Research and Innovation Articles
The Dixmier trace and the Wodzicki residue for global pseudo-differential operators on compact manifolds.
Published 2020-02-14
Keywords
- Dixmier trace,
- non commutative residue,
- global operators,
- representation theory
How to Cite
Cardona, D., & del Corral, C. (2020). The Dixmier trace and the Wodzicki residue for global pseudo-differential operators on compact manifolds. Revista Integración, Temas De matemáticas, 38(1), 67–79. https://doi.org/10.18273/revanu.v38n1-2020006
Abstract
In this note, we announce the results of our investigation on the Dixmier trace and the Wodzicki residue for pseudo-differential operators on compact manifolds. We give formulae for the Dixmier trace and the non-commutative residue (also called Wodzicki’s residue) of invariant pseudo-differential operators on compact manifolds with or without boundary. For every closed manifold, the notion of global symbol for invariant pseudo-differential operators will be based on the Fourier analysis associated to every elliptic and positive operator (developed by M. Ruzhansky, V. Turunen and J. Delgado). In particular, for each compact Lie group we will use its representation theory. For the analysis of operators on compact manifolds with boundary, we will use the non-harmonic analysis associated with boundary valued problems (developed by M. Ruzhansky, N. Tokmagambetov, and J. Delgado).
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References
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[3] Chatzakou M., Delgado J.m and Ruzhansky M., “On a class of anharmonic oscillators”, arXiv:1811.12566.
[4] Cardona D., “Nuclear pseudo-differential operators in Besov spaces on compact Lie groups”, J. Fourier Anal. Appl. 23 (2017), No. 5, 1238–1262.
[5] Cardona D., “A brief description of operators associated to the quantum harmonic oscillator on Schatten-von Neumann classes”, Rev. Integr. temas mat. 36 (2018), No. 1, 49–57.
[6] Cardona D., “On the nuclear trace of Fourier Integral Operators”, Rev. Integr. temas mat. 37 (2019), No. 2, 219–249.
[7] Cardona D., Del Corral C. and Kumar V., “Dixmier traces for discrete pseudo-differential operators”, arXiv:1911.03924.
[8] Cardona D. and Kumar V., “Multilinear analysis for discrete and periodic pseudodifferential operators in Lp spaces”, Rev. Integr. temas mat. 36 (2018), No. 2, 151–164.
[9] Cardona D. and Kumar V., “Lp−boundedness and Lp-nuclearity of multilinear pseudodifferential operators on Zn and the torus T n”, J. Fourier Anal. Appl. 25 (2019), No. 6, 2973–3017.
[10] Cardona D. and Barraza E.S., “Characterization of nuclear pseudo-multipliers associated to the harmonic oscillator”, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys 80 (2018), No. 4, 163–172.
[11] Cardona D. and Del Corral C., “The Dixmier trace and the noncommutative residue for multipliers on compact manifolds”, arXiv:1703.07453.
[12] Connes A., Noncommutative geometry, Academic Press Inc., San Diego, 1994.
[13] Delgado J. and Ruzhansky M., “Schatten-von Neumann classes of integral operators”, arXiv:1709.06446.
[14] Delgado J. and Ruzhansky M., “Fourier multipliers in Hilbert spaces”, arXiv:1707.03062.
[15] Delgado J. and Ruzhansky M., “Lp-nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups”, J. Math. Pures Appl. 102 (2014), No. 1, 153–172.
[16] Delgado J. and RuzhanskyM., “Schatten classes on compact manifolds: Kernel conditions”, J. Funct. Anal. 267 (2014), No. 3, 772–798.
[17] Delgado J. and Ruzhansky M., “Kernel and symbol criteria for Schatten classes and rnuclearity on compact manifolds”, C. R. Acad. Sci. Paris. 352 (2014), No. 10, 779–784.
[18] Delgado J. and Ruzhansky M., “Lp-bounds for pseudo-differential operators on compact Lie groups”, arXiv:1605.07027.
[19] Delgado J. and Ruzhansky M., “Fourier multipliers, symbols and nuclearity on compact manifolds”, arXiv:1404.6479.
