The Quasi-Linear Operator Outer Generalized Inverse with Prescribed Range and Kernel in Banach Spaces

Abstract

Let 𝑋 and 𝑌 be Banach spaces, and let 𝐴∶𝑋→𝑌 be a bounded linear operator. In this paper, we first define and characterize the quasi-linear operator (resp., out) generalized inverse 𝐴^(ℎ)_𝑇,𝑆 (resp., 𝐴^(2,ℎ)_𝑇,𝑆) for the operator 𝐴, where 𝑇⊂𝑋 and 𝑆⊂𝑌 are homogeneous subsets. Then, we further investigate the perturbation problems of the generalized inverses 𝐴^(ℎ)_𝑇,𝑆 and 𝐴^(2,ℎ)_𝑇,𝑆. The results obtained in this paper extend some well-known results for linear operator generalized inverses with prescribed range and kernel.

1. Introduction and Preliminaries

Let 𝑋 and 𝑌 be Banach spaces, let 𝑇∶𝑋→𝑌 be a mapping, and let 𝐷⊂𝑋 be a subset of 𝑋. Recall from [1, 2] that a subset 𝐷 in 𝑋 is called to be homogeneous if for any 𝑥∈𝐷 and 𝜆∈ℝ, we have 𝜆𝑥∈𝐷. If for any 𝑥∈𝒟(𝑇) and 𝜆∈ℝ, we have 𝑇(𝜆𝑥)=𝜆𝑇(𝑥), then we call 𝑇 as a homogeneous operator on 𝒟(𝑇), where 𝒟(𝑇) is the domain of 𝑇; 𝑇 is called a bounded homogeneous operator if 𝑇 maps every bounded set in 𝒟(𝑇) into bounded set in 𝑌. Denote by 𝐻(𝑋,𝑌) the set of all bounded homogeneous operators from 𝑋 to 𝑌. Equipped with the usual linear operations for 𝐻(𝑋,𝑌), and for 𝑇∈𝐻(𝑋,𝑌), the norm is defined by ‖𝑇‖=sup{‖𝑇𝑥‖∣‖𝑥‖=1,𝑥∈𝑋}, and then similar to the space 𝐵(𝑋,𝑌) of all bounded linear operators from 𝑋 to 𝑌, we can easily prove that (𝐻(𝑋,𝑌),‖⋅‖) is a Banach space (cf. [2, 3]). Throughout this paper, we denote by 𝒟(𝑇), 𝒩(𝑇), and ℛ(𝑇) the domain, the null space, and the range of a bounded homogeneous operator 𝑇∈𝐻(𝑋,𝑌), respectively. Obviously, we have 𝐵(𝑋,𝑌)⊂𝐻(𝑋,𝑌).

For an operator 𝐴∈𝐵(𝑋,𝑌), let 𝑇 and 𝑆 be closed subspaces of 𝑋 and 𝑌, respectively. Recall that the out inverse 𝐴⁽²⁾_𝑇,𝑆 with prescribed range 𝑇 and kernel 𝑆 is the unique operator 𝐺∈𝐵(𝑌,𝑋) satisfying 𝐵𝐴𝐵=𝐵,ℛ(𝐵)=𝑇,𝒩(𝐵)=𝑆. It is well known that the important kinds of generalized inverses, the Moore-Penrose inverse, the Drazin inverse, the group inverse, and so on, are all generalized inverse 𝐴⁽²⁾_𝑇,𝑆 (cf. [4, 5]). Researches on the generalized inverse 𝐴⁽²⁾_𝑇,𝑆 of operators or matrices have been actively ongoing for many years (see [5–12], e. g.).

Let 𝐴∈𝐵(𝑋,𝑌) and let 𝑇 and 𝑆 be two homogeneous subsets in 𝑋 and 𝑌, respectively. Motivated by related work on 𝐴⁽²⁾_𝑇,𝑆 in the literature mentioned above and by our own recent research papers [13, 14], in this paper, we will establish the definition of the quasi-linear operator outer generalized inverse 𝐴^(2,ℎ)_𝑇,𝑆 with prescribed range 𝑇 and kernel 𝑆. We give the necessary and sufficient conditions for the existence of the generalized inverses 𝐴^(2,ℎ)_𝑇,𝑆, and we will also study the perturbation problems of the generalized inverse 𝐴^(2,ℎ)_𝑇,𝑆. Similar results on the generalized inverse 𝐴^(ℎ)_𝑇,𝑆 are also given.

2. Definitions and Some Characterizations of 𝐴^(2,ℎ)_𝑇,𝑆 and 𝐴^(ℎ)_𝑇,𝑆

We first give the concepts of quasi-additivity and quasi-linear projectors in Banach spaces, which are important for us to present the main results in this paper.

Definition 1. Let 𝑀 be a subset of 𝑋 and let 𝑇∶𝑋→𝑌 be a mapping. Ones calls 𝑇 as quasi-additive on 𝑀 if 𝑇 satisfies 𝑇(𝑥+𝑧)=𝑇(𝑥)+𝑇(𝑧),∀𝑥∈𝑋,∀𝑧∈𝑀. (1)

Definition 2 (see [3, 15]). Let 𝑋 be a Banach space. A mapping 𝑃∶𝑋→𝑋 is called a quasi-linear projector on 𝑋, if 𝑃 satisfies the following conditions: (1)𝑃is a homogeneous operator; (2)𝑃 is idempotent; that is, 𝑃²=𝑃; (3)𝑃 is quasi-additive on ℛ(𝑃); that is, for any 𝑥∈𝑋 and any 𝑧∈ℛ(𝑃), one has 𝑃(𝑥+𝑧)=𝑃(𝑥)+𝑃(𝑧)=𝑃(𝑥)+𝑧. (2)
If the mapping 𝑃 only satisfies the conditions (1) and (2) above, then we call 𝑃 as a homogeneous projector.

From Definition 2, we see that the quasi-linear projectors include all the usually used projectors, such as the linear projectors in linear spaces, the bounded linear projectors, the metric projectors in Banach spaces, and the orthogonal projectors in Hilbert spaces (cf. [3, 15, 16]). We have the following important property for quasi-linear projector.

Lemma 3 (see [15, Lemma 2.5]). If 𝑃∈𝐻(𝑋,𝑋) is a bounded quasi-linear projector on 𝑋, then the range ℛ(𝑃) of 𝑃 is a closed linear subspace of 𝑋.

