Geodesic
On the equation $\mathbf{h}x - x \mathbf{k}=c$ in Banach algebra $A$ with Hermitian coefficients $\mathbf{h},\mathbf{k}$
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2002), pp. 3-8

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This work investigates the problem of solvability on the equation $\mathbf{h}x - x\mathbf{k}=c$ with Hermitian coefficients $\mathbf{h},\mathbf{k}$ of weak complete Banach algebra $A$. In the article (theorem 1) the criterion of solvability of equation $\mathbf{h}x-x\mathbf{k}=c$ is proved, which implies (theorem 2) the following algebraic criterion of solvability. For solvability in $A$ the equation $\mathbf{h}x-x\mathbf{k}=c$ the similarity of matrixes $\left(\begin {array}{cc}\mathbf{h}, 0 \\ 0, \mathbf{k} \end {array} \right)$ and $\left(\begin {array}{cc}\mathbf{h}, 0 \\ c, \mathbf{k} \end {array} \right)$ is necessary and sufficient.
Keywords: solvability on the equation, Banach algebra.
Mots-clés : Hermitian coefficients 
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     title = {On the equation $\mathbf{h}x - x \mathbf{k}=c$ in {Banach} algebra $A$ with  {Hermitian} coefficients $\mathbf{h},\mathbf{k}$},
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I. M. Karakhanyan. On the equation $\mathbf{h}x - x \mathbf{k}=c$ in Banach algebra $A$ with  Hermitian coefficients $\mathbf{h},\mathbf{k}$. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2002), pp. 3-8. http://geodesic.mathdoc.fr/item/UZERU_2002_1_a0/