Weighted m-Generalized Group Inverse in Banach Algebras

1. Introduction

Let $\mathcal{A}$ be a Banach algebra. An element $a\in \mathcal{A}$ has group inverse provided that there exists $x\in \mathcal{A}$ such that

$$x{a}^2=a,a{x}^2=x,ax=xa.$$

Such x is unique if exists, denoted by $a^{\#}$, and called the group inverse of a (see [14]). As is well known, a square complex matrix A has group inverse if and only if $rank\left(A\right)=rank\left(A^2\right)$.

A Banach algebra is called a Banach *-algebra if there exists an involution $\ast :x\to x^{\ast }$ satisfying ${(x+y)}^{\ast }=x^{\ast }+y^{\ast },{\left(\lambda x\right)}^{\ast }=\overline{\lambda }{x}^{\ast },{\left(xy\right)}^{\ast }=y^{\ast }{x}^{\ast },{\left(x^{\ast }\right)}^{\ast }=x$. The involution * is proper if $x^{\ast }x=0⟹x=0$ for any $x\in \mathcal{A}$, e.g., in a Rickart *-algebra, the involution is always proper. Let ${\mathbb{C}}^{n\times n}$ be the Banach algebra of all $n\times n$ complex matrices, with conjugate transpose * as the involution. Then the involution * is proper. In [21], Zou et al. extended the notion of weak group inverse from complex matrices to elements in a ring with proper involution.

Let $\mathcal{A}$ be a Banach algebra with a proper involution *. An element a in a $\mathcal{A}$ has weak group inverse if there exists $x\in \mathcal{A}$ such that

$$x=a{x}^2,{\left(a^{\ast }{a}^{2}x\right)}^{\ast }=a^{\ast }{a}^{2}x,a^n=x{a}^{n+1}$$

for som $n\in \mathbb{N}$. Such x is unique if it exists and is called the weak group inverse of a. We denote it by $a^{Ⓦ}$ (see [21,22]). A square complex matrix A has weak group inverse X if it satisfies the system of equations:

$$A{X}^2=X,AX=A^{Ⓓ}A.$$

Here, $A^{Ⓓ}$ is the core-EP inverse of A (see [11,13]). Weak group inverse was extensively studied by many authors, e.g., [8,17,20,21,22].

In [1], the authors extended weak group inverse and introduced generalized group inverse in a Banach algebra with proper involution. An element a in $\mathcal{A}$ has generalized group inverse if there exists $x\in \mathcal{A}$ such that

$$x=a{x}^2,{\left(a^{\ast }{a}^{2}x\right)}^{\ast }=a^{\ast }{a}^{2}x,\underset{n\to \infty }{lim}\left|\right|a^n-x{a}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.$$

Such x is unique if it exists and is called the generalized group inverse of a. We denote it by $a^{ⓖ}$. Many properties of generalized group inverse were presented in [1]. Mosić and Zhang introduced and studied weighted weak group inverse for a Hilbert space operator A in $\mathcal{B}\left(X\right)$ (see [17]). Furthermore, the weak group inverse was generalized to the m-weak group inverse (see [11,18,24]). Recently, Gao et al. further introduced and studied the W-weighted m-weak group inverse in [11].

The main purpose of this paper is to extend the concept of W-weighted m-weak group inverse for complex matrices to elements in a Banach *-algebra. This extension is called weighted m-generalized group inverse.

An element $a\in \mathcal{A}$ has generalized w-Drazin inverse x if there exists unique $x\in \mathcal{A}$ such that

$$awx=xwa,xwawx=x\mathrm{and}a-awxwa\in {\mathcal{A}}^{qnil}.$$

We denote x by $a^{d,w}$ (see [19]). Here, ${\mathcal{A}}^{qnil}=\{x\in \mathcal{A}\mid \underset{n\to \infty }{lim}\Vert x^n{\Vert }^{{\displaystyle \frac{1}{n}}}=0\}.$ We denote $a^{d,1}$ by $a^d$. Evidently, $a^{d,w}=x$ if and only if $x=a{\left[{\left(wa\right)}^d\right]}^2$. We introduce a new weighted generalized inverse as follows:

Definition 1.1. 

An element $a\in \mathcal{A}$ has w-weighted m-generalized group inverse if $a\in {\mathcal{A}}^{d,w}$ and there exists $x\in \mathcal{A}$ such that

$$\begin{array}{c}x=a{\left(wx\right)}^2,{\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}wx={\left[{\left(wa\right)}^d\right]}^{\ast }wa,\\ \underset{n\to \infty }{lim}{\left|\right|\left(aw\right)}^n-\left(xw\right){\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.\end{array}$$

The preceding x is called the w-weighted m-generalized group inverse of a, and denoted by $a^{{ⓖ}_m,w}$.

The w-weighted m-generalized group inverse is a natural generalization of the m-generalized group inverse which was introduced in [4]. Let $a^{{ⓖ}_m}$ be the m-generalized group inverse of a. Evidently, $a^{{ⓖ}_m}=a^{{ⓖ}_m,1}$. We list some characterizations of m-generalized group inverse.

Theorem 1.2. 

(see [4] [Theorem 2.3, Theorem 3.1 and Theorem 4.1]) Let $\mathcal{A}$ be a Banach *-algebra, and let $a\in \mathcal{A}$. Then the following are equivalent:

(1)

$a\in {\mathcal{A}}^{{ⓖ}_m}$.

(2)

There exist $x,y\in \mathcal{A}$ such that

$$a=x+y,x^{\ast }{a}^{m-1}y=yx=0,x\in {\mathcal{A}}^{\#},y\in {\mathcal{A}}^{qnil}.$$

(3)

$a\in {\mathcal{A}}^d$ and there exists $x\in \mathcal{A}$ such that

$$x=a{x}^2,{\left(a^d\right)}^{\ast }{a}^{m+1}x={\left(a^d\right)}^{\ast }{a}^m,\underset{n\to \infty }{lim}\left|\right|a^n-x{a}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.$$

(4)

$a\in {\mathcal{A}}^d$ and there exists $x\in \mathcal{A}$ such that

$$x=a{x}^2,{\left(a^d\right)}^{\ast }{a}^{m+1}x={\left(a^d\right)}^{\ast }{a}^m,\underset{n\to \infty }{lim}\left|\right|a^n-x{a}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.$$

(5)

$a\in {\mathcal{A}}^d$ and there exists an idempotent $p\in \mathcal{A}$ such that

$$a+p\in {\mathcal{A}}^{-1},{\left[{\left(a^m\right)}^{\ast }{a}^{m}p\right]}^{\ast }=a^{\ast }apandpa=pap\in {\mathcal{A}}^{qnil}.$$

(6)

$a\in {\mathcal{A}}^d$ and there exists $x\in \mathcal{A}$ such that ${\left(a^d\right)}^{\ast }{a}^{d}x={\left(a^d\right)}^{\ast }{a}^m$.

In Section 2, we investigate elementary properties of w-weighted m-generalized group inverse in a Banach *-algebra. Many new properties of the weak group inverse for a complex matrix and Hilbert space operator are thereby obtained.

Following [2], an element a in $\mathcal{A}$ has generalized w-core-EP inverse if there exist $x\in \mathcal{A}$ such that

$$a{\left(wx\right)}^2=x,{\left(wawx\right)}^{\ast }=wawx,\underset{n\to \infty }{lim}{\left|\right|\left(aw\right)}^n-\left(xw\right){\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.$$

The preceding x is unique if exists, and we denote it by $a^{ⓓ,w}$. We denote $a^{ⓓ,1}$ by $a^{ⓓ}$. Evidently, $a^{ⓓ,w}=x$ if and only if $x=a{\left[{\left(wa\right)}^{ⓓ}\right]}^2$ (see [2] [Theorem 2.1]). In Section 3, we investigate the representations of m-generalized group inverse under weighted generalized core-EP invertibility.

Recall that an element $a\in \mathcal{A}$ has Moore-Penrose inverse if there exist $x\in \mathcal{A}$ such that $axa=a,xax=x,{\left(ax\right)}^{\ast }=ax,{\left(xa\right)}^{\ast }=xa.$ The preceding x is unique if it exists, and we denote it by $a^{†}$. An element a in $\mathcal{A}$ has weak core inverse provided that $a\in {\mathcal{A}}^{Ⓦ}\bigcap {\mathcal{A}}^{†}$ (see [16,23]). In [3], the authors introduced and studied the generalized core inverse. The m-weak core inverse and weighted weak core inverse were investigated in [10,15]. Recently, Ferreyra and Mosić introduced the W-weighted m-weak core inverse for complex matrices which generalized the (weighted) core-EP inverse, the weak group inverse and m-weak core inverse (see [7]). A square complex matrix A has W-weighted m-weak core inverse X if

$$X=A^{{Ⓦ}_m,W}{\left(WA\right)}^m{\left[{\left(WA\right)}^m\right]}^{†}.$$

Here, $A^{{Ⓦ}_m,W}$ is the W-weighted m-weak group inverse of A, i.e., ${\left(WA\right)}^m$ has weak group inverse (see [20]). Let $a,w\in \mathcal{A},m\in \mathbb{N}$. Set $a\in {\mathcal{A}}^{{†}_m,w}$ if ${\left(wa\right)}^m\in {\mathcal{A}}^{†}$. We have

Definition 1.3. 

