Weighted weak group inverse for Hilbert space operators

TL;DR

This paper introduces the weighted weak group inverse for operators on Hilbert spaces, extending the concept from matrices, with new characterizations, representations, and applications to binary relations.

Contribution

It presents the first definition and analysis of the weighted weak group inverse for Hilbert space operators, expanding the theory beyond matrices.

Findings

Defined the weighted weak group inverse for Hilbert space operators

Provided characterizations and representations of the inverse

Applied the inverse to define and analyze binary relations

Abstract

We present the weighted weak group inverse, which is a new generalized inverse of operators between two Hilbert spaces, introduced to extend weak group inverse for square matrices. Some characterizations and representations of the weighted weak group inverse are investigated. We also apply these results to define and study the weak group inverse for a Hilbert space operator. Using the weak group inverse, we define and characterize various binary relations.

Equations

$$AWB=BWA,BWAWB=B\mathrm{and}A-AWBWA\mathrm{is}\mathrm{quasinilpotent}.$$

$$WAWB=P_{R((WA{)}^d)}\mathrm{and}R(B)\subseteq R((AW{)}^d)$$

$$A=\left[\begin{array}[]{cc}A_{1}&A_{2}\\ 0&A_{3}\end{array}\right]:\left[\begin{array}[]{c}R((WA)^{d})\\ N[((WA)^{d})^{*}]\end{array}\right]\rightarrow\left[\begin{array}[]{c}R((AW)^{d})\\ N[((AW)^{d})^{*}]\end{array}\right]$$

$$W=\left[\begin{array}[]{cc}W_{1}&W_{2}\\ 0&W_{3}\end{array}\right]:\left[\begin{array}[]{c}R((AW)^{d})\\ N[((AW)^{d})^{*}]\end{array}\right]\rightarrow\left[\begin{array}[]{c}R((WA)^{d})\\ N[((WA)^{d})^{*}]\end{array}\right],$$

$$A^{d,W}=\left[\begin{array}[]{cc}(W_{1}A_{1}W_{1})^{-1}&W_{1}^{-1}U\\ 0&0\end{array}\right]:\left[\begin{array}[]{c}R((WA)^{d})\\ N[((WA)^{d})^{*}]\end{array}\right]\rightarrow\left[\begin{array}[]{c}R((AW)^{d})\\ N[((AW)^{d})^{*}]\end{array}\right],$$

$$A^{\tiny\tinyd⃝,W}=\left[\begin{array}[]{cc}(W_{1}A_{1}W_{1})^{-1}&0\\ 0&0\end{array}\right]:\left[\begin{array}[]{c}R((WA)^{d})\\ N[((WA)^{d})^{*}]\end{array}\right]\rightarrow\left[\begin{array}[]{c}R((AW)^{d})\\ N[((AW)^{d})^{*}]\end{array}\right],$$

$$(AW)^{d}=\left[\begin{array}[]{cc}(A_{1}W_{1})^{-1}&T\\ 0&0\end{array}\right]:\left[\begin{array}[]{c}R((AW)^{d})\\ N[((AW)^{d})^{*}]\end{array}\right]\rightarrow\left[\begin{array}[]{c}R((AW)^{d})\\ N[((AW)^{d})^{*}]\end{array}\right],$$

$$(WA)^{d}=\left[\begin{array}[]{cc}(W_{1}A_{1})^{-1}&U\\ 0&0\end{array}\right]:\left[\begin{array}[]{c}R((WA)^{d})\\ N[((WA)^{d})^{*}]\end{array}\right]\rightarrow\left[\begin{array}[]{c}R((WA)^{d})\\ N[((WA)^{d})^{*}]\end{array}\right],$$

$$T=\sum _{n=0}^{\infty }(A_1{W}_1{)}^{-(n+2)}(A_1{W}_2+A_2{W}_3)(A_3{W}_3{)}^n$$

$$U=\sum _{n=0}^{\infty }(W_1{A}_1{)}^{-(n+2)}(W_1{A}_2+W_2{A}_3)(W_3{A}_3{)}^n.$$

$$AWBWB=BandAWB=A^{\text{tinyd}⃝,W}WA$$

$$B=\left[\begin{array}[]{cc}(W_{1}A_{1}W_{1})^{-1}&(A_{1}W_{1})^{-2}(A_{2}+W_{1}^{-1}W_{2}A_{3})\\ 0&0\end{array}\right]:\left[\begin{array}[]{c}R((WA)^{d})\\ N[((WA)^{d})^{*}]\end{array}\right]\rightarrow\left[\begin{array}[]{c}R((AW)^{d})\\ N[((AW)^{d})^{*}]\end{array}\right],$$

$$AWB=\left[\begin{array}[]{cc}W_{1}^{-1}&(A_{1}W_{1})^{-1}A_{2}+(W_{1}A_{1}W_{1})^{-1}W_{2}A_{3}\\ 0&0\end{array}\right]=A^{\tiny\tinyd⃝,W}WA$$

$$B=(AWB)WB=A^{\text{tinyd}⃝}W(AWB)=A^{\text{tinyd}⃝,W}W{A}^{\text{tinyd}⃝,W}WA=(A^{\text{tinyd}⃝,W}W{)}^{2}A$$

$$A{B}^2=BandAB=A^{\text{tinyd}⃝}A$$

$$B=\left[\begin{array}[]{cc}A_{1}^{-1}&A_{1}^{-2}A_{2}\\ 0&0\end{array}\right]:\left[\begin{array}[]{c}R(A^{d})\\ N[(A^{d})^{*}]\end{array}\right]\rightarrow\left[\begin{array}[]{c}R(A^{d})\\ N[(A^{d})^{*}]\end{array}\right],$$

