Commuting linear operators and algebraic decompositions

For commuting linear operators $P_0,P_1,\mathrm{.}\mathrm{.}\mathrm{.},P_{\mathrm{\ell }}$ we describe a range of conditions which are weaker than invertibility. When any of these conditions hold we may study the composition $P=P_0{P}_1\mathrm{.}\mathrm{.}\mathrm{.}{P}_{\mathrm{\ell }}$ in terms of the component operators or combinations thereof. In particular the general inhomogeneous problem $Pu=f$ reduces to a system of simpler problems. These problems capture the structure of the solution and range spaces and, if the operators involved are differential, then this gives an effective way of lowering the differential order of the problem to be studied. Suitable systems of operators may be treated analogously. For a class of decompositions the higher symmetries of a composition $P$ may be derived from generalised symmmetries of the component operators $P_i$ in the system.