A $W$-weighted generalization of $\{1,2,3,1^{k}\}$-inverse for rectangular matrices

This paper presents a novel extension of the $\{1,2,3,1^{k}\}$-inverse concept to complex rectangular matrices, denoted as a $W$-weighted $\{1,2,3,1^{k}\}$-inverse (or $\{1',2',3',{1^{k}}'\}$-inverse), where the weight $W \in \mathbb{C}^{n \times m}$. The study begins by introducing a weighted $\{1,2,3\}$-inverse (or $\{1',2',3'\}$-inverse) along with its representations and characterizations. The paper establishes criteria for the existence of $\{1',2',3'\}$-inverses and extends the criteria to $\{1'\}$-inverses. It is further demonstrated that $A\in \mathbb{C}^{m \times n}$ admits a $\{1',2',3',{1^{k}}'\}$-inverse if and only if $r(WAW)=r(A)$, where $r(\cdot)$ is the rank of a matrix. The work additionally establishes various representations for the set $A\{ 1',2',3',{1^{k}}'\}$, including canonical representations derived through singular value and core-nilpotent decompositions. This, in turn, yields distinctive canonical representations for the set $A\{ 1,2,3,{1^{k}}\}$. $\{ 1',2',3',{1^{k}}'\}$-inverse is shown to be unique if and only if it has index $0$ or $1$, reducing it to the weighted core inverse. Moreover, the paper investigates properties and characterizations of $\{1',2',3',{1^{k}}'\}$-inverses, which then results in new insights into the characterizations of the set $A\{ 1,2,3,{1^{k}}\}$.