To a smooth and symmetric function f defined on a symmetric open set
Γ⊂Rn and a real n-dimensional vector space V we
assign an associated operator function F defined on an open subset
Ω⊂L(V) of linear transformations of V, such that for
each inner product g on V, on the subspace
Σg(V)⊂L(V) of g-selfadjoint operators,
Fg=F∣Σg(V) is the isotropic function associated to f, which
means that Fg(A)=f(EV(A)), where 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 Fg to relations between
f and F. By means of an example we show that well known regularity
properties of Fg do not carry over to F.Comment: 13 pages. Added an example to show that loss of regularity is
possible. Extended the bibliography. Comments are welcom