We should note that, although the range ℛ(𝑃) of a bounded quasi-linear projector 𝑃 is a closed linear subspace, but in general, the kernel 𝒩(𝑃) of a bounded quasi-linear projector 𝑃 is not a (closed) subspace. Let 𝑃∈𝐻(𝑋) be a quasi-linear projector. Then by Lemma 3, ℛ(𝑃) is a closed linear subspace of 𝑋 and ℛ(𝐼−𝑃)=𝒩(𝑃). Thus, we can define “the quasi-linearly complement” of a closed linear subspace as follows. Let 𝑉 be a closed subspace of 𝑋. If there exists a bounded quasi-linear projector 𝑃 on 𝑋 such that 𝑉=ℛ(𝑃), then 𝑉 is said to be bounded quasi-linearly complemented in 𝑋 and 𝒩(𝑃) is the bounded quasi-linear complement of 𝑉 in 𝑋. In this case, as usual, we may write 𝑋=𝑉+̇𝒩(𝑃), where 𝒩(𝑃) is a homogeneous subset of 𝑋 and “+̇” means that 𝑉∩𝒩(𝑃)={0} and 𝑋=𝑉+𝒩(𝑃).

Now we establish the definitions of some types of homogeneous (or quasi-linear) operator generalized inverses with prescribed range and kernel as follows.

Definition 4. Let 𝐴∈𝐵(𝑋,𝑌). Let 𝑇 and 𝑆 be homogeneous subsets in 𝑋 and 𝑌, respectively. Let 𝐵∈𝐻(𝑌,𝑋) be a bounded homogeneous operator. Consider the following equations: (1)(2)(3)ℛ(𝐵)(4)𝒩(𝐵)𝐴𝐵𝐴=𝐴,𝐵𝐴𝐵=𝐵,=𝑇,=𝑆. (3)(i)If 𝐵∈𝐻(𝑌,𝑋) satisfies (2)–(4) in (3), then the operator 𝐵 is called to be the homogeneous outer generalized inverse of 𝐴 with prescribed range 𝑇 and kernel 𝑆. We denoted it by 𝐴^(2,𝐻)_𝑇,𝑆. (ii)If 𝐵∈𝐻(𝑌,𝑋) satisfies all equations in (3), then the operator 𝐵 is called to be the homogeneous generalized inverse of 𝐴 with prescribed range 𝑇 and kernel 𝑆. We denoted it by 𝐴^(𝐻)_𝑇,𝑆. (iii)If 𝐵∈𝐻(𝑌,𝑋) is quasi-additive on 𝐴𝑇 and satisfies (2)–(4) in (3), then the operator 𝐵 is called to be the quasi-linear outer generalized inverse of 𝐴 with prescribed range 𝑇 and kernel 𝑆. We denoted it by 𝐴^(2,ℎ)_𝑇,𝑆. (iv)If 𝐵∈𝐻(𝑌,𝑋) is quasi-additive on ℛ(𝐴) and satisfies all equations in (3), then the operator 𝐵 is called to be the quasi-linear generalized inverse of 𝐴 with prescribed range 𝑇 and kernel 𝑆. We denoted it by 𝐴^(ℎ)_𝑇,𝑆.

Remark 5. Let 𝐴∈𝐵(𝑋,𝑌). Assume that 𝒩(𝐴) and ℛ(𝐴) are Chebyshev subspaces in 𝑋 and 𝑌, respectively. Then the Moore-Penrose metric generalized inverse 𝐴^𝑀 of 𝐴 uniquely exists (cf. [3, 17, 18]). Let 𝑆=𝐹_𝑌⁻¹ℛ(𝐴)^⟂,𝑇=𝐹_𝑋⁻¹(𝒩(𝐴))^⟂, (4) where 𝐹_𝑋∶𝑋→𝑋^∗ is the set-valued dual mapping of 𝑋, ℛ(𝐴)^⟂={𝑓∈𝑋^∗∣𝑓(𝑥)=0,∀𝑥∈ℛ(𝐴)} and 𝐹_𝑋⁻¹(ℛ(𝐴)^⟂)={𝑥∈𝑋∣𝐹_𝑋(𝑥)∩ℛ(𝐴)^⟂≠∅}. Then 𝑆 and 𝑇 are homogeneous subsets in 𝑋 and 𝑌, respectively, and 𝐴^𝑀𝐴𝐴^𝑀=𝐴,𝐴^𝑀𝐴𝐴^𝑀=𝐴^𝑀,ℛ𝐴^𝑀=𝑇,𝒩𝐴^𝑀=𝑆. (5)
In this case, we have 𝐴^𝑀=𝐴^(𝐻)_𝑇,𝑆. Furthermore, if 𝐴^𝑀 is quasi-additive on ℛ(𝐴), then 𝐴^𝑀=𝐴^(ℎ)_𝑇,𝑆. Thus, it is meaningful to study the quasi-linear operator (outer) generalized inverses with prescribed range and kernel.

In this paper, we mainly study the generalized inverses 𝐴^(2,ℎ)_𝑇,𝑆 and 𝐴^(ℎ)_𝑇,𝑆. Motivated by the properties of linear operator generalized inverses with prescribed range and kernel, in the rest of this section, we will give some characterizations of the generalized inverses 𝐴^(2,ℎ)_𝑇,𝑆 and 𝐴^(ℎ)_𝑇,𝑆.

Theorem 6. Let 𝐴∈𝐵(𝑋,𝑌). Let 𝑇 and 𝑆 be homogeneous subsets in 𝑋 and 𝑌, respectively. Then the following statements are equivalent: (1)there exists some 𝐵∈𝐻(𝑌,𝑋) such that 𝐵 is quasi-additive on 𝐴𝑇 and 𝐵𝐴𝐵=𝐵, ℛ(𝐵)=𝑇, 𝒩(𝐵)=𝑆; (2)𝒩(𝐴)∩𝑇={0}, 𝑌=𝐴𝑇+̇𝑆 and 𝑇⊂𝑋 is a closed linear subspace.

