# Operator Differentiable Functions

### Gert K. Pedersen

University of Copenhagen, Denmark

## Abstract

We study the Banach $_{∗}$-algebra $C_{op}(I)$ 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 $f(x)=∫e_{itx}f^ _{(}t)dt$, we know that the differential is given by the formula

where $U_{t}=exp(itT)$. Functions of this type are dense in $C_{op}(I)$, and $C_{2}(I)⊂C_{op}(I)⊂C_{1}(I)$, 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 $C_{op}(I)$, where $sp(T)⊂I$, and

## Cite this article

Gert K. Pedersen, Operator Differentiable Functions. Publ. Res. Inst. Math. Sci. 36 (2000), no. 1, pp. 139–157

DOI 10.2977/PRIMS/1195143229