Solvability of Systems of Linear Operator Equations

Let be a semigroup of commuting linear operators on a linear space with the group operation of composition. The solvability of the system of equations <!-- MATH ${l_i}f = {\phi _i},\;i = 1,\, \ldots \,,\,r$ --> , where <!-- MATH ${l_i} \in G$ --> and <!-- MATH ${\phi _i} \in S$ --> , was considered by Dahmen and Micchelli in their studies of the dimension of the kernel space of certain linear operators. The compatibility conditions <!-- MATH ${l_j}{\phi _i} = {l_i}{\phi _j},i \ne j$ --> , are necessary for the system to have a solution in . 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 .