  We extend to infinite dimensional separable Hilbert spaces the Schur
convexity property of eigenvalues of a symmetric matrix with real entries. Our
framework includes both the case of linear, selfadjoint, compact operators, and
that of linear selfadjoint operators that can be approximated by operators of
finite rank and having a countable family of eigenvalues. The abstract results
of the present paper are illustrated by several examples from mechanics or
quantum mechanics, including the Sturm-Liouville problem, the Schr\"{o}dinger
equation, and the harmonic oscillator.
