ON THE EQUATION ${\bf{\it h}}x-x{\bf{\it k}}=c$ IN BANACH ALGEBRA ${\bf{\it A}}$, WITH HERMITIAN COEFFICIENTS ${\bf{\it h}}, {\bf{\it k}}$

Abstract

This work investigates the problem of solvability on the equation ${\bf{\it h}}x-x{\bf{\it k}}=c$ with Hermitian coefficients ${\bf{\it h}}, {\bf{\it k}}$ of weak complete Banach algebra ${\bf{\it A}}$. In the article (theorem 1) the criterion of solvability of equation ${\bf{\it h}}x-x{\bf{\it k}}=c$ is proved which implies (theorem 2) the following algebraic criterion of solvability. For solvability in ${\bf {\it A}}$ the equation ${\bf{\it h}}x-x{\bf{\it k}}=c$ the similarity of matrixes $\Big( \begin{matrix} {\bf{\it h}}, & 0 \\ 0, & {\bf{\it k}}\\ \end{matrix} \Big)$ and  $\Big(\begin{matrix} {\bf{\it h}}, & c\\ 0, & {\bf{\it k}} \\ \end{matrix} \Big)$ is necessary and sufficient.