A Note on the Wodzicki Residue

TL;DR

This paper explores the connection between the Wodzicki residue of elliptic operators and the heat kernel expansion, providing a simplified proof of a known relation for generalized Laplacians on compact manifolds.

Contribution

It offers a straightforward proof linking the Wodzicki residue to heat kernel coefficients for elliptic operators, extending previous results.

Findings

Wodzicki residue relates to heat kernel asymptotics.

Simplified proof for the case of generalized Laplacians.

Residue equals the integral of the second heat kernel coefficient.

Abstract

In this note we explain the relationship of the Wodzicki residue of (certain powers of) an elliptic differential operator $P$\ acting on sections of a complex vector bundle $E$\ over a closed compact manifold $M$\ and the asymptotic expansion of the trace of the corresponding heat operator $e^{-tP}$. In the special case of a generalized laplacian $\triangle$\ and ${{\rm dim}\;M > 2}$, we thereby obtain a simple proof of the fact already shown in [KW], that the Wodzicki residue $res(\triangle^{-{n\over 2}+1} )$\ is the integral of the second coefficient of the heat kernel expansion of $\triangle$\ up to a proportional factor.