Isotropic functions revisited (Q1645357)

To each smooth symmetric function \(f\) defined on the \(n\)-dimensional Euclidean space, the author associates an operator function \(F\) defined on the linear transformations of a real \(n\)-dimensional vector space \(V\), with the key property that, for each inner product \(g\) on \(V\), the restriction \(F_g\) of \(F\) to the subspace of \(g\)-selfadjoint operators, is the isotropic function associated to \(f\). (\(F\) acts on these operators via \(f\) acting on their eigenvalues). The main results generalize some well known relations between the derivatives of \(f\) and each \(F_g\), to relations between \(f\) and \(F\), while also providing new elementary proofs of the known results. By means of an example, he show that well known regularity properties of \(F_g\) do not carry over to \(F\).