We introduce the Frechet differential of operator functions on C*-aIgebras obtained via spectral theory from ordinary differentiate functions. In the finite-dimensional case this differential is expressed in terms of Hadamard products of matrices. A perturbation formula with applications to traces is given.
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Frank Hansen, Gert K. Pedersen, Perturbation Formulas for Traces on <i>C</i>*-algebras. Publ. Res. Inst. Math. Sci. 31 (1995), no. 1, pp. 169–178DOI 10.2977/PRIMS/1195164797