Commuting linear operators and decompositions; applications to Einstein manifolds

Abstract

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.