Proof. (1)⇒(2) Suppose that (1) holds. Put 𝑃=𝐵𝐴 and 𝑄=𝐴𝐵. Since 𝐵𝐴𝐵=𝐵, we have 𝑃=𝑃² and ℛ(𝑃)=ℛ(𝐵𝐴)=ℛ(𝐵)=𝑇. In order to prove  𝑇⊂𝑋  is a closed linear subspace, by Lemma 3, we only need to show that 𝑃 is a bounded quasi-linear projector. Let 𝑥∈𝑋 and 𝑧∈𝑇=ℛ(𝐵𝐴), we get 𝑧=𝐵𝐴𝑧 and 𝐴𝐵𝐴𝑧∈𝐴𝑇. Noting that 𝐵 is quasi-additive on 𝐴𝑇, then 𝑃(𝑥+𝑧)=𝐵𝐴(𝑥+𝐵𝐴𝑧)=𝐵(𝐴𝑥+𝐴𝐵𝐴𝑧)=𝐵𝐴𝑥+𝐵𝐴𝐵𝐴𝑧=𝑃𝑥+𝑧. (6)
Therefore, 𝑃 is a bounded quasi-linear projector and then 𝑇 is a closed linear subspace in 𝑋. Now we show that 𝒩(𝐴)∩𝑇={0}. Let 𝑥∈𝒩(𝐴)∩𝑇. Since ℛ(𝐵)=𝑇, then 𝐴𝑥=0 and 𝑥=𝐵𝑤 for some 𝑤∈𝑌. So 𝑥=𝐵𝑤=𝐵𝐴𝐵𝑤=𝐵𝐴𝑥=0. Therefore, we have 𝒩(𝐴)∩𝑇={0}. Next we will show that 𝑌=𝐴𝑇+̇𝑆. For any 𝑦∈𝑌, we have 𝑦=(𝐼_𝑌−𝐴𝐵)𝑦+𝐴𝐵𝑦. Since we have proved that 𝑇 is a closed linear subspace of 𝑋, we have 𝐴𝐵𝑦∈ℛ(𝑄)=ℛ(𝐴𝐵)=𝐴ℛ(𝐵)=𝐴𝑇. Also noting that 𝐵 is quasi-additive on 𝐴𝑇 and 𝐵𝐴𝐵=𝐵, we get (𝐼_𝑌−𝐴𝐵)𝑦∈𝒩(𝐵)=𝑆. Thus, we get 𝑌=𝐴𝑇+𝑆. From 𝐵𝐴𝐵=𝐵, we can also get 𝑄=𝑄² and 𝑆=𝒩(𝐵)=𝒩(𝐴𝐵)=𝒩(𝑄); therefore, 𝐴𝑇∩𝑆=∅ and then 𝑌=𝐴𝑇+̇𝑆.
(2)⇒(1) Suppose that (2) holds. Since 𝒩(𝐴)∩𝑇={0} and 𝑇 is a closed linear subspace, we see 𝐴|_𝑇∶𝑇→𝐴𝑇 is well defined, which is also a linear bijective mapping, so 𝐴|_𝑇 has a linear inverse (𝐴|_𝑇)⁻¹∶𝐴𝑇→𝑇. Since 𝑌=𝐴𝑇+̇𝑆, then for any 𝑦∈𝑌, it can be uniquely written as 𝑦=𝑦₁+𝑦₂ with 𝑦₁∈𝐴𝑇 and 𝑦₂∈𝑆. Now we can define a mapping 𝐵∶𝑌→𝑋 by 𝐵𝑦=(𝐴|_𝑇)⁻¹𝑦₁. It is easy to see that 𝐵𝐴𝐵=𝐵, ℛ(𝐵)=𝑇 and 𝒩(𝐵)=𝑆 by 𝑌=𝐴𝑇+̇𝑆. Since 𝒩(𝐵)=𝑆 is a homogeneous subsets in 𝑌, we see 𝐵∈𝐻(𝑌,𝑋) is a homogeneous operator. Now we need to prove 𝐵 is quasi-additive on 𝐴𝑇. In fact, for any 𝑦∈𝑌 and 𝑧∈𝐴𝑇, since 𝑌=𝐴𝑇+̇𝑆, we can write 𝑦 as 𝑦=𝑦₁+𝑦₂, where 𝑦₁∈𝐴𝑇 and 𝑦₂∈𝑆. Noting that 𝑦₁+𝑧∈𝐴𝑇 since 𝐴𝑇⊂𝑋 is a linear subspace, then we can check that 𝐵𝑦+𝑧=𝐵𝑦₁+𝑦₂+𝑧=𝐵𝑦₁+𝑧+𝑦₂=𝐴|_𝑇⁻¹𝑦₁+𝑧=(𝐴|_𝑇)⁻¹𝑦₁+(𝐴|_𝑇)⁻¹𝑧=𝐵𝑦+𝐵𝑧. (7)
Therefore, 𝐵 is quasi-additive on 𝐴𝑇.

Theorem 7. Let 𝐴∈𝐵(𝑋,𝑌). Let 𝑇 and 𝑆 be homogeneous subsets in 𝑋 and 𝑌, respectively. Then the following statements are equivalent.(1)There exists some 𝐵∈𝐻(𝑌,𝑋) such that 𝐵 is quasi-additive on ℛ(𝐴) and ℛ(𝐵)(𝐵)𝐴𝐵𝐴=𝐴,𝐵𝐴𝐵=𝐵,=𝑇,𝒩=𝑆. (8)(2)𝑋=𝒩(𝐴)+̇𝑇, 𝑌=ℛ(𝐴)+̇𝑆 and 𝑇⊂𝑋 is a closed linear subspace. (3)𝑌=𝐴𝑇+̇𝑆, 𝒩(𝐴)∩𝑇={0}, ℛ(𝐴)∩𝑆={0} and 𝑇⊂𝑋 is a closed linear subspace.

