Traces and quasi-traces on the Boutet de Monvel algebra
Annales de l'Institut Fourier, Volume 54 (2004) no. 5, p. 1641-1696
We construct an analogue of Kontsevich and Vishik’s canonical trace for pseudodifferential boundary value problems in the Boutet de Monvel calculus on compact manifolds with boundary. For an operator A in the calculus (of class zero), and an auxiliary operator B, formed of the Dirichlet realization of a strongly elliptic second-order differential operator and an elliptic operator on the boundary, we consider the coefficient C 0 (A,B) of (-λ) -N in the asymptotic expansion of the resolvent trace Tr (A(B-λ) -N ) (with N large) in powers and log-powers of λ. This coefficient identifies with the zero-power coefficient in the Laurent series for the zeta function Tr (AB -s ) at s=0, when B is invertible. We show that C 0 (A,B) is in general a quasi-trace, in the sense that it vanishes on commutators [A,A ' ] modulo local terms, and has a specific value independent of the auxiliary operator, modulo local terms. The local “errors” vanish when A is a singular Green operator of noninteger order, or of integer order with a certain parity; then C 0 (A,B) is a trace of A. They do not in general vanish when the interior ps.d.o. part of A is nontrivial.
On construit une fonctionelle analogue à la trace canonique de Kontsevich et Vishik, pour les problèmes aux limites pseudodifférentielles appartenant au calcul de Boutet de Monvel, sur les variétés compactes à bord. Pour un opérateur A de ce calcul (et de classe zéro), avec un opérateur auxiliaire B formé de la réalisation de Dirichlet d’un opérateur différentiel fortement elliptique du second ordre et d’un opérateur elliptique sur le bord, nous considérons le coefficient C 0 (A,B) de (-λ) -N dans le développement asymptotique de la trace de la résolvante Tr (A(B-λ) -N ), avec N grand, dans les puissances et puissances logarithmiques de λ. Ce coefficient s’identifie au coefficient d’ordre zéro dans la série de Laurent pour la fonction zêta Tr (AB -s ) en s=0, quand B est inversible. On montre que C 0 (A,B) est en général une quasi-trace, en ce sens qu’elle s’annule sur les commutateurs [A,A ' ] modulo des termes locaux, ayant une valeur spécifique indépendante de l’opérateur auxiliaire, modulo des termes locaux. Les «erreurs» locales sont nulles quand A est un opérateur de Green singulier d’ordre non entier, ou d’ordre entier avec une certaine parité ; ainsi C 0 (A,B) est une trace sur A dans ces cas. Mais les «erreurs» ne sont en général pas nulles quand la partie intérieure o.ps.d. de A est non-triviale.
DOI : https://doi.org/10.5802/aif.2062
Classification:  58J42,  35S15
@article{AIF_2004__54_5_1641_0,
     author = {Grubb, Gerd and Schrohe, Elmar},
     title = {Traces and quasi-traces on the Boutet de Monvel algebra},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {5},
     year = {2004},
     pages = {1641-1696},
     doi = {10.5802/aif.2062},
     zbl = {1078.58015},
     mrnumber = {2127861},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2004__54_5_1641_0}
}
Grubb, Gerd; Schrohe, Elmar. Traces and quasi-traces on the Boutet de Monvel algebra. Annales de l'Institut Fourier, Volume 54 (2004) no. 5, pp. 1641-1696. doi : 10.5802/aif.2062. http://www.numdam.org/item/AIF_2004__54_5_1641_0/

[B] L. Boutet De Monvel, Boundary problems for pseudo-differential operators, Acta Math 126 (1971) p. 11-51 | MR 407904 | Zbl 0206.39401

[FGLS] B. V. Fedosov, F. Golse, E. Leichtnam & E. Schrohe, The noncommutative residue for manifolds with boundary, J. Funct. Anal 142 (1996) p. 1-31 | MR 1419415 | Zbl 0877.58005

[G1] G. Grubb, Singular Green operators and their spectral asymptotics, Duke Math. J 51 (1984) p. 477-528 | MR 757950 | Zbl 0553.58034

[G2] G. Grubb, Functional calculus of pseudodifferential boundary problems, Second Edition (first edition issued 1986), Progress in Math. vol. 65, Birkhäuser, 1996 | MR 1385196 | Zbl 0844.35002

[G3] G. Grubb, A weakly polyhomogeneous calculus for pseudodifferential boundary problems, J. Funct. Anal 184 (2001) p. 19-76 | MR 1846783 | Zbl 0998.58017

[G4] G. Grubb, A resolvent approach to traces and determinants, AMS Contemp. Math. Proc 366 (2005) p. 67-93 | MR 2114484 | Zbl 1073.58021

[G5] G. Grubb, Spectral boundary conditions for generalizations of Laplace and Dirac operators, Comm. Math. Phys 240 (2003) p. 243-280 | MR 2004987 | Zbl 1039.58027

[G6] G. Grubb, Logarithmic terms in trace expansions of Atiyah-Patodi-Singer problems, Ann. Global Anal. Geom 24 (2003) p. 1-51 | MR 1990084 | Zbl 1048.35095

[GH] G. Grubb & L. Hansen, Complex powers of resolvents of pseudodifferential operators, Comm. Part. Diff. Eq 27 (2002) p. 2333-2361 | MR 1944032 | Zbl 01877187

[GSc] G. Grubb & E. Schrohe, Trace expansions and the noncommutative residue for manifolds with boundary, J. Reine Angew. Math. (Crelle's Journal) 536 (2001) p. 167-207 | MR 1837429 | Zbl 0980.58017

[GS1] G. Grubb & R. Seeley, Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems, Invent. Math. 121 (1995) p. 481-529 | MR 1353307 | Zbl 0851.58043

[GS2] G. Grubb & R. Seeley, Zeta and eta functions for Atiyah-Patodi-Singer operators, J. Geom. Anal. 6 (1996) p. 31-77 | MR 1402386 | Zbl 0858.58050

[Gu] V. Guillemin, A new proof of Weyl's formula on the asymptotic distribution of eigenvalues, Adv. Math 102 (1985) p. 184-201 | Zbl 0559.58025

[H] J. Hadamard, Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques, Hermann, 1932 | JFM 58.0519.16 | Zbl 0006.20501

[KV] M. Kontsevich & S. Vishik, Geometry of determinants of elliptic operators, Progr. Math. 131, Birkhäuser, 1995, p. 173-197 | Zbl 0920.58061

[L] M. Lesch, On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols, Ann. Global Anal. Geom 17 (1999) p. 151-187 | MR 1675408 | Zbl 0920.58047

[MN] R. Melrose & V. Nistor, Homology of pseudodifferential operators I. Manifolds with boundary, e-print manuscript, arXiv:funct-an/9606005

[O] K. Okikiolu, The multiplicative anomaly for determinants of elliptic operators, Duke Math. J. 79 (1995) p. 723-750 | MR 1355182 | Zbl 0851.58048

[S] R. T. Seeley, Complex powers of an elliptic operator, Amer. Math. Soc. Proc. Symp. Pure Math 10 (1967) p. 288-307 | MR 237943 | Zbl 0159.15504

[W] M. Wodzicki, Local invariants of spectral asymmetry, Invent. Math 75 (1984) p. 143-178 | MR 728144 | Zbl 0538.58038