Differentiation of operator functions and perturbation bounds

Given a smooth real function f on the positive half line consider the induced map A --> f(A) on the set of positive Hilbert space operators. Let f((k)) be the k(th) derivative of the real function f and D(k)f the k(th) Frechet derivative of the operator map f. We identify large classes of functions for which //D(k)f(A)// = //f((k))(A)//, for k = 1,2,.... This reduction of a noncommutative problem to a commutative one makes it easy to obtain perturbation bounds for several operator maps. Our techniques serve to illustrate the use of a formalism for "quantum analysis" that is like the one recently developed by M. Suzuki.