[Paper Review] On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols

Matthias Lesch|ArXiv.org|Aug 13, 1997

Advanced Operator Algebra ResearchMathematics13 references19 citations

TL;DR

This paper introduces an algebra of pseudodifferential operators with log-polyhomogeneous symbols—symbols involving logarithmic powers of the cotangent variable—and establishes a generalized noncommutative residue for this class. It proves that while the full algebra lacks nontrivial traces due to higher-order log-terms, a hierarchy of higher noncommutative residue functionals exists, and the Kontsevich–Vishik trace extends naturally to this setting, providing a trace functional via meromorphic continuation of zeta functions.

ABSTRACT

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

Motivation & Objective

• . To define and study a natural algebra of pseudodifferential operators with log-polyhomogeneous symbols, extending classical theory.
• . To generalize the symplectic residue and noncommutative residue to log-polyhomogeneous functions on symplectic cones.
• . To construct higher noncommutative residue functionals indexed by log-power grading.
• . To show that the full algebra admits no nontrivial traces due to higher log-terms in heat expansions.
• . To extend the Kontsevich–Vishik trace to this algebra using meromorphic continuation of generalized zeta functions.

Proposed method

• . Introduces log-polyhomogeneous symbols as expansions in homogeneous components multiplied by powers of log|ξ|.
• . Generalizes the symplectic residue to log-polyhomogeneous functions on symplectic cones, showing it obstructs sums of Poisson brackets.
• . Uses the Mellin transform to relate the generalized ζ-function Tr(AP−s) to heat trace asymptotics with log-terms.
• . Defines higher noncommutative residue functionals Resk on subspaces graded by log-power, proving they vanish on commutators.
• . Proves meromorphic continuation of Tr(AP−s) to C with poles of order up to k+1 for operators in CLa,k.
• . Constructs the Kontsevich–Vishik trace via evaluation of Tr(AP−s) at s=0, showing it satisfies trace and locality properties.

Experimental results

Research questions

• RQ1 . Can the noncommutative residue be generalized to pseudodifferential operators with log-polyhomogeneous symbols, and what are the obstructions to traces in this setting?
• RQ2 . How do higher-order log-terms in heat trace expansions affect the existence of traces on algebras of pseudodifferential operators?
• RQ3 . Is there a hierarchy of residue functionals corresponding to the log-power grading of symbols, and do they vanish on commutators?
• RQ4 . Does the Kontsevich–Vishik trace extend to this algebra, and can it be characterized via zeta function regularization?
• RQ5 . What is the role of the symplectic residue in obstructing log-polyhomogeneous functions from being sums of Poisson brackets?

Key findings

• . The generalized ζ-function Tr(AP−s) for log-polyhomogeneous operators admits meromorphic continuation to C with poles of order up to k+1.
• . The full algebra of log-polyhomogeneous pseudodifferential operators admits no nontrivial traces, due to the presence of higher log-terms.
• . A hierarchy of higher noncommutative residue functionals Resk exists on subspaces graded by log-power, and they vanish on commutators.
• . The Kontsevich–Vishik trace extends to this algebra and is characterized by TR(A) = Tr(AP−s)|s=0 for elliptic P.
• . The residue functional Resk satisfies the relation Resk+1(TR(A(z)))|z=ν = (−1)k+1/(k+1) × Resk(A(ν)) for holomorphic families.
• . The symplectic residue for log-polyhomogeneous functions on symplectic cones obstructs such functions from being sums of Poisson brackets, generalizing the classical case.