The product formula for regularized Fredholm determinants

• Published in: Proceedings of the American Mathematical Society. Series B Vol. 8; no. 4; pp. 42 - 51
• Main Authors: Britz, Thomas, Carey, Alan, Gesztesy, Fritz, Nichols, Roger, Sukochev, Fedor, Zanin, Dmitriy
• Format: Journal Article
• Language: English
• Published: 10-02-2021
• ISSN: 2330-1511; 2330-1511
• DOI: 10.1090/bproc/70

For trace class operators A, B \in \mathcal {B}_1(\mathcal {H}) ( \mathcal {H} a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form \displaystyle {\det }_{\mathcal {H}} ((I_{\mathcal {H}} - A) (I_{\mathcal {H}} ... ...cal {H}} (I_{\mathcal {H}} - A) {\det }_{\mathcal {H}} (I_{\mathcal {H}} - B). When trace class operators are replaced by Hilbert-Schmidt operators A, B \in \mathcal {B}_2(\mathcal {H}) and the Fredholm determinant {\det }_{\mathcal {H}}(I_{\mathcal {H}} - A), A \in \mathcal {B}_1(\mathcal {H}), by the 2nd regularized Fredholm determinant {\det }_{\mathcal {H},2}(I_{\mathcal {H}} - A) = {\det }_{\mathcal {H}} ((I_{\mathcal {H}} - A) \exp (A)), A \in \mathcal {B}_2(\mathcal {H}), the product formula must be replaced by <TD NOWRAP ALIGN="RIGHT">\displaystyle {\det }_{\mathcal {H},2} ((I_{\mathcal {H}} - A) (I_{\mathcal {H}} - B)) <TD NOWRAP ALIGN="LEFT">\displaystyle = {\det }_{\mathcal {H},2} (I_{\mathcal {H}} - A) {\det }_{\mathcal {H},2} (I_{\mathcal {H}} - B) <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> <TD NOWRAP ALIGN="LEFT">\displaystyle \quad \times \exp (- \operatorname {tr}_{\mathcal {H}}(AB)). <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> The product formula for the case of higher regularized Fredholm determinants {\det }_{\mathcal {H},k}(I_{\mathcal {H}} - A), A \in \mathcal {B}_k(\mathcal {H}), k \in \mathbb{N}, k \geqslant 2, does not seem to be easily accessible and hence this note aims at filling this gap in the literature.