# Operator Differentiable Functions

### Gert K. Pedersen

University of Copenhagen, Denmark

## 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*)).