Proof. (1)⇒(2) Suppose that (1) holds, Put 𝑃=𝐵𝐴 and 𝑄=𝐴𝐵. Similar to the proof of Theorem 6, we have 𝑃=𝑃², 𝑄=𝑄² and 𝑇⊂𝑋 is a closed linear subspace. Note that we also have the following relation: ℛ()(𝐵)()(),ℛ()(𝐴)()(),𝒩()()(𝐴)(),𝒩(𝐵)()()(𝐵).𝐵𝐴⊂ℛ=ℛ𝐵𝐴𝐵⊂ℛ𝐵𝐴𝐴𝐵⊂ℛ=ℛ𝐴𝐵𝐴⊂ℛ𝐴𝐵𝐵𝐴⊂𝒩𝐴𝐵𝐴=𝒩⊂𝒩𝐵𝐴⊂𝒩𝐴𝐵⊂𝒩𝐵𝐴𝐵=𝒩 (9)
Thus, we get that ℛ(𝑃)()(𝐵)ℛ(𝑄)()(𝐴),𝒩(𝑃)()(𝐴),𝒩(𝑄)()(𝐵)=ℛ𝐵𝐴=ℛ=𝑇,=ℛ𝐴𝐵=ℛ=𝒩𝐵𝐴=𝒩=𝒩𝐴𝐵=𝒩=𝑆. (10)
From the above equations, we can obtain 𝑋=𝒩(𝐴)+̇𝑇 and 𝑌=ℛ(𝐴)+̇𝑆.
(2)⇒(3) If 𝑋=𝒩(𝐴)+̇𝑇 and 𝑌=ℛ(𝐴)+̇𝑆, then it is obvious that 𝒩(𝐴)∩𝑇={0}, ℛ(𝐴)∩𝑆={0}. We only need to show 𝐴𝑇=ℛ(𝐴). Obviously, we have 𝐴𝑇⊂ℛ(𝐴). Now for any 𝑦∈ℛ(𝐴), we have 𝑦=𝐴𝑥 for some 𝑥∈𝑋. Since 𝑋=𝒩(𝐴)+̇𝑇, we can write 𝑥=𝑥₁+𝑥₂, where 𝑥₁∈𝒩(𝐴) and 𝑥₂∈𝑇. Thus 𝑦=𝐴𝑥=𝐴𝑥₂∈𝐴𝑇. Therefore ℛ(𝐴)⊂𝐴𝑇 and then 𝐴𝑇=ℛ(𝐴).
(3)⇒(1) Suppose that (3) is true, then by Theorem 6, we know that 𝐴^(2,ℎ)_𝑇,𝑆=𝐵 exists, and then ℛ(𝐵)=𝑇, 𝒩(𝐵)=𝑆. We need to show 𝐴𝐵𝐴=𝐴 and 𝐵 is quasi-additive on ℛ(𝐴). Since 𝐴^(2,ℎ)_𝑇,𝑆=𝐵 exists, then 𝐵𝐴𝐵=𝐵 and 𝐵 is quasi-additive on 𝐴𝑇. Noting that for any 𝑥∈𝑋, we have 𝐴𝐵𝐴𝑥∈𝐴𝑇. Thus, 𝐵(𝐴𝐵𝐴−𝐴)𝑥=0 and (𝐴𝐵𝐴−𝐴)𝑥⊂ℛ(𝐴)∩𝒩(𝐵)=ℛ(𝐴)∩𝑆={0}; (11) that is, 𝐴𝐵𝐴=𝐴. Obviously, 𝐴𝑇⊂ℛ(𝐴). On the converse, let 𝑦∈ℛ(𝐴). Since 𝑌=𝐴𝑇+̇𝑆, we have 𝑦=𝑦₁+𝑦₂ with some 𝑦₁∈𝐴𝑇 and 𝑦₂∈𝑆. From 𝐴𝑇⊂ℛ(𝐴), we see 𝑦₁∈ℛ(𝐴). Thus, we have 𝑦₂=𝑦−𝑦₁∈ℛ(𝐴)∩𝑆={0}. So ℛ(𝐴)⊂𝐴𝑇 and then 𝐴𝑇=ℛ(𝐴). Hence, we obtain that 𝐵 is quasi-additive on ℛ(𝐴).

We should indicate that, for Definition 4, we define two different quasi-additivity in (iii) and (iv). But from Theorem 7, we see that if the generalized inverse 𝐴^(ℎ)_𝑇,𝑆 exists, then we must have ℛ(𝐴)=𝐴𝑇. So there is no contradiction in our definition. The following proposition gives the uniqueness of 𝐴^(2,ℎ)_𝑇,𝑆. From this result, it is also easy to see the uniqueness of 𝐴^(ℎ)_𝑇,𝑆, which are similar to the linear (outer) generalized inverses with prescribed range and kernel.

Proposition 8. Let 𝐴∈𝐵(𝑋,𝑌). Let 𝑇 and 𝑆 be homogeneous subsets in 𝑋 and 𝑌, respectively. If the generalized inverse 𝐴^(2,ℎ)_𝑇,𝑆 (or 𝐴^(ℎ)_𝑇,𝑆) exists, then it is unique.

Proof. We only need to prove this proposition for 𝐴^(2,ℎ)_𝑇,𝑆. By Definition 4, we need to prove that there is at most one bounded homogeneous operator 𝐵∈𝐻(𝑌,𝑋) such that 𝐵 is quasi-additive on 𝐴𝑇 and satisfies 𝐵𝐴𝐵=𝐵,ℛ(𝐵)=𝑇,𝒩(𝐵)=𝑆. (12)
Suppose that there is another 𝐶∈𝐻(𝑌,𝑋) such that 𝐶 is quasi-additive on 𝐴𝑇 and satisfies the equations 𝐶𝐴𝐶=𝐶, ℛ(𝐶)=𝑇, and 𝒩(𝐶)=𝑆. Put 𝑄=𝐴𝐵 and 𝑄^′=𝐴𝐶. By using the equations 𝐵𝐴𝐵=𝐵 and 𝐶𝐴𝐶=𝐶, we see that both 𝑄 and 𝑄^′ are bounded homogeneous projectors on 𝑌 and satisfy ℛ(𝑄)=ℛ𝑄^′=𝐴𝑇;𝒩(𝑄)=𝒩𝑄^′=𝑆. (13)
Therefore, we get 𝑄=𝑄^′. Thus, 𝐶=𝐶𝐴𝐶=𝐶𝐴𝐵. On the other hand, if we set 𝑃=𝐵𝐴 and 𝑃^′=𝐶𝐴, then by using the equations 𝐵𝐴𝐵=𝐵 and 𝐶𝐴𝐶=𝐶 again, and also noting that 𝐵 and 𝐶 are both quasi-additive on 𝐴𝑇, similar to the proof (1)⇒(2) in Theorem 6, we see both 𝑃 and 𝑃^′ are bounded quasi-linear projectors and satisfy ℛ(𝑃)=ℛ(𝐵)=ℛ𝑃^′=ℛ(𝐶)=𝑇. (14)
Thus, 𝐵𝐴=𝑃=𝑃^′𝑃=𝐶𝐴𝐵𝐴, and then 𝐵=𝐵𝐴𝐵=𝐶𝐴𝐵𝐴𝐵=𝐶𝐴𝐵=𝐶.

Proposition 9. Let 𝐴∈𝐵(𝑋,𝑌). Let 𝑇 and 𝑆 be homogeneous subsets in 𝑋 and 𝑌, respectively. If the generalized inverse 𝐴^(2,ℎ)_𝑇,𝑆 (or 𝐴^(ℎ)_𝑇,𝑆) exists, then 𝑇 and 𝐴𝑇 are both closed linear subspaces of 𝑋.

Proof. If the generalized inverse 𝐴^(2,ℎ)_𝑇,𝑆 (or 𝐴^(ℎ)_𝑇,𝑆) exists, then from Theorem 6 (or Theorem 7), we see 𝑇 is a closed linear subspace in 𝑋. Let 𝑄=𝐴𝐴^(2,ℎ)_𝑇,𝑆. Then it is easy to check that ℛ(𝑄)=𝐴𝑇. Since 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on 𝐴𝑇, simple computation can show that 𝑄=𝐴𝐴^(2,ℎ)_𝑇,𝑆 is a bounded quasi-linear projector. Then from Lemma 3, we get 𝐴𝑇 is also a closed linear subspace of 𝑋 if 𝐴^(2,ℎ)_𝑇,𝑆 exists. If the generalized inverse 𝐴^(ℎ)_𝑇,𝑆 exists, then 𝐴^(2,ℎ)_𝑇,𝑆 exists and 𝐴^(2,ℎ)_𝑇,𝑆=𝐴^(ℎ)_𝑇,𝑆.