[20] Delgado J., Ruzhansky M. and Tokmagambetov N., “Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary”, arXiv:1505.02261.
[21] Dixmier J., “Existence de traces non normales”, C. R. Acad. Sci. Paris. S´r A-B 262 (1966), 1107–1108.
[22] Fedosov B., Golse F., Leichtnam E. and Schrohe E., “The Noncommutative Residue for Manifolds with Boundary”, J. Funct. Anal. 142 (1996), No. 1, 1–31.
[23] Ghaemi M. B., Jamalpourbirgani M. and Wong M. W., “Characterizations, adjoints and products of nuclear pseudo-differential operators on compact and Hausdorff groups”, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 79 (2017), No. 4, 207–220.
[24] Grubb G. and Schrohe E., “Trace expansions and the noncommutative residue for manifolds with boundary”, J. Reine Angew. Math. 536 (2001), 167–207.
[25] Hörmander L., The Analysis of the linear partial differential operators III, Springer-Verlag, Berlin, 1985.
[26] Kumar V.n and Wong M. W., “C∗-algebras, H∗-algebras and trace ideals of pseudodifferential operators on locally compact, Hausdorff and abelian groups”, J. Pseudo-Differ. Oper. Appl. 10 (2019), No. 2, 269–283.
[27] Nest R. and Schrohe E., “Dixmier’s trace for boundary value problems”, Manuscripta math. Springer-Verlag 96 (1998), No. 2, 203–218.
[28] Pietsch A. “Traces and residues of pseudo-differential operators on the torus“, Integr. Equ. Oper. Theory. 83 (2015), No. 1, 1–23.
[29] Ruzhansky M. and Tokmagambetov N., “Nonharmonic Analysis of Boundary Value Problems“, International Mathematics Research Notices, (2016), No. 12, 3548–3615.
[30] Ruzhansky M., Turunen V., “Pseudo-Differential Operators and Symmetries.”, Background Analysis and Advanced Topics, Birkhäuser-Verlag, Basel, 2010.
[31] Ruzhansky M., Wirth, J., “Lp Fourier multipliers on compact Lie groups.”, Math. Z., 280 (2015), No. 3-4, 621–642.
[32] Schrohe, E., “Noncommutative Residues, Dixmier’s Trace, and Heat Trace Expansions on Manifolds with Boundary.”, Amer. Math. Soc. Providence, Providence, RI, No. 161–186, 1999.
[33] Wodzicki M., “Noncommutative residue. I. Fundamentals.”, Yu. I. Manin (ed.), Lecture Notes in Math., 1289, Springer, Berlin, 1987.
[2] Boutet de Monvel L., “Boundary problems for pseudo-differential operators”, Acta Math. 126 (1971), No. 1–2, 11–51.
[3] Chatzakou M., Delgado J.m and Ruzhansky M., “On a class of anharmonic oscillators”, arXiv:1811.12566.
[4] Cardona D., “Nuclear pseudo-differential operators in Besov spaces on compact Lie groups”, J. Fourier Anal. Appl. 23 (2017), No. 5, 1238–1262.
[5] Cardona D., “A brief description of operators associated to the quantum harmonic oscillator on Schatten-von Neumann classes”, Rev. Integr. temas mat. 36 (2018), No. 1, 49–57.
[6] Cardona D., “On the nuclear trace of Fourier Integral Operators”, Rev. Integr. temas mat. 37 (2019), No. 2, 219–249.
[7] Cardona D., Del Corral C. and Kumar V., “Dixmier traces for discrete pseudo-differential operators”, arXiv:1911.03924.
[8] Cardona D. and Kumar V., “Multilinear analysis for discrete and periodic pseudodifferential operators in Lp spaces”, Rev. Integr. temas mat. 36 (2018), No. 2, 151–164.
[9] Cardona D. and Kumar V., “Lp−boundedness and Lp-nuclearity of multilinear pseudodifferential operators on Zn and the torus T n”, J. Fourier Anal. Appl. 25 (2019), No. 6, 2973–3017.
