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The Dixmier trace and the Wodzicki residue for global pseudo-differential operators on compact manifolds.

  • Cardona, Duván [1] ; del Corral, César [2]
    1. [1] Ghent University

      Ghent University

      Arrondissement Gent, Bélgica

    2. [2] Universidad de Los Andes

      Universidad de Los Andes

      Colombia

  • Localización: Integración: Temas de matemáticas, ISSN 0120-419X, Vol. 38, Nº. 1, 2020, págs. 67-79
  • Idioma: inglés
  • DOI: 10.18273/revanu.v38n1-2020006
  • Títulos paralelos:
    • La traza de Dixmier y el residuo de Wodzicki para operadores pseudodifferenciales globales sobre variedades compactas.
  • Enlaces
  • Resumen
    • español

      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).

    • English

      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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