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We consider Fredholm determinants of the form identity minus product ofspectral projections corresponding to isolated parts of the spectrum of a pairof self-adjoint operators. We show an identity relating such determinants to anintegral over the spectral shift function in the case of a rank-oneperturbation. More precisely, we prove $$ -\ln \left(\det \big(\mathbf{1}-\mathbf{1} _{I}(A) \mathbf{1}_{\mathbb R\backslash I}(B)\mathbf{1}_{I}(A)\big)\right) = \int_I \text{d} x \int_{\mathbb R\backslash I} \text{d} y\,\frac{\xi(x)\xi(y)}{(y-x)^2}, $$ where $\mathbf{1}_J (\cdot)$ denotes thespectral projection of a self-adjoint operator on a set $J\in\text{Borel}(\mathbb R)$. The operators $A$ and $B$ are self-adjoint, boundedfrom below and differ by a rank-one perturbation and $\xi$ denotes thecorresponding spectral shift function. The set $I$ is a union of intervals onthe real line such that its boundary lies in the resolvent set of $A$ and $B$and such that the spectral shift function vanishes there i.e. $I$ containsisolated parts of the spectrum of $A$ and $B$. We apply this formula to thesubspace perturbation problem.

On an integral formula for Fredholm determinants related to pairs of spectral projections