2007/03/29 by Anders Gaarde, Gaarde, Anders, Gerd Grubb +1 · 1 citation
Mathematics · #35S15 #58J42 #Advanced Operator Algebra Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math.AP #math.FA #msc:35S15 #msc:58J42
paper · pdf · doi:10.48550/arxiv.math/0703878
27 pages, minor adjustments and correction of typos. To appear in Math. Scand
openalex publication_date 2007/03/29 · arxiv created 2007/09/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
On a compact manifold with boundary, consider the realization B of an elliptic, possibly pseudodifferential, boundary value problem having a spectral cut (a ray free of eigenvalues), say R-. In the first part of the paper we define and discuss in detail the operator log B; its residue (generalizing the Wodzicki residue) is essentially proportional to the zeta function value at zero, zeta(B,0), and it enters in an important way in studies of composed zeta functions zeta(A,B,s)=Tr(AB-s) (pursued elsewhere). There is a similar definition of the operator logtheta B, when the spectral cut is at a general angle theta. When B has spectral cuts at two angles theta < phi, one can define the sectorial projection Pitheta,phi(B) whose range contains the generalized eigenspaces for eigenvalues with argument in ] theta, phi [; this is studied in the last part of the paper. The operator Pitheta,phi(B) is shown to be proportional to the difference between logtheta B and logphi B, having slightly better symbol properties than they have. We show by examples that it belongs to the Boutet de Monvel calculus in many special cases, but lies outside the calculus in general.