An element $a\in \mathcal{A}$ has w-weighted m-generalized core inverse if $a\in {\mathcal{A}}^{{ⓖ}_m,w}\bigcap {\mathcal{A}}^{{†}_m,w}$.

In Section 4, We present various properties, presentations of such weighted generalized group inverse combined with weighted Moore-Penrose inverse. We extend the properties of generalized core inverse in Banach *-algebra to the general case(see [3]). Many properties of the W-weighted m-weak core inverse are thereby extended to wider cases, e.g. Hilbert operators over an infinitely dimensional space.

Finally, in Section 5, we give the applications of the w-weighted m-generalized group (core) inverse in solving the matrix equations.

Throughout the paper, all Banach algebras are complex with a proper involution *. We use ${\mathcal{A}}^{†},{\mathcal{A}}^{d,w},{\mathcal{A}}^{ⓓ},{\mathcal{A}}^{ⓖ}$ and ${\mathcal{A}}^{Ⓦ}$ to denote the sets of all Moore-Penrose invertible, weighted generalized Drazin invertible, generalized core-EP invertible, generalized group invertible and weak group invertible elements in $\mathcal{A}$, respectively.

2. Weighted $\mathit{m}$-Generalized Group Inverse

In this section we introduce and establish elementary properties of weighted m-generalized group inverse which will be used in the next section. This also extend the concept of w-weighted m-weak group inverse from complex matrices to elements in a Banach algebra (see [11]). We begin with

Theorem 2.1. 

Let $a,w\in \mathcal{A}$. Then the following are equivalent:

(1)

$a\in {\mathcal{A}}^{{ⓖ}_m,w}$.

(2)

$wa\in {\mathcal{A}}^{{ⓖ}_m}$.

In this case, $a^{{ⓖ}_m,w}=a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2.$

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ By hypothesis, we can find $x\in \mathcal{A}$ such that

$$\begin{array}{c}x=a{\left(wx\right)}^2,{\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}wx={\left[{\left(wa\right)}^d\right]}^{\ast }wa,\\ \underset{n\to \infty }{lim}{\left|\right|\left(aw\right)}^{n-1}-\left(xw\right){\left(aw\right)}^n{\left|\right|}^{{\displaystyle \frac{1}{n-1}}}=0.\end{array}$$

Furthermore, we have

$$\begin{array}{cc}& {\left|\right|\left(wa\right)}^n-\left(wx\right){\left(wa\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}\hfill \\ \hfill =& {\left|\right|w\left(aw\right)}^{n-1}a-wxw{\left(aw\right)}^{n}a{\left|\right|}^{{\displaystyle \frac{1}{n}}}\hfill \\ \hfill =& \left|\right|w[{\left(aw\right)}^{n-1}-xw{\left(aw\right)}^n]a{\left|\right|}^{{\displaystyle \frac{1}{n}}}\hfill \\ \hfill \le & {\left|\right|w\left|\right|}^{{\displaystyle \frac{1}{n}}}{\left[\right||\left(aw\right)}^{n-1}-xw{\left(aw\right)}^n{\left|\right|}^{{\displaystyle \frac{1}{n-1}}}{]}^{{\displaystyle \frac{n-1}{n}}}{\left|\right|a\left|\right|}^{{\displaystyle \frac{1}{n}}}.\hfill \end{array}$$

Therefore

$$\underset{n\to \infty }{lim}{\left|\right|\left(wa\right)}^n-\left(wx\right){\left(wa\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.$$

Obviously, $wx=\left(wa\right){\left(wx\right)}^2$. Hence,

$$wa\in {\mathcal{A}}^{{ⓖ}_m}and{\left(wa\right)}^{{ⓖ}_m}=wx.$$

Accordingly,

$$x=a{\left(wx\right)}^2=a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2,$$

as desired.

$\left(2\right)\Rightarrow \left(1\right)$ Let $x=a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2.$ Then $a\in {\mathcal{A}}^{d,w}$ and we verify that

$$\begin{array}{ccc}\hfill a{\left(wx\right)}^2& =\hfill & awa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^{2}wa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2=x.\hfill \end{array}$$

One easily checks that

$$\begin{array}{ccc}\hfill {\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}wx& =\hfill & {\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}wa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2\hfill \\ & =\hfill & {\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}{\left(wa\right)}^{{ⓖ}_m}\hfill \\ & =\hfill & {\left[{\left(wa\right)}^d\right]}^{\ast }wa.\hfill \end{array}$$

Since

$$\begin{array}{ccc}\hfill \left(xw\right){\left(aw\right)}^{n+1}& =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^{2}w{\left(aw\right)}^{n+1}\hfill \\ & =\hfill & {\left(aw\right)}^n-a[{\left(wa\right)}^{n-1}-{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^n]w\hfill \\ & -\hfill & a{\left(wa\right)}^{{ⓖ}_m}[{\left(wa\right)}^n-{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^{n+1}]w,\hfill \end{array}$$

we have

$$\begin{array}{cc}& {\left|\right|\left(aw\right)}^n-\left(xw\right){\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}\hfill \\ \hfill \le & {\left|\right|a\left|\right|}^{{\displaystyle \frac{1}{n}}}{\left|\right|\left(wa\right)}^{n-1}-{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^n{\left|\right|}^{{\displaystyle \frac{1}{n}}}{\left|\right|w\left|\right|}^{{\displaystyle \frac{1}{n}}}\hfill \\ \hfill +& {\left|\right|a\left(wa\right)}^{{ⓖ}_m}{\left|\right|}^{{\displaystyle \frac{1}{n}}}{\left|\right|\left(wa\right)}^n-{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}{\left|\right|w\left|\right|}^{{\displaystyle \frac{1}{n}}}.\hfill \end{array}$$

Therefore

$$\underset{n\to \infty }{lim}{\left|\right|\left(aw\right)}^n-\left(xw\right){\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0,$$

the result follows. □

The preceding unique solution x is called the w-weighted generalized m-group inverse of a, and denote it by $a^{{ⓖ}_m,w}$. That is, $a^{{ⓖ}_m,w}=a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2.$ We use ${\mathcal{A}}^{{ⓖ}_m,w}$ to denote the set of all w-weighted generalized m-group invertible elements in $\mathcal{A}$. By the argument above, we have

Corollary 2.2. 

Let $a,w\in \mathcal{A}$. Then

(1)

$a^{{ⓖ}_m,w}=x$.

(2)

$wa\in {\mathcal{A}}^{{ⓖ}_m}$ and ${\left(wa\right)}^{{ⓖ}_m}=wx$.

Corollary 2.3. 

Let $a,w\in \mathcal{A}$. Then $a\in {\mathcal{A}}^{{ⓖ}_m,w}$ if and only if

(1)

$a\in {\mathcal{A}}^{d,w}$;

(2)

There exists $x\in \mathcal{A}$ such that

$$\begin{array}{c}x=a{\left[wx\right]}^2,{\left[{\left(wa\right)}^{\ast }{\left(wa\right)}^{m+1}wx\right]}^{\ast }={\left(wa\right)}^{\ast }{\left(wa\right)}^{m+1}wx,\\ \underset{n\to \infty }{lim}{\left|\right|\left(aw\right)}^n-\left(xw\right){\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.\end{array}$$

Proof. 

⟹ Obviously, $a\in {\mathcal{A}}^{d,w}$. By hypothesis, there exists $x\in \mathcal{A}$ such that

$$\begin{array}{c}x=a{\left[wx\right]}^2,{\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}wx={\left[{\left(wa\right)}^d\right]}^{\ast }wa,\\ \underset{n\to \infty }{lim}{\left|\right|\left(aw\right)}^n-\left(xw\right){\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.\end{array}$$

In this case, $x=a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2.$ Then

$$\begin{array}{ccc}\hfill {\left(wa\right)}^{\ast }{\left(wa\right)}^{m+1}wx& =\hfill & {\left(wa\right)}^{\ast }{\left(wa\right)}^{m+1}wa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2\hfill \\ & =\hfill & {\left(wa\right)}^{\ast }{\left(wa\right)}^{m+1}{\left(wa\right)}^{{ⓖ}_m},\hfill \\ \hfill {\left({\left(wa\right)}^{\ast }{\left(wa\right)}^{m+1}wx\right)}^{\ast }& =\hfill & {\left(wa\right)}^{\ast }{\left(wa\right)}^{m+1}wx.\hfill \end{array}$$

⟸ By hypothesis, there exists $x\in \mathcal{A}$ such that

$$\begin{array}{c}x=a{\left[wx\right]}^2,{\left[{\left(wa\right)}^{\ast }{\left(wa\right)}^{m+1}wx\right]}^{\ast }={\left(wa\right)}^{\ast }{\left(wa\right)}^{m+1}wx,\\ \underset{n\to \infty }{lim}{\left|\right|\left(aw\right)}^n-\left(xw\right){\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.\end{array}$$

Clearly, $wx=\left(wa\right){\left[wx\right]}^2$. Observing that

$$\begin{array}{ccc}\hfill {\left|\right|\left(wa\right)}^{n+1}-\left(wx\right){\left(wa\right)}^{n+2}\left|\right|& =\hfill & {\left|\right|w\left(aw\right)}^n{-(w\left(xw\right)\left(aw\right)}^{n+1}a\left|\right|\hfill \\ & \le \hfill & {\left|\right|w\left|\right|\left|\right|\left(aw\right)}^n-\left(xw\right){\left(aw\right)}^{n+1}\left|\right|\left|\right|a\left|\right|,\hfill \end{array}$$

we see that

$$\underset{n\to \infty }{lim}{\left|\right|\left(wa\right)}^n-\left(wx\right){\left(wa\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.$$

This implies that $wa\in {\mathcal{A}}^{{ⓖ}_m}$. According to Theorem 2.1, $a\in {\mathcal{A}}^{{ⓖ}_m,w},$ as asserted. □

Theorem 2.4. 

