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Journal of Operators
Volume 2013 (2013), Article ID 204587, 7 pages
http://dx.doi.org/10.1155/2013/204587
Research Article

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

1Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, Henan 453003, China
2Department of Mathematics, East China Normal University, Shanghai 200241, China

Received 20 January 2013; Accepted 4 July 2013

Academic Editor: C. Morales

Copyright © 2013 Jianbing Cao and Yifeng Xue. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let and be Banach spaces, and let be a bounded linear operator. In this paper, we first define and characterize the quasi-linear operator (resp., out) generalized inverse (resp., ) for the operator , where and are homogeneous subsets. Then, we further investigate the perturbation problems of the generalized inverses and . 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 and be Banach spaces, let be a mapping, and let be a subset of . Recall from [1, 2] that a subset in is called to be homogeneous if for any and , we have . If for any and , we have , then we call as a homogeneous operator on , where is the domain of ; is called a bounded homogeneous operator if maps every bounded set in into bounded set in . Denote by the set of all bounded homogeneous operators from to . Equipped with the usual linear operations for , and for , the norm is defined by , and then similar to the space of all bounded linear operators from to , we can easily prove that is a Banach space (cf. [2, 3]). Throughout this paper, we denote by , , and the domain, the null space, and the range of a bounded homogeneous operator , respectively. Obviously, we have .

For an operator , let and be closed subspaces of and , respectively. Recall that the out inverse with prescribed range and kernel is the unique operator satisfying . 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 (cf. [4, 5]). Researches on the generalized inverse of operators or matrices have been actively ongoing for many years (see [512], e. g.).

Let and let and be two homogeneous subsets in and , respectively. Motivated by related work on 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 with prescribed range and kernel . We give the necessary and sufficient conditions for the existence of the generalized inverses , and we will also study the perturbation problems of the generalized inverse . Similar results on the generalized inverse are also given.

2. Definitions and Some Characterizations of and

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 be a subset of and let be a mapping. Ones calls as quasi-additive on if satisfies

Definition 2 (see [3, 15]). Let be a Banach space. A mapping is called a quasi-linear projector on , if satisfies the following conditions: (1)is a homogeneous operator; (2) is idempotent; that is, ; (3) is quasi-additive on ; that is, for any and any , one has
If the mapping only satisfies the conditions (1) and (2) above, then we call 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 is a bounded quasi-linear projector on , then the range of is a closed linear subspace of .

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

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 . Let and be homogeneous subsets in and , respectively. Let be a bounded homogeneous operator. Consider the following equations: (i)If satisfies (2)–(4) in (3), then the operator is called to be the homogeneous outer generalized inverse of with prescribed range and kernel . We denoted it by . (ii)If satisfies all equations in (3), then the operator is called to be the homogeneous generalized inverse of with prescribed range and kernel . We denoted it by . (iii)If is quasi-additive on and satisfies (2)–(4) in (3), then the operator is called to be the quasi-linear outer generalized inverse of with prescribed range and kernel . We denoted it by . (iv)If is quasi-additive on and satisfies all equations in (3), then the operator is called to be the quasi-linear generalized inverse of with prescribed range and kernel . We denoted it by .

Remark 5. Let . Assume that and are Chebyshev subspaces in and , respectively. Then the Moore-Penrose metric generalized inverse of uniquely exists (cf. [3, 17, 18]). Let where is the set-valued dual mapping of , and . Then and are homogeneous subsets in and , respectively, and
In this case, we have . Furthermore, if is quasi-additive on , then . 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 and . 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 and .

Theorem 6. Let . Let and be homogeneous subsets in and , respectively. Then the following statements are equivalent: (1)there exists some such that is quasi-additive on and , , ; (2), and is a closed linear subspace.