[10] Cardona D. and Barraza E.S., “Characterization of nuclear pseudo-multipliers associated to the harmonic oscillator”, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys 80 (2018), No. 4, 163–172.
[11] Cardona D. and Del Corral C., “The Dixmier trace and the noncommutative residue for multipliers on compact manifolds”, arXiv:1703.07453.
[12] Connes A., Noncommutative geometry, Academic Press Inc., San Diego, 1994.
[13] Delgado J. and Ruzhansky M., “Schatten-von Neumann classes of integral operators”, arXiv:1709.06446.
[14] Delgado J. and Ruzhansky M., “Fourier multipliers in Hilbert spaces”, arXiv:1707.03062.
[15] Delgado J. and Ruzhansky M., “Lp-nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups”, J. Math. Pures Appl. 102 (2014), No. 1, 153–172.
[16] Delgado J. and RuzhanskyM., “Schatten classes on compact manifolds: Kernel conditions”, J. Funct. Anal. 267 (2014), No. 3, 772–798.
[17] Delgado J. and Ruzhansky M., “Kernel and symbol criteria for Schatten classes and rnuclearity on compact manifolds”, C. R. Acad. Sci. Paris. 352 (2014), No. 10, 779–784.
[18] Delgado J. and Ruzhansky M., “Lp-bounds for pseudo-differential operators on compact Lie groups”, arXiv:1605.07027.
[19] Delgado J. and Ruzhansky M., “Fourier multipliers, symbols and nuclearity on compact manifolds”, arXiv:1404.6479.
[20] Delgado J., Ruzhansky M. and Tokmagambetov N., “Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary”, arXiv:1505.02261.
[21] Dixmier J., “Existence de traces non normales”, C. R. Acad. Sci. Paris. S´r A-B 262 (1966), 1107–1108.
[22] Fedosov B., Golse F., Leichtnam E. and Schrohe E., “The Noncommutative Residue for Manifolds with Boundary”, J. Funct. Anal. 142 (1996), No. 1, 1–31.
[23] Ghaemi M. B., Jamalpourbirgani M. and Wong M. W., “Characterizations, adjoints and products of nuclear pseudo-differential operators on compact and Hausdorff groups”, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 79 (2017), No. 4, 207–220.
[24] Grubb G. and Schrohe E., “Trace expansions and the noncommutative residue for manifolds with boundary”, J. Reine Angew. Math. 536 (2001), 167–207.
[25] Hörmander L., The Analysis of the linear partial differential operators III, Springer-Verlag, Berlin, 1985.
[26] Kumar V.n and Wong M. W., “C∗-algebras, H∗-algebras and trace ideals of pseudodifferential operators on locally compact, Hausdorff and abelian groups”, J. Pseudo-Differ. Oper. Appl. 10 (2019), No. 2, 269–283.
[27] Nest R. and Schrohe E., “Dixmier’s trace for boundary value problems”, Manuscripta math. Springer-Verlag 96 (1998), No. 2, 203–218.
[28] Pietsch A. “Traces and residues of pseudo-differential operators on the torus“, Integr. Equ. Oper. Theory. 83 (2015), No. 1, 1–23.
[29] Ruzhansky M. and Tokmagambetov N., “Nonharmonic Analysis of Boundary Value Problems“, International Mathematics Research Notices, (2016), No. 12, 3548–3615.
[30] Ruzhansky M., Turunen V., “Pseudo-Differential Operators and Symmetries.”, Background Analysis and Advanced Topics, Birkhäuser-Verlag, Basel, 2010.
[31] Ruzhansky M., Wirth, J., “Lp Fourier multipliers on compact Lie groups.”, Math. Z., 280 (2015), No. 3-4, 621–642.
[32] Schrohe, E., “Noncommutative Residues, Dixmier’s Trace, and Heat Trace Expansions on Manifolds with Boundary.”, Amer. Math. Soc. Providence, Providence, RI, No. 161–186, 1999.
[33] Wodzicki M., “Noncommutative residue. I. Fundamentals.”, Yu. I. Manin (ed.), Lecture Notes in Math., 1289, Springer, Berlin, 1987.