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Register a new userGeneral solutions to equation $axb^*-bx^*a^*=c$ in rings with involution
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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.
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This paper deals with necessary and sufficient condition for consistency of the matrix equation $AXB = C$. We will be concerned with the minimal number of free parameters in Penroses formula $X = A^(1)CB^(1) + Y - A^(1)AYBB^(1)$ for obtaining the general solution of the matrix equation and we will establish the relation between the minimal number of free parameters and the ranks of the matrices A and B. The solution is described in the terms of Rohdes general form of the {1}-inverse of the matrices A and B. We will also use Kronecker product to transform the matrix equation $AXB = C$ into the linear system $(B^T otimes A)vecX = vec C$.
$R$ 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}.
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.
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 introduce an algebraic version of the Katsura $C^*$-algebra of a pair $A,B$ of integer matrices and an algebraic version of the Exel-Pardo $C^*$-algebra of a self-similar action on a graph. We prove a Graded Uniqueness Theorem for such algebras and construct a homomorphism of the latter into a Steinberg algebra that, under mild conditions, is an isomorphism. Working with Steinberg algebras over non-Hausdorff groupoids we prove that in the unital case, our algebraic version of Katsura $C^*$-algebras are all isomorphic to Steinberg algebras.
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