Isotropic functions revisited

To a smooth and symmetric function $f$ defined on a symmetric open set $\Gamma\subset\mathbb{R}^{n}$ and a real $n$-dimensional vector space $V$ we assign an associated operator function $F$ defined on an open subset $\Omega\subset\mathcal{L}(V)$ of linear transformations of $V$, such that for each inner product $g$ on $V$, on the subspace $\Sigma_{g}(V)\subset\mathcal{L}(V)$ of $g$-selfadjoint operators, $F_{g}=F_{|\Sigma_{g}(V)}$ is the isotropic function associated to $f$, which means that $F_{g}(A)=f(\mathrm{EV}(A))$, where $\mathrm{EV}(A)$ denotes the ordered $n$-tuple of real eigenvalues of $A$. We extend some well known relations between the derivatives of $f$ and each $F_{g}$ to relations between $f$ and $F$. By means of an example we show that well known regularity properties of $F_{g}$ do not carry over to $F$.