A Note on the Wodzicki Residue

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.