Fredholm Determinant

The Fredholm determinant det(I+K) is a scalar defined for trace-class (nuclear) operators K on a Hilbert space, constructed as the convergent product ∏_j (1+λ_j) over eigenvalues λ_j of K (counted with algebraic multiplicity), equivalently via det(I+K)=exp(tr log(I+K)). It characterizes invertibility of I+K and analytic dependence on parameters.