Perturbations and operator trace functions

We study the spectral functional for a suitable function f, a self-adjoint operator D having compact resolvent, and a certain class of bounded self-adjoint operators A. Such functionals were introduce by Chamseddine and Connes in the context of noncommutative geometry. Motivated by the physical applications of these functionals, we derive a Taylor expansion of them in terms of Gâteaux derivatives. This involves divided differences of f evaluated on the spectrum of D, as well as the matrix coefficients of A in an eigenbasis of D. This generalizes earlier results to infinite dimensions and to any number of derivatives.