Details
| Original language | English |
|---|---|
| Pages (from-to) | 203-218 |
| Number of pages | 16 |
| Journal | Manuscripta mathematica |
| Volume | 96 |
| Issue number | 2 |
| Publication status | Published - Jun 1998 |
| Externally published | Yes |
Abstract
Let X be a smooth manifold with boundary of dimension n > 1. The operators of order -n and type zero in Boutet de Monvel's calculus form a subset of Dixmier's trace ideal ℒ1,∞(H) for the Hilbert space H = L2(X, E) ⊕ L2(∂X, F) of L2 sections in vector bundles E over X, F over ∂X. We show that, on these operators, Dixmier's trace can be computed in terms of the same expressions that determine the noncommutative residue. In particular it is independent of the averaging procedure. However, the noncommutative residue and Dixmier's trace are not multiples of each other unless the boundary is empty. As a corollary we show how to compute Dixmier's trace for parametrices or inverses of classical elliptic boundary value problems of the form Pu = f; Tu = 0 with an elliptic differential operator P of order n in the interior and a trace operator T. In this particular situation, Dixmier's trace and the noncommutative residue do coincide up to a factor.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Manuscripta mathematica, Vol. 96, No. 2, 06.1998, p. 203-218.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Dixmier's trace for boundary value problems
AU - Nest, Ryszard
AU - Schrohe, Elmar
N1 - Copyright: Copyright 2018 Elsevier B.V., All rights reserved.
PY - 1998/6
Y1 - 1998/6
N2 - Let X be a smooth manifold with boundary of dimension n > 1. The operators of order -n and type zero in Boutet de Monvel's calculus form a subset of Dixmier's trace ideal ℒ1,∞(H) for the Hilbert space H = L2(X, E) ⊕ L2(∂X, F) of L2 sections in vector bundles E over X, F over ∂X. We show that, on these operators, Dixmier's trace can be computed in terms of the same expressions that determine the noncommutative residue. In particular it is independent of the averaging procedure. However, the noncommutative residue and Dixmier's trace are not multiples of each other unless the boundary is empty. As a corollary we show how to compute Dixmier's trace for parametrices or inverses of classical elliptic boundary value problems of the form Pu = f; Tu = 0 with an elliptic differential operator P of order n in the interior and a trace operator T. In this particular situation, Dixmier's trace and the noncommutative residue do coincide up to a factor.
AB - Let X be a smooth manifold with boundary of dimension n > 1. The operators of order -n and type zero in Boutet de Monvel's calculus form a subset of Dixmier's trace ideal ℒ1,∞(H) for the Hilbert space H = L2(X, E) ⊕ L2(∂X, F) of L2 sections in vector bundles E over X, F over ∂X. We show that, on these operators, Dixmier's trace can be computed in terms of the same expressions that determine the noncommutative residue. In particular it is independent of the averaging procedure. However, the noncommutative residue and Dixmier's trace are not multiples of each other unless the boundary is empty. As a corollary we show how to compute Dixmier's trace for parametrices or inverses of classical elliptic boundary value problems of the form Pu = f; Tu = 0 with an elliptic differential operator P of order n in the interior and a trace operator T. In this particular situation, Dixmier's trace and the noncommutative residue do coincide up to a factor.
UR - http://www.scopus.com/inward/record.url?scp=0032375056&partnerID=8YFLogxK
U2 - 10.1007/s002290050062
DO - 10.1007/s002290050062
M3 - Article
AN - SCOPUS:0032375056
VL - 96
SP - 203
EP - 218
JO - Manuscripta mathematica
JF - Manuscripta mathematica
SN - 0025-2611
IS - 2
ER -