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?

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?