$$A=\left[\begin{array}[]{cc}A_{1}&A_{2}\\ 0&A_{3}\end{array}\right]:\left[\begin{array}[]{c}R(A^{d})\\ N((A^{d})^{*})\end{array}\right]\rightarrow\left[\begin{array}[]{c}R(A^{d})\\ N((A^{d})^{*})\end{array}\right],$$

$$A^{\otimes ,W}=(A^{\text{tinyd}⃝,W}W{)}^{2}A.$$

$$A^{\otimes }=(A^{\text{tinyd}⃝}{)}^{2}A.$$

$$WAWB=P_{R(W{A}^{d,W}),N(A^{\text{tinyd}⃝,W}WA)}andR(B)\subseteq R(A^{d,W})$$

$$AB=P_{R(A^d),N(A^{\text{tinyd}⃝}A)}andR(B)\subseteq R(A^d),$$

$$A^{\otimes ,W}WAW{A}^{\otimes ,W}=(A^{\text{tinyd}⃝,W}W{)}^{2}AW{A}^{\text{tinyd}⃝,W}WA=(A^{\text{tinyd}⃝,W}W{)}^{2}A=A^{\otimes ,W}.$$

$$A^{d,W}=A^{\text{tinyd}⃝,W}W{A}^{d,W}WA=A^{\text{tinyd}⃝,W}W{A}^{\text{tinyd}⃝,W}W{A}^{d,W}WAWA=(A^{\text{tinyd}⃝,W}W{)}^2(AW{)}^2{A}^{d,W},$$

$$A^{\text{tinyd}⃝,W}WA=AW(A^{\text{tinyd}⃝,W}W{A}^{\text{tinyd}⃝,W}WA)=AW{A}^{\otimes ,W}$$

$$A^{\text{tinyd}⃝,W}WAWB=BandAWB=A^{\text{tinyd}⃝,W}WA;$$

$$BWAWB=B,AWB=A^{\text{tinyd}⃝,W}WAandBW{A}^{\text{tinyd}⃝,W}=A^{\text{tinyd}⃝,W}W{A}^{\text{tinyd}⃝,W};$$

$$BWAWB=B,AWB=A^{\text{tinyd}⃝,W}WAandBW{A}^{d,W}=A^{d,W}W{A}^{d,W}.$$

$$A^{\text{tinyd}⃝}AB=BandAB=A^{\text{tinyd}⃝}A;$$

$$BAB=B,AB=A^{\text{tinyd}⃝}AandB{A}^{\text{tinyd}⃝}=A^{\text{tinyd}⃝}{A}^{\text{tinyd}⃝};$$

$$BAB=B,AB=A^{\text{tinyd}⃝}AandB{A}^d=(A^d{)}^2.$$

Full text

Key words and phrases: weak group inverse, weighted core–EP inverse, Wg-Drazin inverse, Hilbert space.

2010 Mathematics subject classification: 47A62, 47A05, 15A09.

1 Introduction

Throughout this paper, let ${\cal B}(X,Y)$ be the set of all bounded linear operators from $X$ to $Y$, where $X$ and $Y$ are infinite-dimensional complex Hilbert space. In the case that $X=Y$, we set ${\cal B}(X)={\cal B}(X,X)$. For $A\in{\cal B}(X,Y)$, $A^{*}$, $N(A)$, $R(A)$ and $\sigma(A)$ represent the adjoint of $A$, the null space, the range and the spectrum of $A$, respectively. We call $P\in{\cal B}(X)$ an idempotent if $P^{2}=P$, and an orthogonal projector if $P^{2}=P=P^{*}$. If $L$ and $M$ are closed subspaces, we denote by $P_{L,M}$ an idempotent on $L$ along $M$, and by $P_{L}$ an orthogonal projector onto $L$.

Let $A\in{\cal B}(X,Y)\backslash\{0\}$. There always exists $B\in{\cal B}(Y,X)\backslash\{0\}$ such that $BAB=B$, which is not unique in general and it is called an outer inverse of $A$. The outer inverse is uniquely determined if we fix its range and kernel. For a subspace $T$ of $X$ and a subspace $S$ of $Y$, the unique outer inverse $B$ of $A$ with the prescribed range and the null space $S$ will be denoted by $A^{(2)}_{T,S}$. We now present some special classes of outer inverses.

For a fixed operator $W\in{\cal B}(Y,X)\backslash\{0\}$, an operator $A\in{\cal B}(X,Y)$ is called Wg–Drazin invertible [4] if there exists a unique operator $B\in{\cal B}(X,Y)$ (denoted by $A^{d,w}$) such that

[TABLE]

In the case that $X=Y$ and $W=I$, then $A^{d}=A^{d,W}$ is the generalized Drazin inverse (or the Koliha-Drazin inverse) of $A$ [9]. We use ${\cal B}(X,Y)^{d,W}$ and ${\cal B}(X)^{d}$, respectively, to denote the sets of all Wg–Drazin invertible operators in ${\cal B}(X,Y)$ and generalized Drazin invertible operators in ${\cal B}(X)$.