Proof. (1)⇒(2) Suppose that (1) holds. Put and . Since , we have and . In order to prove   is a closed linear subspace, by Lemma 3, we only need to show that is a bounded quasi-linear projector. Let and , we get and . Noting that is quasi-additive on , then
Therefore, is a bounded quasi-linear projector and then is a closed linear subspace in . Now we show that . Let . Since , then and for some . So . Therefore, we have . Next we will show that . For any , we have . Since we have proved that is a closed linear subspace of , we have . Also noting that is quasi-additive on and , we get . Thus, we get . From , we can also get and ; therefore, and then .
(2)⇒(1) Suppose that (2) holds. Since and is a closed linear subspace, we see is well defined, which is also a linear bijective mapping, so has a linear inverse . Since , then for any , it can be uniquely written as with and . Now we can define a mapping by . It is easy to see that , and by . Since is a homogeneous subsets in , we see is a homogeneous operator. Now we need to prove is quasi-additive on . In fact, for any and , since , we can write as , where and . Noting that since is a linear subspace, then we can check that
Therefore, is quasi-additive on .

Theorem 7. Let . Let and be homogeneous subsets in and , respectively. Then the following statements are equivalent.(1)There exists some such that is quasi-additive on and (2), and is a closed linear subspace. (3), , and is a closed linear subspace.

Proof. (1)⇒(2) Suppose that (1) holds, Put and . Similar to the proof of Theorem 6, we have , and is a closed linear subspace. Note that we also have the following relation:
Thus, we get that
From the above equations, we can obtain and .
(2)⇒(3) If and , then it is obvious that , . We only need to show . Obviously, we have . Now for any , we have for some . Since , we can write , where and . Thus . Therefore and then .
(3)⇒(1) Suppose that (3) is true, then by Theorem 6, we know that exists, and then , . We need to show and is quasi-additive on . Since exists, then and is quasi-additive on . Noting that for any , we have . Thus, and that is, . Obviously, . On the converse, let . Since , we have with some and . From , we see . Thus, we have . So and then . Hence, we obtain that is quasi-additive on .

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 exists, then we must have . So there is no contradiction in our definition. The following proposition gives the uniqueness of . From this result, it is also easy to see the uniqueness of , which are similar to the linear (outer) generalized inverses with prescribed range and kernel.

Proposition 8. Let . Let and be homogeneous subsets in and , respectively. If the generalized inverse (or ) exists, then it is unique.

Proof. We only need to prove this proposition for . By Definition 4, we need to prove that there is at most one bounded homogeneous operator such that is quasi-additive on and satisfies
Suppose that there is another such that is quasi-additive on and satisfies the equations , , and . Put and . By using the equations and , we see that both and are bounded homogeneous projectors on and satisfy
Therefore, we get . Thus, . On the other hand, if we set and , then by using the equations and again, and also noting that and are both quasi-additive on , similar to the proof (1)⇒(2) in Theorem 6, we see both and are bounded quasi-linear projectors and satisfy
Thus, , and then .

Proposition 9. Let . Let and be homogeneous subsets in and , respectively. If the generalized inverse (or ) exists, then and are both closed linear subspaces of .

Proof. If the generalized inverse (or ) exists, then from Theorem 6 (or Theorem 7), we see is a closed linear subspace in . Let . Then it is easy to check that . Since is quasi-additive on , simple computation can show that is a bounded quasi-linear projector. Then from Lemma 3, we get is also a closed linear subspace of if exists. If the generalized inverse exists, then exists and .

Finally, in this section, we will give some representation results of the generalized inverse .

Theorem 10. Let . Let and be homogeneous subsets in and , respectively. Suppose that such that is quasi-additive on , and . Then the following statements are equivalent: (1)the generalized inverse exists; (2) is invertible and is a closed linear subspace. In this case, one has .

Proof. (1)⇒(2) Suppose that (1) holds, then by Proposition 9, we get that is a closed linear subspace. If for some , then for some . Since , by Theorem 6, we have . Noting that , thus, we get and then (2) holds.
(2)⇒(1) Let . Obviously, we have , and , . We show that is quasi-additive on . Let and . Since is quasi-additive on , we get that is a linear operator and then
By Proposition 8, the uniqueness of , we can obtain .

Let . Recall that is group invertible if there exists such that

In this case, denote as usual. For a bounded homogeneous operator , if there exists such that , then we call as a bounded homogeneous inner inverse of or simply say is inner regular.

Theorem 11. Let . Let and be homogeneous subsets in and , respectively. Suppose that such that is quasi-additive on , and , . If has the group inverse , then exists and, moreover, is inner regular and

Proof. Since is quasi-additive on , we see is well defined. Let , we show that . From , we get , and then . The existence of means that
Therefore
Hence . Now, we can compute as follows:
Thus, and . Obviously, we see is quasi-additive on for . So, from Definition 4 and Proposition 8, we see exists and .
Set . Then it is easy to check that
We get is inner regular and . Furthermore, we have
This completes the proof.

