𝒴,𝒳) such that 𝐴𝑋𝐴=𝐴 and that 𝐴 has a {2} (or outer) inverse if there exists an 𝑋∈ℬ(𝒴,𝒳) such that 𝑋𝐴𝑋=𝑋. If 𝐴∈ℬ(𝒳), then we denote its spectrum and spectral radius by 𝜎(𝐴) and 𝜌(𝐴), respectively. Let the symbol 𝐿⊂𝒳 denote that 𝐿 is a subspace of 𝒳. If 𝐴∈ℬ(𝒳,𝒴) and 𝐿⊂𝒳, then the restriction 𝐴|_𝐿 of 𝐴 on 𝐿 is defined by 𝑥↦𝐴𝑥, 𝑥∈𝐿. Let 𝐿,𝑀⊂𝒳 with 𝐿⊕𝑀=𝒳. Then, the symbol 𝑃_𝐿,𝑀 stands for an operator that is called a projection from 𝒳 onto 𝐿 if it is a bounded linear map from 𝒳 onto 𝐿 and 𝑃²_𝐿,𝑀=𝑃_𝐿,𝑀. It is well known that a closed subspace 𝐿 of a Banach space 𝒳 is complemented in 𝒳 if and only if there exists a projection from 𝒳 onto 𝐿.

Let 𝐴∈ℬ(𝒳,𝒴) be close; there exists a unique operator 𝑋∈ℬ(𝒴,𝒳) such that (1)(2)(3)𝐴𝑋𝐴=𝐴𝑋𝐴𝑋=𝑋)(𝐴𝑋^∗=𝐴𝑋(4)(𝑋𝐴)^∗=𝑋𝐴. (1) Then, 𝑋 is called the Moore-Penrose inverse of 𝐴, denoted by 𝑋=𝐴^†. It is well known that 𝐴 is regular ⇔𝑅(𝐴) is closed ⇔𝐴^† exists.

Throughout this paper, let 𝒜 be a complex Banach algebra with the unit 1. The symbols 𝑎𝑛𝑛^𝑙(𝑎) and 𝑎𝑛𝑛^𝑟(𝑎), respectively, stand for the left and right annihilators of 𝑎 in 𝒜. Let 𝑝∈𝒜 be idempotent. Then, 𝑝𝒜𝑝={𝑝𝑎𝑝∶𝑎∈𝒜} is a subalgebra of 𝒜 with unit 𝑝. Thus, for 𝑎∈𝒜, if there exists an element 𝑏∈𝑝𝒜𝑝 such that 𝑎𝑏=𝑏𝑎=𝑝, then we say that 𝑎 is invertible in 𝑝𝒜𝑝, and 𝑏 is denoted by 𝑎|⁻¹_𝑝𝒜𝑝. Recall that an element 𝑏∈𝒜 is the generalized Drazin inverse of 𝑎 (or Koliha-Drazin inverse of 𝑎), if the following hold: 𝑎()𝑏𝑎𝑏=𝑏,𝑏𝑎=𝑎𝑏,1−𝑎𝑏isquasinilpotent. (2) If the generalized Drazin inverse of 𝑎 exists, then it is denoted by 𝑎^𝑑 (see [15] for more details). In particular, if 𝑏=𝑎^𝑑 and 𝑎(1−𝑎𝑏)=0, then 𝑏 is called the group inverse of 𝑎 and is denoted by 𝑎_𝑔.

In [17], W. G. Li and Z. Li defined the iterative formula 𝑋_𝑘+1=𝑋_𝑘𝑘()𝑘𝐼−𝑘−12𝐴𝑋_𝑘+⋯+(−1)^𝑘−1𝐴𝑋_𝑘^𝑘−1,𝑘=2,3,…. (3) In [18], Chen and Wang extended the iterative method (3) proposed by W. G. Li and Z. Li to compute the Moore-Penrose inverse of a matrix. In [19], Liu et al provided the higher-order convergent iterative method (3) in order to calculate the generalized inverse 𝐴⁽²⁾_𝑇,𝑆 of a given matrix. In this paper, we will extend the iterative method proposed by W. G. Li and Z. Li in [17] to compute the {2}-inverse, generalized inverse 𝐴⁽²⁾_𝑇,𝑆 over Banach space and also consider the iterative scheme for computing the generalized Drazin inverse 𝑎^𝑑 in Banach algebra.

