Operator Differentiable Functions

We study the Banach *-algebra C1QP(I) of C1-functions on the compact interval I such that the corresponding Hilbert space operator function T → f(T), for T = T* and sp(T) ⊂ I, is Fréchet differentiable. If f(x) = ∫ eitx f^(t)dt we know that the differential is given by the formula dfT(S) = ∫-∞∞ ∫01 UstSU(1-s)t dsf'^(t) dt, where Ut = exp(itT). Functions of this type are dense in C1QP(I), and C2(I) ⊂ C1QP(I) ⊂ C1(I), so several classical results can be deduced. In particular we show that if T ∈ D(δ), where δ is the generator of a one-parameter group of *-automorphisms of a C*-algebra A (or just a closed *-derivation in A), then f(T) ∈ D(δ) for every f in C1QP(I), sp(T) ⊂ I, and δ(f(T)) = dfT(δ(T)).