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

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 [2–4]). 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 [2–4] 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.