Finally, in this section, we will give some representation results of the generalized inverse 𝐴^(2,ℎ)_𝑇,𝑆.

Theorem 10. Let 𝐴∈𝐵(𝑋,𝑌). Let 𝑇 and 𝑆 be homogeneous subsets in 𝑋 and 𝑌, respectively. Suppose that 𝐺∈𝐻(𝑌,𝑋) such that 𝐺 is quasi-additive on 𝐴𝑇, and ℛ(𝐺)=𝑇,𝒩(𝐺)=𝑆. Then the following statements are equivalent: (1)the generalized inverse 𝐴^(2,ℎ)_𝑇,𝑆 exists; (2)(𝐺𝐴)|_𝑇 is invertible and 𝑇⊂𝑋 is a closed linear subspace. In this case, one has 𝐴^(2,ℎ)_𝑇,𝑆=((𝐺𝐴)|_𝑇)⁻¹𝐺.

Proof. (1)⇒(2) Suppose that (1) holds, then by Proposition 9, we get that 𝑇⊂𝑋 is a closed linear subspace. If 𝐺𝐴𝑥=0 for some 𝑥∈𝑇, then 𝑥=𝐺𝑧 for some 𝑧∈𝑌. Since 𝒩(𝐺)=𝑆, by Theorem 6, we have 𝐴𝑥∈𝐴𝑇∩𝑆={0}. Noting that 𝒩(𝐴)∩𝑇={0}, thus, we get 𝑥=0 and then (2) holds.
(2)⇒(1) Let 𝐵=((𝐺𝐴)|_𝑇)⁻¹𝐺. Obviously, we have 𝐵𝐴𝐵=𝐵, and ℛ(𝐵)=ℛ(𝐺)=𝑇, 𝒩(𝐵)=𝒩(𝐺)=𝑆. We show that 𝐵 is quasi-additive on 𝐴𝑇. Let 𝑦∈𝑌 and 𝑧∈𝐴𝑇. Since 𝐺 is quasi-additive on 𝐴𝑇, we get that ((𝐺𝐴)|_𝑇)⁻¹ is a linear operator and then 𝐵𝑦+𝑧=(𝐺𝐴)|_𝑇⁻¹𝐺=()|𝑦+𝑧𝐺𝐴_𝑇⁻¹𝐺𝑦+𝐺𝑧=𝐵𝑦+𝐵𝑧. (15)
By Proposition 8, the uniqueness of 𝐴^(2,ℎ)_𝑇,𝑆, we can obtain 𝐴^(2,ℎ)_𝑇,𝑆=((𝐺𝐴)|_𝑇)⁻¹𝐺.

Let 𝐴∈𝐵(𝑋,𝑌). Recall that 𝐴 is group invertible if there exists 𝐶∈𝐵(𝑌,𝑋) such that 𝐴𝐶𝐴=𝐴,𝐶𝐴𝐶=𝐶,𝐶𝐴=𝐴𝐶. (16)

In this case, denote 𝐶=𝐴^# as usual. For a bounded homogeneous operator 𝑊∈𝐻(𝑋,𝑌), if there exists 𝑊⁻∈𝐻(𝑌,𝑋) such that 𝑊𝑊⁻𝑊=𝑊, then we call 𝑊⁻ as a bounded homogeneous inner inverse of 𝑊 or simply say 𝑊 is inner regular.

Theorem 11. Let 𝐴∈𝐵(𝑋,𝑌). Let 𝑇 and 𝑆 be homogeneous subsets in 𝑋 and 𝑌, respectively. Suppose that 𝐺∈𝐻(𝑌,𝑋) such that 𝐺 is quasi-additive on ℛ(𝐴), and ℛ(𝐺)=ℛ(𝐺𝐴𝐺)=𝑇, 𝒩(𝐺)=𝑆. If 𝐺𝐴∈𝐵(𝑋,𝑋) has the group inverse (𝐺𝐴)^#, then 𝐴^(2,ℎ)_𝑇,𝑆 exists and, moreover, 𝐺𝐴𝐺∈𝐻(𝑌,𝑋) is inner regular and 𝐴^(2,ℎ)_𝑇,𝑆=(𝐺𝐴)^#𝐺=𝐺(𝐺𝐴𝐺)⁻𝐺. (17)

Proof. Since 𝐺 is quasi-additive on ℛ(𝐴), we see 𝐺𝐴∈𝐵(𝑋) is well defined. Let 𝐵=(𝐺𝐴)^#𝐺, we show that 𝐴^(2,ℎ)_𝑇,𝑆=𝐵. From ℛ(𝐺)=ℛ(𝐺𝐴𝐺)=𝑇, we get ℛ(𝐺𝐴)⊂ℛ(𝐺)=ℛ(𝐺𝐴𝐺)⊂ℛ(𝐺𝐴), and then ℛ(𝐺𝐴)=ℛ(𝐺)=𝑇. The existence of (𝐺𝐴)^# means that 𝒩(𝐺𝐴)∩ℛ(𝐺𝐴)={0},𝐺𝐴𝐺𝐴(𝐺A)^#𝐺−𝐺=0. (18)
Therefore 𝐺𝐴(𝐺𝐴)^#()(𝐺)()()={0}.𝐺−𝐺∈𝒩𝐺𝐴∩ℛ=𝒩𝐺𝐴∩ℛ𝐺𝐴 (19)
Hence 𝐺=𝐺𝐴(𝐺𝐴)^#𝐺. Now, we can compute as follows: 𝐵𝐴𝐵=(𝐺𝐴)^#𝐺𝐴(𝐺𝐴)^#𝐺=(𝐺𝐴)^#(𝐺)()𝐺=𝐵,𝑇=ℛ=ℛ𝐺𝐴𝐺𝐴^#𝐺=ℛ(𝐺𝐴)^#𝐺𝐴𝐺⊂ℛ(𝐺𝐴)^#𝐺=ℛ(𝐵)=ℛ(𝐺𝐴)^#𝐺𝐴(𝐺𝐴)^#𝐺=ℛ𝐺𝐴(𝐺𝐴)^#2𝐺(𝐺)(𝐺)()⊂ℛ=𝑇,𝑆=𝒩⊂𝒩𝐺𝐴^#𝐺=𝒩(𝐵)⊂𝒩𝐺𝐴(𝐺𝐴)^#𝐺=𝒩(𝐺)=𝑆. (20)
Thus, ℛ(𝐵)=𝑇 and 𝒩(𝐵)=𝑆. Obviously, we see 𝐵 is quasi-additive on 𝐴𝑇 for 𝐴𝑇⊂ℛ(𝐴). So, from Definition 4 and Proposition 8, we see 𝐴^(2,ℎ)_𝑇,𝑆 exists and 𝐴^(2,ℎ)_𝑇,𝑆=(𝐺𝐴)^#𝐺.
Set 𝐶=𝐴((𝐺𝐴)^#)². Then it is easy to check that (𝐺𝐴𝐺)𝐶(𝐺𝐴𝐺)=(𝐺𝐴𝐺)𝐴(𝐺𝐴)^#2(𝐺𝐴𝐺)=𝐺𝐴𝐺. (21)
We get 𝐺𝐴𝐺 is inner regular and (𝐺𝐴𝐺)⁻=𝐴((𝐺𝐴)^#)². Furthermore, we have 𝐺(𝐺𝐴𝐺)⁻𝐺=G𝐶𝐺=𝐺𝐴((𝐺𝐴)^#)²𝐺=(𝐺𝐴)^#𝐺=𝐴^(2,ℎ)_𝑇,𝑆. (22)
This completes the proof.

