Calculation of particular solutions of nonhomogeneous linear equationswith two linear operators, of which at least one is almost algebraic,in the case of simple roots of the characteristic equation

The concept of an operator is an almost algebraic with respect to two-sided ideal of the algebra of linear operators in some finite-dimensional linear spaces, it extended to the case when the ideal is left. We prove a theorem on the following equation particular solution \[\sum_{i=0,j=0}^{n,m}a_{i j}A^iB^ju = f,\] where $A$ and $B$ is a linear operator, $f$ is an element of a linear space. The result is applied to the differential-difference equations.