Let $a,w\in \mathcal{A}$. Then $a\in {\mathcal{A}}^{{ⓖ}_m,w}$ if and only if

(1)

$a\in {\mathcal{A}}^{d,w}$;

(2)

There exists $x\in \mathcal{A}$ such that

$$\begin{array}{c}x=a{\left[wx\right]}^2,{\left[{\left({\left(wa\right)}^m\right)}^{\ast }{\left(wa\right)}^{m+1}wx\right]}^{\ast }={\left({\left(wa\right)}^m\right)}^{\ast }{\left(wa\right)}^{m+1}wx,\\ \underset{n\to \infty }{lim}{\left|\right|\left(aw\right)}^n-\left(xw\right){\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.\end{array}$$

Proof. 

⟹ Clearly, $a\in {\mathcal{A}}^{d,w}$. In view of Theorem 2.1, $wa\in {\mathcal{A}}^{{ⓖ}_m}$. According to Theorem 1.2, There exists $z\in \mathcal{A}$ such that

$$\begin{array}{c}z=\left(wa\right)z^2,{\left[{\left({\left(wa\right)}^m\right)}^{\ast }{\left(wa\right)}^{m+1}z\right]}^{\ast }={\left({\left(wa\right)}^m\right)}^{\ast }{\left(wa\right)}^{m+1}z,\\ \underset{n\to \infty }{lim}{\left|\right|\left(wa\right)}^n-z{\left(wa\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.\end{array}$$

Here, $z={\left(wa\right)}^{{ⓖ}_m}=wa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2$. Set $x=a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2$. Then

$$\begin{array}{c}{\left[{\left({\left(wa\right)}^m\right)}^{\ast }{\left(wa\right)}^{m+1}wz\right]}^{\ast }={\left({\left(wa\right)}^m\right)}^{\ast }{\left(wa\right)}^{m+1}wz,\\ \underset{n\to \infty }{lim}{\left|\right|\left(aw\right)}^n-\left(zw\right){\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.\end{array}$$

Moreover, we have

$$wx=wa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2={\left(wa\right)}^{{ⓖ}_m},$$

and then

$$a{\left(wx\right)}^2=a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2=x.$$

In this case, $a^{{ⓖ}_m,w}=x,$ as desired.

⟸ By hypothesis, there exists $x\in \mathcal{A}$ such that

$$\begin{array}{c}x=a{\left(wx\right)}^2,{\left[{\left({\left(wa\right)}^m\right)}^{\ast }{\left(wa\right)}^{m+1}wx\right]}^{\ast }={\left({\left(wa\right)}^m\right)}^{\ast }{\left(wa\right)}^{m+1}wx,\\ \underset{n\to \infty }{lim}{\left|\right|\left(aw\right)}^n-\left(xw\right){\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.\end{array}$$

Then $wx=wa{\left(wx\right)}^2$. In view of Theorem 1.2, $wa\in {\mathcal{A}}^{{ⓖ}_m}$. According to Theorem 2.1, $a\in {\mathcal{A}}^{{ⓖ}_m,w}$, as asserted. □

Corollary 2.5. 

Let $a,w\in \mathcal{A}$. Then $a\in {\mathcal{A}}^{Ⓦ,w}$ if and only if

(1)

$a\in {\mathcal{A}}^{D,w}$;

(2)

There exists $x\in \mathcal{A}$ such that

$$\begin{array}{c}x=a{\left[wx\right]}^2,{\left[{\left(wa\right)}^{\ast }{\left(wa\right)}^{2}wx\right]}^{\ast }={\left(wa\right)}^{\ast }{\left(wa\right)}^{2}wx,\\ \underset{n\to \infty }{lim}{\left|\right|\left(aw\right)}^n-\left(xw\right){\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0.\end{array}$$

Proof. 

This is obvious by Theorem 2.4. □

Set $im\left(x\right)=\{xr\mid r\in \mathcal{A}\}$. We are ready to prove:

Theorem 2.6. 

Let $a,w\in \mathcal{A}$. Then the following are equivalent:

(1)

$a^{{ⓖ}_m,w}=x$.

(2)

$awx=a{\left(wa\right)}^{{ⓖ}_m},a{\left(wx\right)}^2=x.$

(3)

$wawx=wa{\left(wa\right)}^{{ⓖ}_m},im\left(x\right)\subseteq im{\left(aw\right)}^d.$

(4)

$awx=a{\left(wa\right)}^{{ⓖ}_m},im\left(x\right)\subseteq im{\left(aw\right)}^d.$

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ In view of Theorem 2.1, $x=a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2.$ Then $a{\left(wx\right)}^2=x$ and

$$\begin{array}{ccc}\hfill awx& =\hfill & \left(aw\right)a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2\hfill \\ & =\hfill & a\left(wa\right){\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2\hfill \\ & =\hfill & a{\left(wa\right)}^{{ⓖ}_m}.\hfill \end{array}$$

$\left(2\right)\Rightarrow \left(3\right)$ Obviously, $wawx=w\left(awx\right)=w\left[a{\left(wa\right)}^{{ⓖ}_m}\right]=wa{\left(wa\right)}^{{ⓖ}_m}.$ Moreover, we have

$$\begin{array}{ccc}\hfill x& =\hfill & a{\left(wx\right)}^2=\left(awx\right)wx=a{\left(wa\right)}^{{ⓖ}_m}wx\hfill \\ & =\hfill & a{\left(wa\right)}^d\left(wa\right){\left(wa\right)}^{{ⓖ}_m}wx\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^d\right]}^{2}w\left(awa\right){\left(wa\right)}^{{ⓖ}_m}wx\hfill \\ & =\hfill & {\left(aw\right)}^d\left(awa\right){\left(wa\right)}^{{ⓖ}_m}wx.\hfill \end{array}$$

Therefore $im\left(x\right)\subseteq im{\left(aw\right)}^d,$ as desired.

$\left(3\right)\Rightarrow \left(4\right)$ Since $im\left(x\right)\subseteq im{\left(aw\right)}^d,$ we see that

$$\begin{array}{ccc}\hfill awx& =\hfill & aw\left[\left(aw\right){\left(aw\right)}^{d}x\right]={\left(aw\right)}^{d}a\left[wawx\right]\hfill \\ & =\hfill & {\left(aw\right)}^{d}a\left[wa{\left(wa\right)}^{{ⓖ}_m}\right]\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^d\right]}^2{\left(wa\right)}^2{\left(wa\right)}^{{ⓖ}_m}\hfill \\ & =\hfill & {awa)\left(wa\right)}^d{\left(wa\right)}^{{ⓖ}_m}\hfill \\ & =\hfill & a{\left(wa\right)}^{{ⓖ}_m},\hfill \end{array}$$

as desired.

$\left(4\right)\Rightarrow \left(1\right)$ Write $x={\left(aw\right)}^{d}z$ for some $z\in R$. Then

$$\begin{array}{ccc}\hfill x& =\hfill & aw{\left(aw\right)}^{d}x={\left(aw\right)}^d\left(awx\right)\hfill \\ & =\hfill & {\left(aw\right)}^d\left[a{\left(wa\right)}^{{ⓖ}_m}\right]\hfill \\ & =\hfill & {\left(aw\right)}^d\left(aw\right)a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2.\hfill \end{array}$$

This completes the proof by Theorem 2.1. □

Corollary 2.7. 

Let $a\in \mathcal{A}$. Then the following are equivalent:

(1)

$a^{{ⓖ}_m}=x$.

(2)

$ax=a{a}^{{ⓖ}_m},a{x}^2=x.$

(3)

$ax=a{a}^{{ⓖ}_m},im\left(x\right)\subseteq im\left(a^d\right).$

Proof. 

This is a direct consequence of Theorem 2.6. □

We are ready to prove:

Theorem 2.8. 

Let $a\in {\mathcal{A}}^{{ⓖ}_m,w}$. Then $waw{a}^{{ⓖ}_{m+1},w}=w{a}^{{ⓖ}_m,w}wa$.

Proof. 