Theorem 12. Let 𝐴∈𝐵(𝑋,𝑌). Let 𝑇 and 𝑆 be homogeneous subsets in 𝑋 and 𝑌, respectively. Suppose that 𝐺∈𝐻(𝑌,𝑋) such that 𝐺 is quasi-additive on ℛ(𝐴), and ℛ(𝐺)=ℛ(𝐺𝐴𝐺)=𝑇, 𝒩(𝐺)=𝑆. If 𝐺𝐴∈𝐵(𝑋) has the group inverse (𝐺𝐴)^#, then 𝐴^(2,ℎ)_𝑇,𝑆 exists and 𝐴^(2,ℎ)_𝑇,𝑆=lim_{𝜆→0,𝜆∉𝜎(−𝐺𝐴)}(𝜆𝐼_𝑋+𝐺𝐴)⁻¹𝐺.

Proof. By Theorem 11, we know that 𝐴^(2,ℎ)_𝑇,𝑆 exists and 𝐴^(2,ℎ)_𝑇,𝑆=(𝐺𝐴)^#𝐺. Let 𝑃=𝐴^(2,ℎ)_𝑇,𝑆𝐴. Noting that 𝐺 is quasi-additive on ℛ(𝐴), then 𝑃∈𝐵(𝑋) is a projector and 𝑃=(𝐺𝐴)^#𝐺𝐴. Clearly, we see that 𝐺𝐴,(𝐺𝐴)^#∈𝐵(ℛ(𝑃)) and 𝐺𝐴 is invertible in 𝐵(ℛ(𝑃)) with (𝐺𝐴|_{𝐵(ℛ(𝑃))})⁻¹=(𝐺𝐴)^#. Consider the decomposition 𝑋=ℛ(𝑃)⊕ℛ(𝐼_𝑋−𝑃). Then we can write 𝜆𝐼_𝑋+𝐺𝐴 as the following matrix form: 𝜆𝐼_𝑋𝜆𝐼+𝐺𝐴=𝜆𝑃+𝐺𝐴_𝑋−𝑃. (23)
From (23), we see if 𝜆∉𝜎(−𝐺𝐴)∪{0}, then 𝜆𝑃+𝐺𝐴 is invertible in 𝐵(ℛ(𝑃)). Thus, in the case, by (23) we have (𝜆𝐼_𝑋+𝐺𝐴)⁻¹=(𝜆𝑃+𝐺𝐴)⁻¹𝜆⁻¹𝐼_𝑋−𝑃. (24)
From the proof of Theorem 11, we know that ℛ(𝐺𝐴)=ℛ(𝐺)=𝑇. Thus, we get 𝒩(𝐼_𝑋−𝑃)=ℛ(𝑃)=ℛ((𝐺𝐴)^#𝐺𝐴)=𝑇. So, by using (24), we can obtain that (𝜆𝐼_𝑋+𝐺𝐴)⁻¹𝐺=𝑃(𝜆𝑃+𝐺𝐴)⁻¹𝐺. Now we can compute as follows: 𝐴^(2,ℎ)_𝑇,𝑆=𝑃(𝐺𝐴)^#𝐺=lim_{()𝜆→0𝜆∉𝜎−𝐺𝐴}𝑃(𝜆𝑃+𝐺𝐴)⁻¹𝐺=lim_{()𝜆→0𝜆∉𝜎−𝐺𝐴}(𝜆𝐼_𝑋+𝐺𝐴)⁻¹𝐺. (25)
This completes the proof.

3. Perturbations of the Generalized Inverses 𝐴^(2,ℎ)_𝑇,𝑆 and 𝐴^(ℎ)_𝑇,𝑆

In this section, we present some perturbation results for the generalized inverse 𝐴^(2,ℎ)_𝑇,𝑆 and 𝐴^(ℎ)_𝑇,𝑆. The next lemma, which is a generalized Neumman Lemma taken from [19], will be useful for us to improve perturbation bound.

Lemma 13 (see [19]). Let 𝑋 be a Banach space and 𝑃∈𝐵(𝑋). If there exist two constants 𝜆₁,𝜆₂∈[0,1) such that ‖𝑃𝑥‖≤𝜆₁‖𝑥‖+𝜆₂‖(𝐼+𝑃)𝑥‖,(∀𝑥∈𝑋), (26) then 𝐼+𝑃∶𝑋→𝑋 is bijective. Moreover, for any 𝑥∈𝑋1−𝜆₁1+𝜆₂‖𝑥‖≤‖(𝐼+𝑃)𝑥‖≤1+𝜆₁1−𝜆₂‖𝑥‖,1−𝜆₂1+𝜆₁‖𝑥‖≤(𝐼+𝑃)⁻¹𝑥≤1+𝜆₂1−𝜆₁‖𝑥‖. (27)

Let 𝐴,𝛿𝐴∈𝐵(𝑋,𝑌). Suppose that 𝑇⊂𝑋 and 𝑆⊂𝑌 are homogeneous subsets such that 𝐴^(2,ℎ)_𝑇,𝑆 exists. If 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on ℛ(𝛿𝐴), then we see that 𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴 is a bounded linear operator, so 𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴 is also a bounded linear operator on 𝑋. Furthermore, if 𝐴^(2,ℎ)_𝑇,𝑆 and 𝛿𝐴 also satisfy 𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴𝑥≤𝜆₁‖𝑥‖+𝜆₂𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴𝑥,(∀𝑥∈𝑋), (28) where 𝜆₁,𝜆₂∈(−1,1), then by Lemma 13, we get that 𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴 is invertible.

Lemma 14. Let 𝐴∈𝐻(𝑋,𝑌) and 𝐵∈𝐻(𝑌,𝑋) such that 𝐴 is quasi-additive on ℛ(𝐵) and 𝐵 is quasi-additive on ℛ(𝐴); then 𝐼_𝑌+𝐴𝐵 is invertible in 𝐻(𝑌,𝑌) if and only if 𝐼_𝑋+𝐵𝐴 is invertible in 𝐻(𝑋,𝑋).

