Weighted $(b,c)$-core inverses in semigroups with involution

Abstract

Let $S$ be a $*$-moniod and let $a,b,c,v,w\in S$. In this paper, we define the $(v,w)$-weighted $(b,c)$-core inverse of $a$. The element $a$ is called $(v,w)$-weighted $(b,c)$-core invertible if there exists an $x\in S$ such that $xvcawx=x,xvS=bS$ and $Swx=Sc^*$. It is shown that the core inverse, the $w$-core inverse and the $(b,c)$-core inverse are special cases of the defined $(v,w)$-weighted $(b,c)$-core inverse. Several criteria for the $(e,w)$-weighted $(b,c)$-core inverse are given,where $e$ is an invertible Hermitian element. For instance, it is proved that $a$ is $(e,w)$-weighted $(b,c)$-core invertible if and only if there exists some $x\in bS$ such that $xecawb=b$, $cawxec=c$  and $(cawx)^*=cawx$ if and only if $aw$ is $(b,c)$-invertible and $c$ ($ca$ or $cawb$) is $\{e, 1, 3\}$-invertible. The dual $(v,w)$-weighted $(b,c)$-core inverse of $a$ is defined by the existence of $y\in S$ satisfying $yvabwy=y$, $yvS=b^*S$ and $Swy=Sc$. Dual results for the dual $(v,w)$-weighted $(b,c)$-core inverse are also established. Finally, when $S$ is a unital $*$-ring, the (dual) weighted $(b,c)$-core inverse is characterized by the direct sum.