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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 309708, 6 pages
http://dx.doi.org/10.1155/2014/309708
Research Article

On the Covariance of Moore-Penrose Inverses in Rings with Involution

Department of Mathematics, Islamic Azad University, Firoozkooh Branch, Firoozkooh, Iran

Received 17 February 2014; Revised 14 April 2014; Accepted 14 April 2014; Published 5 May 2014

Academic Editor: Sergei V. Pereverzyev

Copyright © 2014 Hesam Mahzoon. 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

We consider the so-called covariance set of Moore-Penrose inverses in rings with an involution. We deduce some new results concerning covariance set. We will show that if is a regular element in a -algebra, then the covariance set of is closed in the set of invertible elements (with relative topology) of -algebra and is a cone in the -algebra.

1. Introduction

Suppose that is a ring with unity . A mapping of into itself is called an involution if for allandin . A ring with an involution is called -ring. Throughout this paper is a -ring.

An element is called regular if it has a generalized inverse (in the sense of von Neumann) in ; that is, there exists such that Note that such is not unique [1, 2].

Definition 1. Let be a -ring and .(i)is called Moore-Penrose invertible if there exists such that (ii) is called Drazin invertible if there exists such that for some nonnegative integer . The least such is the Drazin index of , denoted by .

Obviously, if and only ifis invertible and in this case the Drazin inverses of and coincide. If , then the Drazin inverse is known as the group inverse.

It is well known that the Moore-Penrose inverse (briefly, MP-inverse) and the Drazin inverse are unique if they exist. We reserve the notations and for the MP-inverse and Drazin inverse of , respectively. According to the uniqueness of the notion under consideration, if has a MP-inverse, then and also have MP-inverses. Moreover

In what follows, we will denote by the subset of invertible elements of   and by the set of all MP-invertible elements of . An element in is called idempotent if . A projection satisfies . Note that if , then and are projections. In addition, The commutator of a pair of elements and in is given by Note that if and only if andcommute. Also, it is well known that if , and are in , then

Letbe an element in ; its inverse is covariant with respect to ; that is, for all , we have

In general, the elements of are not covariant under (see [24]). For a given element with MP-inverse we define its covariance set

Schwerdtfeger [4] described the class for the matrices of rank 1 or 2. The characterization of the covariance set for an algebra of matrices was studied by Robinson [2] and some interesting results of were presented by Meenakshi and Chinnadurai [3].

The paper is organized as follows. The endeavour in Section 2 is to show how the results of [3] can be extended to MP-inverses in -rings. Moreover, we show that Drazin inverses are covariant under the group of invertible elements of -rings. In Section 3 we prove that the covariance set is a closed set in and is a cone in . Furthermore, we show that if is a sequence of MP-invertible elements of a -algebra such that their MP-inverses norm is bounded and converges to , then there is some kind of convergence of to .

2. Covariance Set of Moore-Penrose Inverses in -Rings

Many of the results of this section are essentially due to [3], withthe main difference being that in [3] one considers covariance set for matrices. In this section we generalized these results to any -ring.

The next proposition describes a relation between the covariance set and commutators. It was also shown in [24] in the special case of matrices. Here, we include a shorter proof for the sake of completeness.

Proposition 2. Let be -ring and with MP-inverse . Then the following statements are equivalent:(i);(ii) and .

Proof. (i)(ii) Suppose that . Then . Set . Then is projection, so and . From here we get . This implies that . Similarly by putting , we conclude that .
(ii)(i) From the assumptions it is not hard to see that is the MP-inverse of . By the uniqueness of Moore-Penrose inverse we get ; that is, .

From Proposition 2 we deduce the following result.

Corollary 3. Let be -ring and with MP-inverse . Then

Combining the above corollary and Proposition 2, we get the following corollary.

Corollary 4. If is normal, then

We now have some equalities for the covariance sets. See also [3].

Proposition 5. Let be -ring and with MP-inverse . Then

Proof. By replacing with , part (ii) of Proposition 2 does not change so the first equality holds. Since and , Proposition 2 yields the second equality. Also and , again from Proposition 2 we get the last equality.

Note that if is any unitary element in , the ; thus for every This implies that for each .

In the next proposition, we will show that if is Drazin invertible with Drazin inverse , then . For this reason, the notion of covariance sets is not studied to Drazin inverses.

Proposition 6. Suppose that is a -ring and is a Drazin invertible element in . Then is covariant under ; that is,

Proof. Suppose that is the Drazin inverse of and is an arbitrary element in . For simplicity of calculations, set and By hypothesis, , , and ; thus Now the uniqueness of the Drazin inverse implies that ; that is, is covariant under .

In particular, by applying the above proposition, if is group invertible with the group inverse , then is also covariant under .

We reproduce the following definition from [5].

Definition 7. Let be a ring; is called simply polar if it has a commuting generalized inverse (in the sense of von Neumann); that is, if is any generalized inverse of , then .

Some authors used the expression EP instead of simply polar. Indeed, they called with MP-inverse is EP if and only if .

The next remark provides a large class of simply polar elements and some related properties.

Remark 8. Let with MP-inverse .(i)If is self-adjoint, then it is simply polar, since (ii)If is normal, then it is simply polar, since thus . In a similar manner we get . Therefore (iii)It is easy to check that simply polar properties of ,  and are equivalent; that is, if one of them is simply polar, then two others are also simply polar.(iv)If is simply polar, then (v)If is simply polar, then Proposition 5 implies that .
For finding more equivalent statements about the simply polar elements see [1, Theorem 2.3 and final remark].

