Differentiability of scalar functions applied to Hermitian operators - a Fourier transform approach

Abstract

Let g be a (say, sufficiently differentiable) function on the reals. One knows how to apply g to Hermitian elements A of a C* algebra. Yet the question of differentiability of the mapping A to g(A) is not trivial, since in general "A and dA do not commute". However, since this mapping depends linearly on g, one can, via Fourier transform, reduce the case of general g to the case of the exponential function. For the latter one has an explicit formula for the n-th derivative (more complicated than in the scalar case - still "A and dA do not commute"). In this way one bounds the norm of the n-th derivative of (A to g(A)) on the r-ball by a Sobolev norm involving the (n+1)-th derivative of g on [-r,r].