Operator Differentiable Functions

Abstract

We study the Banach ${}^{\ast }$-algebra $C_{\mathrm{op}}^1(I)$ of $C^1$-functions on the compact interval $I$ such that the corresponding Hilbert space operator function $T\to f(T)$, for $T=T^{\ast }$ and $\mathrm{sp}(T)\subset I$, is Fréchet differentiable. If $f(x)=\int e^{itx}{\hat{f}}^{(}t)dt$, we know that the differential is given by the formula

where $U_t=\exp (itT)$. Functions of this type are dense in $C_{\mathrm{op}}^1(I)$, and $C^2(I)\subset C_{\mathrm{op}}^1(I)\subset C^1(I)$, so several classical results can be deduced. In particular we show that if $T\in \mathfrak{D}(\delta )$, where $\delta$ is the generator of a one-parameter group of ${}^{\ast }$-automorphisms of a $C^{\ast }$-algebra $\mathfrak{A}$ (or just a closed ${}^{\ast }$-derivation in $\mathfrak{A}$), then $f(T)\in \mathfrak{D}(\delta )$ for every $f$ in $C_{\mathrm{op}}^1(I)$, where $\mathrm{sp}(T)\subset I$, and