# Traces and quasi-traces on the Boutet de Monvel algebra

Gerd Grubb^{[1]}; Elmar Schrohe

- [1] Copenhagen University, Mathematics Department, Universitetsparken 5, 2100 Copenhagen (Danemark), Institut für Mathematik, Universität Hannover, Welfengarten 1, 30167 Hannover (Allemagne)

Annales de l’institut Fourier (2004)

- Volume: 54, Issue: 5, page 1641-1696
- ISSN: 0373-0956

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topGrubb, Gerd, and Schrohe, Elmar. "Traces and quasi-traces on the Boutet de Monvel algebra." Annales de l’institut Fourier 54.5 (2004): 1641-1696. <http://eudml.org/doc/116155>.

@article{Grubb2004,

abstract = {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 $(-\lambda )^\{-N\}$ in the asymptotic expansion of the resolvent
trace $\{\rm Tr\}(A(B-\lambda )^\{-N\})$ (with $N$ large) in powers and log-powers of
$\lambda $. This coefficient identifies with the zero-power coefficient in the Laurent
series for the zeta function $\{\rm Tr\}(AB^\{-s\})$ 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 $[A,A^\{\prime \}]$ 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.},

affiliation = {Copenhagen University, Mathematics Department, Universitetsparken 5, 2100 Copenhagen (Danemark), Institut für Mathematik, Universität Hannover, Welfengarten 1, 30167 Hannover (Allemagne)},

author = {Grubb, Gerd, Schrohe, Elmar},

journal = {Annales de l’institut Fourier},

keywords = {canonical trace; nonlocal invariant; pseudodifferential boundary value problems; Boutet de Monvel calculus; asymptotic resolvent trace expansions},

language = {eng},

number = {5},

pages = {1641-1696},

publisher = {Association des Annales de l'Institut Fourier},

title = {Traces and quasi-traces on the Boutet de Monvel algebra},

url = {http://eudml.org/doc/116155},

volume = {54},

year = {2004},

}

TY - JOUR

AU - Grubb, Gerd

AU - Schrohe, Elmar

TI - Traces and quasi-traces on the Boutet de Monvel algebra

JO - Annales de l’institut Fourier

PY - 2004

PB - Association des Annales de l'Institut Fourier

VL - 54

IS - 5

SP - 1641

EP - 1696

AB - 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 $(-\lambda )^{-N}$ in the asymptotic expansion of the resolvent
trace ${\rm Tr}(A(B-\lambda )^{-N})$ (with $N$ large) in powers and log-powers of
$\lambda $. This coefficient identifies with the zero-power coefficient in the Laurent
series for the zeta function ${\rm Tr}(AB^{-s})$ 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 $[A,A^{\prime }]$ 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.

LA - eng

KW - canonical trace; nonlocal invariant; pseudodifferential boundary value problems; Boutet de Monvel calculus; asymptotic resolvent trace expansions

UR - http://eudml.org/doc/116155

ER -

## References

top- L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math 126 (1971), 11-51 Zbl0206.39401MR407904
- B. V. Fedosov, F. Golse, E. Leichtnam, E. Schrohe, The noncommutative residue for manifolds with boundary, J. Funct. Anal 142 (1996), 1-31 Zbl0877.58005MR1419415
- G. Grubb, Singular Green operators and their spectral asymptotics, Duke Math. J 51 (1984), 477-528 Zbl0553.58034MR757950
- G. Grubb, Functional calculus of pseudodifferential boundary problems, vol. 65 (1996), Birkhäuser, Boston Zbl0844.35002MR1385196
- G. Grubb, A weakly polyhomogeneous calculus for pseudodifferential boundary problems, J. Funct. Anal 184 (2001), 19-76 Zbl0998.58017MR1846783
- G. Grubb, A resolvent approach to traces and determinants, AMS Contemp. Math. Proc 366 (2005), 67-93 Zbl1073.58021MR2114484
- G. Grubb, Spectral boundary conditions for generalizations of Laplace and Dirac operators, Comm. Math. Phys 240 (2003), 243-280 Zbl1039.58027MR2004987
- G. Grubb, Logarithmic terms in trace expansions of Atiyah-Patodi-Singer problems, Ann. Global Anal. Geom 24 (2003), 1-51 Zbl1048.35095MR1990084
- G. Grubb, L. Hansen, Complex powers of resolvents of pseudodifferential operators, Comm. Part. Diff. Eq 27 (2002), 2333-2361 Zbl1129.58304MR1944032
- G. Grubb, E. Schrohe, Trace expansions and the noncommutative residue for manifolds with boundary, J. Reine Angew. Math. (Crelle's Journal) 536 (2001), 167-207 Zbl0980.58017MR1837429
- G. Grubb, R. Seeley, Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems, Invent. Math. 121 (1995), 481-529 Zbl0851.58043MR1353307
- G. Grubb, R. Seeley, Zeta and eta functions for Atiyah-Patodi-Singer operators, J. Geom. Anal. 6 (1996), 31-77 Zbl0858.58050MR1402386
- V. Guillemin, A new proof of Weyl's formula on the asymptotic distribution of eigenvalues, Adv. Math 102 (1985), 184-201 Zbl0559.58025
- J. Hadamard, Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques, (1932), Hermann, Paris Zbl0006.20501
- M. Kontsevich, S. Vishik, Geometry of determinants of elliptic operators, Functional Analysis on the Eve of the 21'st Century. Vol. I (Rutgers Conference in honor of I. M. Gelfand 1993) 131 (1995), 173-197, Birkhäuser, Boston Zbl0920.58061
- M. Lesch, On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols, Ann. Global Anal. Geom 17 (1999), 151-187 Zbl0920.58047MR1675408
- R. Melrose, V. Nistor, Homology of pseudodifferential operators I. Manifolds with boundary Zbl0898.46060
- K. Okikiolu, The multiplicative anomaly for determinants of elliptic operators, Duke Math. J. 79 (1995), 723-750 Zbl0851.58048MR1355182
- R. T. Seeley, Complex powers of an elliptic operator, Amer. Math. Soc. Proc. Symp. Pure Math 10 (1967), 288-307 Zbl0159.15504MR237943
- M. Wodzicki, Local invariants of spectral asymmetry, Invent. Math 75 (1984), 143-178 Zbl0538.58038MR728144

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