1 Introduction
The core and coreEP inverses of matrices have been intensively studied in recent years to solve a certain type of matrix equations baks ; BakTr14 . Hence, a significant number of papers explored the characterizations of the core inverse and its applications in baskett1969 ; Kurata_2018 ; LiChen18 ; PreMo20 . A few properties of the core inverse and interconnections with different generalized inverses were discussed in baks ; Kurata_2018 ; rakic ; Wang_2014 . The coreEP inverse of matrices, introduced by Prasad and Mohana PrasadMo14 , have significantly impacted for square matrices. Then several characterizations of the coreEP inverse and its extension to rectangular matrices were discussed in ferreyra2018revisiting . In this connection, the authors of gao2018representations have discussed the weighted coreEP inverse and several representations in terms of matrix decomposition. Further, a few characterizations and properties of the coreEP inverse with other inverses are discussed in Gao_2019 ; PrasadMo14 ; PredragKM17 . The vast literature on coreEP, weighted coreEP inverses of matrices along with its multifarious extensions MaH19 ; HaiTing19 ; zhou2019core ; mosic2019 , motivate us to study introduce Eweighted coreEP and Fweighted dual coreEP inverse of matrices.
We mention below a summary of the main points of the discussion.

The notation of weighted coreEP and weighted dual coreEP inverses are proposed. Through these definitions, the existence of generalized weighted MoorePenrose inverse is discussed.

Introduce several explicit expression for the weighted coreEP inverse of matrices through other generalized inverses, like, Drazin inverses, weighted core inverse, and generalized MoorePenrose inverses.

We have discussed additive properties of the weighted coreEP and weighted dual coreEP inverse of matrices.

Introduce star weighted coreEP and weighted coreEP star matrices to solve the system of matrix equations.

A few characterization and representation of star weighted coreEP and weighted coreEP star classes of matrices are discussed.
The main objective of this paper to investigate a new fruitful way for developing the relation of the weighted coreEP inverse of matrices with different generalized inverses, like, Drazin inverses, weighted core inverse, and generalized MoorePenrose inverses. The results of these approaches will help the necessary freedom to deal with different types of inverses and flexibility to choose generalized inverses depending on applications. It is worth to mention the work of Mosić in Mosic20 in which they have introduced the Drazinstar and starDrazin Matrices for solving some kind of system of matrix equations, very recently. The author also discussed the maximal classes of matrices for generating the most general form of this class of matrices. It also motivates us to introduce the star weighted coreEP and weighted coreEP star class of matrices and then provide characterizations of weighted coreEP inverse and weighted dual core inverse of matrices in the form of the outer inverse of the MoorePenrose inverse.
The outline of the paper is as follows. We present some necessary definitions and notation in Section 2. Definition, existence, and several explicit expression for the weighted coreEP inverse of matrices are considered in Section 3. In Section 4, we discuss the new class of matrices (i.e., star weighted coreEP and weighted coreEP star) to solve the system of the matrix equation. In addition to these, we discuss a few characterizations of the new class of matrices. The work is concluded along with a few future perspective problems in Section 5.
2 Preliminaries
For convenience, throughout this paper, stands for the set of matrices over complex numbers. In addition, we assume the matrices and to be invertible, and hermitian. Further, we use the notation , and for the range space, null space and conjugate transpose of respectively. The index of is the smallest nonnegative integer , such that , which is denoted by . The Drazin inverse discussed in cline1980 for a rectangular matrix, however, it was introduced Drazin58 earlier in the context of associative rings and semigroups. Let , the Drazin inverse of is the unique matrix which satisfies the following equations
Let us recall the generalized weighted MoorePenrose inverse sheng2007generalized ; ben of a matrix, as follow.
Definition 2.1.
Let . A matrix satisfying
, , , ,
is called the generalized weighted inverse of and denoted by .
Note that generalized weighted MoorePenrose inverse of a matrix does not exist alwayssheng2007generalized . But the positive definite of and leads to the existence of . The uniqueness of can be verified easily. Recall the definition of weighted core and dual inverse of a matrix as follows.
Definition 2.2.
Let . If a matrix satisfies
(6) , (7) , ,
is called the weighted core inverse of .
This inverse denoted by and the uniqueness of it can found in RJR . At the same time, the weighted dual core inverse of denoted by and defined as follows.
Definition 2.3 (Rjr ).
Let . A matrix is called weighted dual core inverse of if satisfies
(8) , (9) , and .
Theorem 2.4.
(RJR ) Let and . If , then the following five conditions are true.

