Zeta and Fredholm determinants of self-adjoint operators

Let L be a self-adjoint invertible operator in Hilbert space such that − 1 is p -summable. Under certain discrete dimension spectrum assumption on , we study the relation between (regularized) Fredholm determinant, det ( I + z ⋅ ) one hand and zeta regularized ζ other. One of main results formula = exp ⁡ ∑ j ! d log | 0 . We show derivatives can expressed terms values heat trace coefficients Furthermore, give general criterion (and which is, e.g. fulfilled for large classes elliptic operators) guarantees constant term asymptotic expansion equals determinant