Solvability of systems of linear operator equations
Rong Qing Jia, S. D. Riemenschneider, Zuowei Shen · Proceedings of the American Mathematical Society · 1994
Let G G be a semigroup of commuting linear operators on a linear space S S with the group operation of composition. The solvability of the system of equations l i f = ϕ i , i = 1 , … , r {l_i}f = {\phi _i},\;i = 1,\, \ldots \,,\,r , where l i ∈ G {l_i} \in G and ϕ i ∈ S {\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 ϕ i = l i ϕ j , i ≠ j {l_j}{\phi _i} = {l_i}{\phi _j},i e j , are necessary for the system to have a solution in S 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 S .