Complex powers of elliptic operators

Summary

Complex powers $A^s$, $s \in \C$, of a positive elliptic operator $A$ on a closed manifold generalise the powers $\lambda^s$ of a positive number. Seeley showed that they are pseudo-differential operators of order $ms$, where $m$ is the order of $A$, depending holomorphically on $s$. Their traces define the spectral zeta function $\zeta_A(s) = \operatorname{tr} A^{-s}$, which extends meromorphically to $\C$ and is regular at $s = 0$. This is the basis of zeta function regularization of determinants of Laplacians and of analytic torsion.