---
address:
- 'Copenhagen University, Mathematics Department, Universitetsparken 5, 2100 Copenhagen, Denmark. E-mail [grubbmath.ku.dk]{}'
- 'Institut für Mathematik, Universität Hannover, Welfengarten 1, 30167 Hannover, Germany.'
author:
- Gerd Grubb and Elmar Schrohe
title: 'Traces and quasi-traces on the Boutet de Monvel algebra'
---

amstex =1 1200 ‘=11 @ ‘==23.5truecm =16.5truecm

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E-mail [schrohemath.uni-hannover.de]{} Canonical trace — Nonlocal invariant — Pseudodifferential boundary value problems — Boutet de Monvel calculus — Asymptotic resolvent trace expansions 58J42, 35S15

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 $\operatorname{Tr}(A(B-\lambda)^{-N})$ (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 $\operatorname{Tr}(AB^{-s})$ 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 