The W-weighted Drazin inverse is a particular case of the Wg–Drazin inverse for which $A-AWBWA$ is nilpotent. The Drazin inverse is a special case of the generalized Drazin inverse for which $A-A^{2}B$ is nilpotent (or equivalently $A^{k+1}B=A^{k}$, for some non-negative integer $k$). The smallest such $k$ is called the index of $A$ and it is denoted by $ind(A)$. In the case that $ind(A)\leq 1$, $A$ is group invertible and the group inverse $A^{\#}$ of $A$ is a special case of a Drazin inverse. The Drazin inverse is very useful, and its applications in automatics, probability, statistics, mathematical programming, numerical analysis, game theory, econometrics, control theory and so on, can be found in [2, 3]. For more recent results related to generalized Drazin inverse, W-weighted Drazin inverse and Drazin inverse see [16, 19, 20, 21].

Prasad and Mohana [15] introduced the core–EP inverse for a square matrix of arbitrary index, as a generalization of the core inverse restricted to a square matrix of index one [1]. The core–EP inverse was presented for generalized Drazin invertible operators on Hilbert spaces in [14].

As a generalization of the core–EP inverse of a square matrix to a rectangular matrix, the weighted core–EP inverse was given in [6]. In [13], the weighted core–EP inverse was defined for a $W$g-Drazin invertible bounded linear operator between two Hilbert spaces, extending the concepts of the weighted core–EP inverse for a rectangular matrix [6].

Let $W\in{\cal B}(Y,X)\backslash\{0\}$ and let $A\in{\cal B}(X,Y)$ be $W$g-Drazin invertible. Then there exists the unique operator $B$ which satisfies conditions

[TABLE]

and it is called the weighted core–EP inverse of $A$, denoted by $A^{\tiny\tinyd⃝,W}$. If $X=Y$ and $W=I$, then $A^{\tiny\tinyd⃝}=A^{\tiny\tinyd⃝,W}$ is the core–EP inverse of $A$ [14]. In the case that $A\in{\cal B}(X)$ and $ind(A)\leq 1$, the core-EP inverse of $A$ is the core inverse of $A$, denoted by $A^{\tiny\tiny\#⃝}$. Recently, many results concerning the weighted core–EP and core–EP inverse appeared in papers [5, 7, 8, 11, 17, 22].

In [18], the weak group inverse was recently defined for square matrices of an arbitrary index and presented as a generalization of the group inverse.

We extend the definition of the weak group inverse of a square matrix to a Wg–Drazin invertible bounded linear operator between two Hilbert spaces and present a new generalized inverse, named the weighted weak group inverse. We obtain some properties of the weighted weak group inverse, in particular, an operator matrix representation, characterizations and representations of the weighted weak group inverse. As an application of these results, we present and characterize the weak group inverse of a generalized Drazin invertible bounded linear operator on a Hilbert space. Using the weak group inverse, we define and study several binary relations.

2 Weighted weak group inverse

In order to define the weighted weak group inverse of a Wg–Drazin invertible bounded linear operator between two Hilbert spaces as an extension of the weak group inverse of a square matrix, we need following auxiliary result.

Lemma 2.1**.**

[13]* Let $W\in{\cal B}(Y,X)\backslash\{0\}$ and $A\in{\cal B}(X,Y)^{d,W}$. Then*

[TABLE]

and

[TABLE]

*where $A_{1}\in{\cal B}(R((WA)^{d}),R((AW)^{d}))^{-1}$, $W_{1}\in{\cal B}(R((AW)^{d}),R((WA)^{d}))^{-1}$,$A_{3}W_{3}\in{\cal B}(N[((AW)^{d})^{*}])^{qnil}$ and $W_{3}A_{3}\in{\cal B}(N[((WA)^{d})^{*}])^{qnil}$. In addition, *

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

and

[TABLE]

We first give algebraic definition of a new generalized inverse.

Theorem 2.1**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$ and $A\in{\cal B}(X,Y)^{d,W}$. Then the system of equations

[TABLE]

is consistent and it has the unique solution given by

[TABLE]

where $A$ and $W$ are represented as in (1) and (2), respectively.

Proof.

Using (1), (2), (4) and (6), we get

[TABLE]

and $AWBWB=B$, that is, $B$ is a solution of the system (5).

If an operator $B$ satisfies (5), then

[TABLE]

and so $B$ is the unique solution of the system (5). ∎

For $X=Y$ and $W=I$ in Theorem 2.1, we get the next consequence.

Corollary 2.1**.**

Let $A\in{\cal B}(X)^{d}$. Then the system of equations

[TABLE]

is consistent and it has the unique solution given by

[TABLE]

where

[TABLE]

$A_{1}\in{\cal B}(R(A^{d}))$* is invertible and $A_{3}\in{\cal B}[N((A^{d})^{*})]$ is quasinilpotent.*

Definition 2.1**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$ and $A\in{\cal B}(X,Y)^{d,W}$. The weighted weak group inverse of $A$ is defined as

[TABLE]

Definition 2.2**.**

Let $A\in{\cal B}(X)^{d}$. The weak group inverse of $A$ is defined as

[TABLE]

Remark that, by Lemma 2.1 and Theorem 2.1, the weighted weak group inverse is different from the $W$g-Drazin inverse and weighted core–EP inverse. Hence, the weighted weak group and weak group inverses provide new classes of generalized inverses for operators. If $X$ and $Y$ are finite dimensional, then, for $W\in{\cal B}(Y,X)\backslash\{0\}$, every operator $A\in{\cal B}(X,Y)$ has the weighted weak group inverse. For $A\in{\cal B}(X)$ and $ind(A)\leq 1$, the weak group inverse of $A$ is the group inverse of $A$.

We also have a definition of the weighted weak group inverse from a geometrical point of view.

Theorem 2.2**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$ and $A\in{\cal B}(X,Y)^{d,W}$. The system of conditions

[TABLE]

is consistent and it has the unique solution $B=(A^{\tiny\tinyd⃝,W}W)^{2}A$.

