On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols

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dc.creatorLesch, Matthias
dc.date1997-08-13
dc.date1999-02-18
dc.date.accessioned2026-07-07T08:59:14Z
dc.date.available2026-07-07T08:59:14Z
dc.descriptionWe study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion $a\sim \sum_{j=0}^\infty a_{m-j}, a_{m-j}(x,ξ)=\sum_{l=0}^k a_{m-j,l}(x,ξ) \log^l|ξ|,$ where $a_{m-j,l}$ is homogeneous in $ξ$ of degree $m-j$. We will explain why this algebra of pseudodifferential operators is natural. For a pseudodifferential operator in this class, $A$, and a classical elliptic pseudodifferential operator, $P$, we show that the generalized zeta-function $\Tr(AP^{-s})$ has a meromorphic continuation to the whole complex plane, however possibly with higher order poles. Our algebra of operators has a bigrading given by the order and the highest log-power occuring in the symbol expansion. We construct "higher" noncommutative residue functionals on the subspaces given by the log-grading. However, in contrast to the classical case we prove that the whole algebra does not admit any nontrivial traces. Finally we show that the analogue of the Kontsevich-Vishik trace also exists on our algebra. Our method also provides an alternative approach to the Kontsevich-Vishik trace.
dc.descriptionLaTeX2e, 35 pages; v2 16 Sept 1997, section on Kontsevich-Vishik added, Final version, 10 July 1998, minor corrections, to appear in Ann. Glob. Anal. Geom
dc.identifierhttps://arxiv.org/abs/dg-ga/9708010
dc.identifierhttp://arxiv.org/abs/dg-ga/9708010
dc.identifierAnn. Glob. Anal. Geom. 17 (1999), 151-187
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/147637
dc.subjectDifferential Geometry
dc.subject58G15 (Primary), 58G11 (Secondary)
dc.titleOn the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols
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