2001/06/05 by Gerd Grubb, Grubb, Gerd, Elmar Schrohe +1
Mathematics · #35S15 #58J42 #Advanced Operator Algebra Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.AP #math.SP #msc:35S15 #msc:58J42
paper · pdf · doi:10.48550/arxiv.math/0106030
37 pages, to appear in J. Reine Angew. Math
arxiv created 2001/06/05 · openalex publication_date 2001/06/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a pseudodifferential boundary operator A of integer order νand class zero (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Trace(AB-s) where B is an auxiliary system formed of the Dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the boundary. We prove that Trace(AB-s) has a meromorphic extension to the complex plane with poles at the half-integers s = (n+ν-j)/2, j = 0,1,... (possibly double for s<0), and we prove that its residue at zero equals the noncommutative residue of A, as defined by Fedosov, Golse, Leichtnam, and Schrohe by a different method. To achieve this, we establish a full asymptotic expansion of Trace(A(B-λ)-k) in powers of λ-j/2 and log-powers λ-j/2 log λ, where the noncommutative residue equals the coefficient of the highest log-power. There is a related expansion for Trace(A exp(-tB)).