Proof.

For $B=(A^{\tiny\tinyd⃝,W}W)^{2}A$, we have that $WAWB=WA^{\tiny\tinyd⃝,W}WA$ is a projector onto $R(WA^{\tiny\tinyd⃝,W})=R(WA^{d,W})$ along $N(WA^{\tiny\tinyd⃝,W}WA)=N(A^{\tiny\tinyd⃝,W}WA)$ and $R(B)\subseteq R(A^{\tiny\tinyd⃝,W})=R(A^{d,W})$, i.e. $B$ satisfies conditions (10).

Assume that two operators $B_{1}$ and $B_{2}$ satisfy conditions (10). Firstly, $WAW(B_{1}-B_{2})=P_{R(WA^{d,W}),N(A^{\tiny\tinyd⃝,W}WA)}-P_{R(WA^{d,W}),N(A^{\tiny\tinyd⃝,W}WA)}=0$ implies $R(B_{1}-B_{2})\subseteq N(WAW)\subseteq N(A^{d,W}WAW)$. Further, $R(B_{1})\subseteq R(A^{d,W})=R(A^{d,W}WAW)$ and $R(B_{2})\subseteq R(A^{d,W}WAW)$ give $R(B_{1}-B_{2})\subseteq R(A^{d,W}WAW)\cap N(A^{d,W}WAW)=\{0\}$. Therefore, $B_{1}=B_{2}$ and only $B$ satisfies (10). ∎

Consequently, the geometrical approach is given now for the weak group inverse.

Corollary 2.2**.**

Let $A\in{\cal B}(X)^{d}$. The system of conditions

[TABLE]

is consistent and it has the unique solution $B=(A^{\tiny\tinyd⃝})^{2}A$.

We consider some idempotents determined by the weighted weak group inverse and observe that the weighted weak group inverse is an outer inverse.

Lemma 2.2**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$ and $A\in{\cal B}(X,Y)^{d,W}$. Then:

• (i)

$AWA^{\otimes,W}W$* is a projector onto $R(A^{d,W})$ along $N(A^{\tiny\tinyd⃝,W}WAW)$;*

• (ii)

$WAWA^{\otimes,W}$* is a projector onto $R(WA^{d,W})$ along $N(A^{\tiny\tinyd⃝,W}WA)$;*

• (iii)

$A^{\otimes,W}WAW$* is a projector onto $R(A^{d,W})$ along $N(A^{\tiny\tinyd⃝,W}W(AW)^{2})$;*

• (iv)

$WA^{\otimes,W}WA$* is a projector onto $R(WA^{d,W})$ along $N(A^{\tiny\tinyd⃝,W}(WA)^{2})$;*

• (v)

$A^{\otimes,W}=(WAW)^{(2)}_{R(A^{d,W}),N(A^{\tiny\tinyd⃝,W}WA)}$.

Proof.

(i) Notice that $AWA^{\otimes,W}W=A^{\tiny\tinyd⃝,W}WAW$ is a projector onto $R(A^{\tiny\tinyd⃝,W})=R(A^{d,W})$ along $N(A^{\tiny\tinyd⃝,W}WAW)$.

(v) Since $A^{\otimes,W}=(A^{\tiny\tinyd⃝,W}W)^{2}A$ and $AWA^{\otimes,W}=A^{\tiny\tinyd⃝,W}WA$, we obtain

[TABLE]

Hence, $A^{\otimes,W}$ is an outer inverse of $WAW$ with $R(A^{\otimes,W})=R((A^{\tiny\tinyd⃝,W}W)^{2}A)$ and $N(A^{\otimes,W})=N((A^{\tiny\tinyd⃝,W}W)^{2}A)$. From

[TABLE]

we get $R(A^{d,W})\subseteq R((A^{\tiny\tinyd⃝,W}W)^{2}A)\subseteq R(A^{d,W})$ which yields $R(A^{\otimes,W})=R(A^{d,W})$. By (1), (2) and (4), we show that $A(WA^{\tiny\tinyd⃝,W})^{2}=A^{\tiny\tinyd⃝,W}$. Now, we have that

[TABLE]

which gives $N(A^{\otimes,W})\subseteq N(A^{\tiny\tinyd⃝,W}WA)\subseteq N(A^{\otimes,W})$, i.e. $N(A^{\otimes,W})=N(A^{\tiny\tinyd⃝,W}WA)$. Thus, $A^{\otimes,W}=(WAW)^{(2)}_{R(A^{d,W}),N(A^{\tiny\tinyd⃝,W}WA)}$.

Similarly, we verify parts (ii)–(iv). ∎

By Lemma 2.2, notice that the weak group inverse of $A$ is an outer inverse of $A$.

Corollary 2.3**.**

Let $A\in{\cal B}(X)^{d}$. Then:

• (i)

$AA^{\otimes}$* is a projector onto $R(A^{d})$ along $N(A^{\tiny\tinyd⃝}A)$;*

• (ii)

$A^{\otimes}A$* is a projector onto $R(A^{d})$ along $N(A^{\tiny\tinyd⃝}A^{2})$;*

• (iii)

$A^{\otimes}=A^{(2)}_{R(A^{d}),N(A^{\tiny\tinyd⃝}A)}$.

Several characterizations of the weighted weak group inverse are presented now.