The paper is organized as follows. Some lemmas will be presented in the remainder of this section. In Section 2, we consider iterative scheme of [19] to compute the generalized inverses 𝐴⁽²⁾_𝑇,𝑆 in Banach space. In Section 3, we present iterative formulas for computing the generalized Drazin inverse 𝑎^𝑑 of Banach algebra element 𝑎.

The following lemmas are needed in what follows.

Lemma 1 (see [14, Chapter 1]). Let 𝑎∈𝒜. Then(i)𝜎(𝑎) is a nonempty closed subset of ℂ.(ii)(Spectral mapping theorem for polynomials) if 𝑓 is a polynomial, then 𝜎𝑓(𝑎)=𝑓(𝜎(𝑎)). (4)(iii)lim_𝑛→0𝑎^𝑛=0 if and only if 𝜌(𝑎)<1.

Lemma 2 (see [15, Section 4]). Let 𝑋 and 𝑌 be Banach spaces, and let 𝐴∈ℬ(𝒳,𝒴), 𝑇 and 𝑆, respectively, be closed subspaces of 𝑋 and 𝑌. Then, the following statements are equivalent.(i)𝐴 has a {2}-inverse 𝐵∈𝐵(𝑌,𝑋) such that ℛ(𝐵)=𝑇 and 𝒩(𝐵)=𝑆.(ii)𝑇 is a complemented subspace of 𝑋, 𝐴(𝑇) is closed, 𝐴|_𝑇∶𝑇→𝐴(𝑇) is invertible, and 𝐴(𝑇)⊕𝑆=𝑌.
In the case when (𝑖) or (𝑖𝑖) holds, 𝐵 is unique and is denoted by 𝐴⁽²⁾_𝑇,𝑆.

Lemma 3. Suppose that the conditions of Lemma 2 are satisfied. Then, 𝐴𝐴⁽²⁾_𝑇,𝑆=𝑃_{𝐴(𝑇),𝑆} and 𝐴⁽²⁾_𝑇,𝑆𝐴=𝑃_𝑇,𝑇1 where 𝑇₁=𝒩(𝐴⁽²⁾_𝑇,𝑆𝐴). Moreover, for any 𝐺∈ℬ(𝒴,𝒳), 𝑃_𝑇,𝑇1𝐺=𝐺⇔ℛ(𝐺)⊂𝑇; 𝐺𝑃_{𝐴(𝑇),𝑆}=𝐺⇔𝒩(𝐺)⊃𝑆.

2. Higher-Order Convergent Iterative Method for Computing the Generalized Inverse over Banach Spaces

In this section, we will consider higher-order convergent iterative method for computing the generalized inverse 𝐴⁽²⁾_𝑇,𝑆 over Banach spaces. First, we deduce convergent conditions and error bounds of our iterative methods.

Theorem 4. Let 𝐴∈ℬ(𝒳,𝒴), 𝑌∈ℬ(𝒴,𝒳), and let 𝑇⊂𝒳 and 𝑆⊂𝒴 both be complemented subspaces, respectively, with ℛ(𝑌)=𝑇, 𝒩(𝑌)=𝑆. Define the sequence {𝑋_𝑘} in ℬ(𝒴,𝒳) in the following way: 𝑋₀𝑋=𝛼𝑌,_𝑘=𝐶¹_𝑡𝐼−𝐶²_𝑡𝑋_𝑘−1𝐴+⋯+(−1)^𝑡−1𝐶^𝑡_𝑡𝑋_𝑘−1𝐴^𝑡−1𝑋_𝑘−1; (5) it converges to 𝑋_∞ and 𝑋_∞∈𝐴{2} with 𝑅(𝑋_∞)=𝑇 if and only if 𝜌(𝛼𝑌𝐴−𝑃)<1 for some scalar 𝛼∈ℂ⧵{0}, where 𝑡≥2 is an arbitrary positive integer, 𝑋_∞=lim𝑋_𝑘, and 𝑃 is projection from 𝒳 onto 𝑇. Moreover,(i)if 𝒩(𝑋_∞)=𝑆, then 𝐴⁽²⁾_𝑇,𝑆 exists if and only if 𝜌(𝛼𝑌𝐴−𝑃)<1 for 𝛼∈ℂ⧵{0};(ii)if 𝒩(𝑋_∞)=𝑆, then 𝐴⁽²⁾_𝑇,𝑆 exists.
In particular, if 𝐴⁽²⁾_𝑇,𝑆 exists, lim𝑋_𝑘=𝐴⁽²⁾_𝑇,𝑆 and 𝑞=‖𝛼𝑌𝐴−𝑃‖, then 𝐴⁽²⁾_𝑇,𝑆−𝑋_𝑘𝐴⁽²⁾_𝑇,𝑆≤𝑞^𝑡𝑘,𝑘≥0. (6)

