Abstract
Toeplitz operators on strictly pseudo-convex boundaries of complex domains are defined; they behave like pseudo-differential operators. An extension of the Atiyah-Singer formula is proved for elliptic systems of such operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atiyah, M.F.:K-Theory. Amsterdam: Benjamin 1967
Atiyah, M.F., and Singer, I.M.: The index of elliptic operators I. Ann. Math.87, 484–530 (1968)
Atiyah, M., and Singer, I.M.: The index of elliptic operators III. Ann. Math.87, 546–604 (1968)
Atiyah, M.F., and Singer, I.M.: The index of elliptic operators IV. Ann. Math.92, 119–138 (1970)
Bony, J.M., and Schapira, P.: Existence et prolongement des solutions holomorphes des equations aux derivées partielles. Inventiones Math.17, 95–105 (1972)
Boutet de Monvel, L.: Boundary problems for pseudo-differential operators. Acta Math.126, 11–51 (1971)
Boutet de Monvel, L.: Integration des equations de Cauchy-Riemann induites. Seminaire Goulaouic-Schwartz. 1974–75 exposé no 9
Boutet de Monvel, L.: Hypoelliptic operators with Double characteristics and related pseudodifferential operators. Comm. Pure Appl. Math.27, 585–639 (1974)
Boutet de Monvel, L., and Guillemin, V.: Spectral theory of Toeplitz operators. To appear.
Boutet de Monvel, L., and Sjöstrand, J.: Sur la singularité des noyaux de Bergmann et de Szegö. Asterisque34–35, 123–164 (1976)
Burns, D.: On Venugopalkrishna's index problem. Rencontre sur l'analyse complexe à plusiems variable et les systèmes surdeterminés, Montreal 1975.
Dynin, A.S.: An Algebra of pseudo-differential operators on the Heisenberg group. Dokl. Akad. Nauk SSSR 227 (1976) no 4. Soviet Math. Dokl.17, no 2, 508–512 (1976)
Fedosov, B.V.: Analytical formulas for the index of elliptic operators. Trudy Moskov. Mat. Obsc.30 (1974), Transactions Mosc. Math. Soc.30, 159–240 (1974)
Grauert, H.: Analytische Faserungen über holomorph-vollständigen Räumen. Math. Ann.135, 263–273 (1958)
Guillemin, V.: Symplectic spinors and partial Differential equations. Coll. Int. CNRS no 237 Géométrie symplectique et physique mathemtique, 217–252
Hörmander, L.: Fourier integral operators I. Acta Math.127, 79–183 (1971)
Hörmander, L.: The Weyl calculus of pseudo-differential operators. To appear
Kashiwara, M.: Analyse microlocale du noyau de Bergman. Seminaire Goulaouic-Schwartz 1976–1977, exp. no8
Kohn, J.J.: Regularity at the boundary of the\(\bar \partial \) Neumann problem. Proc. Nat. Acad. Sci. USA49, 206–213 (1963)
Melin, A., Sjöstrand, J.: Fourier integral operators with complex valued phase functions — in Fourier integral operators and partial differential equations, Lecture Notes no 459, pp. 120–223. Berlin-Göttingen-Heidelberg: Springer 1975
K-Theory and operator algebras, Lecture Notes no575. Berlin-Heidelberg New York: Springer 1977
Sato, M., Kawai, T., Kashiwara, M.: Mikrofunctions and pseudo-differential equations, Lecture Notes no 287, Chap. II. Berlin-Heidelberg-New York: Springer 1973
Venugopalkrishna, V.: Fredholm operators associated with strongly pseudoconvex domains in ℂn. J. Funct. Anal.9, 349–373 (1972)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
de Monvel, L.B. On the index of Toeplitz operators of several complex variables. Invent Math 50, 249–272 (1978). https://doi.org/10.1007/BF01410080
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01410080