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

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?

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The mathematics here is definitely out of the realm of my understanding, so hopefully someone who knows this stuff will post a better answer.

I believe that Prop 2.8.2 in Segal's Lecture 2 is more or less Proposition 6.4 in "Determinants of elliptic pseudo-differential operators" by Kontsevich and Vishik.

There seems to be an ample literature on zeta determinants of the Laplacian and Dirac operators so you probably can find other sources as well. For example, I actually first found the paper "Zeta Determinants on Manifolds with Boundary" by Scott, which includes a stronger statement as Prop. 2.19.

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