On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols
Abstract
We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion
a∼
∑
∞
j=0
a
m−j
,
a
m−j
(x,ξ)=
∑
k
l=0
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.