Multiple Operator Integrals and Spectral Shift

For a large class of admissible functions f : R 7→ C, the operator derivatives dj dxj f(H0 + xV ), where H0 and V are self-adjoint operators on a separable Hilbert space H, exist and can be represented as multiple operator integrals [1, 14]. Let M be a semi-finite von Neumann algebra acting on H and τ a semi-finite normal faithful trace on M. For H0 = H ∗ 0 affiliated with M and V = V ∗ in the τ -HilbertSchmidt class L2(M, τ) (that is, V ∈ M and τ(|V | ) < ∞), we represent the traces of the derivatives τ [