Traces and quasi-traces on the Boutet de Monvel algebra

Grubb, 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 -