Isotropic functions revisited

Abstract

To a real n-dimensional vector space V and a smooth, symmetric function f defined on the n-dimensional Euclidean space we assign an associated operator function F defined on linear transformations of V. F shall have the property that, for each inner product g on V, its restriction $F_{g}$ to the subspace of g-selfadjoint operators is the isotropic function associated to f. This means that it acts on these operators via f acting on their eigenvalues. We 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 we show that well-known regularity properties of $F_{g}$ do not carry over to F.