In view of Theorem 2.1, we see that

$$\begin{array}{ccc}\hfill waw{a}^{{ⓖ}_{m+1},w}& =\hfill & wawa{\left[{\left(wa\right)}^{{ⓖ}_{m+1}}\right]}^2\hfill \\ & =\hfill & wa{\left(wa\right)}^{{ⓖ}_{m+1}}\hfill \\ \hfill w{a}^{{ⓖ}_m,w}wa& =\hfill & wa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^{2}wa\hfill \\ & =\hfill & {\left(wa\right)}^{{ⓖ}_m}wa.\hfill \end{array}$$

In view of [4] [Corollary 2.4], we have

$${\left(wa\right)}^{{ⓖ}_{m+1}}={\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^{2}wa.$$

Therefore

$$\begin{array}{ccc}\hfill waw{a}^{{ⓖ}_{m+1},w}& =\hfill & wa{\left(wa\right)}^{{ⓖ}_{m+1}}\hfill \\ & =\hfill & wa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^{2}wa\hfill \\ & =\hfill & {\left(wa\right)}^{{ⓖ}_m}wa\hfill \\ & =\hfill & w{a}^{{ⓖ}_m,w}wa.\hfill \end{array}$$

This completes the proof. □

Corollary 2.9. 

Let $a\in {\mathcal{A}}^{{ⓖ}_m}$. Then $a{a}^{{ⓖ}_{m+1}}=a^{{ⓖ}_m}a$.

Proof. 

This is obvious by choosing $w=1$ in Theorem 2.8. □

3. Representations of $\mathit{m}$-Generalized Group Inverse

In this section, we present the representations of m-generalized group inverse under weighted generalized core-EP invertibility.

Theorem 3.1. 

Let $a\in {\mathcal{A}}^{ⓓ,w}$. Then $a\in {\mathcal{A}}^{{ⓖ}_m,w}$ and

$$a^{{ⓖ}_m,w}={\left[a^{ⓓ,w}w\right]}^{m+1}{\left(aw\right)}^{m-1}a.$$

Proof. 

In view of [2] [Theorem 2.1], $a^{ⓓ,w}=a{\left[{\left(wa\right)}^{ⓓ}\right]}^2;$ hence, $w{a}^{ⓓ,w}={\left(wa\right)}^{ⓓ}.$ Then we easily check that

$$\begin{array}{ccc}\hfill {\left[a^{ⓓ,w}w\right]}^{m+1}{\left(aw\right)}^{m-1}a& =\hfill & a^{ⓓ,w}{\left[w{a}^{ⓓ,w}\right]}^{m}w{\left(aw\right)}^{m-1}a\hfill \\ & =\hfill & a^{ⓓ,w}{\left[{\left(wa\right)}^{ⓓ}\right]}^m{\left(wa\right)}^m\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{ⓓ}\right]}^2{\left[{\left(wa\right)}^{ⓓ}\right]}^m{\left(wa\right)}^m\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{ⓓ}\right]}^{m+2}{\left(wa\right)}^m.\hfill \end{array}$$

Thus,

$$w{\left[a^{ⓓ,w}w\right]}^{m+1}{\left(aw\right)}^{m-1}a=wa{\left[{\left(wa\right)}^{ⓓ}\right]}^{m+2}{\left(wa\right)}^m={\left[{\left(wa\right)}^{ⓓ}\right]}^{m+1}{\left(wa\right)}^m.$$

Set $x={\left[{\left(wa\right)}^{ⓓ}\right]}^{m+1}{\left(wa\right)}^m$. Then

$$\begin{array}{ccc}\hfill \left(wa\right)x^2& =\hfill & \left(wa\right){\left[{\left(wa\right)}^{ⓓ}\right]}^{m+1}{\left(wa\right)}^m{\left[{\left(wa\right)}^{ⓓ}\right]}^{m+1}{\left(wa\right)}^m\hfill \\ & =\hfill & \left(wa\right){\left[{\left(wa\right)}^{ⓓ}\right]}^{m+1}{\left(wa\right)}^{ⓓ}{\left(wa\right)}^m\hfill \\ & =\hfill & {\left[{\left(wa\right)}^{ⓓ}\right]}^{m+1}{\left(wa\right)}^m\hfill \\ & =\hfill & x,\hfill \\ \hfill {\left({\left(wa\right)}^d\right)}^{\ast }{\left(wa\right)}^{m+1}x& =\hfill & {\left({\left(wa\right)}^d\right)}^{\ast }{\left(wa\right)}^{m+1}{\left[{\left(wa\right)}^{ⓓ}\right]}^{m+1}{\left(wa\right)}^m\hfill \\ & =\hfill & {\left({\left(wa\right)}^d\right)}^{\ast }\left(wa\right){\left(wa\right)}^{ⓓ}{\left(wa\right)}^m\hfill \\ & =\hfill & {\left({\left(wa\right)}^d\right)}^{\ast }{\left[\left(wa\right){\left(wa\right)}^{ⓓ}\right]}^{\ast }{\left(wa\right)}^m\hfill \\ & =\hfill & {\left[\left(wa\right){\left({\left(wa\right)}^{ⓓ}\right)}^2\right]}^{\ast }{\left(wa\right)}^m\hfill \\ & =\hfill & {\left({\left(wa\right)}^d\right)}^{\ast }{\left(wa\right)}^m,\hfill \\ \hfill \underset{n\to \infty }{lim}{\left|\right|\left(wa\right)}^n-x{\left(wa\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}& =\hfill & 0.\hfill \end{array}$$

This implies that

$${\left(wa\right)}^{{ⓖ}_m}={\left[{\left(wa\right)}^{ⓓ}\right]}^{m+1}{\left(wa\right)}^m.$$

According to Theorem 2.1, we prove that $a\in {\mathcal{A}}^{{ⓖ}_m,w}$ and

$$\begin{array}{ccc}\hfill a^{{ⓖ}_m,w}& =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2\hfill \\ & =\hfill & a{\left[{\left({\left(wa\right)}^{ⓓ}\right)}^{m+1}{\left(wa\right)}^m\right]}^2\hfill \\ & =\hfill & a\left[{\left({\left(wa\right)}^{ⓓ}\right)}^{m+1}{\left(wa\right)}^m\right]\left[{\left({\left(wa\right)}^{ⓓ}\right)}^{m+1}{\left(wa\right)}^m\right]\hfill \\ & =\hfill & a{\left({\left(wa\right)}^{ⓓ}\right)}^{m+1}{\left(wa\right)}^{ⓓ}{\left(wa\right)}^m\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{ⓓ}\right]}^{m+2}{\left(wa\right)}^m\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^{m+1}{\left(aw\right)}^{m-1}a,\hfill \end{array}$$

as required. □

Corollary 3.2. 

Let $a\in {\mathcal{A}}^{ⓓ}$. Then $a\in {\mathcal{A}}^{{ⓖ}_m}$ and

$$a^{{ⓖ}_m}={\left(a^{ⓓ}\right)}^{m+1}{a}^m.$$

Proof. 

This is obvious by choosing $w=1$ in Theorem 3.1. □

We call x is the $(1,3)$-inverse of a if x satisfies the equations $axa=a$ and ${\left(ax\right)}^{\ast }=ax$. We use ${\mathcal{A}}^{(1,3)}$ to denote the set of all $(1,3)$-invertible elements in $\mathcal{A}$. Let $a\in {\mathcal{A}}^{ⓓ,w}$ and $a{\left(wa\right)}^{ⓓ}w\in {\mathcal{A}}^{(1,3)}$. By using [2] [Theorem 2.5], $aw,wa\in {\mathcal{A}}^{ⓓ}$. Let $p=\left(aw\right){\left(aw\right)}^{ⓓ},q=\left(wa\right){\left(wa\right)}^{ⓓ}$. Then $p,q\in \mathcal{A}$ are projections.

Lemma 3.3. 

Let $a\in {\mathcal{A}}^{ⓓ,w}$ and $a{\left(wa\right)}^{ⓓ}w\in {\mathcal{A}}^{(1,3)}$. Then

$$a={\left(\begin{array}{cc}{a}_1& a_2\\ 0& a_3\end{array}\right)}_{p,q},w={\left(\begin{array}{cc}{w}_1& w_2\\ 0& w_3\end{array}\right)}_{q,p},$$

where $a_1\in {\left[p\mathcal{A}q\right]}^{-1},w_1\in {\left[q\mathcal{A}p\right]}^{-1}$ and $a_3{w}_3$ and $w_3{a}_3$ are quasinilpotent.

Proof. 

