Artículo Original
La traza de Dixmier y el residuo de Wodzicki para operadores pseudodiferenciales globales sobre variedades compactas.
Publicado 2020-02-14
Palabras clave
- Traza de Dixmier,
- residuo no conmutativo,
- operador global,
- teoría de representaciones
Cómo citar
Cardona, D., & del Corral, C. (2020). La traza de Dixmier y el residuo de Wodzicki para operadores pseudodiferenciales globales sobre variedades compactas. Revista Integración, Temas De matemáticas, 38(1), 67–79. https://doi.org/10.18273/revanu.v38n1-2020006
Resumen
En esta nota se anuncian los resultados de nuestra investigación sobre la traza de Dixmier y el residuo de Wodzicki para operadores pseudodiferenciales sobre variedades compactas. Se calcula la traza de Dixmier y el residuo no conmutativo (residuo de Wodzicki) de operadores pseudodiferenciales invariantes sobre variedades compactas con o sin borde. Para cada variedad cerrada (suave, compacta y sin borde), se emplea la noción de símbolo global que viene dada por el análisis de Fourier asociado a cada operador elíptico y positivo (desarrollado por M. Ruzhansky and V. Turunen para para grupos de Lie y por M. Ruzhansky, N. Tokmagambetov y J. Delgado para variedades cerradas). En particular, para cada grupo de Lie compacto, se usa su teoría de representación. Respecto al análisis de operadores sobre variedades con borde, se usa el análisis no armónico asociado a problemas con valores de frontera (introducido por M. Ruzhansky, N. Tokmagambetov, y J. Delgado).
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Referencias
[1] Barraza E.S. and Cardona D., “On nuclear Lp multipliers associated to the harmonic oscillator”, in Analysis and Partial Differential Equations: Perspectives from Developing Countries. Springer Proceedings in Mathematics Statistics vol 275 (eds Ruzhansky M. and Delgado J.) Springer, Cham (2019), 31–41.
[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.
[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.