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 t he k(th) Frechet derivative of the operator map f. We identify large c lasses 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 ''quan tum analysis'' that is like the one recently developed by M. Suzuki.