Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 280

On the noncommutative residue for pseudodifferential operators with polyhomogeneous symbols

M. Lesch

We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion $$asim sum_{j=0}^infty a_{m-j},quad a_{m-j}(x,xi)=sum_{l=0}^k a_{m-j,l}(x,xi)log^l|xi|,$$ where $a_{m-j,l}$ is homogeneous in $xi$ of degree $m-j$. We call these symbols polyhomogeneous. We study polyhomogeneous functions on symplectic cones and generalize the symplectic residue of {sc Guillemin} to these functions. Similarly as for homogeneous functions, for a polyhomogeneous function this symplectic residue is an obstruction against being a sum of Poisson brackets. For elliptic pseudodifferential operators with polyhomogeneous symbols we show that the $zeta$--function has a meromorphic continuation to the whole complex plane, however possibly with higher order poles. This 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.

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