 Fredholm determinant

For a trace-class operator $K$ on a Hilbert space, the Fredholm determinant is the convergent product over its eigenvalues, counted with algebraic multiplicity. For self-adjoint $K$, spectral theorem for compact self-adjoint operators reduces the product to a diagonal calculation. For $\Vert K\Vert <1$, $\log \det (I+K)={\sum }_{n\ge 1}(-1{)}^{n+1}\mathrm{tr}(K^n)/n$. This identity exposes the successive cumulants in a Gaussian quadratic exponential moment.

The Fredholm determinant is a mathematical concept that generalizes the notion of a determinant to certain classes of operators, particularly integral operators. It is named after the Swedish mathematician Ivar Fredholm, who studied integral equations and introduced these ideas in the early 20th century. In the context of functional analysis, let \( K \) be a compact operator (often, but not exclusively, an integral operator) acting on a Hilbert space \( \mathcal{H} \).