Operator Differentiable Functions

Abstract

We study the Banach *-algebra _C_1QP(I) of _C_1-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 Ust__SU(1-s)t dsf'^(t) dt,

where Ut = exp(itT). Functions of this type are dense in _C_1QP(I), and _C_2(I) ⊂ _C_1QP(I) ⊂ _C_1(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 _C_1QP(I), sp(T) ⊂ I, and

δ(f(T)) = dfT(δ(T)).