Proposition 9. Let with MP-inverses and , respectively. If and , then .

Proof. The assumptions, after some easy calculations, imply that is the MP-inverse of . Thus . Suppose that . Then Proposition 2 implies that Since and , we have and . From the linearity of commutator we obtain Again by applying Proposition 2, we get .

Corollary 10. Let with MP-inverses and , respectively. If and are self adjoint and , then .

Proof. By assumption and are self adjoint. Thus implies that . The result now follows from Proposition 9.

The next example shows that in Proposition 9 inclusion can be proper.

Example 11. Set and . Then , ,  and  , and is invertible; thus . Now if we set thenis invertible: On the other hand ; therefore From here we conclude that . Thus .

Let and be two subsets of . We recall that

Note that the reverse order rule for the MP-inverse, that is, , is valid under certain conditions on MP-invertible elements; see [6].

Remark 12. Let with MP-inverses and , respectively. One can easily check the following.(i)If and , then .(ii)If , then .(iii)Generally, there is no subset relation between and . For instance, if we put , then which is not a subset of but .(iv)Generally, there is no subset relation between and . Set as Example 11. Then , and so .

Proposition 13. Let with MP-inverses and , respectively. If , then , where .

Proof. By assumption , so there exists in such that . Therefore , and so . In a similar manner we get . Since is projection, .

Corollary 14. Let with MP-inverses and , respectively. If and , then .

Proof. The proof is an immediate consequence of Propositions 5 and 13.

The following corollary was also proved for matrices in [3].

Corollary 15. Let be simply polar and . Then .

According to the above corollary and Remark 8, we have the following.

Corollary 16. If and is simply polar, then for each .

Corollary 17. If and is normal, then = for each .

Note that Example 11 shows that the converses of the two last corollaries do not hold. Indeed, if we set , then is neither simply polar nor normal and but .

We know that if either or , then . One can easily check that if is a -ring with no nonzero nilpotent element, then where and it is an idempotent element of ring. In all cases, we consider that has a group structure. But in general is not a group; see for instance [3]. Our purpose is to find a subset of which has mathematical (group) structure. For this purpose, let be an element in , with MP-inverse . We define (as it is defined in [3] for matrices) by

In the next proposition we collect some interesting properties of .

Proposition 18. Let be an element in with MP-inverse . Then(i)if , then ;(ii);(iii) is a group;(iv) is covariant under ;(v)if such that , then ;(vi)if , then , where is a polynomial in ;(vii)if and , then .

Proof. (i) Assume that . Then and so . By taking the adjoint it follows that . Thus In a similar manner, from , we obtain . Therefore .
(ii) Let by part (i) and definition of ; we have From (8) and (26) we conclude that Therefore .
(iii) Suppose that . Then From (8) and (28) we get This means that . If . Then and so . Multiply this from left and right to ; we obtain . Similarly we have . This means that . Therefore, is subgroup of  .
(iv) It is easy to check that if , then for every , we have
(v) If , by linearity of the commutator we get and . That is, .
(vi) It follows from (ii) and (iv).
(vii) Using (8) and part (i), we see that and ; that is, .

Let be the set of all matrices. It was shown that in [3] is a nonabelian subgroup of if and only if .

Proposition 19. Assume that is an element in with MP-inverse . If is normal, then where is the cyclic group generated by .

Proof. Using Proposition 2, Corollary 4, and induction, we can show that for all integer , .

Note that, in fact if is normal, then , where is a polynomial in .

3. Covariance Set in -Algebras

Given unital -algebras with the nonzero element . We will denote by and the subset of invertible elements and MP-invertible elements of , respectively.

In this section, we find some topological properties for ; for instance, we will show that is a closed set in with respect to the relative topology.

Theorem 20. Suppose that is a -algebra and . Then is closed in with respect to the relative topology.

Proof. Suppose that belongs to the closure of in . Then there exists a sequence such that , from which it follows that . Thus by Proposition 2. Therefore By taking limits in (32) as , we get Since and are in , again Proposition 2 implies that . This means that is closed in with respect to the relative topology.

Note that generally is not a closed set in . For example, if we set and , then for all , but .

We will now reproduce an important theorem of [7] that will be crucial to prove the next result.

Theorem 21 ([see [7]). Let be nonzero elements of such that in . Then the following conditions are equivalent:(i); (ii); (iii); (iv).

The next theorem shows that the covariance set, seen as a multivalued map, has some kind of continuity.

Theorem 22. Let be a sequence of MP-invertible elements in the -algebra such that and the norms are bounded. If and as , then .

Proof. By hypothesis, ’s are MP-invertible, , and . By Theorem 21, is MP-invertible and . Thus Therefore by Proposition 2 Now, letting in (35) we get Again by applying Proposition 2 we conclude that .

We recall that a set is called a cone whenever and .

Proposition 23. Suppose that is a regular element in and is any nonzero scalar. Then if and only if .

Proof. Assume that . Then by Proposition 2, This is true if and only if which is equivalent to Again by Proposition 2, these hold if and only if .

Corollary 24. If is regular in , then is a cone.

Proof. The proof is an immediate consequence of the above proposition.

Proposition 25. Suppose that is a regular element in and is any nonzero scalar. Then .

Proof. By assumption , thus and so By applying Proposition 5 we get

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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