A Jost-Pais-type reduction of (modified) Fredholm determinants for semi-separable operators in infinite dimensions

We study the analog of semi-separable integral kernels in $\mathcal{H}$ of the type $K(x,x')=\begin{cases} F_1(x)G_1(x'), & a<x'< x< b, \\ F_2(x)G_2(x'), & a<x<x'<b, \end{cases}$ where $-\infty\leq a<b\leq \infty$, and for a.e.\ $x \in (a,b)$, $F_j (x) \in \mathcal{B}_2(\mathcal{H}_j,\mathcal{H})$ and $G_j(x) \in \mathcal{B}_2(\mathcal{H},\mathcal{H}_j)$ such that $F_j(\cdot)$ and $G_j(\cdot)$ are uniformly measurable, and $\|F_j(\cdot)\|_{\mathcal{B}_2(\mathcal{H}_j,\mathcal{H})} \in L^2((a,b)), \; \|G_j (\cdot)\|_{\mathcal{B}_2(\mathcal{H},\mathcal{H}_j)} \in L^2((a,b)), \quad j=1,2,$ with $\mathcal{H}$ and $\mathcal{H}_j$, $j=1,2$, complex, separable Hilbert spaces. Assuming that $K(\cdot, \cdot)$ generates a Hilbert-Schmidt operator $\mathbf{K}$ in $L^2((a,b);\mathcal{H})$, we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the modified Fredholm determinant ${\det}_{2, L^2((a,b);\mathcal{H})}(\mathbf{I} - α\mathbf{K})$, $α\in \mathbb{C}$, naturally reduces to appropriate Fredholm determinants in the Hilbert spaces $\mathcal{H}$ (and $\mathcal{H} \oplus \mathcal{H}$). Some applications to Schrödinger operators with operator-valued potentials are provided.

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