Isotropic functions revisited

To a smooth and symmetric function $f$ defined on a symmetric open set $\mathrm{\Gamma }\subset {\mathbb{R}}^n$ and a real $n$-dimensional vector space $V$ we assign an associated operator function $F$ defined on an open subset $\mathrm{\Omega }\subset \mathcal{L}(V)$ of linear transformations of $V$, such that for each inner product $g$ on $V$, on the subspace ${\mathrm{\Sigma }}_g(V)\subset \mathcal{L}(V)$ of $g$-selfadjoint operators, $F_g=F_{\mathrm{\mid }{\mathrm{\Sigma }}_g(V)}$ is the isotropic function associated to $f$, which means that $F_g(A)=f(\mathrm{E}\mathrm{V}(A))$, where $\mathrm{E}\mathrm{V}(A)$ 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 $F$.