2020/07/25 by Britz, Thomas, Carey, Alan, Gesztesy, Fritz +3
#FOS: Mathematics #Primary: 47B10 #Secondary: 47B02 #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2007.12834
For trace class operators A, B ∈ B1(H) (H a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form detH ((IH - A) (IH - B)) = detH (IH - A) detH (IH - B). When trace class operators are replaced by Hilbert--Schmidt operators A, B ∈ B2(H) and the Fredholm determinant detH(IH - A), A ∈ B1(H), by the 2nd regularized Fredholm determinant detH,2(IH - A) = detH ((IH - A) exp(A)), A ∈ B2(H), the product formula must be replaced by detH,2 ((IH - A) (IH - B)) = detH,2 (IH - A) detH,2 (IH - B) exp(- \rm tr(AB)). The product formula for the case of higher regularized Fredholm determinants detH,k(IH - A), A ∈ Bk(H), k ∈ ℕ, k ≥ 2, does not seem to be easily accessible and hence this note aims at filling this gap in the literature.