Proof. From (5), we obtain 𝐶¹_𝑡𝐼−𝐶²_𝑡𝑋_𝑘−1𝐴+⋯+(−1)^𝑡−1𝐶^𝑡_𝑡𝑋_𝑘−1𝐴^𝑡−1𝑋_𝑘−1=𝑋_𝑘−1𝐶¹_𝑡𝐼−𝐶²_𝑡𝐴𝑋_𝑘−1+⋯+(−1)^𝑡−1𝐶^𝑡_𝑡𝐴𝑋_𝑘−1^𝑡−1. (7) Note that ℛ(𝑋_𝑘)⊂ℛ(𝑋_𝑘−1), 𝑘≥1 from (7). Similarly, it is easy to prove that 𝑁(𝑋_𝑘)⊇𝑁(𝑋_𝑘−1), 𝑘≥1.
Since ℛ(𝑋₀)=ℛ(𝛼𝑌)=𝑇 and 𝒩(𝑋₀)=𝒩(𝛼𝑌)=𝑆, then ℛ𝑋_𝑘⊂𝑇,𝒩𝑋_𝑘⊃𝑆, (8) for 𝑘≥0.
From (5), we have 𝑋_𝑘𝐴−𝐼=(−1)^𝑡+1𝑋_𝑘−1𝐴−𝐼^𝑡=(−1)^𝑡+1𝑋₀𝐴−𝐼^𝑡𝑘. (9) By (8), we get 𝑃𝑋_𝑘=𝑋_𝑘. Premultiplying (9) by 𝑃, then (9) yields 𝑋_𝑘𝐴−𝑃=(−1)^𝑡+1𝑋₀𝐴−𝑃^𝑡𝑘. (10)
Next, we will investigate the necessary and sufficient condition for the convergent property of the iterative scheme (5). Assume that lim𝑋_𝑘 exists; denote by 𝑋_∞∈𝐴{2} and ℛ(𝑋_∞𝐴)=𝑇. Then, ℛ(𝑋_∞)=ℛ(𝑋_∞𝐴𝑋_∞)⊂ℛ(𝑋_∞𝐴)⊂ℛ(𝑋_∞). Thus, ℛ(𝑋_∞𝐴)=𝑇 and 𝒳=𝑇⊕𝑁(𝑋_∞𝐴); we obtain 𝑋_∞𝐴=𝑃_{𝑇,𝑁(𝑋∞𝐴)}, a projection from 𝒳 onto 𝑇, and 𝑋_∞𝐴𝑋_𝑘=𝑋_𝑘 by (8).
Since 𝑃_{𝑇,𝑁(𝑋∞𝐴)}𝑋₀=𝑋₀, and 𝑋_𝑘𝐴−𝑃_{𝑇,𝑁(𝑋∞𝐴)}=(−1)^𝑡+1(𝑋₀𝐴−𝑃_{𝑇,𝑁(𝑋∞𝐴)})^𝑡𝑘 by (10). Thus, 0=lim𝑋_𝑘𝐴−𝑋_∞𝐴=lim𝑋_𝑘𝐴−𝑃_{𝑇,𝑁(𝑋∞𝐴)}=lim(−1)^𝑡−1𝑋₀𝐴−𝑃_{𝑇,𝑁(𝑋∞𝐴)}^𝑡𝑘, (11) and then 𝜌(𝛼𝑌𝐴−𝑃_{𝑇,𝑁(𝑋∞𝐴)})<1.
Conversely, suppose that 𝜌(𝛼𝑌𝐴−𝑃)<1 for some scalar 𝛼∈ℂ⧵{0}, where 𝑃 denotes a projection from 𝒳 to 𝑇 and 𝑋 is complement. Then, lim𝑋_𝑘𝐴=𝑃 by (10), and therefore lim_𝑘→∞𝑋_𝑘=(𝐴|_𝑇)⁻¹ and 𝑇=ℛ(𝑃)⊂ℛ(lim𝑋_𝑘).
By (8), 𝑅(lim𝑋_𝑘)⊂𝑇 because 𝑇 is close, and then ℛ(𝑋_∞)=𝑇. Hence, we obtain lim𝑋_𝑘𝐴lim𝑋_𝑘=lim𝑋_𝑘. Thus, lim𝑋_𝑘∈𝐴{2}. It is easy to know that if 𝑁(lim𝑋_𝑘)=𝑆, then lim𝑋_𝑘=𝐴⁽²⁾_𝑇,𝑆. Thus, 𝐴⁽²⁾_𝑇,𝑆 exists.
Assume that 𝐴⁽²⁾_𝑇,𝑆 exists. By (8), 𝑁(lim𝑋_𝑘)⊃𝑆 because 𝑆 is closed complement. If 𝑦∈𝑁(lim𝑋_𝑘)∪𝐴𝑇, then 𝑦=𝐴𝑧 for some 𝑧∈𝑇. Thus, 0=lim𝑋_𝑘𝑦=lim𝑋_𝑘𝐴𝑧=𝑃𝑧=𝑧. Thus, 𝑦=0. Therefore, 𝑁(lim𝑋_𝑘)∪𝐴𝑇={0} and then 𝑁(lim𝑋_𝑘)=𝑆 by Lemma 2. Consequently, lim𝑋_𝑘=𝐴⁽²⁾_𝑇,𝑆.
Since 𝑁(𝑋_𝑘)=𝑆, 𝑋_𝑘𝐴𝐴⁽²⁾_𝑇,𝑆=𝑋_𝑘. Thus, postmultiplying (10) by 𝐴⁽²⁾_𝑇,𝑆 yields 𝑋_𝑘−𝐴⁽²⁾_𝑇,𝑆=(−1)^𝑡+1(𝛼𝑌𝐴−𝑃)^𝑡𝑘𝐴⁽²⁾_𝑇,𝑆. (12) Since 𝐴⁽²⁾_𝑇,𝑆=𝑃𝐴⁽²⁾_𝑇,𝑆, we have 𝐴⁽²⁾_𝑇,𝑆−𝑋_𝑘=(𝛼𝑌𝐴−𝑃)^𝑡𝑘𝐴⁽²⁾_𝑇,𝑆≤‖𝛼𝑌𝐴−𝑃‖^𝑡𝑘𝐴⁽²⁾_𝑇,𝑆=𝑞^𝑡𝑘𝐴⁽²⁾_𝑇,𝑆. (13) Hence, we get (6).

