#### Abstract

We study several algebraic properties of dual covariance and weighted dual covariance sets in rings with involution and -algebras. Moreover, we show that the weighted dual covariance set, seen as a multivalued map, has some kind of continuity. Also, we prove weighed dual covariance set invariant under the bijection multiplicative -functions.

#### 1. Introduction

Suppose is a ring with unity . A mapping of into itself is called an* involution* if
for all and in . Throughout this paper will be a ring with an involution.

An element is called* regular* if it has a generalized inverse in .

It is well known that every regular element in a -algebra has the Moore-Penrose inverse (denoted by MP-inverse from this point on). Generally MP-inverse is uniquely determined in if it exists. We will denote the MP-inverse of by .

In the following, we will denote by and the set of an invertible and MP-invertible elements of , respectively.

Assume that is 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 [1]). For a given element with MP-inverse , we will denote the* covariance set* by and define
For more definitions and notations we refer the interested readers to [2]. Covariance set was studied by [1, 3–5].

We define the* dual covariance set* by reversing the roles of and in and denote it by . In fact,
This notion was studied by Robinson in [6] for matrices.

Note that if with MP-inverse and , then, from (3) and (4), we obtain Also, it should be noted that for every and for each . Moreover, this inclusion can be proper; for instance, but .

The aim of this paper is to investigate the properties of dual covariance and weighted dual covariance set. In Section 2, we define and characterize the weighted dual covariance in terms of commutators. Also we prove that the dual covariance sets of and coincide. Moreover, we collect some interesting properties of and in -algebras and rings with an involution. In addition, we show that the weighted dual covariance set, seen as a multivalued map, has some kind of continuity. In Section 3 we study the relations of and multiplicative -functions. Also, we prove that weighed dual covariance sets are invariant under the bijection multiplicative -functions.

#### 2. Weighted Dual Covariance Set

The weighted Moore-Penrose inverse (weighted MP-inverse from this point on) for matrices was introduced by Chipman in [7]. For some historical notes of weighted MP-inverse see Rao and Mitra [8] and references therein. In the next definition, it will be introduced in -algebras (see also [9]). In what follows, we will only consider unital -algebras. Indeed, and are unital -algebras; the nonzero elements, and , are the units of and , respectively. We will denote by and the subset of invertible elements and MP-invertible elements of , respectively.

*Definition 1. *Let be a -algebra and , two positive elements in . We say that an element has a weighted MP-inverse with weights , if there exists such that

As already observed in [9], if weighted MP-inverse with weights , exists, then it is unique, and so we will denote it by . Every regular element in a -algebra has a weighted MP-inverse [9, Theorem 4] and it can be written as

Next, we extend the definition of dual covariance set to* weighted dual covariance set*.

*Definition 2. *Suppose and , are positive elements in . We define weighted dual covariance set by

In the following theorem we characterize in terms of commutators.

Theorem 3. *Assume and , are positive elements in . Then the following statements are equivalent: *(i)*;*(ii)* and .*

*Proof. *(i)(ii): suppose . Then is a weighted MP-inverse of with weights , . Thus, . Therefore, . This implies that . In a similar manner from we conclude that .

(ii)(i): since is MP-inverse of , it suffices to show that and . By the assumptions . From this we obtain . Hence, . In a similar manner from we get .

The following result is obtained from Theorem 3 by setting for -algebras. However, it is true for a generalized case in rings with involution. In fact, consider the following.

Proposition 4. *Assume that . Then
*

Proposition 5. *Assume that . Then .*

*Proof. *By Proposition 4,
This is equivalent to
Multiply (11) from left and right by ; we get
Again Proposition 4 shows that (12) holds if and only if .

Proposition 6. *Assume that and is normal. Then , where is an integer number.*

*Proof. *Since is normal, the first equality is an immediate consequence of Proposition 5. For proof of the second equality, obviously . For the converse suppose that ; using Proposition 4, normality of , and induction, one can get for all integer . Thus .

The following example shows that the normality hypothesis cannot be omitted from the above proposition.

*Example 7. *Set and . Then since . But , because

We recall that a* cone* is a set of rays; in other words is a cone if implies for each . Also, an element is called simply polar [10] if it has a commuting generalized inverse (in the sense of von Neumann); that is, there exists a generalized inverse of , such that .

In the next result we collect some noteworthy properties of weighted dual covariance set.

