The product formula for regularized Fredholm determinants

Abstract

For trace class operators A , B ∈ B 1 ( H ) A, B \in \mathcal {B}_1(\mathcal {H}) ( H \mathcal {H} a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form \[ det H ( ( I H − A ) ( I H − B ) ) = det H ( I H − A ) det H ( I H − B ) . {\det }_{\mathcal {H}} ((I_{\mathcal {H}} - A) (I_{\mathcal {H}} - B)) = {\det }_{\mathcal {H}} (I_{\mathcal {H}} - A) {\det }_{\mathcal {H}} (I_{\mathcal {H}} - B). \] When trace class operators are replaced by Hilbert–Schmidt operators A , B ∈ B 2 ( H ) A, B \in \mathcal {B}_2(\mathcal {H}) and the Fredholm determinant det H ( I H − A ) {\det }_{\mathcal {H}}(I_{\mathcal {H}} - A) , A ∈ B 1 ( H ) A \in \mathcal {B}_1(\mathcal {H}) , by the 2nd regularized Fredholm determinant det H , 2 ( I H − A ) = det H ( ( I H − A ) exp ⁡ ( A ) ) {\det }_{\mathcal {H},2}(I_{\mathcal {H}} - A) = {\det }_{\mathcal {H}} ((I_{\mathcal {H}} - A) \exp (A)) , A ∈ B 2 ( H ) A \in \mathcal {B}_2(\mathcal {H}) , the product formula must be replaced by det H , 2 ( ( I H − A ) ( I H − B ) ) a m p ; = det H , 2 ( I H − A ) det H , 2 ( I H − B ) a m p ; × exp ⁡ ( − tr H ⁡ ( A B ) ) . \begin{align*} {\det }_{\mathcal {H},2} ((I_{\mathcal {H}} - A) (I_{\mathcal {H}} - B)) &= {\det }_{\mathcal {H},2} (I_{\mathcal {H}} - A) {\det }_{\mathcal {H},2} (I_{\mathcal {H}} - B) \\ & \quad \times \exp (- \operatorname {tr}_{\mathcal {H}}(AB)). \end{align*} The product formula for the case of higher regularized Fredholm determinants det H , k ( I H − A ) {\det }_{\mathcal {H},k}(I_{\mathcal {H}} - A) , A ∈ B k ( H ) A \in \mathcal {B}_k(\mathcal {H}) , k ∈ N k \in \mathbb {N} , k ⩾ 2 k \geqslant 2 , does not seem to be easily accessible and hence this note aims at filling this gap in the literature.