Similarly, we have the dual result as below.

Theorem 5. Let 𝐴∈ℬ(𝒳,𝒴), 𝑌∈ℬ(𝒴,𝒳), and let 𝑇⊂𝒳 and 𝑆⊂𝒴 both be closed, respectively, with ℛ(𝑌)=𝑇, 𝒩(𝑌)=𝑆. Define the sequence {𝑋_𝑘}∈ℬ(𝒴,𝒳) such that 𝑋₀𝑋=𝛼𝑌,_𝑘=𝑋_𝑘−1𝐶¹_𝑡𝐼−𝐶²_𝑡𝐴𝑋_𝑘−1+⋯+(−1)^𝑡−1𝐶^𝑡_𝑡𝐴𝑋_𝑘−1^𝑡−1; (14) it converges to 𝑋_∞ and 𝑋_∞∈𝐴{2} with 𝒩(𝑋_∞)=𝑆 if and only if 𝜌(𝛼𝐴𝑌−𝑄)<1 for some scalar 𝛼∈ℂ⧵{0}, where 𝑡≥2 is an arbitrary positive integer, 𝑋_∞=lim𝑋_𝑘, and 𝑄 is a projection from 𝒴 onto 𝑆. Moreover,(i)if ℛ(Ψ)=𝑇, then 𝐴⁽²⁾_𝑇,𝑆 exists if and only if 𝜌(𝛼𝐴𝑌−𝑄)<1 for 𝛼∈ℂ⧵{0};(ii)if ℛ(Ψ)=𝑇, then 𝐴⁽²⁾_𝑇,𝑆 exists.
In particular, if 𝐴⁽²⁾_𝑇,𝑆 exists, 𝑋_∞=𝐴⁽²⁾_𝑇,𝑆 and 𝑞=‖𝛼𝐴𝑌−𝑄‖, then ‖𝐴⁽²⁾_𝑇,𝑆−𝑋_𝑘‖/‖𝐴⁽²⁾_𝑇,𝑆‖≤𝑞^𝑡𝑘, 𝑘≥0⋯.

