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

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

1997-08-13

arxiv.org/abs/dg-ga/9708010v4

Ann. Glob. Anal. Geom. 17 (1999), 151-187

Mathematics

Differential Geometry

LaTeX2e, 35 pages; v2 16 Sept 1997, section on Kontsevich-Vishik added, Final version, 10 July 1998, minor corrections, to app

Scientific paper

We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion $a\sim \sum_{j=0}^\infty a_{m-j}, a_{m-j}(x,\xi)=\sum_{l=0}^k a_{m-j,l}(x,\xi) \log^l|\xi|,$ where $a_{m-j,l}$ is homogeneous in $\xi$ 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.

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