Derivation of the Trace Formula: The Trace Class Result

Abstract

This chapter is of a technical nature and as its center piece proves that the operator $z\chi _{\varLambda }\mathop{\mathrm{tr}}\nolimits _{N}\big(\left (L^{{\ast}}L + z\right )^{-1} -\left (LL^{{\ast}} + z\right )^{-1}\big)$, $z \in \varrho (-LL^{{\ast}}) \cap \varrho (-L^{{\ast}}L)$, belongs to the trace class $\mathcal{B}_{1}\big(L^{2}(\mathbb{R}^{n})\big)$. Here $\mathop{\mathrm{tr}}\nolimits _{N}$ describes an appropriate internal trace, and $\chi _{\varLambda }$ is the multiplication operator of multiplying with the characteristic function of the ball centered at 0 with radius $\varLambda> 0$ in $\mathbb{R}^{n}$. Moreover, it is shown that the operator $\mathop{\mathrm{tr}}\nolimits _{N}\big(\left (L^{{\ast}}L + z\right )^{-1} -\left (LL^{{\ast}} + z\right )^{-1}\big)$ vanishes in even dimension n, thus, all subsequent index computations can be confined to odd space dimensions n.