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

Abstract

We study the analog of semi-separable integral kernels in $\mathcal{H}$ of the type $$K(x,x^{\prime })=\{\begin{array}{ll}{F}_1(x)G_1(x^{\prime }),& a<x^{\prime }<x<b,\\ F_2(x)G_2(x^{\prime }),& a<x<x^{\prime }<b,\end{array}$$ where $-\infty \le a<b\le \infty$, 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 $$\Vert F_j(\cdot ){\Vert }_{{\mathcal{B}}_2({\mathcal{H}}_j,\mathcal{H})}\in L^2((a,b)),\Vert G_j(\cdot ){\Vert }_{{\mathcal{B}}_2(\mathcal{H},{\mathcal{H}}_j)}\in L^2((a,b)),j=1,2,$$ 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 ${\det }_{2,L^2((a,b);\mathcal{H})}(\mathbf{I}-\alpha \mathbf{K})$, $\alpha \in \mathbb{C}$, 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.

25 pages; typos removed. arXiv admin note: substantial text overlap with arXiv:1404.0739