Theorem 2.3**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$ and $A\in{\cal B}(X,Y)^{d,W}$. Then, for $B\in{\cal B}(X,Y)$, the following statements are equivalent:

• (i)

$B$* is the weighted weak group inverse of $A$;*

• (ii)

$B$* satisfies*

[TABLE]

• (iii)

$B$* satisfies*

[TABLE]

• (iv)

$B$* satisfies*

[TABLE]

Proof.

(i) $\Rightarrow$ (ii) $\wedge$ (iii): The equality $B=(A^{\tiny\tinyd⃝,W}W)^{2}A$ gives $B=A^{\tiny\tinyd⃝,W}WAW(A^{\tiny\tinyd⃝,W}W)^{2}A=A^{\tiny\tinyd⃝,W}WAWB$ and $BWA^{\tiny\tinyd⃝,W}=(A^{\tiny\tinyd⃝,W}W)^{2}AWA^{\tiny\tinyd⃝,W}=A^{\tiny\tinyd⃝,W}WA^{\tiny\tinyd⃝,W}$. The rest is clear.

(ii) $\Rightarrow$ (i): It follows by $B=A^{\tiny\tinyd⃝,W}W(AWB)=A^{\tiny\tinyd⃝,W}WA^{\tiny\tinyd⃝,W}WA=A^{\otimes,W}$.

(iii) $\Rightarrow$ (i): We have that $B=BW(AWB)=(BWA^{\tiny\tinyd⃝,W})WA=A^{\tiny\tinyd⃝,W}WA^{\tiny\tinyd⃝,W}WA=A^{\otimes,W}$.

(iii) $\Leftrightarrow$ (iv): Using $A^{\tiny\tinyd⃝,W}=A^{d,W}WAWA^{\tiny\tinyd⃝,W}$ and $A^{d,W}=A^{\tiny\tinyd⃝,W}WAWA^{d,W}$, we obtain these equivalences. ∎

Applying Theorem 2.3, we can characterize the weak group inverse in the following way.

Corollary 2.4**.**

Let $A\in{\cal B}(X)^{d}$. Then, for $B\in{\cal B}(X)$, the following statements are equivalent:

• (i)

$B$* is the weak group inverse of $A$;*

• (ii)

$B$* satisfies*

[TABLE]

• (iii)

$B$* satisfies*

[TABLE]

• (iv)

$B$* satisfies*

[TABLE]

In the case that $X$ and $Y$ are finite dimensional, the condition $BWA^{d,W}=A^{d,W}WA^{d,W}$ ($BA^{d}=(A^{d})^{2}$) of Theorem 2.3(iv) (Corollary 2.4(iv)) can be replaced with the equivalent condition $B(WA)^{k+1}=(WA)^{k}$ for $k=ind(WA)$ ($BA^{k+1}=A^{k}$ for $k=ind(A)$).

Using idempotents and orthogonal projectors, we present some representations of the weighted weak group inverse of $A$ in the next theorem.

Theorem 2.4**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$ and $A\in{\cal B}(X,Y)^{d,W}$. Then the following statements holds:

• (i)

$A^{\otimes,W}=A^{\tiny\tinyd⃝,W}P_{R(WA^{d,W}),N(A^{\tiny\tinyd⃝,W}WA)}$;

• (ii)

$A^{\otimes,W}=A^{d,W}P_{R(WA^{d,W}),N(A^{\tiny\tinyd⃝,W}WA)}$;

• (iii)

$WP_{R((AW)^{d})}$* is Moore–Penrose invertible, $WA(WA)^{\tiny\tinyd⃝}WA$ is group invertible and*

[TABLE]

• (iv)

$A^{\otimes,W}=[(AW)^{2}]^{\tiny\tinyd⃝}AWA^{\tiny\tinyd⃝,W}WA=[(AW)^{2}]^{\tiny\tinyd⃝}AP_{R(WA^{d,W}),N(A^{\tiny\tinyd⃝,W}WA)}$;

• (v)

$WP_{R((AW)^{d})}$* is Moore–Penrose invertible and $A^{\otimes,W}=[(AW)^{d}]^{2}(WP_{R((AW)^{d})})^{\dagger}WA$;*

• (vi)

$WP_{R((AW)^{d})}$* is Moore–Penrose invertible and $A^{\otimes,W}=(WP_{R((AW)^{d})})^{\dagger}[(WA)^{2}]^{\tiny\tinyd⃝}WA$;*

• (vii)

$(AW)^{3}(AW)^{d}$* is group invertible and $A^{\otimes,W}=[(AW)^{3}(AW)^{d}]^{\tiny\tiny\#⃝}AWA^{\tiny\tinyd⃝,W}WA$.*

Proof.

(i) It follows by $A^{\otimes,W}=(A^{\tiny\tinyd⃝,W}W)^{2}A=A^{\tiny\tinyd⃝,W}P_{R(WA^{d,W}),N(A^{\tiny\tinyd⃝,W}WA)}$.

(ii) We get

[TABLE]

(iii) Let $A$ and $W$ be represented as in (1) and (2), respectively. Notice that the orthogonal projector $P_{R((AW)^{d})}$ has the following representation:

[TABLE]

We observe that WP_{R((AW)^{d})}=\left[\begin{array}[]{cc}W_{1}&0\\ 0&0\end{array}\right] is Moore–Penrose invertible and

[TABLE]

Because $A$ is $W$g-Drazin invertible, then $WA$ is generalized Drazin invertible,

[TABLE]

We now have that WA(WA)^{\tiny\tinyd⃝}WA=\left[\begin{array}[]{cc}W_{1}A_{1}&W_{1}A_{2}+W_{2}A_{3}\\ 0&0\end{array}\right] is group invertible and

[TABLE]

Therefore,

[TABLE]

(iv) Since $A$ is $W$g-Drazin invertible, then $AW$ is generalized Drazin invertible and so $(AW)^{2}$ generalized Drazin invertible too. Hence, $[(AW)^{2}]^{\tiny\tinyd⃝}$ exists and, using (1) and (2),

[TABLE]

By (4), we obtain

[TABLE]

Similarly, we verify parts (v)–(vi). ∎

We have new representations for the weak group inverse by Theorem 2.4.