Remark 6. Now, we consider how to choose a suitable scalar 𝛼∈ℂ⧵{0} for the iterative scheme (5) such that it converges more faster to 𝐴⁽²⁾_𝑇,𝑆.
Since ℛ(𝑌𝐴)⊂𝑇 and for any 𝛼∈ℂ⧵{0}, 𝜌(𝑃−𝛼𝑌𝐴)=𝜌(𝑃−𝛼(𝑌𝐴)|_𝑇)=max|1−𝛼𝜇|(𝜇∈𝜎(𝑌𝐴)|_𝑇). Therefore, 𝜌(𝑃−𝛼𝑌𝐴)<1 if and only if 0∉𝜎((𝑌𝐴)|_𝑇) and max_{𝜇∈(𝑌𝐴)⧵{0}}|1−𝛼𝜇|<1. Thus, there exists 𝜆₀∈(𝑌𝐴)⧵{0} with |1−𝛼𝜆₀|=𝜌(𝑃−𝛼𝑌𝐴).
Let 𝜆₀=|𝜆₀|(cos𝜃+𝑖sin𝜃) and 𝛼=|𝛼|(cos𝜑+𝑖sin𝜑), where 𝜃=arg(𝜆₀), 𝜑=arg(𝛼). Then, 𝜌(𝑃−𝛼𝑌𝐴)=[|𝛼𝜆₀|²+1−2|𝛼𝜆₀|cos(𝜃+𝜑)]^1/2. Thus, 𝜌(𝑃−𝛼𝑌𝐴)<1 if and only if 0<|𝛼𝜆₀|<2cos(𝜃+𝜑) and 0∉𝜎((𝑌𝐴)|_𝑇).
Hence, by 0∉𝜎((𝑌𝐴)|_𝑇) and 𝛼 satisfing 0<|𝛼|<2cos(𝜃+𝜑)/𝜌(𝑌𝐴), we have 𝜌(𝑃−𝛼𝑌𝐴)<1. In practice, once such a 𝜆₀ is determined, 𝛼 is taken to satisfy arg(𝛼)=−arg(𝜆₀) and 0<|𝛼|<2/𝜌(𝑌𝐴). If 𝜎(𝑌𝐴) is a subset of ℝ, then we take 𝛼 satisfying 0<|𝛼|<2/𝜌(𝑌𝐴) and sgn𝛼=sgn𝜆₀, where 𝜆₀∈𝜎(𝑌𝐴), so as to ensure that 𝜌(𝑃−𝛼𝑌𝐴)<1.
Assume that 0∉𝜎((𝑌𝐴)|_𝑇) hold. In the following, we will obtain the best value 𝛼_opt such that 𝜌(𝑃−𝛼𝑌𝐴) minimizes for achieving good convergence. Unfortunately, it may be rather difficult. If 𝜎(𝑌𝐴) is a subset of ℝ and 𝜆_min=min{𝜆∶𝜆∈𝜎(𝑌𝐴)|_𝑇}>0 analogous to [8, Example 4.1], we can have 𝛼_opt=2𝜆_min().+𝜌𝑌𝐴 (15) In practice, because 𝜌(𝑌𝐴) is not easily obtained, we often utilize ‖𝑌𝐴‖ instead of it in the above inequations and (15) to choose 𝛼, which is followed from 𝜌(𝑌𝐴)≤‖𝑌𝐴‖.

3. Higher-Order Convergent Iterative Method for Computing the Generalized Inverse over Banach Algebra

In the section, we will investigate a higher-order convergent iterative method for computing the generalized Drazin inverse 𝑎^𝑑 over Banach algebra.