We easily verify that

$$\begin{array}{ccc}\hfill (1-p)aq& =\hfill & [1-\left(aw\right){\left(aw\right)}^{ⓓ}]a\left(wa\right){\left(wa\right)}^{ⓓ}\hfill \\ & =\hfill & [1-\left(aw\right){\left(aw\right)}^{ⓓ}]awa{\left(wa\right)}^n{\left[{\left(wa\right)}^{ⓓ}\right]}^{n+1}\hfill \\ & =\hfill & [1-\left(aw\right){\left(aw\right)}^{ⓓ}]{\left(aw\right)}^{n+1}a{\left[{\left(wa\right)}^{ⓓ}\right]}^{n+1}\hfill \\ & =\hfill & aw[{\left(aw\right)}^n-{\left(aw\right)}^{ⓓ}{\left(aw\right)}^{n+1}]a{\left[{\left(wa\right)}^{ⓓ}\right]}^{n+1}.\hfill \end{array}$$

Then

$${\left|\right|(1-p)aq\left|\right|}^{{\displaystyle \frac{1}{n}}}\le {\left|\right|aw\left|\right|}^{{\displaystyle \frac{1}{n}}}{\left|\right|\left(aw\right)}^n-{\left(aw\right)}^{ⓓ}{\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}\left|\right|a{\left[{\left(wa\right)}^{ⓓ}\right]}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}.$$

Since $\underset{n\to \infty }{lim}{\left|\right|\left(aw\right)}^n-{\left(aw\right)}^{ⓓ}{\left(aw\right)}^{n+1}{\left|\right|}^{{\displaystyle \frac{1}{n}}}=0$, we see that $\underset{n\to \infty }{lim}{\left|\right|(1-p)aq\left|\right|}^{{\displaystyle \frac{1}{n}}}=0$. This implies that $(1-p)aq=0$. Likewise, we prove that

$$(1-q)wp=[1-\left(wa\right){\left(wa\right)}^{ⓓ}]w\left(aw\right){\left(aw\right)}^{ⓓ}=0.$$

Moreover, we have

$$\begin{array}{cc}& \left[\left(aw\right){\left(aw\right)}^{ⓓ}a\left(wa\right){\left(wa\right)}^{ⓓ}\right]\left[\left(wa\right){\left(wa\right)}^{ⓓ}w{\left(aw\right)}^{ⓓ}\left(aw\right){\left(aw\right)}^{ⓓ}\right]\hfill \\ \hfill =& \left(aw\right){\left(aw\right)}^{ⓓ}\left(aw\right)a{\left(wa\right)}^{ⓓ}w{\left(aw\right)}^{ⓓ}\hfill \\ \hfill =& \left(aw\right)a{\left(wa\right)}^{ⓓ}w{\left(aw\right)}^{ⓓ}\hfill \\ \hfill =& \left(aw\right){\left(aw\right)}^{ⓓ},\hfill \\ & \left[\left(wa\right){\left(wa\right)}^{ⓓ}w{\left(aw\right)}^{ⓓ}\left(aw\right){\left(aw\right)}^{ⓓ}\right]\left[\left(aw\right){\left(aw\right)}^{ⓓ}a\left(wa\right){\left(wa\right)}^{ⓓ}\right]\hfill \\ \hfill =& \left(wa\right){\left(wa\right)}^{ⓓ}w{\left(aw\right)}^{ⓓ}a\left(wa\right){\left(wa\right)}^{ⓓ}\hfill \\ \hfill =& \left(wa\right){\left(wa\right)}^{ⓓ}wa{\left(wa\right)}^{ⓓ}\hfill \\ \hfill =& \left(wa\right){\left(wa\right)}^{ⓓ}.\hfill \end{array}$$

Then $a_1=paq\in {\left[p\mathcal{A}q\right]}^{-1}$. Similarly, $w_1=qwp\in {\left[q\mathcal{A}p\right]}^{-1}$.

Also we easily see that

$$\begin{array}{ccc}\hfill a_3{w}_3& =\hfill & [1-\left(aw\right){\left(aw\right)}^{ⓓ}]a[1-wa{\left(wa\right)}^{ⓓ}]w[1-\left(aw\right){\left(aw\right)}^{ⓓ}]\hfill \\ & \in \hfill & {\mathcal{A}}^{qnil}.\hfill \end{array}$$

Thus, $a_3{w}_3$ is quasinilpotent. By using Cline’s formula, $w_3{a}_3$ is quasinilpotent. This completes the proof. □

Lemma 3.4. 

Let $a\in {\mathcal{A}}^{ⓓ,w}$ and $a{\left(wa\right)}^{ⓓ}w\in {\mathcal{A}}^{(1,3)}$. Then

$$a^{ⓓ,w}={\left(\begin{array}{cc}{\left(w_1{a}_1{w}_1\right)}^{-1}& 0\\ 0& 0\end{array}\right)}_{p,q}.$$

Proof. 

In view of [2] [Theorem 2.1], $a^{ⓓ,w}=a{\left[{\left(wa\right)}^{ⓓ}\right]}^2.$ One easily checks that

$$\begin{array}{ccc}\hfill p{a}^{ⓓ,w}(1-q)& =\hfill & \left(aw\right){\left(aw\right)}^{ⓓ}a{\left[{\left(wa\right)}^{ⓓ}\right]}^2[1-\left(wa\right){\left(wa\right)}^{ⓓ}]\hfill \\ & =\hfill & \left(aw\right){\left(aw\right)}^{ⓓ}a{\left(wa\right)}^{ⓓ}[{\left(wa\right)}^{ⓓ}-{\left(wa\right)}^{ⓓ}\left(wa\right){\left(wa\right)}^{ⓓ}]=0,\hfill \\ \hfill (1-p)a^{ⓓ,w}q& =\hfill & [1-\left(aw\right){\left(aw\right)}^{ⓓ}]a{\left[{\left(wa\right)}^{ⓓ}\right]}^2\left(wa\right){\left(wa\right)}^{ⓓ}\hfill \\ & =\hfill & [1-\left(aw\right){\left(aw\right)}^{ⓓ}]awa{\left[{\left(wa\right)}^{ⓓ}\right]}^3\left(wa\right){\left(wa\right)}^{ⓓ}\hfill \\ & =\hfill & aw[1-{\left(aw\right)}^{ⓓ}aw]a{\left[{\left(wa\right)}^{ⓓ}\right]}^3\left(wa\right){\left(wa\right)}^{ⓓ}=0,\hfill \\ \hfill (1-p)a^{ⓓ,w}(1-q)& =\hfill & [1-\left(aw\right){\left(aw\right)}^{ⓓ}]a{\left[{\left(wa\right)}^{ⓓ}\right]}^2[1-\left(wa\right){\left(wa\right)}^{ⓓ}]=0.\hfill \end{array}$$

Moreover, we see that

$$\begin{array}{ccc}\hfill p{a}^{ⓓ,w}q& =\hfill & \left(aw\right){\left(aw\right)}^{ⓓ}a{\left[{\left(wa\right)}^{ⓓ}\right]}^2\left(wa\right){\left(wa\right)}^{ⓓ}\hfill \\ & =\hfill & \left(aw\right){\left(aw\right)}^{ⓓ}a{\left[{\left(wa\right)}^{ⓓ}\right]}^2\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{ⓓ}\right]}^2\hfill \\ & =\hfill & w_1{a}_1{w}_1\in {\left(p\mathcal{A}q\right)}^{-1},\hfill \end{array}$$

thus yielding the result. □

Theorem 3.5. 

Let $a\in {\mathcal{A}}^{ⓓ,w}$ and $a{\left(wa\right)}^{ⓓ}w\in {\mathcal{A}}^{(1,3)}$. Then

$$a^{{ⓖ}_m,w}={\left(\begin{array}{cc}\alpha & \beta \\ 0& 0\end{array}\right)}_{p,q},$$

where

$$\begin{array}{ccc}\hfill \alpha & =\hfill & {\left(w_1{a}_1{w}_1\right)}^{-1},\hfill \\ \hfill \beta & =\hfill & {\left(w_1{a}_1{w}_1\right)}^{-1}{a}_2+{\left[{\left(w_1{a}_1{w}_1\right)}^{-1}{w}_1\right]}^{m+1}{c}_{m-1}{a}_3+b_{m+1}{\left(a_3{w}_3\right)}^{m-1}{a}_3;\hfill \\ \hfill b_1& =\hfill & {\left(w_1{a}_1{w}_1\right)}^{-1}{w}_2,b_{n+1}={\left(w_1{a}_1{w}_1\right)}^{-1}{w}_1{b}_n,\hfill \\ \hfill c_1& =\hfill & a_1{w}_2+a_2{w}_3,c_{n+1}=a_1{w}_1{c}_n+(a_1{w}_2+a_2{w}_3){\left(a_3{w}_3\right)}^m.\hfill \end{array}$$

Proof. 