Corollary 2.5**.**

Let $A\in{\cal B}(X)^{d}$. Then the following statements holds:

• (i)

$A^{\otimes}=A^{\tiny\tinyd⃝}P_{R(A^{d}),N(A^{\tiny\tinyd⃝}A)}$;

• (ii)

$A^{\otimes}=A^{d}P_{R(A^{d}),N(A^{\tiny\tinyd⃝}A)}$;

• (iii)

$AA^{\tiny\tinyd⃝}A$* is group invertible and $A^{\otimes}=(AA^{\tiny\tinyd⃝}A)^{\#}=(P_{R(A^{d}))}A)^{\#};$*

• (iv)

$A^{\otimes}=(A^{2})^{\tiny\tinyd⃝}AA^{\tiny\tinyd⃝}A=(A^{2})^{\tiny\tinyd⃝}AP_{R(A^{d}),N(A^{\tiny\tinyd⃝}A)}$;

• (v)

$A^{\otimes}=(A^{d})^{2}P_{R(A^{d})}A$;

• (vi)

$A^{\otimes}=(A^{2})^{\tiny\tinyd⃝}A$;

• (vii)

$A^{3}A^{d}$* is group invertible and $A^{\otimes}=(A^{3}A^{d})^{\tiny\tiny\#⃝}AA^{\tiny\tinyd⃝}A=(A^{3}A^{d})^{\tiny\tiny\#⃝}A$.*

Proof.

Because $P_{R(A^{d})}$ is an orthogonal projector, notice that $P_{R(A^{d}))}$ is Moore–Penrose invertible and $(P_{R(A^{d})})^{\dagger}=P_{R(A^{d})}$. Then we can easily verify this result. ∎

Recall that $A\in{\cal B}(X,Y)^{d,W}$ with $A^{d,W}=B\in{\cal B}(X,Y)$ if and only if $AW\in{\cal B}(Y)^{d}$ with $(AW)^{d}=BW$ if and only if $WA\in{\cal B}(X)^{d}$ with $(WA)^{d}=WB$ [4]. For the weighted core–EP inverse, the situation is similar in the case of the operator $WA$, but it is a little different for $AW$ [13]. We see now that the weighted weak group inverse acts as the weighted core–EP inverse.

Theorem 2.5**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$ and let $A\in{\cal B}(X,Y)$.

• (a)

Then the following statements are equivalent:

• (i)

$A$* is weighted weak group invertible with $A^{\otimes,W}=B$;*

• (ii)

$WA$* is weak group invertible with $(WA)^{\otimes}=WB.$*

In addition,

[TABLE]

• (b)

*If $A$ is *$W$g-Drazin invertible, $A$ and $W$ are represented by (1) and (2), respectively, then

• (i)

$(AW)^{\otimes}=A^{\otimes,W}W$* if and only if $W_{2}A_{3}W_{3}=0$;*

• (ii)

$A^{\otimes,W}=[(AW)^{\otimes}]^{2}A$* if and only if $A_{2}W_{3}A_{3}=0$.*

Proof.

(a) (i) $\Rightarrow$ (ii): By [13, Theorem 2.4], $WA^{\tiny\tinyd⃝,W}=(WA)^{\tiny\tinyd⃝}$, which implies $WA^{\otimes,W}=W(A^{\tiny\tinyd⃝,W}W)^{2}A=[(WA)^{\tiny\tinyd⃝}]^{2}WA=(WA)^{\otimes}$.

(ii) $\Rightarrow$ (i): We observe that

[TABLE]

(b) (i) It follows from the equalities

[TABLE]

and

[TABLE]

(ii) Using (6) and

[TABLE]

we get that (ii) holds. ∎

Using the weak group inverses of $AW$ and $WA$, we obtain the next formula for the weighted weak group inverses of $A$.

Theorem 2.6**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$ and $A\in{\cal B}(X,Y)^{d,W}$. Then

[TABLE]

Proof.

Let $A$ and $W$ be given by (1) and (2), respectively. Then

[TABLE]

∎

We also investigate necessary and sufficient conditions for $AWA^{\otimes,W}=A^{\otimes,W}WA$ to hold.

Theorem 2.7**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$ and $A\in{\cal B}(X,Y)^{d,W}$. If $A$ and $W$ are represented by (1) and (2), respectively, then the following statements are equivalent:

• (i)

$AWA^{\otimes,W}=A^{\otimes,W}WA$;

• (ii)

$(W_{1}A_{2}+W_{2}A_{3})W_{3}A_{3}=0$;

• (iii)

$[(WA)^{\otimes}]^{2}=[(WA)^{2}]^{\otimes}$.

In this case, $A^{\otimes,W}=A^{d,W}$.

Proof.

(i) $\Leftrightarrow$ (ii): The equalities

[TABLE]

and

[TABLE]

give $AWA^{\otimes,W}=A^{\otimes,W}WA$ if and only if $(A_{2}+W_{1}^{-1}W_{2}A_{3})W_{3}A_{3}=0$ which is equivalent to $(W_{1}A_{2}+W_{2}A_{3})W_{3}A_{3}=0$.

