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Register a new userCharacterizations of weighted core inverse in rings with involution
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Tingting Li
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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}.
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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.
We show that a von Neumann regular ring with involution is directly finite provided that it admits a representation as a ring of endomorphisms (the involution given by taking adjoints) of a vector space endowed with a non-degenerate orthosymmetric sesquilinear form.
In this paper, we investigate the weighted core-EP inverse introduced by Ferreyra, Levis and Thome. Several computational representations of the weighted core-EP inverse are obtained in terms of singular-value decomposition, full-rank decomposition and QR decomposition. These representations are expressed in terms of various matrix powers as well as matrix product involving the core-EP inverse, Moore-Penrose inverse and usual matrix inverse. Finally, those representations involving only Moore-Penrose inverse are compared and analyzed via computational complexity and numerical examples.
In [Q. Xu et al., The solutions to some operator equations, Linear Algebra Appl.(2008), doi:10.1016/j.laa.2008.05.034], Xu et al. provided the necessary and sufficient conditions for the existence of a solution to the equation $AXB^*-BX^*A^*=C$ in the general setting of the adjointable operators between Hilbert $C^*$-modules. Based on the generalized inverses, they also obtained the general expression of the solution in the solvable case. In this paper, we generalize their work in the more general setting of ring $R$ with involution * and reobtain results for rectangular matrices and operators between Hilbert $C^*$-modules by embedding the rectangles into rings of square matrices or rings of operators acting on the same space.
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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