Commuting Linear Operators and Decompositions; Applications to Einstein Manifolds

For linear operators which factor P=P ₀ P ₁ ??? P _? , 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 operators P _i are polynomial in other commuting operators \(\mathcal{D}_{1},\ldots,\mathcal{D}_{k}\) then we show that, in a suitable sense, generically factorisations algebraically yield decompositions. In the case of operators on a vector space over an algebraically closed field this boils down to elementary algebraic geometry arising from the polynomial formula for P. The results and formulae are independent of the \(\mathcal{D}_{j}\) and so the theory provides a route to studying the solution space and the inhomogenous problem Pu=f without any attempt to “diagonalise” the \(\mathcal{D}_{j}\). Applications include the construction of fundamental solutions (or “Greens functions”) for PDE; analysis of the symmetry algebra for PDE; direct decompositions of Lie group representations into Casimir generalised eigenspaces and related decompositions of vector bundle section spaces on suitable geometries. Operators P polynomial in a single other operator \(\mathcal{D}\) form the simplest case of the general development and here we give universal formulae for the projectors administering the decomposition. As a concrete geometric application, on Einstein manifolds we describe the direct decomposition of the solution space and the general inhomogeneous problem for the conformal Laplacian operators of Graham-Jenne-Mason-Sparling.