Gover, A. R.; Šilhan, Josef Commuting linear operators and decompositions; applications to Einstein manifolds. (English) Zbl 1195.47038 Acta Appl. Math. 109, No. 2, 555-589 (2010). If a linear endomorphism \(D\) of an \(\mathbb{F}\)-vector space is inserted into a polynomial \(P\), then one can study the solutions of the equation \(P(D)u= 0\) of \(P(D)u= f\) by a decomposition of the space into generalized eigenspaces, by analogy with the Jordan normal form, provided that the field \(\mathbb{F}\) is algebraically closed. Moreover, one can study other decompositions of the operator \(P(D)\) as products of mutually commuting factors.In the present paper, general results on this type of decomposition are formulated, for polynomials in a single operator as well as for polynomials in commuting endomorphisms. This leads to applications in various directions. As a concrete geometric application, on Einstein spaces the general inhomogeneous problem for the conformal Laplacian operator of Graham-Jenne-Mason-Sparling (the GJMS-Operator) is studied and solved by this decomposition. Reviewer: Wolfgang Kühnel (Stuttgart) Cited in 3 Documents MSC: 47H60 Multilinear and polynomial operators 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) Keywords:Casimir operator; generalised eigenspace; Laplacian; pseudo-Riemannian Einstein space × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Aubry, E., Guillarmou, C.: Conformal harmonic forms, Branson-Gover operators and Dirichlet problem at infinity. Preprint arXiv:0808.0552 , http://www.arxiv.org · Zbl 1225.53033 [2] Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987). xii+510 · Zbl 0613.53001 [3] Branson, T.: The Functional Determinant. Global Analysis Research Center Lecture Note Series, vol. 4. Seoul National University, Seoul (1993) [4] Branson, T.: Sharp inequalities, the functional determinant, and the complementary series. Trans. Am. Math. Soc. 347, 3671–3742 (1995) · Zbl 0848.58047 · doi:10.2307/2155203 [5] Branson, T., Gover, A.R.: Conformally invariant operators, differential forms, cohomology and a generalisation of Q curvature. Commun. Partial Differ. Equ. 30, 1611–1669 (2005) · Zbl 1226.58011 · doi:10.1080/03605300500299943 [6] Branson, T., Gover, A.R.: Pontrjagin forms and invariant objects related to the Q-curvature. Commun. Contemp. Math. 9, 335–358 (2007) · Zbl 1125.53024 · doi:10.1142/S0219199707002460 [7] Čap, A., Gover, A.R.: Tractor calculi for parabolic geometries. Trans. Am. Math. Soc. 354, 1511–1548 (2002) · Zbl 0997.53016 · doi:10.1090/S0002-9947-01-02909-9 [8] Čap, A., Souček, V.: Curved Casimir operators and the BGG machinery. SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 111, 17, pp. 22 [9] Chang, S.-Y.A., Qing, J., Yang, P.: On the Chern-Gauss-Bonnet integral for conformal metrics on R 4. Duke Math. J. 103, 523–544 (2000) · Zbl 0971.53028 · doi:10.1215/S0012-7094-00-10335-3 [10] Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd edn. Undergraduate Texts in Mathematics. Springer, New York (1997). xiv+536 pp [11] Dirac, P.A.M.: Wave equations in conformal space. Ann. Math. 37, 429–442 (1936) · Zbl 0014.08004 · doi:10.2307/1968455 [12] Djadli, Z., Malchiodi, A.: Existence of conformal metrics with constant Q-curvature. Ann. Math. (to appear). Preprint math.AP/0410141 , http://www.arxiv.org · Zbl 1186.53050 [13] Eastwood, M.: Higher symmetries of the Laplacian. Ann. Math. 161, 1645–1665 (2005) · Zbl 1091.53020 · doi:10.4007/annals.2005.161.1645 [14] Eastwood, M., Leistner, T.: Higher symmetries of the square of the Laplacian. In: Symmetries and Overdetermined Systems of Partial Differential Equations. IMA Vol. Math. Appl., vol. 144, pp. 319–338. Springer, New York (2008) · Zbl 1137.58014 [15] Eschmeier, J.: Local properties of Taylor’s analytic functional calculus. Invent. Math. 68, 103–116 (1982) · Zbl 0503.47014 · doi:10.1007/BF01394269 [16] Fefferman, C., Graham, C.R.: Q-curvature and Poincaré metrics. Math. Res. Lett. 9, 139–151 (2002) · Zbl 1016.53031 [17] Fefferman, C., Graham, C.R.: The ambient metric. Preprint arXiv:0710.0919 , http://www.arxiv.org · Zbl 1243.53004 [18] Gover, A.R.: Laplacian operators and Q-curvature on conformally Einstein manifolds. Math. Ann. 336, 311–334 (2006) · Zbl 1125.53032 · doi:10.1007/s00208-006-0004-z [19] Gover, A.R., Graham, C.R.: CR invariant powers of the sub-Laplacian. J. Reine Angew. Math. 583, 1–27 (2005) · Zbl 1076.53048 · doi:10.1515/crll.2005.2005.583.1 [20] Gover, A.R., Peterson, L.J.: Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus. Commun. Math. Phys. 235, 339–378 (2003) · Zbl 1022.58014 · doi:10.1007/s00220-002-0790-4 [21] Gover, A.R., Šilhan, J.: Commuting linear operators and algebraic decompositions. Arch. Math. 43(5), 373–387 (2007). arXiv:0706.2404 , http://arxiv.org/ · Zbl 1199.53020 [22] Gover, A.R., Šilhan, J.: Conformal operators on forms and detour complexes on Einstein manifolds. Commun. Math. Phys. (to appear). arXiv:0708.3854 , http://arxiv.org/ · Zbl 1158.53039 [23] Graham, C.R.: Private communication [24] Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152, 89–118 (2003) · Zbl 1030.58022 · doi:10.1007/s00222-002-0268-1 [25] Graham, C.R., Jenne, R., Mason, L.J., Sparling, G.A.: Conformally invariant powers of the Laplacian, I: Existence. J. Lond. Math. Soc. 46, 557–565 (1992) · doi:10.1112/jlms/s2-46.3.557 [26] Gromov, M.: Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 9. Springer, Berlin (1986). x+363 pp · Zbl 0651.53001 [27] Müller, V.: Local behaviour of the polynomial calculus of operators. J. Reine Angew. Math. 430, 61–68 (1992) · Zbl 0749.47002 · doi:10.1515/crll.1992.430.61 [28] Müller, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Operator Theory: Advances and Applications, vol. 139. Birkhäuser, Basel (2003). x+381 pp · Zbl 1017.47001 [29] Paneitz, S.: A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008). Paper 036, 3 pp · Zbl 1145.53053 [30] Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20(2), 479–495 (1984) · Zbl 0576.53028 [31] Šilhan, J.: Invariant operators in conformal geometry. Ph.D. thesis, University of Auckland (2006) [32] Taylor, J.L.: A joint spectrum for several commuting operators. J. Funct. Anal. 6, 172–191 (1970) · Zbl 0233.47024 · doi:10.1016/0022-1236(70)90055-8 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.