The product formula for regularized Fredholm determinants

For trace class operators A , B ? B 1 ( mathvariant="script">H stretchy="false">) A, \in \mathcal {B}_1(\mathcal {H}) ( encoding="application/x-tex">\mathcal {H} a complex, separable Hilbert space), the product formula for Fredholm determinants holds in familiar form movablelimits="true" form="prefix">det I ?<!-- ? <mml:mo>= . encoding="application/x-tex">{\det }_{\mathcal {H}} ((I_{\mathcal - A) (I_{\mathcal B)) = {\det B). \] When are replaced by Hilbert–Schmidt 2 2 {B}_2(\mathcal and determinant {H}}(I_{\mathcal A) , encoding="application/x-tex">A 2nd regularized exp exp ?<!-- ? {H},2}(I_{\mathcal \exp (A))</mml:annotation> must be alttext="StartLayout 1st Row Column m p semicolon Blank times period EndLayout"> a m p ; ×<!-- × <mml:mi>tr encoding="application/x-tex">\begin{align*} {H},2} &amp;= B) \\ &amp; \quad \times (- \operatorname {tr}_{\mathcal {H}}(AB)). \end{align*} The case of higher k k {H},k}(I_{\mathcal {B}_k(\mathcal alttext="k double-struck N"> mathvariant="double-struck">N encoding="application/x-tex">k \mathbb {N} greater-than-or-slanted-equals 2"> ?<!-- ? \geqslant 2</mml:annotation> does not seem to easily accessible hence this note aims at filling gap literature.