Construct two series $\left\{b_n\right\}$ and $\left\{c_n\right\}$ by the equalities: Here,

$$\begin{array}{c}{b}_1={\left(w_1{a}_1{w}_1\right)}^{-1}{w}_2,b_{n+1}={\left(w_1{a}_1{w}_1\right)}^{-1}{w}_1{b}_n,\\ c_1=a_1{w}_2+a_2{w}_3,c_{n+1}=a_1{w}_1{c}_n+(a_1{w}_2+a_2{w}_3){\left(a_3{w}_3\right)}^m.\end{array}$$

Then we compute that

$$\begin{array}{c}{\left[\left(\begin{array}{cc}{\left(w_1{a}_1{w}_1\right)}^{-1}& 0\\ 0& 0\end{array}\right)\left(\begin{array}{cc}{w}_1& w_2\\ 0& w_3\end{array}\right)\right]}^{m+1}=\left(\begin{array}{cc}{\left[{\left(w_1{a}_1{w}_1\right)}^{-1}{w}_1\right]}^{m+1}& b_{m+1}\\ 0& 0\end{array}\right),\\ {\left[\left(\begin{array}{cc}{a}_1& a_2\\ 0& a_3\end{array}\right)\left(\begin{array}{cc}{w}_1& w_2\\ 0& w_3\end{array}\right)\right]}^{m-1}=\left(\begin{array}{cc}{\left(a_1{w}_1\right)}^{m-1}& c_{m-1}\\ 0& {\left(a_3{w}_3\right)}^{m-1}\end{array}\right).\end{array}$$

According to Theorem 3.1 and Lemma 3.4, we derive

$$\begin{array}{cc}& a^{{ⓖ}_m,w}={\left[a^{ⓓ,w}w\right]}^{m+1}{\left(aw\right)}^{m-1}a\hfill \\ \hfill =& \left(\begin{array}{cc}{\left[{\left(w_1{a}_1{w}_1\right)}^{-1}{w}_1\right]}^{m+1}& b_{m+1}\\ 0& 0\end{array}\right)\left(\begin{array}{cc}{\left(a_1{w}_1\right)}^{m-1}& c_{m-1}\\ 0& {\left(a_3{w}_3\right)}^{m-1}\end{array}\right)\left(\begin{array}{cc}{a}_1& a_2\\ 0& a_3\end{array}\right)\hfill \\ \hfill =& \left(\begin{array}{cc}\alpha & \beta \\ 0& 0\end{array}\right),\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill \alpha & =\hfill & {\left(w_1{a}_1{w}_1\right)}^{-1},\hfill \\ \hfill \beta & =\hfill & {\left(w_1{a}_1{w}_1\right)}^{-1}{a}_2+{\left[{\left(w_1{a}_1{w}_1\right)}^{-1}{w}_1\right]}^{m+1}{c}_{m-1}{a}_3+b_{m+1}{\left(a_3{w}_3\right)}^{m-1}{a}_3.\hfill \end{array}$$

This completes the proof. □

Corollary 3.6. 

Let $a\in {\mathcal{A}}^{ⓓ}$. Then

$$a^{{ⓖ}_m}={\left(\begin{array}{cc}{a}_1^{-1}& {\left(a_1\right)}^{-(m+1)}{b}_m\\ 0& 0\end{array}\right)}_{s,t},$$

where $b_1=a_2,b_{m+1}=a_1{b}_m+a_2{a}_3^m,s=a{a}^{ⓓ}$ and $t=a^{ⓓ}a$.

Proof. 

This is immediate by choosing $w=1$ in Theorem 3.5. □

4. Weighted $\mathit{m}$-Generalized Core Inverse

The aim of this section is to investigate weighted m-generalized group inverse with weighted Moore-Penrose inverse. We introduce and study weighted m-generalized core inverse in a Banach *-algebra. Let $p_{{\left(wa\right)}^m}={\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}$ be the projection on ${\left(wa\right)}^m$. The following theorem is crucial.

Theorem 4.1. 

Let $a\in {\mathcal{A}}^{{ⓒ}_m,w}$. Then there exists a unique $x\in \mathcal{A}$ such that

$$xwawx=x,awx=aw{a}^{{ⓖ}_m,w}{p}_{{\left(wa\right)}^m},x{\left(wa\right)}^m=a^{{ⓖ}_m,w}{\left(wa\right)}^m.$$

Proof. 

Taking $x=a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}$. Then

$$\begin{array}{ccc}\hfill xwawx& =\hfill & a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}waw{a}^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a^{{ⓖ}_m,w}waw{a}^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & x,\hfill \\ \hfill awx& =\hfill & aw{a}^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}=awa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & aw{a}^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†},\hfill \\ \hfill x{\left(wa\right)}^m& =\hfill & a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}{\left(wa\right)}^m=a^{{ⓖ}_m,w}{\left(wa\right)}^m.\hfill \end{array}$$

Suppose that $x^{\prime }$ satisfies the preceding equations. Then one checks that

$$\begin{array}{ccc}\hfill x^{\prime }& =\hfill & x^{\prime }waw{x}^{\prime }=a^{{ⓖ}_m,w}waw{x}^{\prime }\hfill \\ & =\hfill & a^{{ⓖ}_m,w}waw{a}^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a^{{ⓖ}_m,w}wawa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^{2}wa{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & x,\hfill \end{array}$$

as required. □

We denote the preceding unique x by $a^{{ⓒ}_m,w}$.

Corollary 4.2. 

Let $a\in {\mathcal{A}}^{{ⓒ}_m,w}(m\ge 2)$. Then the following are equivalent:

(1)

$a^{{ⓒ}_m,w}=x$.

(2)

The equation system

$$awx=a{\left(wa\right)}^{{ⓖ}_m}{p}_{{\left(wa\right)}^m},a{\left(wx\right)}^2=x$$

is consistent and its unique solution $x=a^{{ⓒ}_m,w}$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ In view of Theorem 4.1, we have

$$\begin{array}{ccc}\hfill awx& =\hfill & aw{a}^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & awa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}.\hfill \end{array}$$

Moreover, we have

$$\begin{array}{ccc}\hfill a{\left(wx\right)}^2& =\hfill & aw{a}^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}w{a}^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & awa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}wa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & awa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2\left(wa\right){\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & x.\hfill \end{array}$$

$\left(2\right)\Rightarrow \left(1\right)$ Suppose that the equation system

$$awx=a{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†},a{\left(wx\right)}^2=x$$

is consistent. In view of [4] [Corollary 2.4], we have

$$\begin{array}{ccc}\hfill x& =\hfill & \left(awx\right)wx=\left(a{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\right)wx\hfill \\ & =\hfill & a{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}{)wa\left(wx\right)}^2\hfill \\ & =\hfill & a{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}{)\left(wa\right)}^m{\left(wx\right)}^{m+1}\hfill \\ & =\hfill & a{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left(wx\right)}^{m+1}\hfill \\ & =\hfill & a{\left(wa\right)}^{{ⓖ}_m}wx\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_{m-1}}\right]}^{2}w\left(awx\right)\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_{m-1}}\right]}^{2}w\left[a{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\right]\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_{m-1}}\right]}^2\left(wa\right)\left[{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\right]\hfill \\ & =\hfill & a{\left(wa\right)}^{{ⓖ}_m}\left[{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left({\left(wa\right)}^m\right)}^{†}\right]\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a^{{ⓒ}_m,w},\hfill \end{array}$$

as asserted. □

Let $a\in {\mathcal{A}}^{{ⓒ}_m,w}$. In view of Theorem 4.1, $a^{{ⓒ}_m,w}=a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}$. Set $c=a{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m$. We now establish necessary and sufficient conditions under which a has weighted m-generalized core inverse.

Theorem 4.3. 

Let $a\in {\mathcal{A}}^{{ⓒ}_m,w}$. The following are equivalent:

(1)

$a^{{ⓒ}_m,w}=x$.

(2)

$awx=c{\left[{\left(wa\right)}^m\right]}^{†}$ and $x\mathcal{A}\subseteq a^{d,w}\mathcal{A}$.

(3)

$awx=c{\left[{\left(wa\right)}^m\right]}^{†}$ and $a{\left(wx\right)}^2=x$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ In view of Theorem 4.1, we have

$$\begin{array}{ccc}\hfill awx& =\hfill & aw{a}^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & c{\left[{\left(wa\right)}^m\right]}^{†}.\hfill \end{array}$$

By virtue of Theorem 2.1, we have

$$\begin{array}{ccc}\hfill x\mathcal{A}& =\hfill & a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\mathcal{A}\hfill \\ & \subseteq \hfill & a^{{ⓖ}_m,w}\mathcal{A}\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2\mathcal{A}\hfill \\ & \subseteq \hfill & a{\left(wa\right)}^d\mathcal{A}\hfill \\ & \subseteq \hfill & a{\left[{\left(wa\right)}^d\right]}^2\mathcal{A}\hfill \\ & \subseteq \hfill & a^{d,w}\mathcal{A}.\hfill \end{array}$$

$\left(2\right)\Rightarrow \left(1\right)$ Since $awx=c{\left[{\left(wa\right)}^m\right]}^{†}$, we have $awx=a{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}=awa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}=aw{a}^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}$.

Since $x\mathcal{A}\subseteq a^{d,w}\mathcal{A}$, we derive that $a^{d,w}waw{a}^{d,w}=a^{d,w}$. Hence, $a^{d,w}wawx=x$. In view of Theorem 2.1, $a{\left(wa\right)}^{{ⓖ}_m}\subseteq a{\left[{\left(wa\right)}^d\right]}^{2}w\mathcal{A}={\left(aw\right)}^d\mathcal{A}$. Then ${\left(aw\right)}^{d}awa{\left(wa\right)}^{{ⓖ}_m}=a{\left(wa\right)}^{{ⓖ}_m}$. We deduce that

$$\begin{array}{ccc}\hfill x& =\hfill & a^{d,w}w\left(awx\right)=a{\left[{\left(wa\right)}^d\right]}^{2}w\left(awx\right)\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^d\right]}^{2}waw{a}^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & {\left(aw\right)}^{d}aw{a}^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & {\left(aw\right)}^{d}awa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ & =\hfill & a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}.\hfill \end{array}$$

Therefore $a^{{ⓒ}_m,w}=x$, as desired.

