Abstract
In this note we study the analytical index of pseudo-differential operators by using the notion of (infinite dimensional) operator-valued symbols (in the sense of Ruzhansky and Turunen). Our main tools will be the McKean–Singer index formula together with the operator-valued functional calculus developed here.
Similar content being viewed by others
References
Atiyah, M.F., Bott, R.: The Index Problem for Manifolds with Boundary. Differential Analysis, Bombay Colloquium. Oxford University Press, London (1964)
Atiyah, M., Bott, R., Patodi, V.K.: On the heat equation and the index theorem. Invent. Math. 19, 279–330 (1973)
Atiyah, M.F., Singer, I.M.: The index of elliptic operators on compact manifolds. Bull. Am. Math. Soc. 69, 422–433 (1963)
Atiyah, M.F., Singer, I.M.: The index of elliptic operators. I. Ann. Math. (2) 87, 484–530 (1968)
Atiyah, M.F., Segal, G.: The index of elliptic operators. II. Ann. Math. (2) 87, 531–545 (1968)
Atiyah, M.F., Singer, I.M.: The index of elliptic operators. III. Ann. Math. (2) 87, 546–604 (1968)
Atiyah, M.F., Singer, I.M.: Index theory for skew-adjoint Fredholm operators. Inst. Hautes Etudes Sci. Publ. Math. 37, 5–26 (1969)
Atiyah, M.F., Singer, I.M.: The index of elliptic operators. IV. Ann. Math. (2) 93, 119–138 (1971)
Atiyah, M.F., Singer, I.M.: The index of elliptic operators. V. Ann. Math. (2) 93, 139–149 (1971)
Bleecker, D., Booss-Bavnbek, B.: Index Theory with Applications to Mathematics and Physics. International Press, Somerville (2013)
Berline, N., Vergne, M.: A computation of the equivariant index of the Dirac operator. Bull. Soc. Math. France 113(3), 305–345 (1985)
Bott, R.: The index theorem for homogeneous differential operators. In: Cairns, S.S. (ed.) Differential and Combinatorial Topology, pp. 167–186. Princeton University Press, Princeton (1965)
Bronzan, J.B.: Parametrization of SU(3). Phys. Rev. D. 38(6), 1994–1999 (1988)
Delgado, J., Ruzhansky, M.: \(L^p\)-nuclearity, traces, and Grothendieck–Lidskii formula on compact Lie groups. J. Math. Pures Appl. 102(1), 153–172 (2014)
Delgado, J., Ruzhansky, M.: Schatten classes on compact manifolds: kernel conditions. J. Funct. Anal. 267(3), 772–798 (2014)
Delgado, J., Ruzhansk, M.: Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds. C. R. Acad. Sci. Paris Ser. I 352, 779–784 (2014)
Fedosov, B.: Index theorems. In: Partial Differential Equations, VIII, encyclopedia of Mathematical Sciences, vol. 65, pp. 155–251. Springer, Berlin (1996)
Fedosov, B.: Deformation Quantization and Index Theory. Akademie Verlag, New York (1995)
Fegan, H.: Introduction to Compact Lie Groups. Series in Pure Mathematics, vol. 13. World Scientific Publishing Co., Inc., River Edge (1991)
Fischer, V.: Intrinsic pseudo-differential calculus on compact Lie groups. J. Funct. Anal. 268(11), 3404–3477 (2015)
Goette, S.: Equivariant \(\eta \)-invariants on homogeneous spaces. Math. Z. 232(1), 1–42 (1999)
Hörmander, L.: The Analysis of the Linear Partial Differential Operators, vol. III. Springer, Berlin (1985)
Hong, S.: A Lie-algebraic approach to the local index theorem on compact homogeneous spaces. Adv. Math. 296, 127–153 (2016)
Hong, S.: Borel–Weil–Bott theorem via the equivariant McKean–Singer formula. arXiv:1412.3879
Roe, J.: Elliptic Operators, second edn. Addison Wesley, Boston (1998)
Ruzhansky, M., Turunen, V.: Quantization of pseudo-differential operators on the torus. J. Fourier Anal. Appl. 16, 943–982 (2010)
Ruzhansky, M., Turunen, V.: Pseudo-Differential Operators and Symmetries: Background Analysis and Advanced Topics. Birkhäuser-Verlag, Basel (2010)
Ruzhansky, M., Turunen, V., Wirth, J.: Hörmander class of pseudo-differential operators on compact Lie groups and global hypoellipticity. J. Fourier Anal. Appl. 20, 476–499 (2014)
Ruzhansky, M., Turunen, V.: Global quantization of pseudo-differential operators on compact Lie groups, SU(2) and 3-sphere. Int. Math. Res. Not. IMRN 2013(11), 2439–2496 (2013)
Ruzhansky, M., Wirth, J.: Global functional calculus for operators on compact Lie groups. J. Funct. Anal. 267, 144–172 (2014)
Ruzhansky, M., Wirth, J.: \(L^p\) Fourier multipliers on compact Lie groups. Math. Z. 280, 621–642 (2015)
Sabin, A.Y.: On the index of nonlocal elliptic operators for compact Lie groups. Cent. Eur. J. Math. 9(4), 833–850 (2011)
Urakawa, H.: The heat equation on compact Lie group. Osaka J. Math. 12(2), 285–297 (1975)
Weidmann, J.: Linear operators in Hilbert spaces. Translated from the German by Joseph Szücs. Graduate Texts in Mathematics, vol. 68. Springer, New York (1980)
Acknowledgements
I would like to thank Alexander Cardona and Michael Ruzhansky for various discussions. I also would like to thanks Julio Delgado who has suggested me a gap in the proof of Proposition 4.8 in a previous version of this document. The author is indebted with the referee of this paper by his/her asserted suggestions which helped to improve the manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cardona, D. On the index of pseudo-differential operators on compact Lie groups. J. Pseudo-Differ. Oper. Appl. 10, 285–305 (2019). https://doi.org/10.1007/s11868-018-0261-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11868-018-0261-0