Title: Isotropic functions revisited,
Abstract: 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 $F$.