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Register a new userCharacterizations of core and dual core inverses in rings with involution
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Let R be a unital ring with involution, we give the characterizations and representations of the core and dual core inverses of an element in R by Hermitian elements (or projections) and units. For example, let a in R and n is an integer greater than or equal to 1, then a is core invertible if and only if there exists a Hermitian element (or a projection) p such that pa=0, a^n+p is invertible. As a consequence, a is an EP element if and only if there exists a Hermitian element (or a projection) p such that pa=ap=0, a^n+p is invertible. We also get a new characterization for both core invertible and dual core invertible of a regular element by units, and their expressions are shown. In particular, we prove that for n is an integer greater than or equal to 2, a is both Moore-Penrose invertible and group invertible if and only if (a*)^n is invertible along a.
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$R$ is a unital ring with involution. We investigate the characterizations and representations of weighted core inverse of an element in $R$ by idempotents and units. For example, let $ain R$ and $ein R$ be an invertible Hermitian element, $ngeqslant 1$, then $a$ is $e$-core invertible if and only if there exists an element (or an idempotent) $p$ such that $(ep)^{ast}=ep$, $pa=0$ and $a^{n}+p$ (or $a^{n}(1-p)+p$) is invertible. As a consequence, let $e, fin R$ be two invertible Hermitian elements, then $a$ is weighted-$mathrm{EP}$ with respect to $(e, f)$ if and only if there exists an element (or an idempotent) $p$ such that $(ep)^{ast}=ep$, $(fp)^{ast}=fp$, $pa=ap=0$ and $a^{n}+p$ (or $a^{n}(1-p)+p$) is invertible. These results generalize and improve conclusions in cite{Li}.
Let $mathscr{C}$ be an additive category with an involution $ast$. Suppose that $varphi : X rightarrow X$ is a morphism with kernel $kappa : K rightarrow X$ in $mathscr{C}$, then $varphi$ is core invertible if and only if $varphi$ has a cokernel $lambda: X rightarrow L$ and both $kappalambda$ and $varphi^{ast}varphi^3+kappa^{ast}kappa$ are invertible. In this case, we give the representation of the core inverse of $varphi$. We also give the corresponding result about dual core inverse.
In this paper, we introduce two new generalized inverses of matrices, namely, the $bra{i}{m}$-core inverse and the $pare{j}{m}$-core inverse. The $bra{i}{m}$-core inverse of a complex matrix extends the notions of the core inverse defined by Baksalary and Trenkler cite{BT} and the core-EP inverse defined by Manjunatha Prasad and Mohana cite{MM}. The $pare{j}{m}$-core inverse of a complex matrix extends the notions of the core inverse and the ${rm DMP}$-inverse defined by Malik and Thome cite{MT}. Moreover, the formulae and properties of these two new concepts are investigated by using matrix decompositions and matrix powers.
Let $mathscr{C}$ be an additive category with an involution $ast$. Suppose that $varphi : X rightarrow X$ is a morphism of $mathscr{C}$ with core inverse $varphi^{co} : X rightarrow X$ and $eta : X rightarrow X$ is a morphism of $mathscr{C}$ such that $1_X+varphi^{co}eta$ is invertible. Let $alpha=(1_X+varphi^{co}eta)^{-1},$ $beta=(1_X+etavarphi^{co})^{-1},$ $varepsilon=(1_X-varphivarphi^{co})etaalpha(1_X-varphi^{co}varphi),$ $gamma=alpha(1_X-varphi^{co}varphi)beta^{-1}varphivarphi^{co}beta,$ $sigma=alphavarphi^{co}varphialpha^{-1}(1_X-varphivarphi^{co})beta,$ $delta=beta^{ast}(varphi^{co})^{ast}eta^{ast}(1_X-varphivarphi^{co})beta.$ Then $f=varphi+eta-varepsilon$ has a core inverse if and only if $1_X-gamma$, $1_X-sigma$ and $1_X-delta$ are invertible. Moreover, the expression of the core inverse of $f$ is presented. Let $R$ be a unital $ast$-ring and $J(R)$ its Jacobson radical, if $ain R^{co}$ with core inverse $a^{co}$ and $jin J(R)$, then $a+jin R^{co}$ if and only if $(1-aa^{co})j(1+a^{co}j)^{-1}(1-a^{co}a)=0$. We also give the similar results for the dual core inverse.
Let $mathscr{C}$ be a category with an involution $ast$. Suppose that $varphi : X rightarrow X$ is a morphism and $(varphi_1, Z, varphi_2)$ is an (epic, monic) factorization of $varphi$ through $Z$, then $varphi$ is core invertible if and only if $(varphi^{ast})^2varphi_1$ and $varphi_2varphi_1$ are both left invertible if and only if $((varphi^{ast})^2varphi_1, Z, varphi_2)$, $(varphi_2^{ast}, Z, varphi_1^{ast}varphi^{ast}varphi)$ and $(varphi^{ast}varphi_2^{ast}, Z, varphi_1^{ast}varphi)$ are all essentially unique (epic, monic) factorizations of $(varphi^{ast})^2varphi$ through $Z$. We also give the corresponding result about dual core inverse. In addition, we give some characterizations about the coexistence of core inverse and dual core inverse of an $R$-morphism in the category of $R$-modules of a given ring $R$.
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