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On Multiplicative Properties of Determinants
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abstract
Let $A$ be an elliptic pseudodifferential operator of positive order on a compact closed manifold, and let $T$ be a pseudodifferential operator of negative order such that $T^m$ is of trace class. We compute $\log\det(A(I+T))-\log\det A-\log\det_m (I+T)$ where first two determinants are zeta function regularized, and the last one is a regularized Fredholm determinant.
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The determinant of the Dirichlet-to-Neumann map for a surface with boundary and periods of holomorphic differentials on its double
The zeta determinant of the Dirichlet-to-Neumann map of a surface with boundary equals the determinant of the discrete part of its boundary Hilbert transform, a product of period ratios on the double surface.
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