(;

(

;

for any ;

.
Lemma 2.5 (ben ).
Let with . Then has index 1 and for all .
Lemma 2.6 (Proposition 3.2, Rjr ).
For , if , then and .
Lemma 2.7.
Let . If , then .
Proof.
Let . Then , and . Using these, we obtain
∎
Similarly, the following result follows for weighted dual core inverse.
Lemma 2.8.
Let . If , then .
Lemma 2.9 (Rjr ).
Let and with . Then .
Theorem 2.10.
Let and . If and , then .
Proof.
Let . Then there exists a such that . Now the hermitian of follows from the below expression.
(1) 
Using equation (1), we obtain
.
∎
3 Onesided weighted coreEP inverse
In this section, we introduce weighted coreEP inverse and weighted dual coreEP inverses of square matrices. In addition, a few results and characterizations are established. Now we present the definition of the weighted coreEP inverse.
Definition 3.1.
Let and . A matrix is called the weighted coreEP inverse of if it satisfies
, , and .
The weighted coreEP inverse of is denoted by . Next, we define the weighted dual coreEP inverse.
Definition 3.2.
Let and with . A matrix satisfies
, , ,
is called the weighted dual coreEP inverse of and denoted by .
Example 3.3.
Let , and . It is easy to verify ,
and
Now, we discuss a few useful results which will be frequently used in the subsequent sections.
Proposition 3.4.
Let and . Then the following are holds.

If a matrix then for any .

If , then and .

If a matrix then for any .
Proof.
Let . Then .
Let . Then by part , we have
.
Further, from and , we obtain ∎
The characterization of weighted coreEP inverse through the range condition is presented below.
Theorem 3.5.
Let and . Then the following statements are equivalent.

,

, and .
Proof.
By Proposition 3.4, it enough to show . Let . Then
Thus . From
,
we get . Hence .
From , we have , for some . Now
.
Let and . Then there exists such that and . Using these results, we obtain
(2) 
(3) 
In view of Eqns. (2), (3), and invertibility of , we have . Thus
. Hence is the weighted coreEP inverse of .
∎
Similarly, the following result holds for weighted dual coreEP inverse.
Theorem 3.6.
Let and . Then the following statements are equivalent.

,

, and .
The representation of the Drazin inverse is constructed through inverse as follow.
Lemma 3.7.
Let and . For , if , then the following are holds.

.

.
Proof.
The following result can be proved in similar manner for weighed dual coreEP inverse.
Lemma 3.8.
Let and . For , if , then the following are holds.

.

.
We now have the following characterization of the class of inverse.
Lemma 3.9.
Let with . For if , then .
Proof.
Let . Then
, and .
Using these, we obtain
∎
Similarly, we obtain the following result for weighted dual coreEP inverse.
Lemma 3.10.
Let and . For if , then
The uniqueness of weighted coreEP inverse is discussed below.
Theorem 3.11.
Let and . Then is unique.
Proof.
Similarly, the uniqueness of weighted dual coreEP inverse can be verified.
Theorem 3.12.
Let and . Then is unique.
Now we discuss one of our important results, which gives a method of construction of the weighted coreEP inverse using a inverse of matrix.
Theorem 3.13.
Let and . Then , where be any positive integer satisfying and is a inverse of .
Proof.
Let . Then
,
and . ∎
Similarly, construction of the weighted dual coreEP inverse through inverse of a matrix is presented below.
Theorem 3.14.
Let and . Then , where be any positive integer satisfying and is a inverse of .
In view of Theorem 3.13 and 3.14, we state the following as a corollary for construction of weighted coreEP inverse.
Corollary 3.15.
Let and . For , if exists, then

.

, .

.

, .
In conjunction with Lemma 2.9 and Corollary 3.15, we obtain the following representations for the weighted coreEP inverse through weighted core and weighted MoorePenrose inverse.
Proposition 3.16.
Let be positive definite matrices and with . Then

.

,
where is any positive integer satisfying .
Proof.
We next discuss a necessary and sufficient condition in connection to the characterization of and inverses.
Proposition 3.17.
Let . Then is nonempty if and only if for some .
Proof.
Let . Then
, where .
Conversely, let and . Then . Further, and
.
Thus is an inverse of . ∎
Proposition 3.18.
Let . Then is nonempty if and only if for some .
In view of the above Proposition 3.17, we obtain the following necessary and sufficient condition for inverse.
Lemma 3.19.
Let . Then for some if and only if for some .
Proof.
Let . Then by taking , we have . Applying Proposition 3.17, we have . Thus
, where .
Conversely, let for some . Then , where . ∎
At the same time, we get the following result for inverse through the Proposition 3.18.
Lemma 3.20.
Let . Then for some if and only if for some .
In view of Theorem 3.13, Proposition 3.17, and Lemma 3.19, we have the following sufficient condition for weighted coreEP inverse.
Theorem 3.21.
Let and . If for some , then .
Proof.
In view of weighted dual coreEP inverse can be written as follow.
Theorem 3.22.
Let and . If for some , then .
We, next discuss the existence of the power of an weighted coreEP inverse.
Theorem 3.23.
Let and Then exists if and only if exists. Moreover, and .
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