(ii) $\Leftrightarrow$ (iii): Firstly, we observe that

[TABLE]

Furthermore,

[TABLE]

is generalized Drazin invertible and, by Corollary 2.1,

[TABLE]

Thus, $[(WA)^{\otimes}]^{2}=[(WA)^{2}]^{\otimes}$ is equivalent to $(W_{1}A_{2}+W_{2}A_{3})W_{3}A_{3}=0$.

Notice that, by $(W_{1}A_{2}+W_{2}A_{3})W_{3}A_{3}=0$ and (3), we obtain $A^{d,W}=A^{\otimes,W}$. ∎

Applying Theorem 2.7 and Theorem 2.5(b)(ii), we get equivalent conditions for $AA^{\otimes}=A^{\otimes}A$ to be satisfied. We observe that conditions (i)–(iii) appeared for the weak group inverse of a square matrix, but the condition (iv) is new even for matrix case.

Corollary 2.6**.**

Let $A\in{\cal B}(X)^{d}$. If $A$ is represented by (9), then the following statements are equivalent:

• (i)

$AA^{\otimes}=A^{\otimes}A$;

• (ii)

$A_{2}A_{3}=0$;

• (iii)

$(A^{\otimes})^{2}=(A^{2})^{\otimes}$;

• (iv)

$A^{\otimes}=(A^{\otimes})^{2}A$.

In this case, $A^{\otimes}=A^{d}$.

Similarly as Theorem 2.7, we verify the next result.

Theorem 2.8**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$ and $A\in{\cal B}(X,Y)^{d,W}$. If $A$ and $W$ are represented by (1) and (2), respectively, then the following statements are equivalent:

• (i)

$[(AW)^{\otimes}]^{2}=[(AW)^{2}]^{\otimes}$;

• (ii)

$(A_{1}W_{2}+A_{2}W_{3})A_{3}W_{3}=0$.

3 Weighted weak group relations

Various types of partial orders and pre-orders were defined based on various types of generalized inverses [10, 12]. We firstly introduce a new binary relation using the weak group inverse.

Definition 3.1**.**

Let $B\in{\cal B}(X)$ and $A\in{\cal B}(X)^{d}$. Then we say that $A$ is below $B$ under the weak group relation (denoted by $A\leq^{\otimes}B$) if

[TABLE]

Remark that the relation ”$\leq^{\otimes}$” is not a partial order, because it is not antisymmetric. Indeed, if $A,B\in{\cal B}(X)$ are quasinilpotent and $A\neq B$, then $A^{d}=0$ and $B^{d}=0$ imply $A^{\otimes}=0$ and $B^{\otimes}=0$. Hence, $A\leq^{\otimes}B$ and $B\leq^{\otimes}A$, but $A\neq B$.

By the following example, we see that the relation ”$\leq^{\otimes}$” is not transitive and so it is not a pre-order.

Example 3.1. Consider a $3\times 3$ block matrices

[TABLE]

Then

[TABLE]

The equalities $AA^{\otimes}=BA^{\otimes}$, $A^{\otimes}A=A^{\otimes}B$, $BB^{\otimes}=CB^{\otimes}$ and $B^{\otimes}B=B^{\otimes}C$ give $A\leq^{\otimes}B$ and $B\leq^{\otimes}C$. Because $A^{\otimes}A\neq A^{\otimes}C$, the relation $A\leq^{\otimes}C$ is not satisfied and thus ”$\leq^{\otimes}$” is not transitive.

Lemma 3.1**.**

Let $B\in{\cal B}(X)$ and $A\in{\cal B}(X)^{d}$. Then:

• (i)

$AA^{\otimes}=BA^{\otimes}$* $\Leftrightarrow$ $A^{\tiny\tinyd⃝}A=B(A^{\tiny\tinyd⃝})^{2}A$ $\Leftrightarrow$ $A^{\tiny\tinyd⃝}=B(A^{\tiny\tinyd⃝})^{2}$ $\Leftrightarrow$ $A^{d}=BA^{\tiny\tinyd⃝}A^{d}$ $\Leftrightarrow$ $AA^{d}=BA^{d}$;*

• (ii)

$A^{\otimes}A=A^{\otimes}B$* $\Leftrightarrow$ $A^{\tiny\tinyd⃝}A^{2}=A^{\tiny\tinyd⃝}AB$.*

We can get characterizations of the weak group relation combining conditions of Lemma 3.1 from parts (i) and (ii).

For operators between two Hilbert spaces, we consider a weighted operator and define the following binary relations.

Definition 3.2**.**

Let $A,B\in{\cal B}(X,Y)$ and $W\in{\cal B}(Y,X)\backslash\{0\}$. If $A$ is $W$g-Drazin invertible, then we say that

• (i)

$A\leq^{\otimes,W,r}B$ if $AW\leq^{\otimes}BW$,

• (ii)

$A\leq^{\otimes,W,l}B$ if $WA\leq^{\otimes}WB$,

• (iii)

$A\leq^{\otimes,W}B$ if $A\leq^{\otimes,W,r}B$ and $A\leq^{\otimes,W,l}B$,

where $\leq^{\otimes}$ is adequately considered on ${\cal B}(X)$ or ${\cal B}(Y)$.

Several characterizations of the relation $\leq^{\otimes,W,r}$ are presented now.