Theorem 12. Let . Let and be homogeneous subsets in and , respectively. Suppose that such that is quasi-additive on , and , . If has the group inverse , then exists and .

Proof. By Theorem 11, we know that exists and . Let . Noting that is quasi-additive on , then is a projector and . Clearly, we see that and is invertible in with . Consider the decomposition . Then we can write as the following matrix form:
From (23), we see if , then is invertible in . Thus, in the case, by (23) we have
From the proof of Theorem 11, we know that . Thus, we get . So, by using (24), we can obtain that . Now we can compute as follows:
This completes the proof.

3. Perturbations of the Generalized Inverses and

In this section, we present some perturbation results for the generalized inverse and . 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 be a Banach space and . If there exist two constants such that then is bijective. Moreover, for any

Let . Suppose that and are homogeneous subsets such that exists. If is quasi-additive on , then we see that is a bounded linear operator, so is also a bounded linear operator on . Furthermore, if and also satisfy where , then by Lemma 13, we get that is invertible.

Lemma 14. Let and such that is quasi-additive on and is quasi-additive on ; then is invertible in if and only if is invertible in .

Proof. If there is a such that , then
Similarly, we also have . Thus, is invertible on with .
The converse can also be proved by using the same way as above.

Lemma 15. Let . Suppose that and are homogeneous subsets such that exists. If is quasi-additive on and (resp., ) is invertible in (resp., ), then (resp., ) is invertible in (resp., ) and is a bounded homogeneous operator with and .

Proof. By Lemma 14, we obtain that (resp., ) is invertible in (resp., ).
Clearly, is a bounded linear operator and . From the equation and , we get that is a bounded homogeneous operator. Finally, by using (30) and Definition 4, we can obtain that and .

Theorem 16. Let . Suppose that and are homogeneous subsets such that exists. Put . If is quasi-additive on and satisfies (28), then exists and . Moreover, one has

Proof. Since is quasi-additive on and satisfies (28), then from Lemma 13 and Lemma 15, we see is well defined with and . Now, we prove that is quasi-additive on . Let and , then for some . Since is quasi-additive on and , also noting that is a bounded linear operator, then
We get is quasi-additive on . Thus, from Definition 4 and Proposition 8, we see that exists and .
Finally, we show that the estimate formula is valid:
By using Lemma 13, we get
Thus from (34) and (35), we can obtain
This completes the proof.

Corollary 17. Let . Suppose that and are homogeneous subsets such that exists. Put . If is quasi-additive on with , then exists and . Moreover, one has

Proof. Obviously, this corollary follows from Theorem 16 if we take and .

Lemma 18 (Neumann Lemma for quasi-linear operators). Let . If is quasi-additive on with , then exists and

Proof. Noting that is quasi-additive on , 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 . Suppose that and are homogeneous subsets such that exists. Put . If is quasi-additive on with , then (1) is inventible bounded homogeneous operator; (2) exists and .
Moreover, one has

Proof. Since is quasi-additive on , then implies that is quasi-additive on . Obviously, , and it follows from Lemma 18 that is inventible bounded homogeneous operator. Then, by using Lemma 15, similar to the proof of Theorem 16, we can show that exists and . For the estimate formula, we can compute that
Thus, our result follows from Lemma 18 and (40).

Finally, we present two perturbation results for the generalized inverse .

Theorem 20. Let . Suppose that and are homogeneous subsets such that exists. Put . If is quasi-additive on and is invertible in , then if and only if .

Proof. If , then from Theorem 7, obviously, we have . Conversely, suppose that . First, noting that from Lemma 15, we get is well defined, and and . Similar to the proof of Theorem 16, we can show exists and . Noting that is unique if it exists, thus, by using Theorems 6 and 7, we can prove our theorem.

Corollary 21. Let . Suppose that and are homogeneous subsets such that exists. Put . If is quasi-additive on with , then if and only if .

Proof. By our assumption, is quasi-additive on , we get is a bounded linear operator, and then it is well known that is invertible when . So the corollary follows from Theorem 20.

Acknowledgment

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

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