Proof. If there is a Φ∈𝐻(𝑌,𝑌) such that (𝐼_𝑌+𝐴𝐵)Φ=Φ(𝐼_𝑌+𝐴𝐵)=𝐼_𝑌, then 𝐼_𝑋=𝐼_𝑋+𝐵𝐴−𝐵𝐴=𝐼_𝑋+𝐵𝐴−𝑆𝐼_𝑌+𝐴𝐵Φ𝐴=𝐼_𝑋)Φ)𝐴(𝐴)+𝐵𝐴−((𝐵+𝐵𝐴𝐵𝐵quasi-additiveonℛ=𝐼_𝑋+𝐵𝐴−𝐼_𝑋𝐴=𝐼+𝐵𝐴𝑆Φ_𝑋+𝐵𝐴1_𝑋(𝐵).−𝐵Φ𝐴𝐴quasi-additiveonℛ (29)
Similarly, we also have 𝐼_𝑋=(𝐼_𝑋−𝐵Φ𝐴)(𝐼_𝑋+𝐵𝐴). Thus, 𝐼_𝑋+𝐵𝐴 is invertible on 𝑋 with (𝐼_𝑋+𝐵𝐴)⁻¹=(1_𝑋−𝐵Φ𝐴)∈𝐻(𝑋,𝑋).
The converse can also be proved by using the same way as above.

Lemma 15. Let 𝐴,𝛿𝐴∈𝐵(𝑋,𝑌). Suppose that 𝑇⊂𝑋 and 𝑆⊂𝑌 are homogeneous subsets such that 𝐴^(2,ℎ)_𝑇,𝑆 exists. If 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on ℛ(𝛿𝐴) and 𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆 (resp., 𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴) is invertible in 𝐻(𝑌,𝑌) (resp., 𝐻(𝑋,𝑋)), then 𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴 (resp., 𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆) is invertible in 𝐻(𝑋,𝑋) (resp., 𝐻(𝑌,𝑌)) and Φ=𝐴^(2,ℎ)_𝑇,𝑆(𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆)⁻¹=(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝑇)⁻¹𝐴^(2,ℎ)_𝑇,𝑆 (30) is a bounded homogeneous operator with ℛ(Φ)=𝑇 and 𝒩(Φ)=𝑆.

Proof. By Lemma 14, we obtain that 𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆 (resp., 𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴) is invertible in 𝐻(𝑌,𝑌) (resp., 𝐻(𝑋,𝑋)).
Clearly, 𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴 is a bounded linear operator and 𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆∈𝐻(𝑌,𝑌). From the equation 𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆=𝐴^(2,ℎ)_𝑇,𝑆𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆 (31) and 𝐴^(2,ℎ)_𝑇,𝑆∈𝐻(𝑌,𝑋), we get that Φ is a bounded homogeneous operator. Finally, by using (30) and Definition 4, we can obtain that ℛ(Φ)=ℛ(𝐴^(2,ℎ)_𝑇,𝑆)=𝑇 and 𝒩(Φ)=𝒩(𝐴^(2,ℎ)_𝑇,𝑆)=𝑆.

Theorem 16. Let 𝐴,𝛿𝐴∈𝐵(𝑋,𝑌). Suppose that 𝑇⊂𝑋 and 𝑆⊂𝑌 are homogeneous subsets such that 𝐴^(2,ℎ)_𝑇,𝑆 exists. Put 𝐴=𝐴+𝛿𝐴. If 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on ℛ(𝛿𝐴) and satisfies (28), then 𝐴^(2,ℎ)_𝑇,𝑆 exists and 𝐴^(2,ℎ)_𝑇,𝑆=(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴)⁻¹𝐴^(2,ℎ)_𝑇,𝑆. Moreover, one has 𝐴^(2,ℎ)_𝑇,𝑆−𝐴^(2,ℎ)_𝑇,𝑆𝐴^(2,ℎ)_𝑇,𝑆≤1+𝜆₂1−𝜆₁𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴. (32)

Proof. Since 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on ℛ(𝛿𝐴) and satisfies (28), then from Lemma 13 and Lemma 15, we see Φ=𝐴^(2,ℎ)_𝑇,𝑆(𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆)⁻¹=(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝑇)⁻¹𝐴^(2,ℎ)_𝑇,𝑆 is well defined with ℛ(Φ)=𝑇 and 𝒩(Φ)=𝑆. Now, we prove that Φ is quasi-additive on 𝐴𝑇. Let 𝑥∈𝑋 and 𝑧∈𝐴𝑇, then 𝑧=𝐴𝑤=𝐴𝑤+𝛿𝐴𝑤 for some 𝑤∈𝑇. Since 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on ℛ(𝛿𝐴) and 𝐴𝑇, also noting that 𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴 is a bounded linear operator, then Φ(𝑥+𝑧)=𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝑇⁻¹𝐴^(2,ℎ)_𝑇,𝑆(𝑥+𝐴𝑤+𝛿𝐴𝑤)=(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝑇)⁻¹𝐴^(2,ℎ)_𝑇,𝑆(𝑥+𝐴𝑤)+(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝑇)⁻¹𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴w=Φ𝑥+Φ𝐴𝑤+Φ𝛿𝐴𝑤=Φ𝑥+Φ𝑧. (33)
We get Φ is quasi-additive on 𝐴𝑇. Thus, from Definition 4 and Proposition 8, we see that 𝐴^(2,ℎ)_𝑇,𝑆 exists and 𝐴^(2,ℎ)_𝑇,𝑆=(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴)⁻¹𝐴^(2,ℎ)_𝑇,𝑆.
Finally, we show that the estimate formula is valid: 𝐴^(2,ℎ)_𝑇,𝑆−𝐴^(2,ℎ)_𝑇,𝑆𝐴^(2,ℎ)_𝑇,𝑆≤(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝑇)⁻¹𝐴^(2,ℎ)_𝑇,𝑆−𝐴^(2,ℎ)_𝑇,𝑆𝐴^(2,ℎ)_𝑇,𝑆≤𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝑇⁻¹𝐼_𝑋−𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝑇𝐴^(2,ℎ)_𝑇,𝑆𝐴^(2,ℎ)_𝑇,𝑆≤(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝑇)⁻¹𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴. (34)
By using Lemma 13, we get (𝐼_𝑋+𝑇^𝐻𝛿𝑇)⁻¹≤1+𝜆₂1−𝜆₁. (35)
Thus from (34) and (35), we can obtain 𝐴^(2,ℎ)_𝑇,𝑆−𝐴^(2,ℎ)_𝑇,𝑆𝐴^(2,ℎ)_𝑇,𝑆≤1+𝜆₂1−𝜆₁𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴. (36)
This completes the proof.