$\left(1\right)\Rightarrow \left(3\right)$ By the argument above, we have $awx=c{\left[{\left(wa\right)}^m\right]}^{†}$. In view of Theorem 2.1, $x=a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}$. By using Corollary 4.2, we have $a{\left(wx\right)}^2=x$, as required.

$\left(3\right)\Rightarrow \left(1\right)$ Since $awx=c{\left[{\left(wa\right)}^m\right]}^{†}$, we see that $awx=a{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}$. As $a{\left(wx\right)}^2=x$, by virtue of Corollary 4.2, $x=a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}=a^{{ⓒ}_m,w}$, as required. □

Let $X\in {\mathbb{C}}^{n\times n}$. The symbol $\mathcal{R}\left(X\right)$ denote the range space of X. We now derive

Corollary 4.4. 

Let $A\in {\mathbb{C}}^{n\times n}$. The following are equivalent:

(1)

$A^{Ⓦ,†}=X$.

(2)

$AWX=A{A}^{Ⓦ}A{A}^{†}$ and $\mathcal{R}\left(X\right)\subseteq \mathcal{R}\left(A^D\right)$.

(3)

$AWX=A{\left(WA\right)}^{Ⓦ}WA{A}^{†}$ and $A{\left(WX\right)}^2=X$.

Proof. 

Since $A\in {\mathbb{C}}^{n\times n}$, we easily see that $A^{{ⓒ}_1,w}=A^{Ⓦ,†}$. Therefore we complete the proof by Theorem 4.3. □

We are now ready to prove the following.

Theorem 4.5. 

Let $a,w\in \mathcal{A}$. Then the following are equivalent:

(1)

$a^{{ⓒ}_m,w}=x$.

(2)

$xwcwx=x,awx=c{\left[{\left(wa\right)}^m\right]}^{†}$ and $xwc={\left(aw\right)}^{d}c$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ In view of Theorem 4.1, $x=a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}$. By virtue of Theorem 4.3, $awx=c{\left[{\left(wa\right)}^m\right]}^{†}$. Moreover, we verify that

$$\begin{array}{cc}& xwcwx\hfill \\ \hfill =& a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\left[wa{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^{m}w\right]a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ \hfill =& a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\left[wa{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^{m}w\right]{\left[a{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ \hfill =& a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^{2}wa{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ \hfill =& a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ \hfill =& a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}\hfill \\ \hfill =& x,\hfill \\ & xwc\hfill \\ \hfill =& a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}w\left[a{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m\right]\hfill \\ \hfill =& a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}wa{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m\hfill \\ \hfill =& a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^{2}wa{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m\hfill \\ \hfill =& a{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m\hfill \\ \hfill =& a{\left(wa\right)}^d\left(wa\right){\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^2{\left(wa\right)}^m\hfill \\ \hfill =& a{\left(wa\right)}^d{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m\hfill \\ \hfill =& a{\left[{\left(wa\right)}^d\right]}^{2}wa{\left(wa\right)}^{{ⓖ}_m}{\left(wa\right)}^m\hfill \\ \hfill =& {\left(aw\right)}^{d}c,\hfill \end{array}$$

as required.

$\left(2\right)\Rightarrow \left(1\right)$ By hypothesis, we check that

$$\begin{array}{ccc}\hfill x& =\hfill & xwcwx=\left(xwc\right)wx=\left[{\left(aw\right)}^{d}c\right]wx\hfill \\ & =\hfill & a{\left[{\left(wa\right)}^d\right]}^2\left(wcwx\right)\hfill \\ & \in \hfill & a^{d,w}\mathcal{A}.\hfill \end{array}$$

According to Theorem 4.3, we complete the proof. □

Corollary 4.6. 

Let $A\in {\mathbb{C}}^{n\times n}$ and $C=A{\left(WA\right)}^{Ⓦ}WA$. The following are equivalent:

(1)

$A^{Ⓦ,†}=X$.

(2)

$XWCWX=X,AWX=C{\left(WA\right)}^{†}$ and $XWC={\left(AW\right)}^{D}C$.

Proof. 

It is immediate by Theorem 4.5 by choosing $m=1$. □

5. Applications

The purpose of this section is to give the applications of the w-weighted m-generalized group (core) inverse in solving the matrix equations. We consider the following equation in $\mathcal{A}$:

$${\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}wx={\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m}b,\left(5.1\right)$$

where $a,w,b\in \mathcal{A}$ and $m\in \mathbb{N}$.

Theorem 5.1. 

Let $a\in {\mathcal{A}}^{{ⓖ}_m,w}$. Then Eq. (5.1) has solution

$$x=a^{{ⓖ}_m,w}b+[1-a^{{ⓖ}_m,w}waw]y,$$

where $y\in \mathcal{A}$ is arbitrary.

Proof. 

Let $x=a^{{ⓖ}_m,w}b+[1-a^{{ⓖ}_m,w}waw]y,$ where $y\in \mathcal{A}$. Then

$$\begin{array}{ccc}\hfill wx& =\hfill & wa{\left[{\left(wa\right)}^{{ⓖ}_m}\right]}^{2}b+w[1-a{\left({\left(wa\right)}^{{ⓖ}_m}\right)}^{2}waw]y\hfill \\ & =\hfill & {\left(wa\right)}^{{ⓖ}_m}b+[w-{\left(wa\right)}^{{ⓖ}_m}waw]y.\hfill \end{array}$$

Since ${\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}{\left(wa\right)}^{{ⓖ}_m}={\left(wa\right)}^m,$ we verify that

$$\begin{array}{cc}& {\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}wx\hfill \\ \hfill =& {\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}{\left(wa\right)}^{{ⓖ}_m}b+{\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}[w-{\left(wa\right)}^{{ⓖ}_m}waw]y\hfill \\ \hfill =& {\left(wa\right)}^d{]}^{\ast }{\left(wa\right)}^{m}b+{\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}w-{\left(wa\right)}^d{]}^{\ast }{\left(wa\right)}^{m}waw]y\hfill \\ \hfill =& {\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m}b,\hfill \end{array}$$

as asserted. □

Corollary 5.2. 

Let $a\in {\mathcal{A}}^{ⓓ,w}$. Then the general solution of Eq. (5.1) is

$$x=a^{{ⓖ}_m,w}b+[1-a^{{ⓖ}_m,w}waw]y,$$

where $y\in \mathcal{A}$ is arbitrary.

Proof. 

Let x be the solution of the Eq. (5.1). Then

$${\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}wx={\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m}b.$$

In view of Theorem 3.1, $a^{{ⓖ}_m,w}={\left[a^{ⓓ,w}w\right]}^{m+1}{\left(aw\right)}^{m-1}a$. Then

$$\begin{array}{ccc}\hfill a^{{ⓖ}_m,w}wawx& =\hfill & {\left[a^{ⓓ,w}w\right]}^{m+1}{\left(aw\right)}^{m-1}awawx\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^{m+1}{\left(aw\right)}^{m+1}x\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^m\left[a{\left({\left(wa\right)}^{ⓓ}\right)}^{2}w\right]{\left(aw\right)}^{m+1}x\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^m\left[a{\left(wa\right)}^{ⓓ}\right]\left[{\left(wa\right)}^{ⓓ}wa{\left(wa\right)}^{ⓓ}\right]{\left(wa\right)}^{m+1}wx\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^m\left[a{\left(wa\right)}^{ⓓ}\right]{\left(wa\right)}^{ⓓ}\left[wa{\left(wa\right)}^{ⓓ}\right]{\left(wa\right)}^{m+1}wx\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^m\left[a{\left(wa\right)}^{ⓓ}\right]{\left(wa\right)}^{ⓓ}{\left[wa{\left(wa\right)}^{ⓓ}\right]}^{\ast }{\left(wa\right)}^{m+1}wx\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^m\left[a{\left(wa\right)}^{ⓓ}\right]{\left(wa\right)}^{ⓓ}{\left[{\left(wa\right)}^d{\left(wa\right)}^2{\left(wa\right)}^{ⓓ}\right]}^{\ast }{\left(wa\right)}^{m+1}wx\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^m\left[a{\left(wa\right)}^{ⓓ}\right]{\left(wa\right)}^{ⓓ}{\left[{\left(wa\right)}^2{\left(wa\right)}^{ⓓ}\right]}^{\ast }\left[{\left({\left(wa\right)}^d\right)}^{\ast }{\left(wa\right)}^{m+1}wx\right]\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^m\left[a{\left(wa\right)}^{ⓓ}\right]{\left(wa\right)}^{ⓓ}{\left[{\left(wa\right)}^2{\left(wa\right)}^{ⓓ}\right]}^{\ast }\left[{\left({\left(wa\right)}^d\right)}^{\ast }{\left(wa\right)}^{m}b\right]\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^m\left[a{\left(wa\right)}^{ⓓ}\right]{\left(wa\right)}^{ⓓ}{\left[{\left(wa\right)}^d{\left(wa\right)}^2{\left(wa\right)}^{ⓓ}\right]}^{\ast }{\left(wa\right)}^{m}b\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^m\left[a{\left(wa\right)}^{ⓓ}\right]\left[{\left(wa\right)}^{ⓓ}wa{\left(wa\right)}^{ⓓ}\right]{\left(wa\right)}^{m}b\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^m\left[a{\left({\left(wa\right)}^{ⓓ}\right)}^2\right]{\left(wa\right)}^{m}b\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^m\left[a{\left({\left(wa\right)}^{ⓓ}\right)}^{2}w\right]{\left(aw\right)}^{m-1}ab\hfill \\ & =\hfill & {\left[a^{ⓓ,w}w\right]}^{m+1}{\left(aw\right)}^{m-1}ab\hfill \\ & =\hfill & a^{{ⓖ}_m,w}b.\hfill \end{array}$$