Theorem 7. Let 𝑎∈𝒜, 𝑝∈𝒜 be idempotents with 𝑎𝑝=𝑝𝑎, and 𝑦∈𝒜 with (1−𝑝)𝑦=𝑦(1−𝑝)=𝑦. Define the sequence {𝑥_𝑘} in 𝒜 such that 𝑥₀=𝛼𝑦,∀𝑥₀𝑥∈𝒜,_𝑘=𝐶¹_𝑡−𝐶²_𝑡𝑥_𝑘−1𝑎+⋯+(−1)^𝑡−1𝐶^𝑡_𝑡𝑥_𝑘−1𝑎^𝑡−1𝑥_𝑘−1, (16) where 𝛼∈ℂ⧵{0} and 𝑡≥2. Then the iteration (16) converges to lim𝑥_𝑘 and 𝑝𝑥₀=0 if and only if 𝜌(1−𝑝−𝛼𝑦𝑎)<1. In this case, assume that 𝑎𝑛𝑛^𝑙(𝑦)∩(1−𝑝)𝒜(1−𝑝)={0}. Then(i)𝑎^𝑑 exists and the iteration (16) converges to 𝑎^𝑑 if and only if 𝑎𝑝 is quasinilpotent in 𝒜;(ii)if 𝑞=(1−𝑝−𝛼𝑦𝑎)<1, then ‖𝑎^𝑑−𝑥_𝑘‖≤𝑞^𝑡𝑘‖𝑦‖⋅‖𝛼‖/‖𝑝+𝛼𝑦𝑎‖.

Proof. (i) By (1−𝑝)𝑦=𝑦(1−𝑝)=𝑦 and 𝑥₀=𝛼𝑦, it implies that (1−𝑝)𝑥₀=𝑥₀. By induction on 𝑘, we have 1−𝑝𝑥_𝑘=1−𝑝𝐶¹_𝑡−𝐶²_𝑡𝑥_𝑘−1𝑎+⋯+(−1)^𝑡−1𝐶^𝑡_𝑡𝑥_𝑘−1𝑎^𝑡−1𝑥_𝑘−1=𝑥_𝑘. (17) By (16), we obtain 𝑥_𝑘𝑎−1=(−1)^𝑡−1𝑥_𝑘−1𝑎−1^𝑡=(−1)^{(𝑡−1)𝑘}𝑥₀𝑎−1^𝑡𝑘. (18) From (17) and (18), we get 1−𝑝𝑥_𝑘𝑎−1=𝑥_𝑘=()𝑎−1−𝑝−1^{(𝑡−1)𝑘}𝑥₀𝑎−1−𝑝^𝑡𝑘. (19)
The right-hand side of the last equality of (19) implies that 0=lim_𝑡→∞(−1)^𝑡−1𝑥₀𝑎−(1−𝑝)^𝑡𝑘. (20) By (20), we easily have 𝜌(𝑥₀𝑎−(1−𝑝))=𝜌(1−𝑝−𝛼𝑦𝑎)<1.
Conversely, assume that 𝜌(1−𝑝−𝛼𝑦𝑎)<1. Since 𝑝𝑎=𝑎𝑝 and (1−𝑝)𝑦=𝑦(1−𝑝)=𝑦, (1−𝑝−𝛼𝑦𝑎)∈(1−𝑝)𝒜(1−𝑝), and then 𝑦𝑎 is invertible in (1−𝑝)𝒜(1−𝑝). We will show that 𝑎𝑦 is invertible in (1−𝑝)𝒜(1−𝑝). Clearly, 𝑎𝑦∈(1−𝑝)𝒜(1−𝑝), if 𝑎𝑦𝑐=0 for some 𝑐∈(1−𝑝)𝒜(1−𝑝), then 𝑦𝑐=[(𝑦𝑎)]⁻¹_{(1−𝑝)𝒜}𝑦𝑎𝑐. Hence, 𝑐∈𝑎𝑛𝑛^𝑟(𝑦)∩(1−𝑝)𝒜(1−𝑝)={0} and 𝑐=0. Hence, 0∉[(𝑎𝑦)]_{(1−𝑝)𝒜(1−𝑝)}, and then 𝑎𝑦 is invertible in (1−𝑝)𝒜(1−𝑝).(i) In the following, we will consider the result (i). It is similar to the deduction of (10), we can write (16) as 𝑥_𝑘𝑎=1−𝑝+(−1)^𝑡−1𝑥₀𝑎−1−𝑝^𝑡𝑘. (21)Thus, postmultiplying (21) by 𝑦 yields to 𝑥_𝑘𝑎𝑦=1−𝑝𝑦+(−1)^𝑡−1𝑥₀𝑎−1−𝑝^𝑡𝑘𝑦. (22) By Lemma 1 and (22), we prove that 𝑥_𝑘 converges to 𝑦[(𝑎𝑦)]⁻¹_{(1−𝑝)𝒜(1−𝑝)} and is denoted by 𝑥_∞=𝑦[(𝑎𝑦)]⁻¹_{(1−𝑝)𝒜(1−𝑝)}. Therefore, 𝑦[(𝑎𝑦)]⁻¹=[(𝑦𝑎)]⁻¹𝑦 in (1−𝑝)𝒜(1−𝑝); then 𝑥_∞𝑎=𝑦(𝑎𝑦)_𝒜⁻¹_{1−𝑝1−𝑝}𝑎=𝑦𝑎_𝒜⁻¹_{1−𝑝1−𝑝}𝑦𝑎=1−𝑝=𝑎𝑦(𝑎𝑦)_𝒜⁻¹_{1−𝑝1−𝑝}=𝑎𝑥_∞. (23) Thus, we obtain 𝑎−𝑎²𝑥_∞=𝑎𝑝. Since 𝑥_∞𝑎𝑥_∞=𝑎𝑦[(𝑎𝑦)]⁻¹_{(1−𝑝)𝒜(1−𝑝)}𝑥_∞=𝑥_∞, we have that 𝑥_∞=𝑎^𝑑 if and only if 𝑎𝑝 is quasinilpotent in 𝒜.(ii) Since 𝑝 is idempotent and 𝑎𝑝=𝑝𝑎, and 𝑝,1−𝑝𝑦=𝑦1−𝑝=𝑦,𝑝+𝛼𝑎𝑦𝑎𝑦=𝑝𝑝+𝛼𝑦𝑎 (24)then 𝛼𝑝+𝛼𝑎𝑦⁻¹𝑎𝑦=1−𝑝+𝛼𝑎𝑦⁻¹𝑝=1−𝑝=1−𝑝𝑝+𝛼𝑎𝑦=𝛼𝑎𝑦𝑝+𝛼𝑎𝑦⁻¹. (25) Therefore, we obtain (𝑎𝑦)⁻¹=𝛼(𝑝+𝛼𝑎𝑦)⁻¹ in (1−𝑝)𝒜(1−𝑝). By (10), we have 𝑥_𝑘𝑎𝑦=1−𝑝+(−1)^𝑡+1𝛼𝑎𝑦−1−𝑝^𝑡𝑘𝑦. (26) Hence, by the argument in (i) and (26), we have 𝑎^𝑑−𝑥_𝑘=𝑥_∞−𝑥_𝑘=𝑦(𝑎𝑦)⁻¹_{(1−𝑝)𝒜(1−𝑝)}−1−𝑝+(−1)^𝑡+1𝛼𝑎𝑦−(1−𝑝)^𝑡𝑘𝑦×(𝑎𝑦)⁻¹_{(1−𝑝)𝒜1−𝑝}=(−1)^𝑡+2𝛼𝑎𝑦−1−𝑝^𝑡𝑘𝑦×(𝑎𝑦)⁻¹_{(1−𝑝)𝒜(1−𝑝)}. (27) Taking limit in (27), then it reduces to (ii).

