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 $F$.

13 pages. Added an example to show that loss of regularity is possible. Extended the bibliography. Comments are welcome

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