  For linear operators which factor with suitable assumptions concerning
commutativity of the factors, we introduce several notions of a decomposition.
When any of these hold then questions of null space and range are subordinated
to the same questions for the factors, or certain compositions thereof. When
the factors are polynomial in other commuting operators then we show that, in a
suitable sense, generically factorisations algebraically yield decompositions.
In the case of operators on a space over an algebraically closed field this
boils down to elementary algebraic geometry arising from the polynomial formula
for the orginal operator. Applied to operators P polynomial in single other
operator D this shows that the solution space for P decomposes directly into a
sum of generalised eigenspaces for D. We give universal formulae for the
projectors administering the decomposition. In the generic setting the
inhomogenous problems for P reduce to an equivalent inhomogeneous problem for
an operator linear in D. These results are independent of the operator D, and
so provide a route to progressing such questions when functional calculus is
unavailable. Related generalising results are obtained as well as a treatment
for operators on vector spaces over arbitrary fields. We introduce and discuss
symmetry algebras for such operators. As a motivating example application we
treat, on Einstein manifolds, the conformal Laplacian operators of
Graham-Jenne-Mason-Sparling.