Similarly, we have the following.

Theorem 8. Let 𝑎∈𝒜,𝑝∈𝒜 be idempotents with 𝑎𝑝=𝑝𝑎, and 𝑦∈𝒜 with (1−𝑝)𝑦=𝑦(1−𝑝)=𝑦. Define the sequence {𝑥_𝑘} in 𝒜 such that 𝑥₀=𝛼𝑦,∀𝑥₀𝑥∈𝒜,_𝑘=𝑥_𝑘−1𝐶¹_𝑡−𝐶²_𝑡𝑎𝑥_𝑘−1+⋯+(−1)^𝑡−1𝐶^𝑡_𝑡𝑎𝑥_𝑘−1^𝑡−1, (28) where 𝛼∈ℂ⧵{0} and 𝑡≥2. Then, the iteration (28) converges to lim𝑥_𝑘 and 𝑝𝑥₀=0 if and only if 𝜌(1−𝑝−𝑎𝛼𝑦)<1. In this case, assume that 𝑎𝑛𝑛^𝑙(𝑦)∩(1−𝑝)𝒜(1−𝑝)={0}. Then(i)𝑎^𝑑 exists and the iteration (28) converges to 𝑎^𝑑 if and only if 𝑎𝑝 is quasinilpotent in 𝒜;(ii)if 𝑞=(1−𝑝−𝛼𝑦𝑎)<1, then ‖𝑎^𝑑−𝑥_𝑘‖≤𝑞^𝑡𝑘‖𝑦‖⋅‖𝛼‖/‖𝑝+𝛼𝑦𝑎‖.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11061005) and the Ministry of Education Science and Grants (HCIC201103) of Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis Open Fund.