Proposition 8. *Assume that . Then, the following statements are equivalent: *(i)*;*(ii)*;*(iii)*;*(iv)* and ;*(v)* for any nonzero scalar . Moreover, if is simply polar, then the following statements are equivalent to the above statements:*(vi)*;*(vii)*.*

*Proof. *First we show that (i)(ii) by Theorem 3 if and only if
Since , thus (14) is equivalent to
Again, Theorem 3 shows that (15) holds if and only if . (i)(iii): in a similar manner, since and by applying Theorem 3, we get . (i)(iv): if and only if (14) holds. Since and , thus (14) is equivalent to
This implies that . Similarly we get
Thus, .

For the proof of (iv)(i), it is easy to verify that (iv) is satisfied if and only if (16) and (17) hold. These together imply (14); that is, (i) holds. (i)(v): since , . Now applying Theorem 3 we obtain the result. (vi)(vii): since is simply polar, thus and so (vi) and (vii) are equivalent. (iv)(vi) is obvious.

Corollary 9. *If , then is a cone.*

It is well known that every normal element is simply polar. Hence, consider the following.

Corollary 10. *If is normal, then
*

Corollary 11. *Assume that . Then for each ,
*

Proposition 12. *Assume that and is any scalar. Then .*

*Proof. *By Theorem 3, if and only if (14) is satisfied which is equivalent to
This holds if and only if .

Corollary 13. *If and is a nonzero scalar, then .*

Proposition 14. *Let with MP-inverses and , respectively. Assume that and . If , then .*

*Proof. *It is easy to verify that is the MP-inverse of . Then, . Since ,
By using the linearity of commutator and the assumptions, from (21), we conclude that
Now, Theorem 3 implies that .

Corollary 15. *Let with MP-inverses and , respectively. Assume that . If are self-adjoint, then .*

Corollary 16. *Let with MP-inverses and , respectively. Assume that . If and , then .*

It should be noted that is not closed under addition even . For example, set , , and ; then , , and are invertible. Thus . But and . This means that .

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

Theorem 17. *Assume that is a sequence in and . Let be such that . If , then .*

*Proof. *By assumption, . Therefore [11, Theorem 1.6] implies that , , and . Since ,
Letting in (23), we obtain
Now, Theorem 3 implies that .

#### 3. Weighted Dual Covariance and -Functions

Let us start with a definition.

*Definition 18. *Let and be unital -algebras. A multiplicative -function is a map satisfying (i) for all and in ;(ii) for all in ;(iii).

Note that every -algebras homomorphism is a linear multiplicative -function.

It is easy to verify that if is a positive element, then is positive. In addition, if is positive with MP-inverse , then is positive because Moreover, formula (7) shows that if is weighted MP-invertible with the weights and , then is positive and so is positive.

Proposition 19. *Let and be unital -algebras and suppose that is a multiplicative -function. *(i)*If is a regular element in , then is regular in .*(ii)*If is weighted MP-invertible in with weights , , then is weighted MP-invertible in with weights , . Moreover
*

*Proof. *The first assertion is a consequence of the second one. So we only prove (ii). Let be the weighted MP-inverse of in with weights , . Then by definition,
Since is a multiplicative -function, so (27) implies that
and similarly
Therefore is weighted MP-inverse of with weights , . Now by the uniqueness of weighted MP-inverse we get

Proposition 20. *Let and be unital -algebras and . *(i)*If is a multiplicative -function, then .*(ii)*If is a bijection multiplicative -function, then .*

*Proof. *(i) Suppose that . Then and so is weighted MP-inverse of with weights . By applying Proposition 19 we get .

(ii) Since is bijection multiplicative -function, is also multiplicative -function. From here and part (i) we obtain the desired assertion.

Corollary 21. *If is a multiplicative -function, then . Moreover, if is bijection, then .*

Corollary 22. *Assume that . Then, *(i)*if is a unitary, then ;*(ii)*if , then .*

*Proof. *Since the maps and are bijection multiplicative -functions, the results follow from Proposition 20.

Corollary 23. *If is a unitary, then , and if , then .*

*Remark 24. *The notion of dual covariance is not studied to Drazin inverses because it is easy to see that, for every Drazin invertible element and for every , we have
In fact, for any we have where is the set of all Drazin invertible elements of .

#### Conflict of Interests

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