Publication details

Commuting Linear Operators and Decompositions; Applications to Einstein Manifolds



Year of publication 2008
Type Article in Periodical
Magazine / Source Acta Applicandae Mathematicae
MU Faculty or unit

Faculty of Science

Field General mathematics
Keywords Commuting linear operators; Conformally invariant operators; Einstein manifolds; Symmetries of differential operators
Description For linear operators which factor P=P0 P1 ... Pp , 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 Pi are polynomial in other commuting operators D1,...,Dk then we show that, in a suitable sense, generically such factorisation of Pi yield decompositions algebraically. 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 Dj and so the theory provides a route to studying the solution space and the inhomogenous problem Pu=f without any attempt to 'diagonalise' the Dj. 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 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.
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