SOLVABILITY OF SYSTEMS OF LINEAR OPERATOR-EQUATIONS

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,..., r, where l(i) is-an-element-of G and phi(i) is-an-element-of 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 not-equal 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 .