Isotropic functions revisited |
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Authors: | Julian Scheuer |
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Institution: | 1.Mathematisches Institut,Albert-Ludwigs-Universit?t,Freiburg,Germany |
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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. |
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