Theorem 3.1**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$, $A\in{\cal B}(X,Y)^{d,W}$ and $B\in{\cal B}(X,Y)$. Then the following statements are equivalent:

• (i)

$A\leq^{\otimes,W,r}B$;

• (ii)

$AW(AW)^{\otimes}=BW(AW)^{\otimes}$* and $(AW)^{\otimes}AW=(AW)^{\otimes}BW$;*

• (iii)

the following matrix representations with respect to the orthogonal sums $X=R((WA)^{d})\oplus N[((WA)^{d})^{*}]$ and $Y=R((AW)^{d})\oplus N[((AW)^{d})^{*}]$ hold

[TABLE]

where $A_{1}\in{\cal B}(R((WA)^{d}),R((AW)^{d}))^{-1}$, $W_{1}\in{\cal B}(R((AW)^{d}),R((WA)^{d}))^{-1}$, $(A_{2}-B_{2})W_{3}+(A_{1}W_{1})^{-1}(A_{1}W_{2}+A_{2}W_{3})(A_{3}-B_{3})W_{3}=0$, $A_{3}W_{3}\in{\cal B}(N[((AW)^{d})^{*}])^{qnil}$ and $W_{3}A_{3}\in{\cal B}(N[((WA)^{d})^{*}])^{qnil}$.

Proof.

(i) $\Leftrightarrow$ (ii): By the definition of the relation $\leq^{\otimes,W,r}$, this is clear.

(ii) $\Rightarrow$ (iii): Let $A$ and $W$ be given by (1) and (2), respectively. Suppose that

[TABLE]

Then

[TABLE]

[TABLE]

and

[TABLE]

Therefore, $AW(AW)^{\otimes}=BW(AW)^{\otimes}$ is equivalent to $B_{1}=A_{1}$ and $B_{4}=0$. Further, the equalities

[TABLE]

[TABLE]

and $(AW)^{\otimes}AW=(AW)^{\otimes}BW$ give $(A_{2}-B_{2})W_{3}+(A_{1}W_{1})^{-1}(A_{1}W_{2}+A_{2}W_{3})(A_{3}-B_{3})W_{3}=0$.

(iii) $\Rightarrow$ (ii): This part can be checked by direct computations. ∎

In an analogy way, we characterize of the relation $\leq^{\tiny\tinyd⃝,W,l}$.

Theorem 3.2**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$, $A\in{\cal B}(X,Y)^{d,W}$ and $B\in{\cal B}(X,Y)$. Then the following statements are equivalent:

• (i)

$A\leq^{\otimes,W,l}B$;

• (ii)

$WA(WA)^{\otimes}=WB(WA)^{\otimes}$* and $(WA)^{\otimes}WA=(WA)^{\otimes}WB$;*

• (iii)

$WAWA^{\otimes,W}=WBWA^{\otimes,W}$* and $WA^{\otimes,W}WA=WA^{\otimes,W}WB$;*

• (iv)

the following matrix representations with respect to the orthogonal sums $X=R((WA)^{d})\oplus N[((WA)^{d})^{*}]$ and $Y=R((AW)^{d})\oplus N[((AW)^{d})^{*}]$ hold

[TABLE]

where $A_{1}\in{\cal B}(R((WA)^{d}),R((AW)^{d}))^{-1}$ and $W_{1}\in{\cal B}(R((AW)^{d}),R((WA)^{d}))^{-1}$, $W_{3}B_{4}=0$, $B_{2}=A_{2}+W_{1}^{-1}W_{2}(A_{3}-B_{3})+(W_{1}A_{1}W_{1})^{-1}(W_{1}A_{2}+W_{2}A_{3})W_{3}(A_{3}-B_{3})$, $A_{3}W_{3}\in{\cal B}(N[((AW)^{d})^{*}])^{qnil}$ and $W_{3}A_{3}\in{\cal B}(N[((WA)^{d})^{*}])^{qnil}$.

Combining Theorem 3.1 and Theorem 3.2, we get the next results.

Corollary 3.1**.**

Let $W\in{\cal B}(Y,X)\backslash\{0\}$, $A\in{\cal B}(X,Y)^{d,W}$ and $B\in{\cal B}(X,Y)$. Then the following statements are equivalent:

• (i)

$A\leq^{\otimes,W}B$;

• (ii)

the following matrix representations with respect to the orthogonal sums $X=R((WA)^{d})\oplus N[((WA)^{d})^{*}]$ and $Y=R((AW)^{d})\oplus N[((AW)^{d})^{*}]$ hold

[TABLE]

where $A_{1}\in{\cal B}(R((WA)^{d}),R((AW)^{d}))^{-1}$, $W_{1}\in{\cal B}(R((AW)^{d}),R((WA)^{d}))^{-1}$, $B_{2}=A_{2}+W_{1}^{-1}W_{2}(A_{3}-B_{3})+(W_{1}A_{1}W_{1})^{-1}(W_{1}A_{2}+W_{2}A_{3})W_{3}(A_{3}-B_{3})$, $W_{2}A_{3}W_{3}(A_{3}-B_{3})W_{3}=0$, $A_{3}W_{3}\in{\cal B}(N[((AW)^{d})^{*}])^{qnil}$ and $W_{3}A_{3}\in{\cal B}(N[((WA)^{d})^{*}])^{qnil}$.

Corollary 3.2**.**

Let $A\in{\cal B}(X)^{d}$ and $B\in{\cal B}(X)$. Then the following statements are equivalent:

• (i)

$A\leq^{\otimes}B$;

• (ii)

the following matrix representations with respect to the orthogonal sum $X=R(A^{d})\oplus N[(A^{d})^{*}]$ hold

[TABLE]

where $A_{1}\in{\cal B}(R(A^{d}))$ is invertible, $A_{3}\in{\cal B}(N[(A^{d})^{*}])$ is quasinilpotent.