Solvability of systems of linear operator equations

Abstract:

Let $G$ be a semigroup of commuting linear operators on a linear space $S$ with the group operation of composition. The solvability of the system of equations ${l_i}f = {\phi _i},\;i = 1, \ldots , r$, where ${l_i} \in G$ and ${\phi _i} \in S$, was considered by Dahmen and Micchelli in their studies of the dimension of the kernel space of certain linear operators. The compatibility conditions ${l_j}{\phi _i} = {l_i}{\phi _j},i \ne j$, are necessary for the system to have a solution in $S$. However, in general, they do not provide sufficient conditions. We discuss what kinds of conditions on operators will make the compatibility sufficient for such systems to be solvable in $S$.