Accordingly,

$$x=a^{{ⓖ}_m,w}b+[1-a^{{ⓖ}_m,w}waw]x.$$

By using Theorem 5.1, we complete the proof. □

Corollary 5.3. 

Let $a\in {\mathcal{A}}^{{ⓖ}_m,w}$. If x is the solution of Eq. (5.1) and $im\left(x\right)\subseteq im\left({\left(aw\right)}^d\right)$, then

$$x=a^{{ⓖ}_m,w}b.$$

Proof. 

By virtue of Theorem 5.1, $a^{{ⓖ}_m,w}b$ is a solution of Eq. (5.1). Let $x_1,x_2\in \mathcal{A}$ be the solutions of Eq. (5.1) and satisfy $im\left(x_i\right)\subseteq im\left({\left(aw\right)}^d\right)$. Write $x_1={\left(aw\right)}^d{y}_1$ and $x_2={\left(aw\right)}^d{y}_2$. Then $x_1-x_2={\left[{\left(aw\right)}^d\right]}^{2}a\left(waw\right)(x_1-x_2)$. Hence, $im(x_1-x_2)\subseteq im\left({\left(aw\right)}^d\right)$. By hypothesis, we have

$${\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}w{x}_i={\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m}b$$

for $i=1,2$. Then ${\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}w(x_1-x_2)=0$; and so

$${\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}w{\left[{\left(aw\right)}^d\right]}^{2}a\left(waw\right)(x_1-x_2)=0.$$

By using Cline’s formula, we have $w{\left[{\left(aw\right)}^d\right]}^{2}a={\left(wa\right)}^d$, and then

$${\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^d{\left(wa\right)}^{m+2}w(x_1-x_2)=0.$$

Since the involution is proper, we have ${\left(wa\right)}^d{\left(wa\right)}^{m+2}w(x_1-x_2)=0$; whence, $\left(aw\right){\left(aw\right)}^d(x_1-x_2)=0$. Thus, $x_1=aw{\left(aw\right)}^d{x}_1=aw{\left(aw\right)}^d{x}_2=x_2$. Therefore $x=a^{{ⓖ}_m,w}b$ is the unique solution of Eq. (5.1). □

Consider the following matrix equation:

$${\left[{\left(WA\right)}^D\right]}^{\ast }{\left(WA\right)}^{m+1}WX={\left[{\left(WA\right)}^D\right]}^{\ast }{\left(WA\right)}^{m}B,\left(5.2\right)$$

where $A\in {\mathbb{C}}^{q\times n},W\in {\mathbb{C}}^{n\times q},B\in {\mathbb{C}}^{n\times p}$ and $m\in \mathbb{N}$.

Corollary 5.4. 

$\left(1\right)$ The general solution of Eq. (5.2) is

$$X=A^{{Ⓦ}_m,W}B+[I_n-A^{{Ⓦ}_m,W}WAW]Y,$$

where $Y\in {\mathbb{C}}^{n\times p}$ is arbitrary.

$\left(2\right)$ If X is the solution of Eq. (5.2) and $\mathcal{R}\left(X\right)\subseteq \mathcal{R}\left({\left(AW\right)}^D\right)$, then

$$X=A^{{Ⓦ}_m,W}B.$$

Proof. 

This is obvious by Corollary 5.2 and Corollary 5.3. □

Let $a\in {\mathcal{A}}^{{ⓒ}_m,w}$. We now come to consider the following equation in $\mathcal{A}$:

$${\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}wx={\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{2m}{\left[{\left(wa\right)}^m\right]}^{†}b,\left(5.3\right)$$

where $a,w,b\in \mathcal{A}$ and $m\in \mathbb{N}$. The following lemma is crucial.

Lemma 5.5. 

Let $a\in {\mathcal{A}}^{{ⓒ}_m,w}$. Then $a\in {\mathcal{A}}^{ⓓ,w}$.

Proof. 

By hypothesis, $a\in {\mathcal{A}}^{{ⓖ}_m,w}\bigcap {\mathcal{A}}^{{†}_m,w}.$ In light of [4] [Theorem 2.1], ${\left(wa\right)}^m\in {\mathcal{A}}^{ⓖ}\bigcap {\mathcal{A}}^{†}$. By virtue of [3] [Theorem 3.1], ${\left(wa\right)}^m\in {\mathcal{A}}^{ⓓ}$. Then $wa\in {\mathcal{A}}^{ⓓ}$. Evidently, ${\left(wa\right)}^{ⓓ}={\left(wa\right)}^{m-1}{\left[{\left(wa\right)}^m\right]}^{ⓓ}$. Accordingly, $a\in {\mathcal{A}}^{ⓓ,w}$ by [2] [Theorem 2.1]. □

We are ready to prove:

Theorem 5.6. 

Let $a\in {\mathcal{A}}^{{ⓒ}_m,w}$. Then the general solution of Eq. (5.3) is

$$x=a^{{ⓒ}_m,w}b+[1-a^{{ⓖ}_m,w}waw]y,$$

where $y\in \mathcal{A}$ is arbitrary.

Proof. 

Let $x=a^{{ⓒ}_m,w}b+[1-a^{{ⓖ}_m,w}waw]y,$ where $y\in \mathcal{A}$. In view of Theorem 4.1, $a^{{ⓒ}_m,w}=a^{{ⓖ}_m,w}{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}$. Then

$$x=a^{{ⓖ}_m,w}\left[{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}b\right]+[1-a^{{ⓖ}_m,w}waw]y.$$

By virtue of Theorem 5.1, x is the solution of Eq. (5.3).

In light of Lemma 5.5, $a\in {\mathcal{A}}^{ⓓ,w}$. By using Corollary 5.2,

$$x=a^{{ⓖ}_m,w}\left[{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}b\right]+[1-a^{{ⓖ}_m,w}waw]y$$

is the general solution of Eq. (5.3), as required. □

Corollary 5.7. 

Let $a\in {\mathcal{A}}^{{ⓒ}_m,w}$. If x is the solution of Eq. (5.3) and $im\left(x\right)\subseteq im\left({\left(aw\right)}^d\right)$, then

$$x=a^{{ⓒ}_m,w}b.$$

Proof. 

By virtue of Theorem 5.6, $a^{{ⓒ}_m,w}b$ is a solution of Eq. (5.3). Let $x_1,x_2\in \mathcal{A}$ be the solutions of Eq. (5.3) and satisfy $im\left(x_i\right)\subseteq im\left({\left(aw\right)}^d\right)$. Then they are solutions of the equation:

$${\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^{m+1}wx={\left[{\left(wa\right)}^d\right]}^{\ast }{\left(wa\right)}^m\left[{\left(wa\right)}^m{\left[{\left(wa\right)}^m\right]}^{†}b\right],$$

as desired. □

Consider the following matrix equation:

$${\left[{\left(WA\right)}^D\right]}^{\ast }{\left(WA\right)}^{m+1}WX={\left[{\left(WA\right)}^D\right]}^{\ast }{\left(WA\right)}^{2m}{\left[{\left(WA\right)}^m\right]}^{†}B,\left(5.4\right)$$

where $A\in {\mathbb{C}}^{q\times n},W\in {\mathbb{C}}^{n\times q},B\in {\mathbb{C}}^{n\times p}$ and $m\in \mathbb{N}$.

Corollary 5.8. 

$\left(1\right)$ The general solution of Eq. (5.4) is

$$X=A^{{\overline{)\#}}_m,W}B+[I_n-A^{{\overline{)\#}}_m,W}WAW]Y,$$

where $Y\in {\mathbb{C}}^{n\times p}$ is arbitrary.

$\left(2\right)$ If X is the solution of Eq. (5.4) and $\mathcal{R}\left(X\right)\subseteq \mathcal{R}\left({\left(AW\right)}^D\right)$, then

$$X=A^{{\overline{)\#}}_m,W}B.$$

Proof. 

This is obvious by Theorem 5.5 and Corollary 5.6. □