For a trace-class operator on a Hilbert space, the Fredholm determinant is the convergent product over its eigenvalues, counted with algebraic multiplicity. For self-adjoint , spectral theorem for compact self-adjoint operators reduces the product to a diagonal calculation. For , . This identity exposes the successive cumulants in a Gaussian quadratic exponential moment.
For bounded operators with trace-class commutator, the multiplicative commutator differs from identity by a trace-class operator. Along , the trace of its logarithmic derivative is the constant . This follows by writing as the integral of conjugates of and using trace invariance under bounded similarities. Integrating the Fredholm determinant derivative from proves the formula. Individual traces of and need not exist.
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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 \( \mathcal{H} \).