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

الأصل · EN

We study the analog of semi-separable integral kernels in H of the type K(x,x')=cases F₁(x)G₁(x'), & a<x'< x< b, F₂(x)G₂(x'), & a<x<x'<b, cases where -∞≤ a<b≤ ∞, and for a.e.x ∈ (a,b), Fⱼ (x) ∈ B₂(Hⱼ,H) and Gⱼ(x) ∈ B₂(H,Hⱼ) such that Fⱼ(·) and Gⱼ(·) are uniformly measurable, and Fⱼ(·)B₂₍ₕⱼ,ₕ₎ ∈ L²((a,b)), Gⱼ (·)B₂₍ₕ,ₕⱼ₎ ∈ L²((a,b)), j=1,2, with H and Hⱼ, j=1,2, complex, separable Hilbert spaces. Assuming that K(·, ·) generates a Hilbert-Schmidt operator K in L²((a,b);H), we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the modified Fredholm determinant ₂, ₗ₂₍₍ₐ,b₎;ₕ₎(I - αK), α∈ C, naturally reduces to appropriate Fredholm determinants in the Hilbert spaces H (and H ⊕ H). Some applications to Schrödinger operators with operator-valued potentials are provided.