Abstract
In this paper we define trace functionals on the algebra of pseudo-differential operators with cone-shaped exits to infinity. Furthermore, we improve the Weyl formula on the asymptotic distribution of eigenvalues and make use of it in order to establish inclusion relations between the interpolation normed ideals of compact operators in L 2(R n) and the above operator classes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Boggiatto, P., Buzano, E. and Rodino, L.: Global Hypoellipticity and Spectral Theory, Akademie-Verlag, Berlin, 1996.
Boggiatto, P. and Nicola, F.: Non-commutative residues for anisotropic pseudo-differential operators in ℝn, 2001, submitted to J. Funct. Anal.
Connes, A.: The action functional in non-commutative geometry, Comm. Math. Phys. 117 (1988), 673–683.
Connes, A.: Noncommutative Geometry, Academic Press, New York, 1994.
Cordes, H. O.: A global parametrix for pseudo-differential operators over ℝn, with applications, Reprint, SFB 72, Univetsität Bonn, 1976.
Cordes, H. O.: The Technique of Pseudodifferential Operators, Cambridge Univ. Press, 1995.
Dixmier, J.: Existence de traces non normales, C.R. Acad. Sci. Paris, Sér. A 262 (1966), 1107–1108.
Fedosov, B. V., Golse, F., Leichtnam, E. and Schrohe, E.: Le résidue non commutatif pour les variétés à bord, C.R. Acad. Sci. Paris Sér. I 320 (1995), 669–674.
Fedosov, B. V., Golse, F., Leichtnam, E. and Schrohe, E.: The noncommutative residue for manifolds with boundary, J. Funct. Anal. 142 (1996), 1–31.
Feygin, V. I.: Two algebras of pseudodifferential operators in ℝn and some applications, Trudy Moskov. Mat. Obshch. 36 (1977), 155–194.
Gohberg, I. C. and Krein, M. G.: Introduction to the Theory of Non-selfadjoint Operators, Moscow, 1985.
Guillemin, V.: Residue traces for certain algebras of Fourier integral operators, J. Funct. Anal. 115 (1993), 391–417.
Grushin, V. V.: Pseudodifferential operators in ℝn with bounded symbols, Funktsional. Anal. i Prilozhen. 3 (1970), 37–50.
Hörmander, L.: On the asymptotic distribution of the eigenvalues of pseudodifferential operators in ℝn, Ark. Mat. 17 (1979), 297–313.
Hörmander, L.: The Analysis of Linear Partial Differential Operators III, Springer, Berlin, 1985.
Kassel, C.: Le residue non commutatif [d'apres M. Wodzicki], Astérisque 177-178 (1989), 199–229; Séminaire Bourbaki, 41ème année, Expos No. 708, 1988-89.
Lauter, R. and Moroianu, S.: Homology of pseudo-differential operators on manifolds with fibered boundaries, J. Reine Angew. Math., to appear.
Maniccia, L. and Panarese, P.: Eigenvalues asymptotics for a class of elliptic ψdo's on manifold with cylindrical exits, I, Preprint, 1998.
Melrose, R.: The eta invariant and families of pseudodifferential operators, Math. Res. Lett. 2(5) (1995), 541–561.
Melrose, R. and Nistor, V.: Homology of pseudodifferential operators I, Manifolds with boundary, Preprint, MIT 1996.
Nilsson, N.: Asymptotic estimates for spectral function connected with hypoelliptic differential operators, Ark. Mat. 5 (1965), 527–540.
Parenti, C.: Operatori pseudo-differentiali in ℝn e applicazioni, Ann. Mat. Pura Appl. 93 (1972), 359–389.
Schrohe, E.: Spaces of weighted symbols and weighted Sobolev spaces on manifolds, In: Lecture Notes in Math. 1256, Springer, New York, 1987, pp. 360–377.
Schrohe, E.: Traces on the cone algebra with asymptotics, Actes des Journées de Saint Jean de Monts, Journées Equations aux Dérivées Partielles 1996, Ecole Polytechnique, Palaiseau, 1996.
Schrohe, E.: Noncommutative residues and manifold with conical singularities, J. Funct. Anal. 150 (1997), 146–174.
Schrohe, E.: Wodzicki's noncommutative residue and traces for operator algebras on manifolds with conical singularities, In: L. Rodino (ed.), Microlocal Analysis and Spectral Theory, Kluwer Acad. Publ., Dordrecht, 1997, pp. 227–250.
Schulze, B. W.: Boundary Value Problems and Singular Pseudo-differential Operators, Wiley, Chichester, 1998.
Seeley, R. T.: Complex powers of an elliptic operator, In: Singular Integrals, Proc. Sympos. 10, Pure Math., Amer. Math. Soc., Providence, 1967, pp. 288–307.
Shubin, M. A.: Pseudodifferential operators in ℝn, Dokl. Akad. Nauk SSSR 196 (1971), 316–319.
Shubin, M. A.: Pseudodifferential Operators and Spectral Theory, Springer, Berlin, 1987.
Wodzicki, M.: Spectral asymmetry and noncommutative residue, Thesis, Stekhlov Inst. Math., Moscow, 1984.
Wodzicki, M.: Noncommutative residue, Chapter I. Fundamentals, In: Manin, Yu. I. (ed.), K-theory, Arithmetic and Geometry, Lecture Notes in Math. 1289, New York, 1987, pp. 320–399.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nicola, F. Trace Functionals for a Class of Pseudo-Differential Operators in R n . Mathematical Physics, Analysis and Geometry 6, 89–105 (2003). https://doi.org/10.1023/A:1022421819602
Issue Date:
DOI: https://doi.org/10.1023/A:1022421819602