A Jost-Pais-Type Reduction of Fredholm Determinants and Some Applications

We study the analog of semi-separable integral kernels in of the type where , and for a.e. , and such that F (j) (center dot) and G (j) (center dot) are uniformly measurable, and with and , j = 1, 2, complex, separable Hilbert spaces. Assuming that K(center dot, center dot) generates a trace class operator K in , we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the Fredholm determinant (I - alpha K), , naturally reduces to appropriate Fredholm determinants in the Hilbert spaces (and ). Explicit applications of this reduction theory to Schrodinger operators with suitable bounded operator-valued potentials are made. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator.