2014/04/03 by Fritz Gesztesy, Gesztesy, Fritz, Roger Nichols +1
Mathematics · Physics and Astronomy · #34L40 #47G10 #Advanced Operator Algebra Research #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Primary: 47B10 #Secondary: 34B27 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.FA #math.MP #math.SP #msc:34B27 #msc:34L40 #msc:47B10 #msc:47G10
paper · pdf · doi:10.48550/arxiv.1404.1074
25 pages; typos removed. arXiv admin note: substantial text overlap with arXiv:1404.0739
openalex publication_date 2014/04/03 · arxiv created 2014/08/29 · arxiv updated 2014/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the analog of semi-separable integral kernels in H of the type K(x,x')=\begincases F1(x)G1(x'), a<x'< x< b,
F2(x)G2(x'), a<x<x'<b, \endcases 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 det2, 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.