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 syste m 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 Micch elli 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 di scuss what kinds of conditions on operators will make the compatibilit y sufficient for such systems to be solvable in S .