Derivations, derivatives and chain rules

AbstractFor a smooth function f on the space of bounded operators in a Hilbert space, we obtain formulas for the nth order commutator [[[f(A),X],X],…,X] in terms of the Fréchet derivatives Dmf(A). We illustrate the use of these formulas in obtaining bounds for norms of generalised commutators f(A)X−Xf(B) and their higher order analogues