Corollary 17. Let 𝐴,𝛿𝐴∈𝐵(𝑋,𝑌). Suppose that 𝑇⊂𝑋 and 𝑆⊂𝑌 are homogeneous subsets such that 𝐴^(2,ℎ)_𝑇,𝑆 exists. Put 𝐴=𝐴+𝛿𝐴. If 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on ℛ(𝛿𝐴) with ‖𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴‖<1, then 𝐴^(2,ℎ)_𝑇,𝑆 exists and 𝐴^(2,ℎ)_𝑇,𝑆=(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴)⁻¹𝐴^(2,ℎ)_𝑇,𝑆. Moreover, one has 𝐴^(2,ℎ)_𝑇,𝑆−𝐴^(2,ℎ)_𝑇,𝑆𝐴^(2,ℎ)_𝑇,𝑆≤𝐴^(2,ℎ)_𝑇,𝑆𝛿𝑇1−𝐴^(2,ℎ)_𝑇,𝑆.𝛿𝐴 (37)

Proof. Obviously, this corollary follows from Theorem 16 if we take 𝜆₁=‖𝐴^(2,ℎ)_𝑇,𝑆𝛿𝑇‖ and 𝜆₂=0.

Lemma 18 (Neumann Lemma for quasi-linear operators). Let 𝐴∈𝐻(𝑋,𝑋). If 𝐴 is quasi-additive on ℛ(𝐴) with ‖𝐴‖<1, then (𝐼+𝐴)⁻¹ exists and (𝐼+𝐴)⁻¹=^∞_𝑛=0(−𝐴)^𝑛,(𝐼+𝐴)⁻¹≤1‖𝐴‖.1− (38)

Proof. Noting that 𝐴 is quasi-additive on ℛ(𝐴), then the proof is almost the same to the classical Neumann Lemma for bounded linear operators. Here we omit it, and we refer to [20, Lemma 4.2.6] for more information.

Theorem 19. Let 𝐴,𝛿𝐴∈𝐵(𝑋,𝑌). Suppose that 𝑇⊂𝑋 and 𝑆⊂𝑌 are homogeneous subsets such that 𝐴^(2,ℎ)_𝑇,𝑆 exists. Put 𝐴=𝐴+𝛿𝐴. If 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on ℛ(𝛿𝐴) with ‖𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆‖<1, then (1)𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆 is inventible bounded homogeneous operator; (2)𝐴^(2,ℎ)_𝑇,𝑆 exists and 𝐴^(2,ℎ)_𝑇,𝑆=𝐴^(2,ℎ)_𝑇,𝑆(𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆)⁻¹.
Moreover, one has 𝐴^(2,ℎ)_𝑇,𝑆−𝐴^(2,ℎ)_𝑇,𝑆𝐴^(2,ℎ)_𝑇,𝑆≤𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆1−𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆. (39)

Proof. Since 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on ℛ(𝛿𝐴), then ℛ(𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆)⊂ℛ(𝛿𝐴) implies that 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on ℛ(𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆). Obviously, 𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆∈𝐻(𝑌), and it follows from Lemma 18 that 𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆 is inventible bounded homogeneous operator. Then, by using Lemma 15, similar to the proof of Theorem 16, we can show that 𝐴^(2,ℎ)_𝑇,𝑆 exists and 𝐴^(2,ℎ)_𝑇,𝑆=𝐴^(2,ℎ)_𝑇,𝑆(𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆)⁻¹. For the estimate formula, we can compute that 𝐴^(2,ℎ)_𝑇,𝑆−𝐴^(2,ℎ)_𝑇,𝑆𝐴^(2,ℎ)_𝑇,𝑆≤𝐴^(2,ℎ)_𝑇,𝑆(𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆)⁻¹−𝐴^(2,ℎ)_𝑇,𝑆𝐴^(2,ℎ)_𝑇,𝑆=𝐴^(2,ℎ)_𝑇,𝑆−𝐴^(2,ℎ)_𝑇,𝑆𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆(𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆)⁻¹𝐴^(2,ℎ)_𝑇,𝑆≤𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆(𝐼_𝑌+𝛿𝐴𝐴^(2,ℎ)_𝑇,𝑆)⁻¹. (40)
Thus, our result follows from Lemma 18 and (40).

Finally, we present two perturbation results for the generalized inverse 𝐴^(ℎ)_𝑇,𝑆.

Theorem 20. Let 𝐴,𝛿𝐴∈𝐵(𝑋,𝑌). Suppose that 𝑇⊂𝑋 and 𝑆⊂𝑌 are homogeneous subsets such that 𝐴^(2,ℎ)_𝑇,𝑆 exists. Put 𝐴=𝐴+𝛿𝐴. If 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on ℛ(𝛿𝐴) and 𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴 is invertible in 𝐻(𝑋,𝑋), then 𝐴^(ℎ)_𝑇,𝑆=(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴)⁻¹𝐴^(2,ℎ)_𝑇,𝑆 if and only if ℛ(𝐴)∩𝑆={0}.

Proof. If 𝐴^(ℎ)_𝑇,𝑆=(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴)⁻¹𝐴^(2,ℎ)_𝑇,𝑆, then from Theorem 7, obviously, we have ℛ(𝐴)∩𝑆={0}. Conversely, suppose that ℛ(𝐴)∩𝑆={0}. First, noting that from Lemma 15, we get Φ=(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴)⁻¹𝐴^(2,ℎ)_𝑇,𝑆 is well defined, and ℛ(Φ)=𝑇 and 𝒩(Φ)=𝑆. Similar to the proof of Theorem 16, we can show 𝐴^(2,ℎ)_𝑇,𝑆 exists and 𝐴^(2,ℎ)_𝑇,𝑆=Φ. Noting that 𝐴^(ℎ)_𝑇,𝑆 is unique if it exists, thus, by using Theorems 6 and 7, we can prove our theorem.

Corollary 21. Let 𝐴,𝛿𝐴∈𝐵(𝑋,𝑌). Suppose that 𝑇⊂𝑋 and 𝑆⊂𝑌 are homogeneous subsets such that 𝐴^(2,ℎ)_𝑇,𝑆 exists. Put 𝐴=𝐴+𝛿𝐴. If 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on ℛ(𝛿𝐴) with ‖𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴‖<1, then 𝐴^(ℎ)_𝑇,𝑆=(𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴)⁻¹𝐴^(2,ℎ)_𝑇,𝑆 if and only if ℛ(𝐴)∩𝑆={0}.

Proof. By our assumption, 𝐴^(2,ℎ)_𝑇,𝑆 is quasi-additive on ℛ(𝛿𝐴), we get 𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴 is a bounded linear operator, and then it is well known that 𝐼_𝑋+𝐴^(2,ℎ)_𝑇,𝑆𝛿𝐴 is invertible when ‖𝐴^(2,ℎ)_T,𝑆𝛿𝐴‖<1. So the corollary follows from Theorem 20.

Acknowledgment

The authors thank the referees for their helpful comments and suggestions.