[Recently][1], someone asked on MO about lecture notes from Graeme Segal's "Stanford lectures" on TQFT, and the answer was to check [here][2]. 

When scrolling over the notes, I stumpled of Prop. 2.8.2 in lecture 2. It says (modulo typos) that if $P$ is a Dirac operator and $S$ is an invertible operator such that $S-\mathrm{id}$ is trace-class, then
$$\det\nolimits_\zeta(PS) = \det(S)\det\nolimits_\zeta(P)),$$
where $\det\nolimits_\zeta$ denotes the zeta-regularized determinant and $\det$ denotes the usual determinant.

However, there are neither references nor proofs there. Does anyone know of a proof?

  [1]: http://mathoverflow.net/questions/216210/segals-1999-stanford-lecture-notes-on-tqft-where-to-find-them
  [2]: http://web.archive.org/web/20000901075112/http://www.cgtp.duke.edu/ITP99/segal/